diff --git a/-tAzT4oBgHgl3EQfvf0n/content/tmp_files/2301.01706v1.pdf.txt b/-tAzT4oBgHgl3EQfvf0n/content/tmp_files/2301.01706v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5869be4c88ee9382b510e3d9c95c5f14adb0cdc1 --- /dev/null +++ b/-tAzT4oBgHgl3EQfvf0n/content/tmp_files/2301.01706v1.pdf.txt @@ -0,0 +1,1697 @@ +On-chip Hong-Ou-Mandel interference from separate quantum dot emitters in an +integrated circuit +�Lukasz Dusanowski,1, 2, ∗ Dominik K¨ock,1 Christian Schneider,1, 3 and Sven H¨ofling1 +1Technische Physik and W¨urzburg-Dresden Cluster of Excellence ct.qmat, +University of W¨urzburg, Physikalisches Institut and Wilhelm-Conrad-R¨ontgen-Research +Center for Complex Material Systems, Am Hubland, D-97074 W¨urzburg, Germany +2currently at: Department of Electrical and Computer Engineering, +Princeton University, Princeton, NJ 08544, USA +3Institute of Physics, University of Oldenburg, D-26129 Oldenburg, Germany +(Dated: January 5, 2023) +Scalable quantum photonic technologies require low-loss integration of many identical single- +photon sources with photonic circuitry on a chip. Relatively complex quantum photonic circuits +have already been demonstrated; however, sources used so far relied on parametric-down-conversion. +Hence, the efficiency and scalability are intrinsically limited by the probabilistic nature of the sources. +Quantum emitter-based single-photon sources are free of this limitation, but frequency matching of +multiple emitters within a single circuit remains a challenge. In this work, we demonstrate a key +component in this regard in the form of a fully monolithic GaAs circuit combing two frequency- +matched quantum dot single-photon sources interconnected with a low-loss on-chip beamsplitter +connected via single-mode ridge waveguides. This device enabled us to perform a two-photon inter- +ference experiment on-chip with visibility reaching 66%, limited by the coherence of the emitters. +Our device could be further scaled up, providing a clear path to increase the complexity of quantum +circuits toward fully scalable integrated quantum technologies. +Optical quantum computing and communication ap- +plications with single photons and linear optics rely crit- +ically on the quantum interference of two photons on +a beamsplitter [1]. +This process, known as Hong-Ou- +Mandel (HOM) effect, occurs when two identical single +photons enter a 50:50 beamsplitter, one in each input +port. When the photons are indistinguishable, they will +coalesce into a two-photon Fock state [2], in which they +exit the same but random output port. This process un- +derlines the simplest non-trivial path-entangled NOON +state generation and introduces an optical non-linearity +which is the base for the implementation of more-complex +photonic gates and protocols. +Consequently, scalable optical quantum information +technologies will require integrating many identical in- +distinguishable single-photon sources with reliable pho- +tonic circuits consisting of beamsplitters. Utilizing well- +developed integrated photonics technology is particularly +appealing in this regard, as it dramatically reduces the +footprint of quantum devices. Furthermore, it allows con- +trolling photon states with high fidelity due to the in- +trinsic sub-wavelength stability of the path-lengths, low- +losses, and near-perfect mode overlap at an integrated +beamsplitter for high-fidelity quantum interference [3–5]. +Advances in integrated photonic technology allowed +already realizations of relatively complex quantum cir- +cuits demonstrating CNOT-gate operation [6, 7], boson +sampling [7, 8], quantum walks [9], some simple quan- +tum algorithms [7, 10] and chip-to-chip quantum tele- +portation [11]. A combination of integrated photonic cir- +cuits with spontaneous four-wave mixing photon sources +has also been achieved [11–13]. +However, due to the +used sources’ probabilistic nature, their efficiency and +scalability are intrinsically limited. Quantum emitters- +based single-photon sources are free of this limitation +and recently have been shown to outperform spontaneous +four-wave mixing and down-conversion photon sources +in simultaneously reaching high levels of photons indis- +tinguishability and brightness [14–18]. +Moreover, re- +mote interference between two quantum emitters was al- +ready demonstrated using trapped ions [19, 20], quan- +tum dots [21–30], organic molecules [31, 32] or vacancy +centers in diamond [33–36]. The vast majority of those +experiments have been performed in free space as proof- +of-principle demonstrations. +Performing similar exper- +iments on-chip, taking advantage of the photonic cir- +cuit consisting of fully integrated quantum emitters and +beamsplitter, has not been performed yet, and it is still +a missing component towards scaling-up aforementioned +quantum technologies. +In this work, we demonstrate a crucial component in +this regard in the form of a fully monolithic GaAs cir- +cuit combing two frequency-matched quantum dot single- +photon sources interconnected with on-chip beamsplitter +via single-mode ridge waveguides. This device enabled +performing two-photon interference experiments on-chip +with visibility limited by the coherence of our emitters. +Our semiconductor photonic device is schematically +presented in Fig. 1a and b. It is based on InAs/GaAs +distributed Bragg-reflector ridge waveguides, which have +been proven to facilitate high optical quality quantum +dot single-photon sources [37]. The central part of the de- +vice consists of the single-mode directional coupler (DC), +which is the integrated optical analog of the bulk beam- +arXiv:2301.01706v1 [quant-ph] 4 Jan 2023 + +2 +QD1 +QD2 +DC +Output Arm 1 +Output Arm 2 +a +c +QD1 +QD1 +QD2 +QD2 +Input Arm 1 +Input Arm 2 +QDs +5x DBR +24x DBR +0.6 µm +1.3 µm +λ/2 cav. +b +… +1k +2k +3k +4k +5k +1.3930 +1.3935 +1k +2k +3k +4k +5k +Output Arm 2 +QD2 +PL intensity (arb. units) +QD1 +1.3930 +1.3935 +Energy (eV) +Output Arm 1 +FIG. 1. On-chip two-photon interference circuit and beam splitting operation. a, Schematic representation of the +photonic circuit based on a directional coupler (DC) interconnected with two input waveguides with coupled quantum dots and +two output arms with inverted tapers for photons collection. b, Ridge waveguide cross-section with marked layer structure. +c, Demonstration of beam splitting operation for fabricated DC. The photoluminescence signal from QD1 and QD2 is recorded +for Output Arms 1 and 2, respectively. QD1 and QD2 are frequency matched with a precision of 5 µeV. +splitter. In the two input arms of the DC, two frequency- +matched quantum dots (QDs) are located. For single- +photon generation, the QDs are excited non-resonantly +from the top using two separated picosecond pulsed laser +beams. +Photons interfered on the DC are finally col- +lected off the chip using inverse taper out-couplers. Spec- +tral filtering and detection are performed off-chip using a +monochromator and two superconducting single-photon +detectors. +To find two quantum dots with matching optical tran- +sition energies, the position of the excitation beam spot +on each input arm of the DC was scanned using an au- +tomatized translation stage. +Within such a scanning +routine, we localized two matching emission lines at +1.3931 eV energy originating from the QDs located in +two individual input arms of the DC and separated spa- +tially by around 200 µm. In Fig. 1b, photoluminescence +(PL) spectra from QD1 and QD2 recorded from DC out- +put arms 1 and 2 are presented at a temperature of 4.5 K. +Single well-resolved emission lines matching within 5 µeV +fit precision are visible. Comparing amplitudes of QD1 +and QD2 emission peaks visible within both output arms, +a beam splitting ratio of 48:52 is derived (including un- +even transmission through out-coupling arms - more de- +tails in Supplementary Section 8). +To show that optical excitation of our QDs leads to +the generation of single photons, we analyzed the photon +emission statistics of separate QDs by performing sec- +ond order-correlation experiments in Hanbury Brown and +Twiss (HBT) configuration. For that purpose, QDs have +been excited non-resonantly from the top by an 813 nm +wavelength train of picosecond pulses at a repetition rate +of 76 HMz. Photons emitted by the QDs were then cou- +pled into the circuit input arm waveguides and guided +into the directional coupler, where the signal was divided +between two output arms. Next, photons were collected +off-chip from the side of the sample using out-couplers +and subsequently filtered spectrally by a monochroma- +tor (70 µeV width) and coupled into two single-mode +fibres connected with superconducting single-photon de- +tectors (SSPD). Finally, the photon correlation statistics +were acquired by a multichannel picosecond event timer. +Data have been recorded under excitation powers corre- +sponding to half of the QD saturation intensity. +Fig. 2a and c present the second-order autocorrelation +function g(2) +HBT (τ) measurement recorded for each QD in- +dividually. In the case of both QDs, a clear suppression of +the central peak counts is visible, proving single-photon +emission. To quantitatively evaluate the probability of +multi-photon emission, g(2) +HBT (0) values were calculated +by integrating residual counts of the zero delay peak +with respect to the neighboring six peaks, resulting in +g(2) +HBT (0) = 0.35 ± 0.08 and g(2) +HBT (0) = 0.15 ± 0.02 for +QD1 and QD2, respectively. +In Fig. 2b and d, time- +resolved photoluminescence traces of the QD1 and QD2 +emission are shown. In this case, the repetition rate of +the laser was reduced to 19 MHz using a pulse picker. +Clear bi-exponential signal decays are visible, with a fast +and slow time constant of 720±5 ps and 12±1 ns for QD1 +and 600±5 ps and 22±1 ns for QD2. We attribute the +fast decay to the spontaneous recombination of electron- +hole pairs in QD (T1) and the slow one, which corre- +sponds to about 2% (1.2%) of the total QD1 (QD2) line +intensity, is tentatively interpreted as the recapturing of +the carriers by the QD. Using fit parameters obtained +from the time-resolved experiments, g(2) +HBT (τ) correla- +tion histograms have been fitted with double-sided bi- +exponential decay convoluted with 80 ps width Gaussian + +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 + experiment + fit +20 +40 +60 +80 +100 +120 +140 +160 +180 +Raw coincidences +0 +5 +10 15 20 +101 +102 +103 +PL (coincidences) +Time (ns) +Afast/Aslow = 85 +-45 -30 -15 +0 +15 30 45 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +g(2) +HBT(t) +Delay time (ns) +20 +40 +60 +80 +100 +120 +140 +160 +101 +102 +103 + experiment + bi-exp fit +Afast/Aslow = 50 +QD1 +QD2 +a +b +c +d +FIG. 2. +Single-photon generation and emission dy- +namics. a,c Second order auto-correlation histograms of QD1 +and QD2 emission under pulsed 76 MHz repetition rate excita- +tion. Data have been recorded in HBT configuration using an +on-chip beamsplitter. b,d Time-resolved PL traces revealing +bi-exponential decays with fast (slow) time constant of 720 ps +(12 ns) and 600 ps (22 ns) for QD1 and QD2, respectively. +instrumental response function (black dashed lines). +To demonstrate on-chip two-photon interference, the +single QDs in both input arms of the DC are excited us- +ing the picosecond pulses. For that, the laser beam is di- +vided into two independently controllable optical excita- +tion axis and synchronized in advance to ensure optimal +temporal overlap of emitted photons on the DC. It is per- +formed by sending the emission from each QD separately +through the on-chip DC and using time-resolved detec- +tion to eliminate the time delay difference between inde- +pendently generated single photons. The same technique +is used to introduce an intentional 0.5 ns time delay for +reference measurements. The excitation laser powers for +each QD are adjusted such that their emission intensities +are the same (around half of QD1 saturated intensity). +As we utilize on-chip beam splitting operation using DC +with single-mode inputs and outputs, we expect a very +high spatial mode overlap of our interferometer. To test +this, we send the continuous wave laser simultaneously +into both DC input arms and record classical interference +fringes with 98±1% visibility. In earlier experiments per- +formed on ridge waveguide structures, we observed that +the QD emission couples into the well-defined transverse- +electric mode of the WG with close to unity degree of +linear polarization [37]. In the case of the investigated +device, the polarization of the emitted photons was an- +alyzed after passing the whole circuit consisting of bend +regions, DC itself, and out-couplers (more details in Sup- +plementary Section 6). We found that for both QDs, the +degree of polarization is above 95%, suggesting optimal +polarization alignment for interference experiments. +Within the above-mentioned prerequisites, the two- +photon interference should be mainly limited by the co- +herence of our single-photon emitters. To get access to +the coherence times of our QDs, we performed a high- +resolution measurement of the emission linewidths us- +ing a scanning Fabry-Perot interferometer. We extract +the full-width at half-maximum of 13.5±2.5 µeV and +3.0±0.2 µeV by Lorentzian fit for QD1 and QD2, re- +spectively (see Supplementary Section 7). +The coher- +ence times calculated based on observed broadenings are +T QD1 +2 += 100±20 ps and T QD2 +2 += 440±30 ps. As the mea- +surements are performed on the tens of seconds timescale, +we speculate that recorded coherence times might be lim- +ited by charge and spin noise [38, 39]. Following Ref. [40], +we calculated the expected interference visibility of our +two independent emitters and derived the theoretical vis- +ibility in the range of Vtheory = 10-15%. +Figure 3a shows two-photon interference data in the +form of second-order HOM cross-correlation between +photons exiting the two output arms of the on-chip beam- +splitter. The height of the central peak is clearly below +the half intensity of the neighboring peaks, proving that +photons emitted by two separate QDs indeed interfere +on the DC. Another interference signature is the pres- +ence of the coincidences dip superimposed on the central +peak around zero time delay. The depth of this dip con- +stitutes to the interference events where photons arrive +simultaneously at the DC, giving rise to the narrow-time +window post-selected coalescence. In our case, the exact +value of g(2) +HOM(τ) at τ = 0 is equal to 0.17 in the case +of background-corrected data and 0.31 for as-measured +data. The same type of time post-selected interference +can be observed for cw HOM correlations. +Figure 3b +shows the non-corrected cw, and pulsed HOM interfer- +ence histograms overlapped on each other (correspond- +ing cw g(2) +HBT (τ) graphs are shown in Supplementary Sec- +tion 10). Similar to the pulsed case, the cw correlation +shows clear suppression of coincident counts at zero time +delay, with time post-selected g(2) +HOM(0) of 0.35, close to +the pulsed as-measured value of 0.31. +To evaluate photons full wave-packet interference +probability (non-post-selected), we calculate the pulsed +HOM correlation central peak area normalized by +the +average +area +of +the +neighboring +six +peaks. +For +integration +window +∆t +of +3 +ns, +we +obtain +g(2) +HOM(0, ∆t) = 0.459±0.002 for background corrected +data and g(2) +HOM(0, ∆t) = 0.587±0.002 for raw data, +where uncertainty is based on the standard deviation +of non-central peaks areas. In the case of background- +corrected data, we reach a value below the 0.5 classical +limit. It needs to be noted that derived g(2) +HOM(0, ∆t) and +g(2) +HOM(0) values are partially influenced by the non-zero +multi-photon emission extend observed in HBT measure- +ments. + +4 +-45 +-30 +-15 +0 +15 +30 +45 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +g(2) +HOM(t) +Delay time (ns) +100 +200 +300 +400 +500 +600 +Raw coincidences +-5 -4 -3 -2 -1 +0 +1 +2 +3 +4 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +HOM peak area - g(2) +HOM(t,Dt) +Peak number + indist. (sync.) + dist. (0.5 ns delay) +Dtint = 3 ns +V = 17.8±0.7% +g(2) +HOM(0,Dt) = 0.459±0.002 +-4 +-2 +0 +2 +4 +0.0 +0.2 +0.4 +Dtint +-6 +-4 +-2 +0 +2 +4 +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 + cw + pulsed +g(2) +HOM(t) +Delay time (ns) +a +b +c +QD1 +QD2 +FIG. 3. +On-chip two-photon interference from separate quantum emitters. +a, Two-photon Hong-Ou-Mandel +interference measurement between QD1 and QD2 showing the normalized HOM coincidences versus the delay time. +The +central peak area is suppressed with respect to neighboring peaks. Inset: Magnified view of the central peak area. b, Raw +HOM interference measurement recorded under cw (red points) and pulsed (blue points) excitation. c, Integrated counts of +the central eleven peaks (∆t = 3 ns integration window) of the HOM correlation in case of synchronized (blue bars) and 0.5 ns +delayed (red bars) photons from QD1 and QD2. All presented data are recorded using an on-chip beamsplitter. +As it has been recently pointed out in Ref. [41, 42], +to estimate two-photon interference visibility for remote +emitters correctly, it is necessary to perform reference +HOM measurements for distinguishable photons, due to +the possible blinking effect. Since within the fabricated +circuit polarization rotation is impossible to unambigu- +ously confirm the two-photon interference and properly +evaluate visibility, photons were made distinguishable by +introducing a 0.5 ns time delay between excitation pulses. +Such delay should be sufficient to lose the temporal pho- +tons overlap on the DC within the emitters coherence +times and record reference data. +Figure 3c demonstrates the normalized histogram of +the central eleven peaks areas (∆t = 3 ns) of the +HOM second-order cross-correlation in case of synchro- +nized - indistinguishable (red bars) and 0.5 ns de- +layed - distinguishable (grey bars) photons. +The cen- +tral peak area in the case of unsynchronized pho- +tons is equal to g(2) +HOMd(0, ∆t) = 0.558±0.002, which +is slightly above the theoretically expected 0.5 value. +We relate this discrepancy with non-zero multi-photon +emission extend. +Finally, we calculate remote sources +two-photon interference visibility V +following V += +[g(2) +HOMd(0, ∆t) − g(2) +HOM(0, ∆t)]/g(2) +HOMd(0, ∆t), resulting +in V = 17.8±0.7% for background corrected data. This +value is relatively close to the theoretically expected vis- +ibility and even partially exceeds it, suggesting that it is +limited solely by the coherence of the emitters (more de- +tails Supplementary Section 9). Using background cor- +rected data for the pulsed case, give post-selected visi- +bility of V ′ +p = 66%. The probability of the time post- +selected interference is known to depend on the ratio of +the emitters coherence times to the setup timing reso- +lution [21, 22, 43], thus possibly even higher V ′ values +could be potentially achieved with faster detectors. +While our results provide clear scientific evidence for +on-chip generation and interference of on-demand sin- +gle photons in a circuit, the recorded visibility values +need to be improved for future practical applications. In +the current device architecture, the indistinguishability +of interfered photons is limited by T2/(2T1) of particu- +lar QDs. We propose a few strategies for improving this +ratio. Firstly, the QD charge environment could be stabi- +lized via passivation [44], weak optical illumination [45], +or gating [15, 46]. At the same time, by embedding QDs +into optical cavities, the Purcell effect might be used to +enhance the radiative emission rate 1/T1 [14–16, 46, 47]. +Recently, we demonstrated a QD circuit with ring cav- +ities allowing to significantly increase the QD coupling +efficiency into the WG mode and decrease the T1 be- +low 200 ps [48]. +Finally, by applying a resonant exci- +tation, the photons emission time-jitter could be mini- +mized, and strong suppression of multi-photon emission +events achieved [14–16, 37, 46, 48]. +Within such cir- +cuit and excitation improvements, two-photon interfer- +ence with near-unity visibility seems to be within reach. +To realize circuits combining multiple QD sources cou- +pled to cavities, deterministic fabrication technologies +such as in-situ electron-beam lithography or imaging will +be required. This will allow to preselect emitters with +identical spectral characteristics, build cavities around +them and combine them within a single functional pho- +tonic circuit. In principle, since QD imaging could be +performed in an automatized manner, a very large num- +ber of emitters could be combined on a single chip. At +such a stage of complexity, separate control over QD +emission energies might also be desired. +This could +be directly implemented by a local laser drive via AC +Stark effect [49] or adapting the circuit for the electric +field [21, 46] and strain [22, 50–52] control. Ultimately, a +practical quantum photonic chip will require the presence +of additional functionalities such as single-photon detec- +tors and phase-shifter. +Fortunately, the GaAs circuits +are compatible with superconducting detectors technol- +ogy [53–55] and thanks to the large χ2 nonlinear coeffi- + +5 +DC +Circuit +Ring +cavity +QD emitters +Phase shifter +Detectors +FIG. 4. +Envisioned fully integrated quantum photonic circuit. +Draft of the possible circuit design with multiple +quantum dot-based single photon sources coupled to ring cavities, interconnected with ridge waveguides, directional couplers, +phase shifters, and superconducting detectors. +cient of the GaAs, electro-optical phase shifters have al- +ready been demonstrated [56]. Such an envisioned fully +functional QDs-GaAs circuit is shown schematically in +Fig. 4. +In conclusion, we have shown that two identical QD +single-photon sources can be integrated monolithically in +a waveguide circuit and made to interfere with visibility +limited by the coherence of those sources. We pointed +out the potential strategies to improve the QDs perfor- +mance by employing deterministic fabrication and cavity +enhancement. The implemented integrated system could +be potentially further extended to facilitate more com- +plex circuits and fully on-chip operation. Results shown +in this article, along with a clearly outlined path for fu- +ture improvements, take us one step closer to scalable +integrated quantum circuits based on quantum emitters +capable of generating and manipulating large photonic +states. +Methods +Sample description. +To fabricate our integrated +single-photon source waveguide device, we use a semi- +conductor sample that contains self-assembled In(Ga)As +QDs grown by the Stranski-Krastanow method at the +center of a planar GaAs microcavity. +The lower +and upper cavity mirrors contain 24 and 5 pairs of +Al0.9Ga0.1As/GaAs λ/4-layers, respectively, yielding a +quality factor of ∼200. +A δ-doping layer of Si donors +with a surface density of roughly ∼1010 cm−2 was grown +10 nm below the layer of QDs to dope them probabilis- +tically. +To fabricate ridge waveguides devices, the top +mirror layer along with the GaAs cavity is etched down, +forming the ridge with a width of ∼0.6 µm and a height +of ∼1.3 µm. +The cross-section of the WG with layer +structure is shown in Figure 1b (see also Supplementary +Section 1). Ridges have been defined by e-beam lithog- +raphy and reactive ion etching. +After processing, the +sample was cleaved perpendicularly to the WGs, around +30 µm away from the tapered out-coupler edges to get +clear side access. +Integrated circuit design We designed and fabricated +GaAs directional couplers with different coupling lengths +and gaps. A directional coupler with a near 50:50 cou- +pling ratio at around 1.3931 eV was obtained when the +gap distance was set to 120 nm and the coupling length to +30 µm (see Supplementary Section 2 for layout scheme). +The total length of the device was about 1 mm, including +four S-bends with a radius of 60 µm and the input/output +waveguides. +Experimental setup. +For all experiments, the sam- +ple is kept in a low-vibrations closed-cycle cryostat (at- +toDry800) at temperatures of ∼4.5 K. The cryostat is +equipped with two optical windows allowing for access +from the side and the top of the sample. +A spectro- +scopic setup consisting of two independent perpendicu- +larly aligned optical paths is employed (see Supplemen- +tary Section 3 for more details). +QDs embedded into +WGs are excited from the top through a first microscope +objective with NA = 0.26, while the emission signal is +detected from a side facet of the WG with a second ob- +jective with NA = 0.4. The photoluminescence signal, +simultaneously collected from both output arms of the +DC, is then passed through a Dove prim to rotate the +sample image plane from a horizontal into a vertical di- +rection to fit the monochromator slit orientation. +For +PL analysis, the signal is then spectrally dispersed by a +75 cm focal length monochromator and focused on a low- +noise liquid-nitrogen-cooled CCD camera (around 40 µeV +spectral resolution), allowing to resolve signal from both +DC output arms spatially. For HBT and HOM experi- +ments, the monochromator serves as a spectral filter with +70 µeV width, and the signal from both DC outputs is +introduced into separate single-mode optical fibres con- +nected with superconducting single-photon counting de- +tectors (30 ps time-response). +Integrated beamsplitter visibility. To test the clas- +sical visibility of the DC device, we simultaneously send +the continuous wave laser light tuned to the energy of +QDs transitions, using circular reflectors placed on the +ends of the input arms waveguides (see Supplementary +Section 8). +The power of the laser coupled into both + +O6 +arms was adjusted such that the intensity from both in- +put arms was the same. Next, we focused on the signal +passing through DC and out coupled from one output +arm. We observed intensity modulation in the function of +time, related to small path-length difference fluctuation, +allowing us to see interference pattern and calculate the +interferometer visibility. In the case of the investigated +device, the classical visibility of 98±1% was extracted. +Correlation histograms analysis. For the time post- +selected visibility V ′ analysis, we assume that for distin- +guishable photons g(2) +HOMd(0) is equal to 0.5, as reference +measurement is not possible. It allows to calculate V ′ ac- +cording to V ′ = [0.5 − g(2) +HOM(0)]/0.5. Data from Fig.3b +lead to raw visibility of V ′ +cw = 30% and V ′ +p = 38% for +cw and pulsed excitation, respectively. For g(2) +HBT (τ) and +g(2) +HOM(τ) correlation functions evaluation we take into +account a presence of the time-independent background +offset in recorded histograms (it constitutes to around +15-20% of the coincidences), which we relate to the dark +counts of the SSPDs (100-500 cps). +Non-background- +corrected HOM graphs can be found in Fig. 3c and the +Supplementary Section 9. +The authors thank Silke Kuhn for fabricating the +structures. +�L.D. acknowledges the financial support +from the Alexander von Humboldt Foundation. We ac- +knowledge financial support by the German Ministry +of Education and Research (BMBF) within the project +”Q.Link.X” (FKZ: 16KIS0871). +We are furthermore +grateful for the support by the State of Bavaria. +∗ lukaszd@princeton.edu +[1] Kok, P.; Lovett, B. Introduction to Optical Quantum In- +formation Processing; Cambridge University Press, 2010. +[2] Hong, C. K.; Ou, Z. Y.; Mandel, L. 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Optics Communications 2014, +327, 49–55. + +Supporting Information: +On-chip Hong-Ou-Mandel interference from separate quantum +dot emitters in an integrated circuit +�Lukasz Dusanowski,1, 2, ∗ Dominik K¨ock,1 Christian Schneider,1, 3 and Sven H¨ofling1 +1Technische Physik and W¨urzburg-Dresden Cluster of Excellence ct.qmat, +University of W¨urzburg, Physikalisches Institut and +Wilhelm-Conrad-R¨ontgen-Research Center for Complex Material Systems, +Am Hubland, D-97074 W¨urzburg, Germany +2currently at: Department of Electrical and Computer Engineering, +Princeton University, Princeton, NJ 08544, USA +3Institute of Physics, University of Oldenburg, D-26129 Oldenburg, Germany +(Dated: January 5, 2023) +1 +arXiv:2301.01706v1 [quant-ph] 4 Jan 2023 + +S1: SAMPLE STRUCTURE +The full layer structure of the sample is shown in Figure S1. +etched +etched +600 nm +1250 nm +Bottom DBR mirror: +24x Al0.9Ga0.1As/GaAs +Top DBR mirror: +5x Al0.9Ga0.1As/GaAs +GaAs substrate +In(Ga)As QDs and WL +GaAs +λ-cavity +1 µm +FIG. S1. Planar sample scanning electron microscope cross-section image with visible layers and +schematically marked areas for etching. +The quantum dot layer is placed inside a center of a +λ cavity sandwiched between two distributed Bragg Reflectors consisting of the 5/24 alternating +λ/4-thick layers of Al0.9Ga0.1As and GaAs. +S2: INTEGRATED CIRCUIT LAYOUT +Figure S2 shows the layout scheme of the fabricated GaAs device. It is based on 600 nm +width single-mode ridge waveguides. The central part of the circuit is a directional cou- +pler (DC) formed by two WGs separated by a 120 nm gap along a 30 µm long coupling +region. WGs were brought together using two circular bend regions with a radius of 60 µm. +Waveguides on the right end of the circuit are terminated by inverse taper (30 µm length) +out-couplers, minimizing reflection and optimized for better light extraction out of the chip. +The left side of the DC consists of 1.2 mm long straight WG sections designated for search- +ing two quantum dots with the same transition frequencies. WGs on the left side of the +circuit are terminated with circular Bragg grating mirrors optimized for increased reflectiv- +ity (around 80% expected at 900 nm). Figure S3 shows the scanning electron microscope +images of the fabricated integrated circuits. +2 + +350nm +30µm +30µm +Gap=120nm +W=600nm +70µm +R=60µm +R=60µm +1200µm +730nm +FIG. S2. Integrated circuit layout. Scheme of the fabricated GaAs directional coupler including +circular Bragg reflectors located at the ends of the input WG arms and inverse taper out-couplers +at the end of the output arms. Bending regions are based on circular profiles with a radius of +60 µm. +a +b +c +d +e +f +120 µm +60 µm +30 µm +3 µm +3 µm +3 µm +FIG. S3. +Scanning electron microscope images of the fabricated devices. +a,b, The DCs with +different coupling lengths. c,d, The DC with 30 µm length coupling region. e, An inverse taper +outcoupler. f, A circular Bragg grating reflector. +S3: OPTICAL SET-UP +For all experiments, the sample is kept in a low-vibrations closed-cycle cryostat (at- +toDry800) at temperatures of ∼4.5 K. The cryostat is equipped with two optical windows +allowing for access from both side and top of the sample. A spectroscopic setup consisting of +two independent perpendicularly aligned optical paths is employed as shown schematically +3 + +SSPD1 +SSPD2 +Ti:Si pulsed +laser 813nm +76 MHz +Pulse picker +19MHz +(for TRPL) +BS 50:50 +CW 660nm laser +Obj. +x10 +BS 92:8 +Dove prism +Obj. x20 +L1 +L2 +L3 +CCD +Slit +Slit +Knife edge +mirror +Monochromator +Sample in cryostat +Time-tagger +L4 +SM fiber +SM fiber +SM fiber +0.5 ns delay +fiber +ND filter +excitation +XYZ position +control +HWP+LP +90 deg image rotation +Dove prism +Mirror +FIG. S4. Optical setup. Scheme of the experimental configuration used for top excitation (blue +path) and side detection (red path) photoluminescence and resonance fluorescence measurements. +In the case of two-photon interference experiments, a QD was excited twice every laser pulse cycle +with a delay of 3 ns, and the subsequently emitted photons, spatially and temporally overlapped +in an unbalanced Mach-Zehnder interferometer (dashed lines) utilizing polarization maintaining +(PM) fibers and beam-splitters (BS). For signal detection, two avalanche photo-diodes (APD) with +350 ps response time were used. For polarization control in free space, a half-wave-plate (HWP) +combined with a linear polarizer (LP) was used, while for polarization rotation (PR) in the fiber- +based HOM interferometer ceramic sleeve connectors between two fiber facets were used allowing +to align fast and slow axis at the desired angle. +in Figure S4. Additionally, the excitation path allows for the separate routing of two laser +beams for the simultaneous excitation of two spots on the sample. For HOM and HBT +experiments, a tunable Ti:Si picosecond pulsed laser is used. QDs embedded into the two +input arms of the DC are excited from the top through a first microscope objective with +x10 magnification and NA = 0.26, while the emission signal from both DC output arms +is detected simultaneously from the side facet of the sample with a second objective with +x20 magnification and NA = 0.4. Photoluminescence signal from both arms is then passed +through a spatial filter (lenses L1 and L2) and polarization optics. For light polarization +analysis, a half-wave plate (HWP) combined with a linear polarizer (LP) is used. The sam- +ple image plane is rotated from a horizontal into a vertical direction using a Dove prism, +which allows simultaneous coupling signals from both DC output arms into the monochro- +4 + +mator. The collected light is analyzed by a high-resolution monochromator equipped with +a liquid nitrogen-cooled low-noise charge-coupled device detector (CCD), featuring a spec- +tral resolution of ∼40 µeV. Taking advantage of the spatial separation of DC output arms, +the spectrum from both WG arms is resolved spatially on the CCD camera. +For HBT +and HOM experiments, the monochromator is used as a spectral filter with 70 µeV width. +Next, the signal from both arms is separated spatially using a knife-edge mirror and cou- +pled into single-mode fibres interconnected with superconducting single-photon detectors +(SSPD). The time-correlated measurements are acquired using a stand-alone time-tagger. +S4: POWER-RESOLVED PL +In Fig. S5 QD1 and QD2 emission intensity as a function of excitation power is shown. +An almost linear dependence of the emission intensity on excitation power suggests that the +analyzed lines originate from the recombination of neutral or charged excitonic complexes. +a +b +0.01 +0.1 +1 +0.01 +0.1 +1 +PL intensity (arb. units) +Power (P/Psat) +QD1 +I~ P0.90±0.05 +QD2 +I~ P0.93±0.05 +PL intensity (arb. units) +Power (P/Psat) +FIG. S5. +Photoluminescence intensity vs incident excitation power. Solid red/blue curve: fit with +a power function revealing linear dependence of the emission intensity on excitation power. +5 + +S5: WAVEGUIDE TRANSMISSION LOSSES +To estimate the quality of the etched ridge waveguides, the optical WG transmission +losses were determined. For that purpose, the sample was excited with very high pumping +power, allowing us to observe spectrally broad QD ensemble emission. The beam spot was +scanned along the DC input arm 1/2, and emission was detected from the side through the +waveguide arm 1. Figure S6a and b show the corresponding attenuation of the measured +intensities at 890 nm plotted as a function of the distance to the DC bends for input arms 1 +and 2, respectively. Input arm 1 exhibits transmission losses on the level of 6.5±0.5 dB/mm +and arm 2 of 5.0±0.6 dB/mm. Waveguide transmission characteristics are limited by ridge +sidewall imperfections, which could be potentially further improved by optimizing the etch- +ing process. +a +b +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +8 +7 +6 +5 +4 +3 +2 +1 +0 +-1 +0.0 +0.2 +0.4 +0.6 +0.8 +5 +4 +3 +2 +1 +0 +-1 +Input Arm 1 +Losses: 6.5±0.5 dB/mm + experiment + fit +Attenuation (dB) +Distance (mm) +Input Arm 2 +Losses: 5.0±0.6 dB/mm + experiment + fit +Attenuation (dB) +Distance (mm) +FIG. S6. Waveguides transmission losses. Attenuation of the side detected ensemble PL signal in +function of the distance from DC bend regions. +6 + +S6: POLARIZATION-RESOLVED PL +Side detected emission from both studied emission lines show a high degree of linear +polarization (DOLP) of around 95%, oriented in the sample plane as shown in Fig. S7. A +high DOLP and its direction are related to the QDs dipole moments, which are mainly +in-plane oriented and thus emitted photons mostly couple to and propagate in the TE +waveguide mode. It needs to be noted that high DOLP is maintained after passing the +whole circuits consisting of bends, DC, and out-coupler, which could potentially spoil the +detected polarization contrast. The same polarization level is observed for both output arms +of the DC. +QD1 +QD2 +0 +30 +60 +90 +120 +150 +180 +210 +240 +270 +300 +330 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +30 +60 +90 +120 +150 +180 +210 +240 +270 +300 +330 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +30 +60 +90 +120 +150 +180 +210 +240 +270 +300 +330 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +30 +60 +90 +120 +150 +180 +210 +240 +270 +300 +330 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Norm. PL intensity (arb. units) + experiment + Sine fit +DOLP = 95±2% +DOLP = 95±2% +DOLP = 94±2% +DOLP = 95±2% +Output Arm 1 +Output Arm 2 +FIG. S7. Polarization characteristics of the QD1 and QD2 emission coupled to input arms of the +DC and detected from side facet output arms 1 and 2. Both QDs PL emission is strongly linearly +polarized with around 95% degree of linear polarization. +7 + +S7: EMITTERS TRANSITION LINEWIDTHS +Figure S8 shows a high-resolution spectrum of the QD1 and QD2 emission recorded under +cw 660 nm excitation. Measurements are performed using a scanning Fabry-Perot interfer- +ometer with a 3 µeV Lorentzian profile linewidth. Both spectra are fit using Lorentzian +functions with full-width at half-maximum (FWHM) of 16.5±2.5 µeV and 6.0±0.2 µeV, for +QD1 and QD2, respectively (error related to fit precision). It can be shown that the convo- +lution of two Lorentzian profiles with FWHM1 and FWHM2 is also a Lorentzian profile with +broadening of FWHM1+FWHM2. Following the above, we can correct the recorded optical +linewidths for the finite resolution of our setup by simply subtracting its linewidth. The +deconvoluted linewidths are 13.5±2.5 µeV and 3.0±0.2 µeV for QD1 and QD2, respectively. +a +b +-60 +-40 +-20 +0 +20 +40 +60 +80 +-60 +-40 +-20 +0 +20 +40 +60 +QD1 +FWHMfit= 16.5±1.5 meV +FWHMdec.= 13.5±1.5 meV + experiment + Lorentz fit +Intensity (arb. units) +Detuning (meV) +QD2 +FWHMfit = 6.0±0.2 meV +FWHMdec.= 3.0±0.2 meV + experiment + Lorentz fit +Intensity (arb. units) +Detuning (meV) +FIG. S8. A high-resolution PL spectrum of QD1 and QD2, obtained using a home-built Fabry- +Perot scanning cavity with a resolution of 3 µeV (FWHM), and free spectral range of 140 µeV at +890 nm. Solid lines are fits with the Lorentz function. +S8: DIRECTIONAL COUPLER CHARACTERISTICS +To extract the DC splitting ratio r : t accounting for the different performance of both +output arms due to the fabrication imperfections, the following procedure has been used. +First, QD located in input arm 1 was excited, and PL signal from output arm 1 II1 +O1 and +arm 2 II1 +O2 collected. Next, the same measurement has been repeated on QD located in arm +2, revealing II2 +O1 and II2 +O2. If the uneven outcoupling efficiency between the two output arms +8 + +is quantified as a constant x (transmission ratio between two arms), the following set of +equations applies +r + t = 1, +xr +t = II1 +O1 +II1 +O2 +, +x t +r = II2 +O1 +II2 +O2 +. +(1) +Based on that, the DC splitting ratio r : t can be derived, accounting for the imbalanced +outcoupling +r : t = +� +II1 +O1 +II1 +O2 +II2 +O2 +II2 +O1 +. +(2) +In the case of the investigated DC with II1 +O1/II1 +O2 of 51:49 and II2 +O2/II2 +O1 of 46:54 we ended up +with the r:t ratio of 48:52, which is very close to the desired 50:50. +0 +50 +100 +150 +200 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Intensity (cps) +Time frame +laser coupled to: + Input Arm 1 + Input Arm 2 + Arm 1+2 (interference) +classical visibility at 890 nm +V = (Imax-Imin)/(Imax+Imin) = 98±1% +FIG. S9. Intensity fluctuations of the cw laser light transmitted simultaneously through two input +arms of the DC. Fluctuations in time are related to the off-chip path-length difference fluctuations +of signal interfered on the directional coupler. +To test the classical visibility of the DC device, we simultaneously send cw laser light +tuned to the energy of QDs transitions (890 nm), using circular reflectors placed on the +ends of the input arms waveguides. The power of the laser coupled into both arms was +adjusted, so the intensity from both input arms was the same. Next, we focused on the +9 + +signal passing through the DC and out-coupled from one of the output arms. We observed +intensity modulation in the function of time, related to the small path-length difference +fluctuation, allowing us to record interference pattern as shown in Fig. S8. Classical DC +visibility was obtained by calculating the intensity contrast +VDC = Imax − Imin +Imax + Imin +. +(3) +In the case of the investigated DC device, a classical visibility of 98±1% was extracted. +10 + +S9: RAW HOM CORRELATION DATA +Figure S10a shows a non-corrected two-photon Hong-Ou-Mandel interference experiment +result between QD1 and QD2 performed under 76MHz and 19MHz pulsed laser repetition +rate. In the case of both graphs, the same time-independent background offset is visible, +corresponding to around 15% of the normalized peak intensity. Since the background level +is the same for the HOM graphs at different laser repetition rates, we exclude the emit- +ters long-decay contribution to the observable background. We attribute the observed cw +offset to the SSPDs dark counts (100-500 cps). Figure S10b shows the non-corrected nor- +malized histogram of the central eleven peaks areas (∆ = 3 ns integration window) of the +HOM second-order cross-correlation in case of synchronized - indistinguishable (red bars) +and 0.5 ns delayed - distinguishable (grey bars) photons. The non-corrected central peak +area in case of synchronized photons is equal to g(2) +HOM(0, ∆t) = 0.587±0.002, while in case +of unsynchronized photons g(2) +HOMd(0, ∆t) = 0.680±0.002. In this case, the non-corrected +two-photon interference visibility yields Vraw = 12.1±0.3% in correspondence to 17.8±0.7% +visibility obtained for background-corrected graphs (see Fig.3c in the main text). +a +b +-50 +-25 +0 +25 +50 +75 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Raw data +Raw norm. HOM counts - g(2) +HOM(t) +Delay time (ns) + 19 MHz + 76 MHz +cw bck +-5 -4 -3 -2 -1 +0 +1 +2 +3 +4 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 Dtint = 3 ns +Vraw = 12.1±0.3% +g(2) +HOM(0,Dt) = 0.587±0.002 +Raw norm. HOM +peak area - g(2) +HOM(t,Dt) +Peak number + indist. (sync.) + dist. (0.5 ns delay) +bck +FIG. S10. +a, Two-photon Hong-Ou-Mandel interference measurement between QD1 and QD2 +performed using on-chip beamsplitter showing the normalized coincidences versus the delay time +with no background counts correction. An experiment performed under 76MHz and 19MHz pulsed +laser repetition rate yields the same cw background. b, Non-corrected integrated counts of the +central eleven peaks (3 ns integration window) of the HOM correlation in case of synchronized (blue +bars) and 0.5 ns delayed (red bars) photons from QD1 and QD2 under pulsed 76MHz excitation. +11 + +S10: SINGLE PHOTON EMISSION UNDER CW EXCITATION +In Figure S11, HBT second-order correlation histograms recorded under cw (660 nm) +excitation for QD1 and QD2 are shown. The data in Fig. S11 are fit with the function +g(2) +HBT(τ) = (1 − g(2) +HBT(0))exp(−|τ|/τd) convoluted with 80 ps width Gaussian instrumental +response function, where τ is the delay time between detection events, g(2) +HBT(0) is the prob- +ability of two-photon emission events, τd is decay time constant corresponding to the sum +of the spontaneous emission rate 1/T1 and pump rate G of the source. +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +g(2) +HBT(t) +Delay time (ns) +0 +10 +20 +30 +40 +50 +60 +70 +Raw coincidences +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +g(2)(0) = 0.08±0.01 +g(2) +HBT(t) +Delay time (ns) +g(2)(0) = 0.16±0.01 +0 +10 +20 +30 +40 +50 +60 +70 +Raw coincidences +a +b +QD1 +QD2 +FIG. S11. Second order auto-correlation histograms of a QD1 and b QD2 emission under non- +resonant (660 nm) cw excitation. Data have been recorded in HBT configuration using an on-chip +beamsplitter. Presented data are shown as measured with no background subtraction or other +corrections. +12 + diff --git a/-tAzT4oBgHgl3EQfvf0n/content/tmp_files/load_file.txt b/-tAzT4oBgHgl3EQfvf0n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b506a68783e8ccfc28caa6716daa17bb62544ea --- /dev/null +++ b/-tAzT4oBgHgl3EQfvf0n/content/tmp_files/load_file.txt @@ -0,0 +1,1505 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf,len=1504 +page_content='On-chip Hong-Ou-Mandel interference from separate quantum dot emitters in an integrated circuit �Lukasz Dusanowski,1, 2, ∗ Dominik K¨ock,1 Christian Schneider,1, 3 and Sven H¨ofling1 1Technische Physik and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='qmat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' University of W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Physikalisches Institut and Wilhelm-Conrad-R¨ontgen-Research Center for Complex Material Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Am Hubland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' D-97074 W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Germany 2currently at: Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' NJ 08544,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' USA 3Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' University of Oldenburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' D-26129 Oldenburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Germany (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 2023) Scalable quantum photonic technologies require low-loss integration of many identical single- photon sources with photonic circuitry on a chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Relatively complex quantum photonic circuits have already been demonstrated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' however, sources used so far relied on parametric-down-conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Hence, the efficiency and scalability are intrinsically limited by the probabilistic nature of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Quantum emitter-based single-photon sources are free of this limitation, but frequency matching of multiple emitters within a single circuit remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In this work, we demonstrate a key component in this regard in the form of a fully monolithic GaAs circuit combing two frequency- matched quantum dot single-photon sources interconnected with a low-loss on-chip beamsplitter connected via single-mode ridge waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' This device enabled us to perform a two-photon inter- ference experiment on-chip with visibility reaching 66%, limited by the coherence of the emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Our device could be further scaled up, providing a clear path to increase the complexity of quantum circuits toward fully scalable integrated quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Optical quantum computing and communication ap- plications with single photons and linear optics rely crit- ically on the quantum interference of two photons on a beamsplitter [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' This process, known as Hong-Ou- Mandel (HOM) effect, occurs when two identical single photons enter a 50:50 beamsplitter, one in each input port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' When the photons are indistinguishable, they will coalesce into a two-photon Fock state [2], in which they exit the same but random output port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' This process un- derlines the simplest non-trivial path-entangled NOON state generation and introduces an optical non-linearity which is the base for the implementation of more-complex photonic gates and protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Consequently, scalable optical quantum information technologies will require integrating many identical in- distinguishable single-photon sources with reliable pho- tonic circuits consisting of beamsplitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Utilizing well- developed integrated photonics technology is particularly appealing in this regard, as it dramatically reduces the footprint of quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Furthermore, it allows con- trolling photon states with high fidelity due to the in- trinsic sub-wavelength stability of the path-lengths, low- losses, and near-perfect mode overlap at an integrated beamsplitter for high-fidelity quantum interference [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Advances in integrated photonic technology allowed already realizations of relatively complex quantum cir- cuits demonstrating CNOT-gate operation [6, 7], boson sampling [7, 8], quantum walks [9], some simple quan- tum algorithms [7, 10] and chip-to-chip quantum tele- portation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' A combination of integrated photonic cir- cuits with spontaneous four-wave mixing photon sources has also been achieved [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' However, due to the used sources’ probabilistic nature, their efficiency and scalability are intrinsically limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Quantum emitters- based single-photon sources are free of this limitation and recently have been shown to outperform spontaneous four-wave mixing and down-conversion photon sources in simultaneously reaching high levels of photons indis- tinguishability and brightness [14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Moreover, re- mote interference between two quantum emitters was al- ready demonstrated using trapped ions [19, 20], quan- tum dots [21–30], organic molecules [31, 32] or vacancy centers in diamond [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The vast majority of those experiments have been performed in free space as proof- of-principle demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Performing similar exper- iments on-chip, taking advantage of the photonic cir- cuit consisting of fully integrated quantum emitters and beamsplitter, has not been performed yet, and it is still a missing component towards scaling-up aforementioned quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In this work, we demonstrate a crucial component in this regard in the form of a fully monolithic GaAs cir- cuit combing two frequency-matched quantum dot single- photon sources interconnected with on-chip beamsplitter via single-mode ridge waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' This device enabled performing two-photon interference experiments on-chip with visibility limited by the coherence of our emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Our semiconductor photonic device is schematically presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 1a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' It is based on InAs/GaAs distributed Bragg-reflector ridge waveguides, which have been proven to facilitate high optical quality quantum dot single-photon sources [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The central part of the de- vice consists of the single-mode directional coupler (DC), which is the integrated optical analog of the bulk beam- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='01706v1 [quant-ph] 4 Jan 2023 2 QD1 QD2 DC Output Arm 1 Output Arm 2 a c QD1 QD1 QD2 QD2 Input Arm 1 Input Arm 2 QDs 5x DBR 24x DBR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 µm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3 µm λ/2 cav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' b … 1k 2k 3k 4k 5k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3930 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3935 1k 2k 3k 4k 5k Output Arm 2 QD2 PL intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' units) QD1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3930 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3935 Energy (eV) Output Arm 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' On-chip two-photon interference circuit and beam splitting operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a, Schematic representation of the photonic circuit based on a directional coupler (DC) interconnected with two input waveguides with coupled quantum dots and two output arms with inverted tapers for photons collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' b, Ridge waveguide cross-section with marked layer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' c, Demonstration of beam splitting operation for fabricated DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The photoluminescence signal from QD1 and QD2 is recorded for Output Arms 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' QD1 and QD2 are frequency matched with a precision of 5 µeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In the two input arms of the DC, two frequency- matched quantum dots (QDs) are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For single- photon generation, the QDs are excited non-resonantly from the top using two separated picosecond pulsed laser beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Photons interfered on the DC are finally col- lected off the chip using inverse taper out-couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Spec- tral filtering and detection are performed off-chip using a monochromator and two superconducting single-photon detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To find two quantum dots with matching optical tran- sition energies, the position of the excitation beam spot on each input arm of the DC was scanned using an au- tomatized translation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Within such a scanning routine, we localized two matching emission lines at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3931 eV energy originating from the QDs located in two individual input arms of the DC and separated spa- tially by around 200 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 1b, photoluminescence (PL) spectra from QD1 and QD2 recorded from DC out- put arms 1 and 2 are presented at a temperature of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Single well-resolved emission lines matching within 5 µeV fit precision are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Comparing amplitudes of QD1 and QD2 emission peaks visible within both output arms, a beam splitting ratio of 48:52 is derived (including un- even transmission through out-coupling arms - more de- tails in Supplementary Section 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To show that optical excitation of our QDs leads to the generation of single photons, we analyzed the photon emission statistics of separate QDs by performing sec- ond order-correlation experiments in Hanbury Brown and Twiss (HBT) configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For that purpose, QDs have been excited non-resonantly from the top by an 813 nm wavelength train of picosecond pulses at a repetition rate of 76 HMz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Photons emitted by the QDs were then cou- pled into the circuit input arm waveguides and guided into the directional coupler, where the signal was divided between two output arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Next, photons were collected off-chip from the side of the sample using out-couplers and subsequently filtered spectrally by a monochroma- tor (70 µeV width) and coupled into two single-mode fibres connected with superconducting single-photon de- tectors (SSPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Finally, the photon correlation statistics were acquired by a multichannel picosecond event timer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Data have been recorded under excitation powers corre- sponding to half of the QD saturation intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 2a and c present the second-order autocorrelation function g(2) HBT (τ) measurement recorded for each QD in- dividually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In the case of both QDs, a clear suppression of the central peak counts is visible, proving single-photon emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To quantitatively evaluate the probability of multi-photon emission, g(2) HBT (0) values were calculated by integrating residual counts of the zero delay peak with respect to the neighboring six peaks, resulting in g(2) HBT (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='08 and g(2) HBT (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='02 for QD1 and QD2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 2b and d, time- resolved photoluminescence traces of the QD1 and QD2 emission are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In this case, the repetition rate of the laser was reduced to 19 MHz using a pulse picker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Clear bi-exponential signal decays are visible, with a fast and slow time constant of 720±5 ps and 12±1 ns for QD1 and 600±5 ps and 22±1 ns for QD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We attribute the fast decay to the spontaneous recombination of electron- hole pairs in QD (T1) and the slow one, which corre- sponds to about 2% (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2%) of the total QD1 (QD2) line intensity, is tentatively interpreted as the recapturing of the carriers by the QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Using fit parameters obtained from the time-resolved experiments, g(2) HBT (τ) correla- tion histograms have been fitted with double-sided bi- exponential decay convoluted with 80 ps width Gaussian 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 experiment fit 20 40 60 80 100 120 140 160 180 Raw coincidences 0 5 10 15 20 101 102 103 PL (coincidences) Time (ns) Afast/Aslow = 85 45 -30 -15 0 15 30 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 g(2) HBT(t) Delay time (ns) 20 40 60 80 100 120 140 160 101 102 103 experiment bi-exp fit Afast/Aslow = 50 QD1 QD2 a b c d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Single-photon generation and emission dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a,c Second order auto-correlation histograms of QD1 and QD2 emission under pulsed 76 MHz repetition rate excita- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Data have been recorded in HBT configuration using an on-chip beamsplitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' b,d Time-resolved PL traces revealing bi-exponential decays with fast (slow) time constant of 720 ps (12 ns) and 600 ps (22 ns) for QD1 and QD2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' instrumental response function (black dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To demonstrate on-chip two-photon interference, the single QDs in both input arms of the DC are excited us- ing the picosecond pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For that, the laser beam is di- vided into two independently controllable optical excita- tion axis and synchronized in advance to ensure optimal temporal overlap of emitted photons on the DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' It is per- formed by sending the emission from each QD separately through the on-chip DC and using time-resolved detec- tion to eliminate the time delay difference between inde- pendently generated single photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The same technique is used to introduce an intentional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 ns time delay for reference measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The excitation laser powers for each QD are adjusted such that their emission intensities are the same (around half of QD1 saturated intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' As we utilize on-chip beam splitting operation using DC with single-mode inputs and outputs, we expect a very high spatial mode overlap of our interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To test this, we send the continuous wave laser simultaneously into both DC input arms and record classical interference fringes with 98±1% visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In earlier experiments per- formed on ridge waveguide structures, we observed that the QD emission couples into the well-defined transverse- electric mode of the WG with close to unity degree of linear polarization [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In the case of the investigated device, the polarization of the emitted photons was an- alyzed after passing the whole circuit consisting of bend regions, DC itself, and out-couplers (more details in Sup- plementary Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We found that for both QDs, the degree of polarization is above 95%, suggesting optimal polarization alignment for interference experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Within the above-mentioned prerequisites, the two- photon interference should be mainly limited by the co- herence of our single-photon emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To get access to the coherence times of our QDs, we performed a high- resolution measurement of the emission linewidths us- ing a scanning Fabry-Perot interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We extract the full-width at half-maximum of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 µeV and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 µeV by Lorentzian fit for QD1 and QD2, re- spectively (see Supplementary Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The coher- ence times calculated based on observed broadenings are T QD1 2 = 100±20 ps and T QD2 2 = 440±30 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' As the mea- surements are performed on the tens of seconds timescale, we speculate that recorded coherence times might be lim- ited by charge and spin noise [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' [40], we calculated the expected interference visibility of our two independent emitters and derived the theoretical vis- ibility in the range of Vtheory = 10-15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Figure 3a shows two-photon interference data in the form of second-order HOM cross-correlation between photons exiting the two output arms of the on-chip beam- splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The height of the central peak is clearly below the half intensity of the neighboring peaks, proving that photons emitted by two separate QDs indeed interfere on the DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Another interference signature is the pres- ence of the coincidences dip superimposed on the central peak around zero time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The depth of this dip con- stitutes to the interference events where photons arrive simultaneously at the DC, giving rise to the narrow-time window post-selected coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In our case, the exact value of g(2) HOM(τ) at τ = 0 is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='17 in the case of background-corrected data and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='31 for as-measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The same type of time post-selected interference can be observed for cw HOM correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Figure 3b shows the non-corrected cw, and pulsed HOM interfer- ence histograms overlapped on each other (correspond- ing cw g(2) HBT (τ) graphs are shown in Supplementary Sec- tion 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Similar to the pulsed case, the cw correlation shows clear suppression of coincident counts at zero time delay, with time post-selected g(2) HOM(0) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='35, close to the pulsed as-measured value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To evaluate photons full wave-packet interference probability (non-post-selected), we calculate the pulsed HOM correlation central peak area normalized by the average area of the neighboring six peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For integration window ∆t of 3 ns, we obtain g(2) HOM(0, ∆t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='459±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='002 for background corrected data and g(2) HOM(0, ∆t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='587±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='002 for raw data, where uncertainty is based on the standard deviation of non-central peaks areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In the case of background- corrected data, we reach a value below the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' It needs to be noted that derived g(2) HOM(0, ∆t) and g(2) HOM(0) values are partially influenced by the non-zero multi-photon emission extend observed in HBT measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 4 45 30 15 0 15 30 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 g(2) HOM(t) Delay time (ns) 100 200 300 400 500 600 Raw coincidences 5 -4 -3 -2 -1 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 HOM peak area - g(2) HOM(t,Dt) Peak number indist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' (sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=') dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 ns delay) Dtint = 3 ns V = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='7% g(2) HOM(0,Dt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='459±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='002 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 Dtint 6 4 2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 cw pulsed g(2) HOM(t) Delay time (ns) a b c QD1 QD2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' On-chip two-photon interference from separate quantum emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a, Two-photon Hong-Ou-Mandel interference measurement between QD1 and QD2 showing the normalized HOM coincidences versus the delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The central peak area is suppressed with respect to neighboring peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Inset: Magnified view of the central peak area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' b, Raw HOM interference measurement recorded under cw (red points) and pulsed (blue points) excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' c, Integrated counts of the central eleven peaks (∆t = 3 ns integration window) of the HOM correlation in case of synchronized (blue bars) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 ns delayed (red bars) photons from QD1 and QD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' All presented data are recorded using an on-chip beamsplitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' As it has been recently pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' [41, 42], to estimate two-photon interference visibility for remote emitters correctly, it is necessary to perform reference HOM measurements for distinguishable photons, due to the possible blinking effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Since within the fabricated circuit polarization rotation is impossible to unambigu- ously confirm the two-photon interference and properly evaluate visibility, photons were made distinguishable by introducing a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 ns time delay between excitation pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Such delay should be sufficient to lose the temporal pho- tons overlap on the DC within the emitters coherence times and record reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Figure 3c demonstrates the normalized histogram of the central eleven peaks areas (∆t = 3 ns) of the HOM second-order cross-correlation in case of synchro- nized - indistinguishable (red bars) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 ns de- layed - distinguishable (grey bars) photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The cen- tral peak area in the case of unsynchronized pho- tons is equal to g(2) HOMd(0, ∆t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='558±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='002, which is slightly above the theoretically expected 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We relate this discrepancy with non-zero multi-photon emission extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Finally, we calculate remote sources two-photon interference visibility V following V = [g(2) HOMd(0, ∆t) − g(2) HOM(0, ∆t)]/g(2) HOMd(0, ∆t), resulting in V = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='7% for background corrected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' This value is relatively close to the theoretically expected vis- ibility and even partially exceeds it, suggesting that it is limited solely by the coherence of the emitters (more de- tails Supplementary Section 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Using background cor- rected data for the pulsed case, give post-selected visi- bility of V ′ p = 66%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The probability of the time post- selected interference is known to depend on the ratio of the emitters coherence times to the setup timing reso- lution [21, 22, 43], thus possibly even higher V ′ values could be potentially achieved with faster detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' While our results provide clear scientific evidence for on-chip generation and interference of on-demand sin- gle photons in a circuit, the recorded visibility values need to be improved for future practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In the current device architecture, the indistinguishability of interfered photons is limited by T2/(2T1) of particu- lar QDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We propose a few strategies for improving this ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Firstly, the QD charge environment could be stabi- lized via passivation [44], weak optical illumination [45], or gating [15, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' At the same time, by embedding QDs into optical cavities, the Purcell effect might be used to enhance the radiative emission rate 1/T1 [14–16, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Recently, we demonstrated a QD circuit with ring cav- ities allowing to significantly increase the QD coupling efficiency into the WG mode and decrease the T1 be- low 200 ps [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Finally, by applying a resonant exci- tation, the photons emission time-jitter could be mini- mized, and strong suppression of multi-photon emission events achieved [14–16, 37, 46, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Within such cir- cuit and excitation improvements, two-photon interfer- ence with near-unity visibility seems to be within reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To realize circuits combining multiple QD sources cou- pled to cavities, deterministic fabrication technologies such as in-situ electron-beam lithography or imaging will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' This will allow to preselect emitters with identical spectral characteristics, build cavities around them and combine them within a single functional pho- tonic circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In principle, since QD imaging could be performed in an automatized manner, a very large num- ber of emitters could be combined on a single chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' At such a stage of complexity, separate control over QD emission energies might also be desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' This could be directly implemented by a local laser drive via AC Stark effect [49] or adapting the circuit for the electric field [21, 46] and strain [22, 50–52] control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Ultimately, a practical quantum photonic chip will require the presence of additional functionalities such as single-photon detec- tors and phase-shifter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Fortunately, the GaAs circuits are compatible with superconducting detectors technol- ogy [53–55] and thanks to the large χ2 nonlinear coeffi- 5 DC Circuit Ring cavity QD emitters Phase shifter Detectors FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Envisioned fully integrated quantum photonic circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Draft of the possible circuit design with multiple quantum dot-based single photon sources coupled to ring cavities, interconnected with ridge waveguides, directional couplers, phase shifters, and superconducting detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' cient of the GaAs, electro-optical phase shifters have al- ready been demonstrated [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Such an envisioned fully functional QDs-GaAs circuit is shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In conclusion, we have shown that two identical QD single-photon sources can be integrated monolithically in a waveguide circuit and made to interfere with visibility limited by the coherence of those sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We pointed out the potential strategies to improve the QDs perfor- mance by employing deterministic fabrication and cavity enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The implemented integrated system could be potentially further extended to facilitate more com- plex circuits and fully on-chip operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Results shown in this article, along with a clearly outlined path for fu- ture improvements, take us one step closer to scalable integrated quantum circuits based on quantum emitters capable of generating and manipulating large photonic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Methods Sample description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To fabricate our integrated single-photon source waveguide device, we use a semi- conductor sample that contains self-assembled In(Ga)As QDs grown by the Stranski-Krastanow method at the center of a planar GaAs microcavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The lower and upper cavity mirrors contain 24 and 5 pairs of Al0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='9Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='1As/GaAs λ/4-layers, respectively, yielding a quality factor of ∼200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' A δ-doping layer of Si donors with a surface density of roughly ∼1010 cm−2 was grown 10 nm below the layer of QDs to dope them probabilis- tically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To fabricate ridge waveguides devices, the top mirror layer along with the GaAs cavity is etched down, forming the ridge with a width of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 µm and a height of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The cross-section of the WG with layer structure is shown in Figure 1b (see also Supplementary Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Ridges have been defined by e-beam lithog- raphy and reactive ion etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' After processing, the sample was cleaved perpendicularly to the WGs, around 30 µm away from the tapered out-coupler edges to get clear side access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Integrated circuit design We designed and fabricated GaAs directional couplers with different coupling lengths and gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' A directional coupler with a near 50:50 cou- pling ratio at around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3931 eV was obtained when the gap distance was set to 120 nm and the coupling length to 30 µm (see Supplementary Section 2 for layout scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The total length of the device was about 1 mm, including four S-bends with a radius of 60 µm and the input/output waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For all experiments, the sam- ple is kept in a low-vibrations closed-cycle cryostat (at- toDry800) at temperatures of ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The cryostat is equipped with two optical windows allowing for access from the side and the top of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' A spectro- scopic setup consisting of two independent perpendicu- larly aligned optical paths is employed (see Supplemen- tary Section 3 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' QDs embedded into WGs are excited from the top through a first microscope objective with NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='26, while the emission signal is detected from a side facet of the WG with a second ob- jective with NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The photoluminescence signal, simultaneously collected from both output arms of the DC, is then passed through a Dove prim to rotate the sample image plane from a horizontal into a vertical di- rection to fit the monochromator slit orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For PL analysis, the signal is then spectrally dispersed by a 75 cm focal length monochromator and focused on a low- noise liquid-nitrogen-cooled CCD camera (around 40 µeV spectral resolution), allowing to resolve signal from both DC output arms spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For HBT and HOM experi- ments, the monochromator serves as a spectral filter with 70 µeV width, and the signal from both DC outputs is introduced into separate single-mode optical fibres con- nected with superconducting single-photon counting de- tectors (30 ps time-response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Integrated beamsplitter visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To test the clas- sical visibility of the DC device, we simultaneously send the continuous wave laser light tuned to the energy of QDs transitions, using circular reflectors placed on the ends of the input arms waveguides (see Supplementary Section 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The power of the laser coupled into both O6 arms was adjusted such that the intensity from both in- put arms was the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Next, we focused on the signal passing through DC and out coupled from one output arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We observed intensity modulation in the function of time, related to small path-length difference fluctuation, allowing us to see interference pattern and calculate the interferometer visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In the case of the investigated device, the classical visibility of 98±1% was extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Correlation histograms analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For the time post- selected visibility V ′ analysis, we assume that for distin- guishable photons g(2) HOMd(0) is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5, as reference measurement is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' It allows to calculate V ′ ac- cording to V ′ = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 − g(2) HOM(0)]/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Data from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3b lead to raw visibility of V ′ cw = 30% and V ′ p = 38% for cw and pulsed excitation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For g(2) HBT (τ) and g(2) HOM(τ) correlation functions evaluation we take into account a presence of the time-independent background offset in recorded histograms (it constitutes to around 15-20% of the coincidences), which we relate to the dark counts of the SSPDs (100-500 cps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Non-background- corrected HOM graphs can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 3c and the Supplementary Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The authors thank Silke Kuhn for fabricating the structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' �L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' acknowledges the financial support from the Alexander von Humboldt Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We ac- knowledge financial support by the German Ministry of Education and Research (BMBF) within the project ”Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='Link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='X” (FKZ: 16KIS0871).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We are furthermore grateful for the support by the State of Bavaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' ∗ lukaszd@princeton.' metadata={'source': 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photons by inter- ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Physical Review Letters 1987, 59, 2044–2046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' [3] Bonneau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Silverstone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': 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quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Nature Photonics 2019, [6] Crespi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Ramponi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Osellame, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Sahin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Jahanmirine- jad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Frucci, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Mattioli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Leoni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Beetz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Ler- mer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Kamp, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' H¨ofling, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Sanjines, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Fiore, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Waveguide superconducting single-photon detectors for integrated quantum photonic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Applied Physics Letters 2011, 99, 181110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' [54] Reithmaier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Kaniber, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Flassig, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Lichtman- necker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' M¨uller, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Andrejew, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Vuˇckovi´c, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Gross, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Finley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' On-Chip Generation, Routing, and Detection of Resonance Fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Nano Letters 2015, 15, 5208–5213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' [55] Schwartz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Schmidt, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Rengstl, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Hornung, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Hepp, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Portalupi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Llin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Jetter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Siegel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Michler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Fully On-Chip Single-Photon Hanbury-Brown and Twiss Experiment on a Monolithic Semiconductor–Superconductor Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Nano Letters 2018, 18, 6892–6897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' [56] Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Gallium arsenide (GaAs) quantum pho- tonic waveguide circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Optics Communications 2014, 327, 49–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Supporting Information: On-chip Hong-Ou-Mandel interference from separate quantum dot emitters in an integrated circuit �Lukasz Dusanowski,1, 2, ∗ Dominik K¨ock,1 Christian Schneider,1, 3 and Sven H¨ofling1 1Technische Physik and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='qmat, University of W¨urzburg, Physikalisches Institut and Wilhelm-Conrad-R¨ontgen-Research Center for Complex Material Systems, Am Hubland, D-97074 W¨urzburg, Germany 2currently at: Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544, USA 3Institute of Physics, University of Oldenburg, D-26129 Oldenburg, Germany (Dated: January 5, 2023) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='01706v1 [quant-ph] 4 Jan 2023 S1: SAMPLE STRUCTURE The full layer structure of the sample is shown in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' etched etched 600 nm 1250 nm Bottom DBR mirror: 24x Al0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='9Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='1As/GaAs Top DBR mirror: 5x Al0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='9Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='1As/GaAs GaAs substrate In(Ga)As QDs and WL GaAs λ-cavity 1 µm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Planar sample scanning electron microscope cross-section image with visible layers and schematically marked areas for etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The quantum dot layer is placed inside a center of a λ cavity sandwiched between two distributed Bragg Reflectors consisting of the 5/24 alternating λ/4-thick layers of Al0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='9Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='1As and GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S2: INTEGRATED CIRCUIT LAYOUT Figure S2 shows the layout scheme of the fabricated GaAs device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' It is based on 600 nm width single-mode ridge waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The central part of the circuit is a directional cou- pler (DC) formed by two WGs separated by a 120 nm gap along a 30 µm long coupling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' WGs were brought together using two circular bend regions with a radius of 60 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Waveguides on the right end of the circuit are terminated by inverse taper (30 µm length) out-couplers, minimizing reflection and optimized for better light extraction out of the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The left side of the DC consists of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 mm long straight WG sections designated for search- ing two quantum dots with the same transition frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' WGs on the left side of the circuit are terminated with circular Bragg grating mirrors optimized for increased reflectiv- ity (around 80% expected at 900 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Figure S3 shows the scanning electron microscope images of the fabricated integrated circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 2 350nm 30µm 30µm Gap=120nm W=600nm 70µm R=60µm R=60µm 1200µm 730nm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Integrated circuit layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Scheme of the fabricated GaAs directional coupler including circular Bragg reflectors located at the ends of the input WG arms and inverse taper out-couplers at the end of the output arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Bending regions are based on circular profiles with a radius of 60 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a b c d e f 120 µm 60 µm 30 µm 3 µm 3 µm 3 µm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Scanning electron microscope images of the fabricated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a,b, The DCs with different coupling lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' c,d, The DC with 30 µm length coupling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' e, An inverse taper outcoupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' f, A circular Bragg grating reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S3: OPTICAL SET-UP For all experiments, the sample is kept in a low-vibrations closed-cycle cryostat (at- toDry800) at temperatures of ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The cryostat is equipped with two optical windows allowing for access from both side and top of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' A spectroscopic setup consisting of two independent perpendicularly aligned optical paths is employed as shown schematically 3 SSPD1 SSPD2 Ti:Si pulsed laser 813nm 76 MHz Pulse picker 19MHz (for TRPL) BS 50:50 CW 660nm laser Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' x10 BS 92:8 Dove prism Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' x20 L1 L2 L3 CCD Slit Slit Knife edge mirror Monochromator Sample in cryostat Time-tagger L4 SM fiber SM fiber SM fiber 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 ns delay fiber ND filter excitation XYZ position control HWP+LP 90 deg image rotation Dove prism Mirror FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Optical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Scheme of the experimental configuration used for top excitation (blue path) and side detection (red path) photoluminescence and resonance fluorescence measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In the case of two-photon interference experiments, a QD was excited twice every laser pulse cycle with a delay of 3 ns, and the subsequently emitted photons, spatially and temporally overlapped in an unbalanced Mach-Zehnder interferometer (dashed lines) utilizing polarization maintaining (PM) fibers and beam-splitters (BS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For signal detection, two avalanche photo-diodes (APD) with 350 ps response time were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For polarization control in free space, a half-wave-plate (HWP) combined with a linear polarizer (LP) was used, while for polarization rotation (PR) in the fiber- based HOM interferometer ceramic sleeve connectors between two fiber facets were used allowing to align fast and slow axis at the desired angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' in Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Additionally, the excitation path allows for the separate routing of two laser beams for the simultaneous excitation of two spots on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For HOM and HBT experiments, a tunable Ti:Si picosecond pulsed laser is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' QDs embedded into the two input arms of the DC are excited from the top through a first microscope objective with x10 magnification and NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='26, while the emission signal from both DC output arms is detected simultaneously from the side facet of the sample with a second objective with x20 magnification and NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Photoluminescence signal from both arms is then passed through a spatial filter (lenses L1 and L2) and polarization optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For light polarization analysis, a half-wave plate (HWP) combined with a linear polarizer (LP) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The sam- ple image plane is rotated from a horizontal into a vertical direction using a Dove prism, which allows simultaneous coupling signals from both DC output arms into the monochro- 4 mator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The collected light is analyzed by a high-resolution monochromator equipped with a liquid nitrogen-cooled low-noise charge-coupled device detector (CCD), featuring a spec- tral resolution of ∼40 µeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Taking advantage of the spatial separation of DC output arms, the spectrum from both WG arms is resolved spatially on the CCD camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For HBT and HOM experiments, the monochromator is used as a spectral filter with 70 µeV width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Next, the signal from both arms is separated spatially using a knife-edge mirror and cou- pled into single-mode fibres interconnected with superconducting single-photon detectors (SSPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The time-correlated measurements are acquired using a stand-alone time-tagger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S4: POWER-RESOLVED PL In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S5 QD1 and QD2 emission intensity as a function of excitation power is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' An almost linear dependence of the emission intensity on excitation power suggests that the analyzed lines originate from the recombination of neutral or charged excitonic complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='1 1 PL intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' units) Power (P/Psat) QD1 I~ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='05 QD2 I~ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='05 PL intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' units) Power (P/Psat) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Photoluminescence intensity vs incident excitation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Solid red/blue curve: fit with a power function revealing linear dependence of the emission intensity on excitation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 5 S5: WAVEGUIDE TRANSMISSION LOSSES To estimate the quality of the etched ridge waveguides, the optical WG transmission losses were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' For that purpose, the sample was excited with very high pumping power, allowing us to observe spectrally broad QD ensemble emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The beam spot was scanned along the DC input arm 1/2, and emission was detected from the side through the waveguide arm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Figure S6a and b show the corresponding attenuation of the measured intensities at 890 nm plotted as a function of the distance to the DC bends for input arms 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Input arm 1 exhibits transmission losses on the level of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 dB/mm and arm 2 of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 dB/mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Waveguide transmission characteristics are limited by ridge sidewall imperfections, which could be potentially further improved by optimizing the etch- ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 8 7 6 5 4 3 2 1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 5 4 3 2 1 0 1 Input Arm 1 Losses: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 dB/mm experiment fit Attenuation (dB) Distance (mm) Input Arm 2 Losses: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 dB/mm experiment fit Attenuation (dB) Distance (mm) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Waveguides transmission losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Attenuation of the side detected ensemble PL signal in function of the distance from DC bend regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 6 S6: POLARIZATION-RESOLVED PL Side detected emission from both studied emission lines show a high degree of linear polarization (DOLP) of around 95%, oriented in the sample plane as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' A high DOLP and its direction are related to the QDs dipole moments, which are mainly in-plane oriented and thus emitted photons mostly couple to and propagate in the TE waveguide mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' It needs to be noted that high DOLP is maintained after passing the whole circuits consisting of bends, DC, and out-coupler, which could potentially spoil the detected polarization contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The same polarization level is observed for both output arms of the DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' QD1 QD2 0 30 60 90 120 150 180 210 240 270 300 330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0 30 60 90 120 150 180 210 240 270 300 330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' PL intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' units) experiment Sine fit DOLP = 95±2% DOLP = 95±2% DOLP = 94±2% DOLP = 95±2% Output Arm 1 Output Arm 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Polarization characteristics of the QD1 and QD2 emission coupled to input arms of the DC and detected from side facet output arms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Both QDs PL emission is strongly linearly polarized with around 95% degree of linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 7 S7: EMITTERS TRANSITION LINEWIDTHS Figure S8 shows a high-resolution spectrum of the QD1 and QD2 emission recorded under cw 660 nm excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Measurements are performed using a scanning Fabry-Perot interfer- ometer with a 3 µeV Lorentzian profile linewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Both spectra are fit using Lorentzian functions with full-width at half-maximum (FWHM) of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 µeV and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 µeV, for QD1 and QD2, respectively (error related to fit precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' It can be shown that the convo- lution of two Lorentzian profiles with FWHM1 and FWHM2 is also a Lorentzian profile with broadening of FWHM1+FWHM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Following the above, we can correct the recorded optical linewidths for the finite resolution of our setup by simply subtracting its linewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The deconvoluted linewidths are 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 µeV and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 µeV for QD1 and QD2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a b 60 40 20 0 20 40 60 80 60 40 20 0 20 40 60 QD1 FWHMfit= 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 meV FWHMdec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='= 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 meV experiment Lorentz fit Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' units) Detuning (meV) QD2 FWHMfit = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 meV FWHMdec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 meV experiment Lorentz fit Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' units) Detuning (meV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' A high-resolution PL spectrum of QD1 and QD2, obtained using a home-built Fabry- Perot scanning cavity with a resolution of 3 µeV (FWHM), and free spectral range of 140 µeV at 890 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Solid lines are fits with the Lorentz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S8: DIRECTIONAL COUPLER CHARACTERISTICS To extract the DC splitting ratio r : t accounting for the different performance of both output arms due to the fabrication imperfections, the following procedure has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' First, QD located in input arm 1 was excited, and PL signal from output arm 1 II1 O1 and arm 2 II1 O2 collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Next, the same measurement has been repeated on QD located in arm 2, revealing II2 O1 and II2 O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' If the uneven outcoupling efficiency between the two output arms 8 is quantified as a constant x (transmission ratio between two arms), the following set of equations applies r + t = 1, xr t = II1 O1 II1 O2 , x t r = II2 O1 II2 O2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' (1) Based on that, the DC splitting ratio r : t can be derived, accounting for the imbalanced outcoupling r : t = � II1 O1 II1 O2 II2 O2 II2 O1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' (2) In the case of the investigated DC with II1 O1/II1 O2 of 51:49 and II2 O2/II2 O1 of 46:54 we ended up with the r:t ratio of 48:52, which is very close to the desired 50:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 0 50 100 150 200 0 500 1000 1500 2000 2500 3000 3500 4000 Intensity (cps) Time frame laser coupled to: Input Arm 1 Input Arm 2 Arm 1+2 (interference) classical visibility at 890 nm V = (Imax-Imin)/(Imax+Imin) = 98±1% FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Intensity fluctuations of the cw laser light transmitted simultaneously through two input arms of the DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Fluctuations in time are related to the off-chip path-length difference fluctuations of signal interfered on the directional coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' To test the classical visibility of the DC device, we simultaneously send cw laser light tuned to the energy of QDs transitions (890 nm), using circular reflectors placed on the ends of the input arms waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The power of the laser coupled into both arms was adjusted, so the intensity from both input arms was the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Next, we focused on the 9 signal passing through the DC and out-coupled from one of the output arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We observed intensity modulation in the function of time, related to the small path-length difference fluctuation, allowing us to record interference pattern as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Classical DC visibility was obtained by calculating the intensity contrast VDC = Imax − Imin Imax + Imin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' (3) In the case of the investigated DC device, a classical visibility of 98±1% was extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 10 S9: RAW HOM CORRELATION DATA Figure S10a shows a non-corrected two-photon Hong-Ou-Mandel interference experiment result between QD1 and QD2 performed under 76MHz and 19MHz pulsed laser repetition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In the case of both graphs, the same time-independent background offset is visible, corresponding to around 15% of the normalized peak intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Since the background level is the same for the HOM graphs at different laser repetition rates, we exclude the emit- ters long-decay contribution to the observable background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' We attribute the observed cw offset to the SSPDs dark counts (100-500 cps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Figure S10b shows the non-corrected nor- malized histogram of the central eleven peaks areas (∆ = 3 ns integration window) of the HOM second-order cross-correlation in case of synchronized - indistinguishable (red bars) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 ns delayed - distinguishable (grey bars) photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The non-corrected central peak area in case of synchronized photons is equal to g(2) HOM(0, ∆t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='587±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='002, while in case of unsynchronized photons g(2) HOMd(0, ∆t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='680±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' In this case, the non-corrected two-photon interference visibility yields Vraw = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3% in correspondence to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='7% visibility obtained for background-corrected graphs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3c in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a b 50 25 0 25 50 75 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 Raw data Raw norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' HOM counts - g(2) HOM(t) Delay time (ns) 19 MHz 76 MHz cw bck 5 -4 -3 -2 -1 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='3% g(2) HOM(0,Dt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='587±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='002 Raw norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' HOM peak area - g(2) HOM(t,Dt) Peak number indist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' (sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=') dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 ns delay) bck FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' a, Two-photon Hong-Ou-Mandel interference measurement between QD1 and QD2 performed using on-chip beamsplitter showing the normalized coincidences versus the delay time with no background counts correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' An experiment performed under 76MHz and 19MHz pulsed laser repetition rate yields the same cw background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' b, Non-corrected integrated counts of the central eleven peaks (3 ns integration window) of the HOM correlation in case of synchronized (blue bars) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='5 ns delayed (red bars) photons from QD1 and QD2 under pulsed 76MHz excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 11 S10: SINGLE PHOTON EMISSION UNDER CW EXCITATION In Figure S11, HBT second-order correlation histograms recorded under cw (660 nm) excitation for QD1 and QD2 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' The data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S11 are fit with the function g(2) HBT(τ) = (1 − g(2) HBT(0))exp(−|τ|/τd) convoluted with 80 ps width Gaussian instrumental response function, where τ is the delay time between detection events, g(2) HBT(0) is the prob- ability of two-photon emission events, τd is decay time constant corresponding to the sum of the spontaneous emission rate 1/T1 and pump rate G of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 20 15 10 5 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='4 g(2)(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='01 g(2) HBT(t) Delay time (ns) g(2)(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content='01 0 10 20 30 40 50 60 70 Raw coincidences a b QD1 QD2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Second order auto-correlation histograms of a QD1 and b QD2 emission under non- resonant (660 nm) cw excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Data have been recorded in HBT configuration using an on-chip beamsplitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' Presented data are shown as measured with no background subtraction or other corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfvf0n/content/2301.01706v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 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Netherlands +2Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands +fahad.sarfraz@navinfo.eu, elahe.arani@tue.nl, bahram.zonooz@gmail.com +Abstract +Efficient continual learning in humans is enabled by a rich set +of neurophysiological mechanisms and interactions between +multiple memory systems. The brain efficiently encodes in- +formation in non-overlapping sparse codes, which facilitates +the learning of new associations faster with controlled inter- +ference with previous associations. To mimic sparse coding +in DNNs, we enforce activation sparsity along with a dropout +mechanism which encourages the model to activate simi- +lar units for semantically similar inputs and have less over- +lap with activation patterns of semantically dissimilar inputs. +This provides us with an efficient mechanism for balancing +the reusability and interference of features, depending on the +similarity of classes across tasks. Furthermore, we employ +sparse coding in a multiple-memory replay mechanism. Our +method maintains an additional long-term semantic memory +that aggregates and consolidates information encoded in the +synaptic weights of the working model. Our extensive eval- +uation and characteristics analysis show that equipped with +these biologically inspired mechanisms, the model can fur- +ther mitigate forgetting1. +1 +Introduction +The ability to continually acquire, consolidate, and retain +knowledge is a hallmark of intelligence. Particularly, as we +look to deploy deep neural networks (DNNs) in the real +world, it is essential that learning agents continuously inter- +act and adapt to the ever-changing environment. However, +standard DNNs are not designed for lifelong learning and +exhibit catastrophic forgetting of previously learned knowl- +edge (McCloskey and Cohen 1989) when required to learn +tasks sequentially from a stream of data (McCloskey and +Cohen 1989). +The core challenge in continual learning (CL) in DNNs +is to maintain an optimal balance between plasticity and +the stability of the model. Ideally, the model should be sta- +ble enough to retain previous knowledge while also plastic +enough to acquire and consolidate new knowledge. Catas- +trophic forgetting in DNNs can be attributed to the lack of +stability, and multiple approaches have been proposed to ad- +dress it. Among them, Rehearsal-based methods, (Riemer +*These authors contributed equally. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +1Code available at https://github.com/NeurAI-Lab/SCoMMER +et al. 2018; Aljundi et al. 2019b) which aim to reduce for- +getting by continual rehearsal of previously seen tasks, have +proven to be an effective approach in challenging CL tasks +(Farquhar and Gal 2018). They attempt to approximate the +joint distribution of all the observed tasks by saving samples +from previous tasks in a memory buffer and intertwine the +training of the new task with samples from memory. How- +ever, due to the limited buffer size, it is difficult to approx- +imate the joint distribution with the samples alone. There +is an inherent imbalance between the samples of previous +tasks and the current task. This results in the network update +being biased towards the current task, leading to forgetting +and recency bias in predictions. Therefore, more informa- +tion from the previous state of the model is needed to better +approximate the joint distribution and constrain the update +of the model to preserve the learned knowledge. However, it +is still an open question what the optimal information is for +replay and how to extract and preserve it. +The human brain provides an existence proof for success- +ful CL in complex dynamic environments without intransi- +gence or forgetting. Therefore, it can provide insight into +the design principles and mechanisms that can enable CL +in DNNs. The human brain maintains a delicate balance +between stability and plasticity through a complex set of +neurophysiological mechanisms (Parisi et al. 2019; Zenke +et al. 2017) and the effective use of multiple memory sys- +tems (Hassabis et al. 2017). In particular, evidence suggests +that the brain employs Sparse Coding, that the neural code is +characterized by strong activations of a relatively small set +of neurons. The efficient utilization of sparsity for informa- +tion representation enables new associations to be learned +faster with controlled interference with previous associa- +tions while maintaining sufficient representation capacity. +Furthermore, complementary learning systems (CLS) the- +ory posits that effective learning requires two complemen- +tary learning systems. The hippocampus rapidly encodes +episodic information into non-overlapping representations, +which are then gradually consolidated into the structural +knowledge representation in the neocortex through the re- +play of neural activities. +Inspired by these mechanisms in the brain, we hypothe- +size that employing a mechanism to encourage sparse cod- +ing in DNNs and mimic the interplay of multiple memory +systems can be effective in maintaining a balance between +arXiv:2301.05058v1 [cs.NE] 28 Dec 2022 + +Long-term +Memory + + + + +Working +Memory +Episodic +Memory +Consolidation +Data Stream +Rehearsal +Layer + c: 1 2 3 4 +Activation Count +Semantic Dropout +Knowledge Retrieval +Activation +k-WTA +Figure 1: SCoMMER employs sparse coding in a multi-memory experience replay mechanism. In addition to the instance-based +episodic memory, we maintain a long-term memory that consolidates the learned knowledge in the working memory throughout +training. The long-term memory interacts with the episodic memory to enforce consistency in the functional space of working +memory through the knowledge retrieval loss. To mimic sparse coding in the brain, we enforce activation sparsity along with +semantic dropout, whereby the model tracks the class-wise activations during training and utilizes them to enforce sparse code, +which encourages the model to activate similar units for semantically similar inputs. Schematic shows how the activations from +layer l are propagated to the next layer. Darker shades indicate higher values. Given a sample from class 4, semantic dropout +retains the units with higher activation counts for the class, and top-k remaining (here 2) units with higher activations are +propagated to the next layer. This enables the network to form semantically conditioned subnetworks and mitigate forgetting. +stability and plasticity. To this end, we propose a multi- +memory experience replay mechanism that employs sparse +coding, SCoMMER. We enforce activation sparsity along +with a complementary dropout mechanism, which encour- +ages the model to activate similar units for semantically sim- +ilar inputs while reducing the overlap with activation pat- +terns of semantically dissimilar inputs. The proposed se- +mantic dropout provides us with an efficient mechanism to +balance the reusability and interference of features depend- +ing on the similarity of classes across tasks. Furthermore, +we maintain additional long-term semantic memory that ag- +gregates the information encoded in the synaptic weights +of the working memory. Long-term memory interacts with +episodic memory to retrieve structural knowledge from pre- +vious tasks and facilitates information consolidation by en- +forcing consistency in functional space. +Our empirical evaluation on challenging CL settings and +characteristic analysis show that equipping the model with +these biologically inspired mechanisms can further mitigate +forgetting and effectively consolidate information across the +tasks. Furthermore, sparse activations in conjunction with +semantic dropout in SCoMMER leads to the emergence of +subnetworks, enables efficient utilization of semantic mem- +ory, and reduces the bias towards recent tasks. +2 +Related Work +The different approaches to address the problem of catas- +trophic forgetting in CL can be broadly divided into three +categories: Regularization-based methods regularize the up- +date of the model in the parameter space (Farajtabar et al. +2020; Kirkpatrick et al. 2017; Ritter et al. 2018; Zenke et al. +2017) or the functional space (Rannen et al. 2017; Li and +Hoiem 2017), Dynamic architecture expands the network +to dedicate a distinct set of parameters to each task, and +Rehearsal-based methods (Riemer et al. 2018; Aljundi et al. +2019b) mitigate forgetting by maintaining an episodic mem- +ory buffer and continual rehearsal of samples from previous +tasks. Among these, our method focuses on rehearsal-based +methods, as it has proven to be an effective approach in +challenging continual learning scenarios (Farquhar and Gal +2018). The base method, Experience Replay (ER) (Riemer +et al. 2018) interleaves the training of the current task with +the memory sample to train the model on the approximate +joint distribution of tasks. Several studies focus on the differ- +ent aspects of rehearsal: memory sample selection (Lopez- +Paz and Ranzato 2017; Isele and Cosgun 2018), sample re- +trieval from memory (Aljundi et al. 2019a) and what infor- +mation to extract and replay from the previous model (Li and +Hoiem 2017; Ebrahimi et al. 2020; Bhat et al. 2022). +Dark Experience Replay (DER++) samples the output +logits along with the samples in the memory buffer through- +out the training trajectory and applies a consistency loss on +the update of the model. Recently, CLS theory has inspired +a number of approaches that utilize multiple memory sys- +tems (Wang et al. 2022a,b; Pham et al. 2021) and show the +benefits of multiple systems in CL. CLS-ER (Arani et al. +2022) mimics the interplay between fast and slow learning +systems by maintaining two additional semantic memories +that aggregate the weights of the working model at differ- +ent timescales using an exponential moving average. Our +method enforces sparse coding for efficient representation +and utilization of multiple memories. +3 +Methodology +We first provide an overview of motivation from biologi- +cal systems before formally introducing the different com- +ponents of the proposed approach. + +3.1 +Continual Learning in the Biological System +Effective CL in the brain is facilitated by a complex set of +mechanisms and multiple memory systems. Information in +the brain is represented by neural activation patterns, which +form a neural code (Foldiak and Endres 2008). Specifically, +evidence suggests that the brain employs Sparse Coding, in +which sensory events are represented by strong activations +of a relatively small set of neurons. A different subset of +neurons is used for each stimulus (Foldiak 2003; Barth and +Poulet 2012). There is a correlation between these sparse +codes (Lehky et al. 2021) that could capture the similar- +ity between different stimuli. Sparse codes provide several +advantages: they enable faster learning of new associations +with controlled interference with previous associations and +allow efficient maintenance of associative memory while re- +taining sufficient representational capacity. +Another salient feature of the brain is the strong differ- +entiation and specialization of the nervous systems (Had- +sell et al. 2020). There is evidence for modularity in bio- +logical systems, which supports functional specialization of +brain regions (Kelkar and Medaglia 2018) and reduces in- +terference between different tasks. Furthermore, the brain +is believed to utilize multiple memory systems (Atkinson +and Shiffrin 1968; McClelland et al. 1995). Complementary +learning systems (CLS) theory states that efficient learning +requires at least two complementary systems. The instance- +based hippocampal system rapidly encodes new episodic +events into non-overlapping representations, which are then +gradually consolidated into the structured knowledge repre- +sentation in the parametric neocortical system. Consolida- +tion of information is accompanied by replay of the neural +activities that accompanied the learning event. +The encoding of information into efficient sparse codes, +the modular and dynamic processing of information, and +the interplay of multiple memory systems might play a cru- +cial role in enabling effective CL in the brain. Therefore, our +method aims to incorporate these components in ANNs. +3.2 +Sparse coding in DNNs +The sparse neural codes in the brain are in stark contrast +to the highly dense connections and overlapping representa- +tions in standard DNNs which are prone to interference. In +particular, for CL, sparse representations can reduce the in- +terference between different tasks and therefore result in less +forgetting, as there will be fewer task-sensitive parameters +or fewer effective changes to the parameters (Abbasi et al. +2022; Iyer et al. 2021). Activation sparsity can also lead to +the natural emergence of modules without explicitly impos- +ing architectural constraints (Hadsell et al. 2020). Therefore, +to mimic sparse coding in DNNs, we enforce activation spar- +sity along with a complementary semantic dropout mecha- +nism which encourages the model to activate similar units +for semantically similar samples. +Sparse Activations: +To enforce the sparsity in activations, +we employ the k-winner-take-all (k-WTA) activation func- +tion (Maass 2000). k-WTA only retains the top-k largest val- +ues of an N × 1 input vector and sets all the others to zero +before propagating the vector to the next layer of the net- +work. Importantly, we deviate from the common implemen- +tation of k-WTA in convolutional neural networks (CNNs) +whereby the activation map of a layer (C × H × W ten- +sor where C is the number of channels and H and W are +the spatial dimensions) is flattened into a long CHW × 1 +vector input and the k-WTA activation is applied similar +to the fully connected network (Xiao et al. 2019; Ahmad +and Scheinkman 2019). We believe that this implementation +does not take into account the functional integrity of an in- +dividual convolution filter as an independent feature extrac- +tor and does not lend itself to the formation of task-specific +subnetworks with specialized feature extractors. Instead, we +assign an activation score to each filter in the layer by taking +the absolute sum of the corresponding activation map and +select the top-k filters to propagate to the next layer. +Given the activation map, we flatten the last two dimen- +sions and assign a score to each filter by taking the absolute +sum of the activations. Based on the sparsity ratio for each +layer, the activation maps of the filters with higher scores are +propagated to the next layers, and the others are set to zero. +This enforces global sparsity, whereby each stimulus is pro- +cessed by only a selected set of convolution filters in each +layer, which can be considered as a subnetwork. We also +consider each layer’s role when setting the sparsity ratio. +The earlier layers have a lower sparsity ratio as they learn +general features, which can enable higher reusability, and +forward transfer to subsequent tasks use a higher sparsity for +later layers to reduce the interference between task-specific +features. +Semantic Dropout: +While the k-WTA activation function +enforces the sparsity of activation for each stimulus, it does +not encourage semantically similar inputs to have similar ac- +tivation patterns and reduce overlap with semantically dis- +similar inputs. To this end, we employ a complementary +Semantic Dropout mechanism, which controls the degree +of overlap between neural activations between samples be- +longing to different tasks while also encouraging the sam- +ples belonging to the same class to utilize a similar set of +units. We utilize two sets of activation trackers: global ac- +tivity counter, Ag ∈ RN, counts the number of times each +unit has been activated throughout training, whereas class- +wise activity counter, As ∈ RC×N, tracks the number of +times each unit has been active for samples belonging to a +particular class. N and C denote the total number of units +and classes, respectively. For each subsequent task, we first +employ Heterogeneous Dropout (Abbasi et al. 2022) to en- +courage the model to learn the new classes by using neu- +rons that have been less active for previously seen classes by +setting the probability of a neuron being dropped to be in- +versely proportional to its activation counts. Concretely, let +[Al +g]j denote the number of times that the unit j in layer l has +been activated after learning t sequential tasks. For learning +the new classes in task t+1, the probability of retaining this +unit is given by: +[P l +h]j = exp( +−[Al +g]j +maxi [Alg]i +πh) +(1) + +Algorithm 1: SCoMMER Algorithm for Sparse Coding in Multiple-Memory Experience Replay System +Input: data stream D; learning rate η; consistency weight γ; update rate r and decay parameter α, dropout rates πh and πs +Initialize: θs = θw +M ←− {} +1: for Dt ∈ D do +2: +while Training do +3: +Sample training data: (xt, yt) ∼ Dt and (xm, ym) ∼ M, and interleave x ← (xt, xm) +4: +Retrieve structural knowledge: Zs ← f(xm; θs) +5: +Evaluate overall loss loss: L = Lce(f(x; θw), y) + γLkr(f(xm; θw), Zs) (Eq. 4) +6: +Update working memory: θw ←− θw − η∇θwL +7: +Aggregate knowledge: θs ← αθs + (1 − α) θw, +if r > a ∼ U(0, 1) (Eq. 3) +8: +Update episodic memory: M ←− Reservoir(M, (xt, yt)) +9: +After Eh epochs, update semantic dropout probabilities at the end of each epoch: Ps +(Eq. 2) +10: +Update heterogeneous dropout probabilities: Ph +(Eq. 1) +return θs +where πh controls the strength of dropout with larger val- +ues leading to less overlap between representations. We then +allow the network to learn with the new task with hetero- +geneous dropout in place of a fixed number of epochs, Eh. +During this period, we let the class-wise activations emerge +and then employ Semantic Dropout. It encourages the model +to utilize the same set of units by setting the probability of +retention of a unit for each class c as proportional to the +number of times it has been activated for that class so far: +[P l +s]c,j = 1 − exp( +−[Al +s]c,j +maxi [Als]c,i +πs) +(2) +where πs controls the strength of dropout. The probabilities +for semantic dropout are updated at the end of each epoch +to enforce the emerging pattern. This provides us with an +efficient mechanism for controlling the degree of overlap +in representations as well as enabling context-specific pro- +cessing of information which facilitates the formation of se- +mantically conditioned subnetworks. Activation sparsity, to- +gether with semantic dropout, also provides us with an effi- +cient mechanism for balancing the reusability and interfer- +ence of features depending on the similarity of classes across +the tasks. +3.3 +Multiple Memory Systems +Inspired by the interaction of multiple memory systems in +the brain, in addition to a fixed-size instance-based episodic +memory, our method builds a long-term memory that aggre- +gates the learned information in the working memory. +Episodic Memory: +Information consolidation in the brain +is facilitated by replaying the neural activation patterns that +accompanied the learning event. To mimic this mechanism, +we employ a fixed-size episodic memory buffer, which can +be thought of as a very primitive hippocampus. The memory +buffer is maintained with Reservoir Sampling, (Vitter 1985) +which aims to match the distribution of the data stream by +assigning an equal probability to each incoming sample. +Long-Term Memory: +We aim to build a long-term +semantic memory that can consolidate and accumulate +the structural knowledge learned in the working memory +throughout the training trajectory. The knowledge acquired +in DNNs resides in the learned synaptic weights (Krishnan +et al. 2019). Hence, progressively aggregating the weights +of the working memory (θw) as it sequentially learns tasks +allows us to consolidate the information efficiently. To this +end, we build long-term memory (θs) by taking the expo- +nential moving average of the working memory weights +in a stochastic manner (which is more biologically plausi- +ble (Arani et al. 2021)), similar to (Arani et al. 2022): +θs ← αθs + (1 − α) θw, +if r > a ∼ U(0, 1) +(3) +where α is the decay parameter and r is the update rate. +Long-term memory builds structural representations for +generalization and mimics the slow acquisition of struc- +tured knowledge in the neocortex, which can generalize well +across tasks. The long-term memory then interacts with the +instance-level episodic memory to retrieve structural rela- +tional knowledge (Sarfraz et al. 2021) for the previous tasks +encoded in the output logits. Consolidated logits are then +utilized to enforce consistency in the functional space of the +working model. This facilitates the consolidation of infor- +mation by encouraging the acquisition of new knowledge +while maintaining the functional relation of previous knowl- +edge and aligning the decision boundary of working mem- +ory with long-term memory. +3.4 +Overall Formulation +Given a continuous data stream D containing a sequence of +tasks (D1, D2, .., DT ), the CL task is to learn the joint dis- +tribution of all the observed tasks without the availability of +task labels at test time. Our proposed method, SCoMMER, +involves training a working memory θw, and maintains an +additional long-term memory θs and an episodic memory +M. The long-term memory is initialized with the same pa- +rameters as the working memory and has the same spar- +sity constraints. Therefore, long-term memory aggregates +the weights of working memory. We initialize heterogeneous +dropout probabilities πh randomly to set the probability of +retention of a fraction of units to 1 and others to 0 so that the +first task is learned using a few, but sufficient units and the +remaining can be utilized to learn subsequent tasks. + +Table 1: Comparison on different CL settings. The baseline results for S-CIFAR100 and GCIL are from (Arani et al. 2022). +Buffer +Method +S-CIFAR10 +S-CIFAR100 +GCIL +Class-IL +Task-IL +Class-IL +Task-IL +Unif +Longtail +– +JOINT +92.20±0.15 +98.31±0.12 +70.62±0.64 +86.19±0.43 +58.36±1.02 +56.94±1.56 +SGD +19.62±0.05 +61.02±3.33 +17.58±0.04 +40.46±0.99 +12.67±0.24 +22.88±0.53 +200 +ER +44.79±1.86 +91.19±0.94 +21.40±0.22 +61.36±0.39 +16.40±0.37 +19.27±0.77 +DER++ +64.88±1.17 +91.92±0.60 +29.60±1.14 +62.49±0.78 +18.84±0.60 +26.94±1.27 +CLS-ER +66.19±0.75 +93.90±0.60 +35.23±0.86 +67.34±0.79 +25.06±0.81 +28.54±0.87 +SCoMMER +69.19±0.61 +93.20±0.10 +40.25±0.05 +69.39±0.43 +30.84±0.80 +29.08±0.31 +500 +ER +57.74±0.27 +93.61±0.27 +28.02±0.31 +68.23±0.16 +28.21±0.69 +20.30±0.63 +DER++ +72.70±1.36 +93.88±0.50 +41.40±0.96 +70.61±0.11 +32.92±0.74 +25.82±0.83 +CLS-ER +75.22±0.71 +94.94±0.53 +47.63±0.61 +73.78±0.86 +36.34±0.59 +28.63±0.68 +SCoMMER +74.97±1.05 +94.36±0.06 +49.63±1.43 +75.49±0.43 +36.87±0.36 +35.20±0.21 +T1 +T2 +T3 +T4 +T5 +After T1 +After T2 +After T3 +After T4 +After T5 +98.0 +88.3 +85.4 +86.2 +38.3 +88.5 +81.5 +29.9 +42.0 +96.8 +47.4 +45.0 +55.5 +70.0 +94.9 +Working Memory +T1 +T2 +T3 +T4 +T5 +98.6 +92.0 +84.8 +87.7 +57.5 +79.7 +85.5 +49.0 +64.5 +86.7 +70.0 +52.0 +60.8 +79.2 +86.5 +Long-Term Memory +Figure 2: Task-wise performance of working memory and +the long-term memory. The long-term memory effectively +aggregates knowledge encoded in the working memory and +generalizes well across the tasks. +During each training step, we interleave the batch of sam- +ples from the current task xt ∼ Dt, with a random batch of +exemplars from episodic memory xm ∼ M. Working mem- +ory is trained with a combination of cross-entropy loss on +the interleaved batch x ← (xt, xb), and knowledge retrieval +loss on the exemplars. Thus, the overall loss is given by: +L = Lce(f(x; θw), y) + γLkr(f(xm; θw), f(xm; θs)) (4) +where γ controls the strength of the enforcement of con- +sistency, and mean-squared error loss is used for Lkr. The +training step is followed by stochastically updating the long- +term memory (Eq. 3). The semantic dropout and heteroge- +neous dropout probabilities are updated at the end of each +epoch and task, respectively (using Eqs. 1 and 3). We use +long-term memory for inference, as it aggregates knowledge +and generalizes well across tasks (cf. Figure 2). Agorithm 1 +provides further training details. +4 +Evaluation Protocol +To gauge the effectiveness of SCoMMER in tackling the dif- +ferent challenges faced by a lifelong learning agent, we con- +sider multiple CL settings that test different aspects of the +model. +Class-IL presents a challenging CL scenario where each +task presents a new set of disjoint classes, and the model +must learn to distinguish between all the classes seen so +far without the availability of task labels at the test time. +It requires the model to effectively consolidate information +across tasks and learn generalizable features that can be +reused to acquire new knowledge. Generalized Class-IL +(GCIL) (Mi et al. 2020) extends the Class-IL setting to more +realistic scenarios where the agent has to learn an object over +multiple recurrences spread across tasks and tackle the chal- +lenges of class imbalance and a varying number of classes +in each task. GCIL utilizes probabilistic modeling to sam- +ple the number of classes, the appearing classes, and their +sample sizes. Details of the datasets used in each setting are +provided in the Appendix. Though our method does not uti- +lize separate classification heads or subnets, for completion, +we also evaluate the performance under the Task-IL setting, +where the model has access to the task labels at inference. +In this setting, we use the task label to select the subset of +output logits to select from. +5 +Empirical Evaluation +We compare SCoMMER with state-of-the-art rehearsal- +based methods across different CL settings under uniform +experimental settings (details provided in Appendix). SGD +provides the lower bound with standard training on sequen- +tial tasks, and JOINT gives the upper bound on performance +when the model is trained on the joint distribution. +Table 1 shows that SCoMMER provides performance +gains in the majority of the cases and demonstrates the ef- +fectiveness of our approach under varying challenging CL +settings. In particular, it provides considerable improve- +ment under low buffer size settings, which suggests that our +method is able to mitigate forgetting with fewer samples +from previous tasks. The performance gains over CLS-ER, +which employs two semantic memories, show that sparse +coding in our method enables the effective utilization of a +single semantic memory. In particular, the gains in the GCIL +setting, where the agent has to face the challenges of class +imbalance and learn over multiple occurrences of objects, +alludes to several advantages of our method. Our proposed +semantic dropout in conjunction with sparse activations en- +ables the model to reuse the sparse code associated with the + +T1 +T2 +T3 +T4 +T5 +After T1 +After T2 +After T3 +After T4 +After T5 +98.8 +67.0 +92.1 +54.0 +16.9 +95.8 +55.9 +13.1 +22.9 +98.7 +15.8 +15.1 +36.0 +73.8 +98.2 +ER +T1 +T2 +T3 +T4 +T5 +98.2 +89.2 +87.3 +82.0 +50.0 +90.0 +79.3 +33.4 +63.0 +94.8 +58.4 +29.5 +67.5 +81.1 +95.7 +DER++ +T1 +T2 +T3 +T4 +T5 +98.7 +89.0 +89.5 +78.2 +53.5 +89.0 +81.2 +42.4 +76.3 +87.5 +69.2 +41.5 +76.8 +83.3 +41.1 +CLS-ER +T1 +T2 +T3 +T4 +T5 +98.6 +92.0 +84.8 +87.7 +57.5 +79.7 +85.5 +49.0 +64.5 +86.7 +70.0 +52.0 +60.8 +79.2 +86.5 +SCoMMER +Figure 3: Task-wise performance of different methods. The heatmaps provide the test set of each task (x-axis) evaluated at the +end of each sequential learning task (y-axis). SCoMMER retains the performance of earlier tasks better without compromising +on the current task. +Table 2: Ablation Study: Effect of systematically removing +different components of SCoMMER on the performance of +the models on S-CIFAR10. All components contribute to the +performance gain. +Sparse +Long-Term +Semantic +Accuracy +Activations +Memory +Dropout + + + +69.19±0.61 + + + +67.38±1.51 + + + +61.88±2.43 + + + +49.44±5.43 + + + +44.79±1.86 +recurring object and learn better representations with the ad- +ditional samples by adapting the corresponding subset of fil- +ters. Furthermore, compared to the dense activations in CLS- +ER, the sparse coding in SCoMMER leads to the emergence +of subnetworks that provide modularity and protection to +other parts of the network since the entire network is not +updated for each input image. This increases the robustness +of the model to the class imbalance. +Overall, our method provides an effective approach to em- +ploy sparse coding in DNN and enables better utilization of +long-term memory, which can effectively consolidate infor- +mation across tasks and further mitigate forgetting. +6 +Ablation Study +To gain further insight into the contribution of each com- +ponent of our method, we systematically remove them and +evaluate the performance of the model in Table 2. The results +show that all components of SCoMMER contribute to the +performance gains. The drop in performance from remov- +ing semantic dropout suggests that it is effective in enforc- +ing sparse coding on the representations of the model, which +reduces the interference between tasks and allows semanti- +cally similar classes to share information. We also observe +the benefits of multiple memory systems in CL. Additional +long-term memory provides considerable performance im- +provement and suggests that the EMA of the learned synap- +tic weights can effectively consolidate knowledge across +tasks. Furthermore, we observe that sparsity is a critical +component for enabling CL in DNNs. Sparse activations +alone significantly improve ER performance and also en- +able efficient utilization of semantic memory. We highlight +that these individual components complement each other +and that the combined effect leads to the observed perfor- +mance improvement in our method. +7 +Characteristics Analysis +We look at different characteristics of the model to under- +stand what enables the performance gains in our method. +We analyze the models trained on S-CIFAR100 with a buffer +size of 200. +7.1 +Stability-Plasticity Dilemma +To better understand how well different methods maintain +a balance between stability and plasticity, we look at how +task-wise performance evolves as the model learns tasks se- +quentially. The diagonal of the heatmap shows the plastic- +ity of the model as it learns the new task, whereas the dif- +ference between the accuracy of the task when it was first +learned and at the end of the training indicates the stabil- +ity of the model. Figure 3 shows that SCoMMER is able to +maintain a better balance and provides a more uniform per- +formance on tasks compared to baselines. While CLS-ER +provides better stability than DER++, it comes at the cost of +the model’s performance on the last task, which could be due +to the lower update rate of the stable model. SCoMMER, on +the other hand, retains performance on the earlier tasks (T1 +and T2) and provides good performance on the recent task. +We also compare the long-term semantic and working mem- +ory performance in Figure 2. Long-term memory effectively +aggregates the learned knowledge into the synaptic weights +of working memory and generalizes well across tasks. +7.2 +Emergence of Subnetworks +To evaluate the effectiveness of activation sparsity and se- +mantic dropout in enforcing sparse coding in the model, we +look at the average activity of the units in the penultimate +layer. The emerging sparse code for each class is tracked +during training using the class-wise activity counter and en- +forced using semantic dropout probabilities (Equation 2). + +Figure 4: Class-wise activation counts of the filters in the penultimate layer of the model trained on S-CIFAR10 with 200 buffer +size. Comparison of the activation counts on the test set with the learned class-wise probabilities, Ps, during training shows the +effectiveness of semantic dropout in enforcing sparse coding. Right plot shows the cosine similarities between the activation +counts of different classes. Semantically similar classes have higher correlation in activations. Darker color shows higher values. +Given a test sample from class c, ideally, we would want +the model to use the subset of neurons that had higher activ- +ity for the training samples from class c without providing +any task information. Concretely, we track the class-wise +activity on the test set and plot the normalized activation +counts for a set of neurons next to their class-wise proba- +bilities at the end of training. Figure 4 shows a high correla- +tion between the test set activation counts and the semantic +dropout probabilities at the end of training, particularly for +recent classes. The activation counts also hint at the natu- +ral emergence of semantically conditioned subnets, as the +model utilizes a different set of units for different classes. +Furthermore, we observe that semantically similar classes +have a higher degree of correlation between their activation +patterns. For instance, cat and dog share the most active neu- +rons, a similar pattern is observed between horse and deer, +and car and truck. The cosine similarities between the ac- +tivation counts of the different classes further supports the +observation. This is even more remarkable given that these +classes are observed in different tasks, particularly for cars +and trucks, which are observed in the first and last tasks. +7.3 +Task Recency Bias +A major challenge in CL is the recency bias, in which the +update of the model on new task samples biases its predic- +tions toward the current task (Wu et al. 2019). This leads to +considerable forgetting of earlier tasks. To compare the de- +gree to which SCoMMER tackles this issue, we evaluate the +probabilities of predicting each task by aggregating the soft- +max output of samples from the test set of all seen tasks and +averaging the probabilities of classes in each task. Figure +5 shows that SCoMMER provides more uniform probabili- +ties to predict each task. CLS-ER is able to mitigate the bias +towards the last task, which can be attributed to the aggrega- +tion of knowledge in the semantic memories; however, CLS- +ER reduces the probability of predicting the last task, which +explains the low performance. SCoMMER effectively mit- +igates recency bias and provides uniform prediction proba- +ER +DER++ +CLS-ER +SCoMMER +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Task Probability +Task 1 +Task 2 +Task 3 +Task 4 +Task 5 +Figure 5: Average probabilities of predicting classes from +each tasks at the end of training. SCoMMER provides more +uniform probabilities across the tasks. +bilities across tasks without any explicit regularization. +8 +Conclusion +Motivated by the mechanisms for information representation +and utilization of multiple memory systems in the brain, we +proposed a novel approach to employ sparse coding in mul- +tiple memory systems. SCoMMER enforces activation spar- +sity along with a complementary semantic dropout mecha- +nism, which encourages the model to activate similar units +for semantically similar inputs and reduce the overlap with +dissimilar inputs. Additionally, it maintains long-term mem- +ory, which consolidates the learned knowledge in working +memory. Our empirical evaluation shows the effectiveness +of the approach in mitigating forgetting in challenging CL +scenarios. Furthermore, sparse coding enables efficient con- +solidation of knowledge in the long-term memory, reduces +the bias towards recent tasks, and leads to the emergence +of semantically conditioned subnetworks. We hope that our +study inspires further research in this promising direction. + +References +Abbasi, A.; Nooralinejad, P.; Braverman, V.; Pirsiavash, H.; +and Kolouri, S. 2022. 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Enhancing ad- +versarial defense by k-winners-take-all. +arXiv preprint +arXiv:1905.10510. +Zenke, F.; Poole, B.; and Ganguli, S. 2017. Continual learn- +ing through synaptic intelligence. Proceedings of machine +learning research, 70: 3987. + +A Appendix +B +Experimental Setting +For a fair comparison with different CL methods in uni- +form experimental settings, we extended the Mammoth +framework (Buzzega et al. 2020). To disentangle the per- +formance improvement of the algorithm from the training +regimes (Mirzadeh et al. 2020), we use the same network +(ResNet-18), optimizer (SGD), batch size for task data and +memory buffer (32), data augmentations (random crop and +random horizontal flip), and the number of epochs (50) for +all our experiments. +For hyperparameter tuning, we use a small held-out val- +idation set and perform a grip search on activation sparsity, +γ, dropout strengths, πh and πs, and the update frequency +for long-term memory r. Table S1 provides the selected hy- +perparameters for each setting. Note that our method does +not require an extensive hyperparameter search for differ- +ent buffer sizes, and sensitivity to hyperparameters section +shows that the different parameters are complementary in +nature and the model performs well for a number of differ- +ent combinations. Therefore, majority of parameters can be +fixed, which reduces the search space of hyperparameters +significantly. We report the average and one standard devia- +tion of three different seeds. +C +Continual Learning Datasets +We consider the Class-IL and Generalized Class-IL setting +for our empirical evaluation to extensively assess the ver- +satility of our approach. Here, we provide details of the +datasets used in each of the settings. +C.1 +Class Incremental Learning (Class-IL) +Class-IL (van de Ven and Tolias 2019) requires the agent +to learn a new disjoint set of classes with each task, and +the agent has to distinguish between all the classes seen so +far without the availability of task labels at the test time. +We consider the split variants of the benchmark datasets S- +CIFAR10 and S-CIFAR100 where the classes are split into +5 tasks with 2 and 20 classes each, respectively. The order +of the classes in the experiments remains fixed, whereby for +CIFAR10 the first task includes the first two classes, and so +forth. +C.2 +Generalized Class Incremental Learning +(GCIL) +GCIL (Mi et al. 2020) extends the Class-IL setting to more +realistic scenarios. In addition to avoiding forgetting, the +model has to tackle the challenges of class imbalance, learn- +ing an object over multiple recurrences. GCIL utilizes prob- +abilistic modeling to sample three characteristics of a task: +the number of classes, the classes that appear, and their sam- +ple sizes. Similarly to (Arani et al. 2022), we consider GCIL +on the CIFAR100 dataset with 20 tasks, each with 1000 sam- +ples, and the maximum number of classes in a single task set +to 50. To disentangle the effect of class imbalance from +the ability of the model to learn from recurring classes un- +der non-uniform task lengths, we evaluate the model on uni- +form (Unif) and longtail data distributions. we set the GCIL +dataset seed to 1993 for all the experiments. +D +Implementation Details +Here, we provide more details on the implementation of +k-WTA activation for CNNs and the proposed semantic +dropout mechanism. +E +k-WTA for Convolutional Neural +Networks +The common implementation of k-WTA in convolutional +neural networks involves flattening the activation map into a +long CHW ×1 vector and applying the activation of k-WTA +in a way similar to that of the fully connected network (Xiao +et al. 2019; Ahmad and Scheinkman 2019). This translates to +setting some spatial dimensions of a filter to zero while prop- +agating others. However, this implementation does not take +into account the functional integrity of an individual con- +volution filter as an independent feature extractor and does +not enable the formation of task-specific subnetworks with +specialized feature extractors. Different tasks cannot utilize +a different subset of filters, and we cannot track the activity +of an individual filter. +Our implementation, on the other hand, assigns an activa- +tion score to each filter in the layer by taking the absolute +sum of the corresponding activation map. Given the activa- +tion map of the layer l, Al, (C × W × H) where C is the +number of filters, W and H are the width and height, we +flatten the spatial dimensions, C × WH), and the activation +score for each filter j is given by the absolute sum of its ac- +tivations, [Cscore]j = �HW +i=1 |[Al]j,i|. We then find the value +k for the layer using the activation sparsity (% of the active +filters in the layer), k ← %k × N l +filters where N l +filters is +the number of filters in the layer l. The kth highest value of +the filter activation scores vector, Cscore ∈ RC×1 gives the +threshold value used to apply a mask to the input activation +map, which only propagates the activations of filters with a +score above threshold by setting the others to zero. Finally, +the ReLU activation function is applied to the masked acti- +vations. Algorithm 2 provides more details. +For the ResNet-18 network in our method, we set the ac- +tivation sparsity for each ResNet block, for example % k = +[0.9, 0.9, 0.9, 0.8] enforces the activation sparsity of 0.9 in +the first three ResNet blocks, that is 90% of the filters in each +convolutional layer are active for a given stimulus and 80% +in the convolutional layers of the last ResNet block. + +Table S1: Selected parameters for SCoMMER. For each of our experiments, we apply Heterogeneous and Semantic dropout +only on the output of the last residual block in ResNet-18, the decay parameter for long-term memory is set to 0.999, the batch +size of 32 is used for both the current task and the memory buffer, and the models are train for 50 epochs. For the first three +ResNet blocks, we use an activation sparsity of 0.9 and vary the sparsity ratio for the last block. +Dataset +Buffer +size +Activation +Sparsity +η +πh +πs +γ +r +S-CIFAR10 +200 +0.8 +0.1 +0.5 +2.0 +0.15 +0.5 +500 +0.8 +0.1 +0.5 +2.0 +0.15 +0.7 +S-CIFAR100 +200 +0.9 +0.1 +0.5 +3.0 +0.15 +0.1 +500 +0.9 +0.1 +0.5 +3.0 +0.15 +0.1 +GCIL - Unif +200 +0.9 +0.05 +0.5 +3.0 +0.2 +0.6 +500 +0.9 +0.05 +0.5 +3.0 +0.2 +0.6 +GCIL - Longtail +200 +0.9 +0.05 +0.5 +2.0 +0.2 +0.5 +500 +0.9 +0.05 +0.5 +3.0 +0.2 +0.6 +Algorithm 2: Global k-WTA for CNNs +Input: Activation map A; activation ratio %k; number +of filters Nfilters +Evaluate activation scores: +1: Flatten the spatial dimensions: +2: Cscore ← Reshape(Cscore, C × HW) +3: Assign score to each filter: +4: Cscore = abs sum(Cscore, dim=1) +Calculate threshold: +5: Get k value for the layer: +6: k ← %k × Nfilters +7: Return kth largest value +8: Cthresh = kth value(Cscore, k) +Mask Activation Map: +9: Initialize mask with zeros +10: M ← Zeros(C × H × W) +11: Set filter mask with score above threshold to 1 +12: M[Cscore > Cthresh] = 1 +13: Apply mask +14: A ← M · A +Apply ReLU activation function: +15: A ← ReLU(A) +return A +E.1 +Semantic Dropout +At the beginning of training, we initialize the heterogeneous +dropout probabilities Ph so that for each layer l the prob- +ability of (1.1 × %kl × N l +filters) filters is set to 1 and the +remaining set to 0. This is done to ensure that the learning +of the first task does not utilize all filters while having the +flexibility to learn a different subset of units for the classes +in the first task. The semantic dropout probabilities Ps are +updated at the end of each epoch, once the epoch num- +ber e for the task is higher than the heterogeneous dropout +warm-up period Eh to allow the emergence of class-wise ac- +tivity patterns before it is explicitly enforced with seman- +tic dropout. Note that to ensure that we have enough ac- +tive filters before applying k-WTA activation, when apply- +ing heterogeneous, we use the probabilities Ph to sample the +(1.1 × %kl × N l +filters) filters for the given layer before ap- +plying k-WTA activation. The 1.1 factor is arbitrarily chosen +and works well in practice; however, a different value can be +selected. Further details of the method are provided in Algo- +rithm 3. +Importantly, we disable the dropout activation counter up- +date for the buffer samples so that the sparse code is learned +during task training. Also, dropout is applied only to work- +ing memory as it is learned with gradient decent, whereas +the long-term memory aggregates the weights of working +memory. Our analysis shows that the learned sparse coding +is effectively transferred to long-term memory through ema. +For the ResNet-18 model used in our experiments, we ap- +ply dropout at the output of each ResNet block. Although +our method provides the flexibility to apply different dropout +strengths for each block, we observe empirically that it +works better if applied only at the output of the last ResNet +block. This allows the model to learn features in the earlier +layers that can generalize well across the tasks and to learn +specialized features for the classes in later layers. +F +Performance of working memory +To gain a better understanding of the performance of the +different memories, Table S2 provides the performance of +both working memory and long-term memory in the settings +considered. Long-term memory consistently provides better +generalization across tasks, especially in the Class-IL set- +ting. This shows the benefits of using multiple memory sys- +tems in CL. Furthermore, it demonstrates the effectiveness +of the exponential moving average of the working memory +weights as an efficient approach for aggregating the learned +knowledge. + +Algorithm 3: Semantic Dropout +Input: Activation map A; class labels y; activation ratio %k; number of filters Nfilters; dropout probabilities Ph and Ps +Get the Heterogeneous Dropout Mask: +1: Initialize Heterogeneous dropout mask with zeros +2: Hmask ← Zeros(C × H × W) +3: Calculate the sampling probabilities so that they sum to zero +Psample = Ph / sum(Ph) +4: Get the indices of retained filters +Nretain = 1.1 × %k × Nfilters +idx = Sample(range=Nfilters, #samples=Nretain, prob=Psample, replace=False) +5: Set the mask at retained indices to 1 +Hmask[idx] = 1 +Get the Heterogeneous Dropout Mask: +6: Initialize Semantic dropout mask with zeros +7: Smask ← Zeros(C × H × W) +8: Use the semantic dropout probabilities to select units +retain = N ∼ U(0, 1) ≤ Ps +9: Set the mask at retained indices to 1 +Smask[retain] = 1 +Select the mask for each input sample +10: For each sample, select Semantic dropout mask if available for the class label, otherwise use Heterogeneous dropout mask: +11: M = Smask if Ps[y] > 0, otherwise Hmask +Mask Activation Map: +12: A ← M · A +return A +G +Sensitivity to Hyperparameters +SCoMMER employs sparse coding in a multiple-memory +replay mechanism. Therefore, the setting of two sets of pa- +rameters is required: sparse coding (activation sparsity %k +and dropout strength πs and πh) and the aggregation of in- +formation in long-term memory (r, α). We show the effect +of different sets of hyperparameters in Table S3. We can +see that the different components are complementary in na- +ture and therefore different combinations of parameters can +provide similar performance. Interestingly, we observe that +increasing the semantic dropout strength considerably in- +creases the performance of the working model, but the long- +term memory performance remains quite stable. The method +is not highly sensitive to a particular set of parameters, and +often we can fix the majority of parameters and fine-tune +only a few, which significantly reduces the search space. + +Table S2: Performance of working memory and long-term memory in different settings. Long-term memory consistently pro- +vides better performance. +Buffer +Memory +S-CIFAR10 +S-CIFAR100 +GCIL +Class-IL +Task-IL +Class-IL +Task-IL +Unif +Longtail +200 +Working +58.03±5.17 +92.58±0.56 +30.07±0.71 +67.18±0.16 +27.64±0.30 +27.06±0.97 +Long-Term +69.19±0.61 +93.20±0.10 +40.25±0.05 +69.39±0.43 +30.84±0.80 +29.08±0.31 +500 +Working +66.10±3.60 +93.59±0.09 +41.36±1.07 +73.52±0.37 +34.34±0.88 +33.39±0.74 +Long-Term +74.97±1.05 +94.36±0.06 +49.63±1.43 +75.49±0.43 +36.87±0.36 +35.20±0.21 +Table S3: Sensitivity to different hyperparameters. We pro- +vide the performance of Working memory and Long-term +memory of models trained on S-CIFAR-10 with 200 buffer +size. For all experiments γ = 0.15, lr = 0.1, decay parameter += 0.999, πh = 0.5, and the model is trained for 50 epochs. +For the first three ResNet blocks, we use an activation spar- +sity of 0.9 and vary the sparsity ratio for the last block (%k) +r +%k +πs +Memory +Working +Long-Term +0.4 +0.7 +1.0 +56.65 +69.58 +2.0 +59.46 +68.30 +3.0 +59.89 +68.93 +0.8 +1.0 +50.25 +67.19 +2.0 +58.01 +69.89 +3.0 +56.91 +68.72 +0.9 +1.0 +51.26 +67.49 +2.0 +56.58 +68.32 +3.0 +56.87 +66.89 +0.5 +0.7 +1.0 +57.01 +66.80 +2.0 +59.61 +69.26 +3.0 +60.51 +69.00 +0.8 +1.0 +49.09 +67.36 +2.0 +58.03 +69.19 +3.0 +60.37 +67.99 +0.9 +1.0 +49.38 +66.27 +2.0 +60.47 +68.16 +3.0 +57.64 +67.88 +0.6 +0.7 +1.0 +56.91 +67.85 +2.0 +61.2 +67.64 +3.0 +62.44 +67.94 +0.8 +1.0 +51.11 +65.97 +2.0 +58.61 +66.55 +3.0 +61.01 +69.36 +0.9 +1.0 +49.26 +66.93 +2.0 +58.35 +67.44 +3.0 +60.18 +67.90 + diff --git a/0tE4T4oBgHgl3EQfZwwm/content/tmp_files/load_file.txt b/0tE4T4oBgHgl3EQfZwwm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6aa470571a8c8864ad24b539205f1ef44ae913cb --- /dev/null +++ b/0tE4T4oBgHgl3EQfZwwm/content/tmp_files/load_file.txt @@ -0,0 +1,1265 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf,len=1264 +page_content='Sparse Coding in a Dual Memory System for Lifelong Learning Fahad Sarfraz*1, Elahe Arani*1,2, Bahram Zonooz1,2 1Advanced Research Lab, NavInfo Europe, The Netherlands 2Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands fahad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='sarfraz@navinfo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='eu, elahe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='arani@tue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='nl, bahram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='zonooz@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='com Abstract Efficient continual learning in humans is enabled by a rich set of neurophysiological mechanisms and interactions between multiple memory systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The brain efficiently encodes in- formation in non-overlapping sparse codes, which facilitates the learning of new associations faster with controlled inter- ference with previous associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' To mimic sparse coding in DNNs, we enforce activation sparsity along with a dropout mechanism which encourages the model to activate simi- lar units for semantically similar inputs and have less over- lap with activation patterns of semantically dissimilar inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This provides us with an efficient mechanism for balancing the reusability and interference of features, depending on the similarity of classes across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, we employ sparse coding in a multiple-memory replay mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Our method maintains an additional long-term semantic memory that aggregates and consolidates information encoded in the synaptic weights of the working model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Our extensive eval- uation and characteristics analysis show that equipped with these biologically inspired mechanisms, the model can fur- ther mitigate forgetting1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 1 Introduction The ability to continually acquire, consolidate, and retain knowledge is a hallmark of intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Particularly, as we look to deploy deep neural networks (DNNs) in the real world, it is essential that learning agents continuously inter- act and adapt to the ever-changing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' However, standard DNNs are not designed for lifelong learning and exhibit catastrophic forgetting of previously learned knowl- edge (McCloskey and Cohen 1989) when required to learn tasks sequentially from a stream of data (McCloskey and Cohen 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The core challenge in continual learning (CL) in DNNs is to maintain an optimal balance between plasticity and the stability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Ideally, the model should be sta- ble enough to retain previous knowledge while also plastic enough to acquire and consolidate new knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Catas- trophic forgetting in DNNs can be attributed to the lack of stability, and multiple approaches have been proposed to ad- dress it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Among them, Rehearsal-based methods, (Riemer These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 1Code available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='com/NeurAI-Lab/SCoMMER et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019b) which aim to reduce for- getting by continual rehearsal of previously seen tasks, have proven to be an effective approach in challenging CL tasks (Farquhar and Gal 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' They attempt to approximate the joint distribution of all the observed tasks by saving samples from previous tasks in a memory buffer and intertwine the training of the new task with samples from memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' How- ever, due to the limited buffer size, it is difficult to approx- imate the joint distribution with the samples alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' There is an inherent imbalance between the samples of previous tasks and the current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This results in the network update being biased towards the current task, leading to forgetting and recency bias in predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Therefore, more informa- tion from the previous state of the model is needed to better approximate the joint distribution and constrain the update of the model to preserve the learned knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' However, it is still an open question what the optimal information is for replay and how to extract and preserve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The human brain provides an existence proof for success- ful CL in complex dynamic environments without intransi- gence or forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Therefore, it can provide insight into the design principles and mechanisms that can enable CL in DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The human brain maintains a delicate balance between stability and plasticity through a complex set of neurophysiological mechanisms (Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Zenke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2017) and the effective use of multiple memory sys- tems (Hassabis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' In particular, evidence suggests that the brain employs Sparse Coding, that the neural code is characterized by strong activations of a relatively small set of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The efficient utilization of sparsity for informa- tion representation enables new associations to be learned faster with controlled interference with previous associa- tions while maintaining sufficient representation capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, complementary learning systems (CLS) the- ory posits that effective learning requires two complemen- tary learning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The hippocampus rapidly encodes episodic information into non-overlapping representations, which are then gradually consolidated into the structural knowledge representation in the neocortex through the re- play of neural activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Inspired by these mechanisms in the brain, we hypothe- size that employing a mechanism to encourage sparse cod- ing in DNNs and mimic the interplay of multiple memory systems can be effective in maintaining a balance between arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='05058v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='NE] 28 Dec 2022 Long-term Memory Working Memory Episodic Memory Consolidation Data Stream Rehearsal Layer c: 1 2 3 4 Activation Count Semantic Dropout Knowledge Retrieval Activation k-WTA Figure 1: SCoMMER employs sparse coding in a multi-memory experience replay mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' In addition to the instance-based episodic memory, we maintain a long-term memory that consolidates the learned knowledge in the working memory throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The long-term memory interacts with the episodic memory to enforce consistency in the functional space of working memory through the knowledge retrieval loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' To mimic sparse coding in the brain, we enforce activation sparsity along with semantic dropout, whereby the model tracks the class-wise activations during training and utilizes them to enforce sparse code, which encourages the model to activate similar units for semantically similar inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Schematic shows how the activations from layer l are propagated to the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Darker shades indicate higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Given a sample from class 4, semantic dropout retains the units with higher activation counts for the class, and top-k remaining (here 2) units with higher activations are propagated to the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This enables the network to form semantically conditioned subnetworks and mitigate forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' stability and plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' To this end, we propose a multi- memory experience replay mechanism that employs sparse coding, SCoMMER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We enforce activation sparsity along with a complementary dropout mechanism, which encour- ages the model to activate similar units for semantically sim- ilar inputs while reducing the overlap with activation pat- terns of semantically dissimilar inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The proposed se- mantic dropout provides us with an efficient mechanism to balance the reusability and interference of features depend- ing on the similarity of classes across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, we maintain additional long-term semantic memory that ag- gregates the information encoded in the synaptic weights of the working memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Long-term memory interacts with episodic memory to retrieve structural knowledge from pre- vious tasks and facilitates information consolidation by en- forcing consistency in functional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Our empirical evaluation on challenging CL settings and characteristic analysis show that equipping the model with these biologically inspired mechanisms can further mitigate forgetting and effectively consolidate information across the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, sparse activations in conjunction with semantic dropout in SCoMMER leads to the emergence of subnetworks, enables efficient utilization of semantic mem- ory, and reduces the bias towards recent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2 Related Work The different approaches to address the problem of catas- trophic forgetting in CL can be broadly divided into three categories: Regularization-based methods regularize the up- date of the model in the parameter space (Farajtabar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Ritter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Zenke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2017) or the functional space (Rannen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Li and Hoiem 2017), Dynamic architecture expands the network to dedicate a distinct set of parameters to each task, and Rehearsal-based methods (Riemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019b) mitigate forgetting by maintaining an episodic mem- ory buffer and continual rehearsal of samples from previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Among these, our method focuses on rehearsal-based methods, as it has proven to be an effective approach in challenging continual learning scenarios (Farquhar and Gal 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The base method, Experience Replay (ER) (Riemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2018) interleaves the training of the current task with the memory sample to train the model on the approximate joint distribution of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Several studies focus on the differ- ent aspects of rehearsal: memory sample selection (Lopez- Paz and Ranzato 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Isele and Cosgun 2018), sample re- trieval from memory (Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019a) and what infor- mation to extract and replay from the previous model (Li and Hoiem 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Ebrahimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Bhat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Dark Experience Replay (DER++) samples the output logits along with the samples in the memory buffer through- out the training trajectory and applies a consistency loss on the update of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Recently, CLS theory has inspired a number of approaches that utilize multiple memory sys- tems (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2022a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2021) and show the benefits of multiple systems in CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' CLS-ER (Arani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2022) mimics the interplay between fast and slow learning systems by maintaining two additional semantic memories that aggregate the weights of the working model at differ- ent timescales using an exponential moving average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Our method enforces sparse coding for efficient representation and utilization of multiple memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 3 Methodology We first provide an overview of motivation from biologi- cal systems before formally introducing the different com- ponents of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 Continual Learning in the Biological System Effective CL in the brain is facilitated by a complex set of mechanisms and multiple memory systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Information in the brain is represented by neural activation patterns, which form a neural code (Foldiak and Endres 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Specifically, evidence suggests that the brain employs Sparse Coding, in which sensory events are represented by strong activations of a relatively small set of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' A different subset of neurons is used for each stimulus (Foldiak 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Barth and Poulet 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' There is a correlation between these sparse codes (Lehky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2021) that could capture the similar- ity between different stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Sparse codes provide several advantages: they enable faster learning of new associations with controlled interference with previous associations and allow efficient maintenance of associative memory while re- taining sufficient representational capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Another salient feature of the brain is the strong differ- entiation and specialization of the nervous systems (Had- sell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' There is evidence for modularity in bio- logical systems, which supports functional specialization of brain regions (Kelkar and Medaglia 2018) and reduces in- terference between different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, the brain is believed to utilize multiple memory systems (Atkinson and Shiffrin 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' McClelland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Complementary learning systems (CLS) theory states that efficient learning requires at least two complementary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The instance- based hippocampal system rapidly encodes new episodic events into non-overlapping representations, which are then gradually consolidated into the structured knowledge repre- sentation in the parametric neocortical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Consolida- tion of information is accompanied by replay of the neural activities that accompanied the learning event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The encoding of information into efficient sparse codes, the modular and dynamic processing of information, and the interplay of multiple memory systems might play a cru- cial role in enabling effective CL in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Therefore, our method aims to incorporate these components in ANNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 Sparse coding in DNNs The sparse neural codes in the brain are in stark contrast to the highly dense connections and overlapping representa- tions in standard DNNs which are prone to interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' In particular, for CL, sparse representations can reduce the in- terference between different tasks and therefore result in less forgetting, as there will be fewer task-sensitive parameters or fewer effective changes to the parameters (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Activation sparsity can also lead to the natural emergence of modules without explicitly impos- ing architectural constraints (Hadsell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Therefore, to mimic sparse coding in DNNs, we enforce activation spar- sity along with a complementary semantic dropout mecha- nism which encourages the model to activate similar units for semantically similar samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Sparse Activations: To enforce the sparsity in activations, we employ the k-winner-take-all (k-WTA) activation func- tion (Maass 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' k-WTA only retains the top-k largest val- ues of an N × 1 input vector and sets all the others to zero before propagating the vector to the next layer of the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Importantly, we deviate from the common implemen- tation of k-WTA in convolutional neural networks (CNNs) whereby the activation map of a layer (C × H × W ten- sor where C is the number of channels and H and W are the spatial dimensions) is flattened into a long CHW × 1 vector input and the k-WTA activation is applied similar to the fully connected network (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Ahmad and Scheinkman 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We believe that this implementation does not take into account the functional integrity of an in- dividual convolution filter as an independent feature extrac- tor and does not lend itself to the formation of task-specific subnetworks with specialized feature extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Instead, we assign an activation score to each filter in the layer by taking the absolute sum of the corresponding activation map and select the top-k filters to propagate to the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Given the activation map, we flatten the last two dimen- sions and assign a score to each filter by taking the absolute sum of the activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Based on the sparsity ratio for each layer, the activation maps of the filters with higher scores are propagated to the next layers, and the others are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This enforces global sparsity, whereby each stimulus is pro- cessed by only a selected set of convolution filters in each layer, which can be considered as a subnetwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We also consider each layer’s role when setting the sparsity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The earlier layers have a lower sparsity ratio as they learn general features, which can enable higher reusability, and forward transfer to subsequent tasks use a higher sparsity for later layers to reduce the interference between task-specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Semantic Dropout: While the k-WTA activation function enforces the sparsity of activation for each stimulus, it does not encourage semantically similar inputs to have similar ac- tivation patterns and reduce overlap with semantically dis- similar inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' To this end, we employ a complementary Semantic Dropout mechanism, which controls the degree of overlap between neural activations between samples be- longing to different tasks while also encouraging the sam- ples belonging to the same class to utilize a similar set of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We utilize two sets of activation trackers: global ac- tivity counter, Ag ∈ RN, counts the number of times each unit has been activated throughout training, whereas class- wise activity counter, As ∈ RC×N, tracks the number of times each unit has been active for samples belonging to a particular class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' N and C denote the total number of units and classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' For each subsequent task, we first employ Heterogeneous Dropout (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2022) to en- courage the model to learn the new classes by using neu- rons that have been less active for previously seen classes by setting the probability of a neuron being dropped to be in- versely proportional to its activation counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Concretely, let [Al g]j denote the number of times that the unit j in layer l has been activated after learning t sequential tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' For learning the new classes in task t+1, the probability of retaining this unit is given by: [P l h]j = exp( −[Al g]j maxi [Alg]i πh) (1) Algorithm 1: SCoMMER Algorithm for Sparse Coding in Multiple-Memory Experience Replay System Input: data stream D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' learning rate η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' consistency weight γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' update rate r and decay parameter α, dropout rates πh and πs Initialize: θs = θw M ←− {} 1: for Dt ∈ D do 2: while Training do 3: Sample training data: (xt, yt) ∼ Dt and (xm, ym) ∼ M, and interleave x ← (xt, xm) 4: Retrieve structural knowledge: Zs ← f(xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' θs) 5: Evaluate overall loss loss: L = Lce(f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' θw), y) + γLkr(f(xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' θw), Zs) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 4) 6: Update working memory: θw ←− θw − η∇θwL 7: Aggregate knowledge: θs ← αθs + (1 − α) θw, if r > a ∼ U(0, 1) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 3) 8: Update episodic memory: M ←− Reservoir(M, (xt, yt)) 9: After Eh epochs, update semantic dropout probabilities at the end of each epoch: Ps (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2) 10: Update heterogeneous dropout probabilities: Ph (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 1) return θs where πh controls the strength of dropout with larger val- ues leading to less overlap between representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We then allow the network to learn with the new task with hetero- geneous dropout in place of a fixed number of epochs, Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' During this period, we let the class-wise activations emerge and then employ Semantic Dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' It encourages the model to utilize the same set of units by setting the probability of retention of a unit for each class c as proportional to the number of times it has been activated for that class so far: [P l s]c,j = 1 − exp( −[Al s]c,j maxi [Als]c,i πs) (2) where πs controls the strength of dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The probabilities for semantic dropout are updated at the end of each epoch to enforce the emerging pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This provides us with an efficient mechanism for controlling the degree of overlap in representations as well as enabling context-specific pro- cessing of information which facilitates the formation of se- mantically conditioned subnetworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Activation sparsity, to- gether with semantic dropout, also provides us with an effi- cient mechanism for balancing the reusability and interfer- ence of features depending on the similarity of classes across the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='3 Multiple Memory Systems Inspired by the interaction of multiple memory systems in the brain, in addition to a fixed-size instance-based episodic memory, our method builds a long-term memory that aggre- gates the learned information in the working memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Episodic Memory: Information consolidation in the brain is facilitated by replaying the neural activation patterns that accompanied the learning event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' To mimic this mechanism, we employ a fixed-size episodic memory buffer, which can be thought of as a very primitive hippocampus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The memory buffer is maintained with Reservoir Sampling, (Vitter 1985) which aims to match the distribution of the data stream by assigning an equal probability to each incoming sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Long-Term Memory: We aim to build a long-term semantic memory that can consolidate and accumulate the structural knowledge learned in the working memory throughout the training trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The knowledge acquired in DNNs resides in the learned synaptic weights (Krishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Hence, progressively aggregating the weights of the working memory (θw) as it sequentially learns tasks allows us to consolidate the information efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' To this end, we build long-term memory (θs) by taking the expo- nential moving average of the working memory weights in a stochastic manner (which is more biologically plausi- ble (Arani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2021)), similar to (Arani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2022): θs ← αθs + (1 − α) θw, if r > a ∼ U(0, 1) (3) where α is the decay parameter and r is the update rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Long-term memory builds structural representations for generalization and mimics the slow acquisition of struc- tured knowledge in the neocortex, which can generalize well across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The long-term memory then interacts with the instance-level episodic memory to retrieve structural rela- tional knowledge (Sarfraz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2021) for the previous tasks encoded in the output logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Consolidated logits are then utilized to enforce consistency in the functional space of the working model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This facilitates the consolidation of infor- mation by encouraging the acquisition of new knowledge while maintaining the functional relation of previous knowl- edge and aligning the decision boundary of working mem- ory with long-term memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='4 Overall Formulation Given a continuous data stream D containing a sequence of tasks (D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='., DT ), the CL task is to learn the joint dis- tribution of all the observed tasks without the availability of task labels at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Our proposed method, SCoMMER, involves training a working memory θw, and maintains an additional long-term memory θs and an episodic memory M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The long-term memory is initialized with the same pa- rameters as the working memory and has the same spar- sity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Therefore, long-term memory aggregates the weights of working memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We initialize heterogeneous dropout probabilities πh randomly to set the probability of retention of a fraction of units to 1 and others to 0 so that the first task is learned using a few, but sufficient units and the remaining can be utilized to learn subsequent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Table 1: Comparison on different CL settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The baseline results for S-CIFAR100 and GCIL are from (Arani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Buffer Method S-CIFAR10 S-CIFAR100 GCIL Class-IL Task-IL Class-IL Task-IL Unif Longtail – JOINT 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='15 98.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='43 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='36 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='21 T1 T2 T3 T4 T5 After T1 After T2 After T3 After T4 After T5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 Working Memory T1 T2 T3 T4 T5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 Long-Term Memory Figure 2: Task-wise performance of working memory and the long-term memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The long-term memory effectively aggregates knowledge encoded in the working memory and generalizes well across the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' During each training step, we interleave the batch of sam- ples from the current task xt ∼ Dt, with a random batch of exemplars from episodic memory xm ∼ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Working mem- ory is trained with a combination of cross-entropy loss on the interleaved batch x ← (xt, xb), and knowledge retrieval loss on the exemplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Thus, the overall loss is given by: L = Lce(f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' θw), y) + γLkr(f(xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' θw), f(xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' θs)) (4) where γ controls the strength of the enforcement of con- sistency, and mean-squared error loss is used for Lkr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The training step is followed by stochastically updating the long- term memory (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The semantic dropout and heteroge- neous dropout probabilities are updated at the end of each epoch and task, respectively (using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 1 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We use long-term memory for inference, as it aggregates knowledge and generalizes well across tasks (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Agorithm 1 provides further training details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 4 Evaluation Protocol To gauge the effectiveness of SCoMMER in tackling the dif- ferent challenges faced by a lifelong learning agent, we con- sider multiple CL settings that test different aspects of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Class-IL presents a challenging CL scenario where each task presents a new set of disjoint classes, and the model must learn to distinguish between all the classes seen so far without the availability of task labels at the test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' It requires the model to effectively consolidate information across tasks and learn generalizable features that can be reused to acquire new knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Generalized Class-IL (GCIL) (Mi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2020) extends the Class-IL setting to more realistic scenarios where the agent has to learn an object over multiple recurrences spread across tasks and tackle the chal- lenges of class imbalance and a varying number of classes in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' GCIL utilizes probabilistic modeling to sam- ple the number of classes, the appearing classes, and their sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Details of the datasets used in each setting are provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Though our method does not uti- lize separate classification heads or subnets, for completion, we also evaluate the performance under the Task-IL setting, where the model has access to the task labels at inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' In this setting, we use the task label to select the subset of output logits to select from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 5 Empirical Evaluation We compare SCoMMER with state-of-the-art rehearsal- based methods across different CL settings under uniform experimental settings (details provided in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' SGD provides the lower bound with standard training on sequen- tial tasks, and JOINT gives the upper bound on performance when the model is trained on the joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Table 1 shows that SCoMMER provides performance gains in the majority of the cases and demonstrates the ef- fectiveness of our approach under varying challenging CL settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' In particular, it provides considerable improve- ment under low buffer size settings, which suggests that our method is able to mitigate forgetting with fewer samples from previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The performance gains over CLS-ER, which employs two semantic memories, show that sparse coding in our method enables the effective utilization of a single semantic memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' In particular, the gains in the GCIL setting, where the agent has to face the challenges of class imbalance and learn over multiple occurrences of objects, alludes to several advantages of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Our proposed semantic dropout in conjunction with sparse activations en- ables the model to reuse the sparse code associated with the T1 T2 T3 T4 T5 After T1 After T2 After T3 After T4 After T5 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='7 DER++ T1 T2 T3 T4 T5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='3 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 CLS-ER T1 T2 T3 T4 T5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 SCoMMER Figure 3: Task-wise performance of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The heatmaps provide the test set of each task (x-axis) evaluated at the end of each sequential learning task (y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' SCoMMER retains the performance of earlier tasks better without compromising on the current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Table 2: Ablation Study: Effect of systematically removing different components of SCoMMER on the performance of the models on S-CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' All components contribute to the performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Sparse Long-Term Semantic Accuracy Activations Memory Dropout \x13 \x13 \x13 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='61 \x13 \x13 \x17 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='38±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='51 \x17 \x13 \x17 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='88±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='43 \x13 \x17 \x17 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='44±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='43 \x17 \x17 \x17 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='79±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='86 recurring object and learn better representations with the ad- ditional samples by adapting the corresponding subset of fil- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, compared to the dense activations in CLS- ER, the sparse coding in SCoMMER leads to the emergence of subnetworks that provide modularity and protection to other parts of the network since the entire network is not updated for each input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This increases the robustness of the model to the class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Overall, our method provides an effective approach to em- ploy sparse coding in DNN and enables better utilization of long-term memory, which can effectively consolidate infor- mation across tasks and further mitigate forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 6 Ablation Study To gain further insight into the contribution of each com- ponent of our method, we systematically remove them and evaluate the performance of the model in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The results show that all components of SCoMMER contribute to the performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The drop in performance from remov- ing semantic dropout suggests that it is effective in enforc- ing sparse coding on the representations of the model, which reduces the interference between tasks and allows semanti- cally similar classes to share information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We also observe the benefits of multiple memory systems in CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Additional long-term memory provides considerable performance im- provement and suggests that the EMA of the learned synap- tic weights can effectively consolidate knowledge across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, we observe that sparsity is a critical component for enabling CL in DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Sparse activations alone significantly improve ER performance and also en- able efficient utilization of semantic memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We highlight that these individual components complement each other and that the combined effect leads to the observed perfor- mance improvement in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 7 Characteristics Analysis We look at different characteristics of the model to under- stand what enables the performance gains in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We analyze the models trained on S-CIFAR100 with a buffer size of 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 Stability-Plasticity Dilemma To better understand how well different methods maintain a balance between stability and plasticity, we look at how task-wise performance evolves as the model learns tasks se- quentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The diagonal of the heatmap shows the plastic- ity of the model as it learns the new task, whereas the dif- ference between the accuracy of the task when it was first learned and at the end of the training indicates the stabil- ity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Figure 3 shows that SCoMMER is able to maintain a better balance and provides a more uniform per- formance on tasks compared to baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' While CLS-ER provides better stability than DER++, it comes at the cost of the model’s performance on the last task, which could be due to the lower update rate of the stable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' SCoMMER, on the other hand, retains performance on the earlier tasks (T1 and T2) and provides good performance on the recent task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We also compare the long-term semantic and working mem- ory performance in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Long-term memory effectively aggregates the learned knowledge into the synaptic weights of working memory and generalizes well across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 Emergence of Subnetworks To evaluate the effectiveness of activation sparsity and se- mantic dropout in enforcing sparse coding in the model, we look at the average activity of the units in the penultimate layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The emerging sparse code for each class is tracked during training using the class-wise activity counter and en- forced using semantic dropout probabilities (Equation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Figure 4: Class-wise activation counts of the filters in the penultimate layer of the model trained on S-CIFAR10 with 200 buffer size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Comparison of the activation counts on the test set with the learned class-wise probabilities, Ps, during training shows the effectiveness of semantic dropout in enforcing sparse coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Right plot shows the cosine similarities between the activation counts of different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Semantically similar classes have higher correlation in activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Darker color shows higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Given a test sample from class c, ideally, we would want the model to use the subset of neurons that had higher activ- ity for the training samples from class c without providing any task information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Concretely, we track the class-wise activity on the test set and plot the normalized activation counts for a set of neurons next to their class-wise proba- bilities at the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Figure 4 shows a high correla- tion between the test set activation counts and the semantic dropout probabilities at the end of training, particularly for recent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The activation counts also hint at the natu- ral emergence of semantically conditioned subnets, as the model utilizes a different set of units for different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, we observe that semantically similar classes have a higher degree of correlation between their activation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' For instance, cat and dog share the most active neu- rons, a similar pattern is observed between horse and deer, and car and truck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The cosine similarities between the ac- tivation counts of the different classes further supports the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This is even more remarkable given that these classes are observed in different tasks, particularly for cars and trucks, which are observed in the first and last tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='3 Task Recency Bias A major challenge in CL is the recency bias, in which the update of the model on new task samples biases its predic- tions toward the current task (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This leads to considerable forgetting of earlier tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' To compare the de- gree to which SCoMMER tackles this issue, we evaluate the probabilities of predicting each task by aggregating the soft- max output of samples from the test set of all seen tasks and averaging the probabilities of classes in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Figure 5 shows that SCoMMER provides more uniform probabili- ties to predict each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' CLS-ER is able to mitigate the bias towards the last task, which can be attributed to the aggrega- tion of knowledge in the semantic memories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' however, CLS- ER reduces the probability of predicting the last task, which explains the low performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' SCoMMER effectively mit- igates recency bias and provides uniform prediction proba- ER DER++ CLS-ER SCoMMER 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='6 Task Probability Task 1 Task 2 Task 3 Task 4 Task 5 Figure 5: Average probabilities of predicting classes from each tasks at the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' SCoMMER provides more uniform probabilities across the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' bilities across tasks without any explicit regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 8 Conclusion Motivated by the mechanisms for information representation and utilization of multiple memory systems in the brain, we proposed a novel approach to employ sparse coding in mul- tiple memory systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' SCoMMER enforces activation spar- sity along with a complementary semantic dropout mecha- nism, which encourages the model to activate similar units for semantically similar inputs and reduce the overlap with dissimilar inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Additionally, it maintains long-term mem- ory, which consolidates the learned knowledge in working memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Our empirical evaluation shows the effectiveness of the approach in mitigating forgetting in challenging CL scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, sparse coding enables efficient con- solidation of knowledge in the long-term memory, reduces the bias towards recent tasks, and leads to the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Ye, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' and Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Large scale incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 374–382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Xiao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Zhong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' and Zheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Enhancing ad- versarial defense by k-winners-take-all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='10510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Zenke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Poole, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' and Ganguli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Continual learn- ing through synaptic intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Proceedings of machine learning research, 70: 3987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' A Appendix B Experimental Setting For a fair comparison with different CL methods in uni- form experimental settings, we extended the Mammoth framework (Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' To disentangle the per- formance improvement of the algorithm from the training regimes (Mirzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2020), we use the same network (ResNet-18), optimizer (SGD), batch size for task data and memory buffer (32), data augmentations (random crop and random horizontal flip), and the number of epochs (50) for all our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' For hyperparameter tuning, we use a small held-out val- idation set and perform a grip search on activation sparsity, γ, dropout strengths, πh and πs, and the update frequency for long-term memory r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Table S1 provides the selected hy- perparameters for each setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Note that our method does not require an extensive hyperparameter search for differ- ent buffer sizes, and sensitivity to hyperparameters section shows that the different parameters are complementary in nature and the model performs well for a number of differ- ent combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Therefore, majority of parameters can be fixed, which reduces the search space of hyperparameters significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We report the average and one standard devia- tion of three different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' C Continual Learning Datasets We consider the Class-IL and Generalized Class-IL setting for our empirical evaluation to extensively assess the ver- satility of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Here, we provide details of the datasets used in each of the settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 Class Incremental Learning (Class-IL) Class-IL (van de Ven and Tolias 2019) requires the agent to learn a new disjoint set of classes with each task, and the agent has to distinguish between all the classes seen so far without the availability of task labels at the test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We consider the split variants of the benchmark datasets S- CIFAR10 and S-CIFAR100 where the classes are split into 5 tasks with 2 and 20 classes each, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The order of the classes in the experiments remains fixed, whereby for CIFAR10 the first task includes the first two classes, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 Generalized Class Incremental Learning (GCIL) GCIL (Mi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2020) extends the Class-IL setting to more realistic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' In addition to avoiding forgetting, the model has to tackle the challenges of class imbalance, learn- ing an object over multiple recurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' GCIL utilizes prob- abilistic modeling to sample three characteristics of a task: the number of classes, the classes that appear, and their sam- ple sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Similarly to (Arani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2022), we consider GCIL on the CIFAR100 dataset with 20 tasks, each with 1000 sam- ples, and the maximum number of classes in a single task set to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' To disentangle the effect of class imbalance from the ability of the model to learn from recurring classes un- der non-uniform task lengths, we evaluate the model on uni- form (Unif) and longtail data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' we set the GCIL dataset seed to 1993 for all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' D Implementation Details Here, we provide more details on the implementation of k-WTA activation for CNNs and the proposed semantic dropout mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' E k-WTA for Convolutional Neural Networks The common implementation of k-WTA in convolutional neural networks involves flattening the activation map into a long CHW ×1 vector and applying the activation of k-WTA in a way similar to that of the fully connected network (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Ahmad and Scheinkman 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This translates to setting some spatial dimensions of a filter to zero while prop- agating others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' However, this implementation does not take into account the functional integrity of an individual con- volution filter as an independent feature extractor and does not enable the formation of task-specific subnetworks with specialized feature extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Different tasks cannot utilize a different subset of filters, and we cannot track the activity of an individual filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Our implementation, on the other hand, assigns an activa- tion score to each filter in the layer by taking the absolute sum of the corresponding activation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Given the activa- tion map of the layer l, Al, (C × W × H) where C is the number of filters, W and H are the width and height, we flatten the spatial dimensions, C × WH), and the activation score for each filter j is given by the absolute sum of its ac- tivations, [Cscore]j = �HW i=1 |[Al]j,i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We then find the value k for the layer using the activation sparsity (% of the active filters in the layer), k ← %k × N l filters where N l filters is the number of filters in the layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The kth highest value of the filter activation scores vector, Cscore ∈ RC×1 gives the threshold value used to apply a mask to the input activation map, which only propagates the activations of filters with a score above threshold by setting the others to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Finally, the ReLU activation function is applied to the masked acti- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Algorithm 2 provides more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' For the ResNet-18 network in our method, we set the ac- tivation sparsity for each ResNet block, for example % k = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8] enforces the activation sparsity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 in the first three ResNet blocks, that is 90% of the filters in each convolutional layer are active for a given stimulus and 80% in the convolutional layers of the last ResNet block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Table S1: Selected parameters for SCoMMER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' For each of our experiments, we apply Heterogeneous and Semantic dropout only on the output of the last residual block in ResNet-18, the decay parameter for long-term memory is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='999, the batch size of 32 is used for both the current task and the memory buffer, and the models are train for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' For the first three ResNet blocks, we use an activation sparsity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 and vary the sparsity ratio for the last block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Dataset Buffer size Activation Sparsity η πh πs γ r S-CIFAR10 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='7 S-CIFAR100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 GCIL - Unif 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='6 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='6 GCIL - Longtail 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='6 Algorithm 2: Global k-WTA for CNNs Input: Activation map A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' activation ratio %k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' number of filters Nfilters Evaluate activation scores: 1: Flatten the spatial dimensions: 2: Cscore ← Reshape(Cscore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' C × HW) 3: Assign score to each filter: 4: Cscore = abs sum(Cscore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' dim=1) Calculate threshold: 5: Get k value for the layer: 6: k ← %k × Nfilters 7: Return kth largest value 8: Cthresh = kth value(Cscore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' k) Mask Activation Map: 9: Initialize mask with zeros 10: M ← Zeros(C × H × W) 11: Set filter mask with score above threshold to 1 12: M[Cscore > Cthresh] = 1 13: Apply mask 14: A ← M · A Apply ReLU activation function: 15: A ← ReLU(A) return A E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 Semantic Dropout At the beginning of training, we initialize the heterogeneous dropout probabilities Ph so that for each layer l the prob- ability of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 × %kl × N l filters) filters is set to 1 and the remaining set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This is done to ensure that the learning of the first task does not utilize all filters while having the flexibility to learn a different subset of units for the classes in the first task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The semantic dropout probabilities Ps are updated at the end of each epoch, once the epoch num- ber e for the task is higher than the heterogeneous dropout warm-up period Eh to allow the emergence of class-wise ac- tivity patterns before it is explicitly enforced with seman- tic dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Note that to ensure that we have enough ac- tive filters before applying k-WTA activation, when apply- ing heterogeneous, we use the probabilities Ph to sample the (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 × %kl × N l filters) filters for the given layer before ap- plying k-WTA activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 factor is arbitrarily chosen and works well in practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' however, a different value can be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Further details of the method are provided in Algo- rithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Importantly, we disable the dropout activation counter up- date for the buffer samples so that the sparse code is learned during task training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Also, dropout is applied only to work- ing memory as it is learned with gradient decent, whereas the long-term memory aggregates the weights of working memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Our analysis shows that the learned sparse coding is effectively transferred to long-term memory through ema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' For the ResNet-18 model used in our experiments, we ap- ply dropout at the output of each ResNet block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Although our method provides the flexibility to apply different dropout strengths for each block, we observe empirically that it works better if applied only at the output of the last ResNet block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This allows the model to learn features in the earlier layers that can generalize well across the tasks and to learn specialized features for the classes in later layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' F Performance of working memory To gain a better understanding of the performance of the different memories, Table S2 provides the performance of both working memory and long-term memory in the settings considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Long-term memory consistently provides better generalization across tasks, especially in the Class-IL set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' This shows the benefits of using multiple memory sys- tems in CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Furthermore, it demonstrates the effectiveness of the exponential moving average of the working memory weights as an efficient approach for aggregating the learned knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Algorithm 3: Semantic Dropout Input: Activation map A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' class labels y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' activation ratio %k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' number of filters Nfilters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' dropout probabilities Ph and Ps Get the Heterogeneous Dropout Mask: 1: Initialize Heterogeneous dropout mask with zeros 2: Hmask ← Zeros(C × H × W) 3: Calculate the sampling probabilities so that they sum to zero Psample = Ph / sum(Ph) 4: Get the indices of retained filters Nretain = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='1 × %k × Nfilters idx = Sample(range=Nfilters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' #samples=Nretain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' prob=Psample,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' replace=False) 5: Set the mask at retained indices to 1 Hmask[idx] = 1 Get the Heterogeneous Dropout Mask: 6: Initialize Semantic dropout mask with zeros 7: Smask ← Zeros(C × H × W) 8: Use the semantic dropout probabilities to select units retain = N ∼ U(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' 1) ≤ Ps 9: Set the mask at retained indices to 1 Smask[retain] = 1 Select the mask for each input sample 10: For each sample,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' select Semantic dropout mask if available for the class label,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' otherwise use Heterogeneous dropout mask: 11: M = Smask if Ps[y] > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' otherwise Hmask Mask Activation Map: 12: A ← M · A return A G Sensitivity to Hyperparameters SCoMMER employs sparse coding in a multiple-memory replay mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Therefore, the setting of two sets of pa- rameters is required: sparse coding (activation sparsity %k and dropout strength πs and πh) and the aggregation of in- formation in long-term memory (r, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We show the effect of different sets of hyperparameters in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We can see that the different components are complementary in na- ture and therefore different combinations of parameters can provide similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Interestingly, we observe that increasing the semantic dropout strength considerably in- creases the performance of the working model, but the long- term memory performance remains quite stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' The method is not highly sensitive to a particular set of parameters, and often we can fix the majority of parameters and fine-tune only a few, which significantly reduces the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Table S2: Performance of working memory and long-term memory in different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Long-term memory consistently pro- vides better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' Buffer Memory S-CIFAR10 S-CIFAR100 GCIL Class-IL Task-IL Class-IL Task-IL Unif Longtail 200 Working 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='03±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='17 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='56 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='71 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='16 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='30 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='06±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='43 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='80 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='31 500 Working 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='10±3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='21 Table S3: Sensitivity to different hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' We pro- vide the performance of Working memory and Long-term memory of models trained on S-CIFAR-10 with 200 buffer size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content=' For all experiments γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} +page_content='15, lr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE4T4oBgHgl3EQfZwwm/content/2301.05058v1.pdf'} 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XX, NO. XX, JANUARY 2023 +1 +Small Moving Object Detection Algorithm Based +on Motion Information +Ziwei Sun, Zexi Hua, and Hengcao Li, Fellow, IEEE +Abstract—A Samll Moving Object Detection algorithm Based +on Motion Information (SMOD-BMI) was proposed to detect +small moving objects with low Signal-to-Noise Ratio (SNR). +Firstly, To capture suspicious moving objects, a ConvLSTM- +SCM-PAN model structure was designed, in which the Convo- +lutional Long and Short Time Memory (ConvLSTM) network +fused temporal and spatial information, the Selective Concatenate +Module (SCM) was selected to solve the problem of channel +unbalance during feature fusion, and the Path Aggregation +Network (PAN) located the suspicious moving objects. Then, an +object tracking algorithm is used to track suspicious moving +objects and calculate their Motion Range (MR). At the same time, +according to the moving speed of the suspicious moving objects, +the size of their MR is adjusted adaptively (To be specific, if the +objects move slowly, we expand their MR according their speed +to ensure the contextual environment information) to obtain their +Adaptive Candidate Motion Range (ACMR), so as to ensure that +the SNR of the moving object is improved while the necessary +context information is retained adaptively. Finally, a LightWeight +SCM U-Shape Net (LW-SCM-USN) based on ACMR with a +SCM module is designed to classify and locate small moving +objects accurately and quickly. In this paper, the moving bird in +surveillance video is used as the experimental dataset to verify +the performance of the algorithm. The experimental results show +that the proposed small moving object detection method based +on motion information can effectively reduce the missing rate +and false detection rate, and its performance is better than the +existing moving small object detection method of SOTA. +Index Terms—Object Detection; Small Moving Objects; Mo- +tion Information; Motion Range; Low Signal-to-Noise Ratio +I. INTRODUCTION +T +HE intelligent video analysis technology can reduce the +work intensity of the monitoring center staff and reduce +the false positives and missing positives caused by manual +monitoring. And moving object detection is one of the basic +tasks of intelligent video analysis technology [1], [2]. Through +moving object detection technology, information such as the +category, location, size and motion speed of moving objects +can be obtained, which can provide basic data support for in- +telligent video analysis technology such as behavior prediction +and trajectory tracking of moving objects in the next step. +For the detection of small moving objects, there are two +main challenges. +• The object has a low SNR. For the general unattended +monitoring scene, the monitoring area is usually a room +or an outdoor area. If a mouse or bird intrudes into the +Manuscript received January 4, 2023. +Ziwei Sun, Zexi Hua and Hengcao Li are with the School of Information +Science and Technology, Southwest JiaoTong University, chengdu 611756, +China. +monitoring area, the number of pixels is usually small, +as shown by Bird A in Fig. 1. +• The moving object may blur. Since most of the low-cost +surveillance cameras do not have the ability of low-delay +photography, the moving object captured has a certain +trailing phenomenon, which may lead the moving object +blur, as shown by Bird B in Fig. 1. +Fig. 1: Small and blurred moving birds in the surveillance +area. The Bird A is small but clear, the Bird B is small and +blur. +To solve the above problems, researchers mainly use the +motion information (spatio-temporal features). Of course, like +other vision tasks, the detection method of moving small +objects has also experienced the development from traditional +methods based on knowledge-driven to deep learning methods +based on data-driven. +At present, the knowledge-driven moving object detection +algorithms mainly include frame difference method [3], back- +ground difference method [4], robust principal component +Analysis method [5] and optical flow method [6]. In the early +stage, the frame difference method, background difference +method, and robust principal component analysis method +were only suitable for the situation that the background was +static and there was no more complex interference (such as +illumination change, branches and leaves swaggling, water +waves and so on). The optical flow method was suitable for the +situation of moving background, but it still could not overcome +some interference such as illumination change, the object stop +or slow motion. However, through the continuous efforts of +researchers, the traditional methods can accurately extract the +moving object to a certain extent [7], [8], [9]. However, the +traditional methods can only extract the pixels of the moving +object at most, can not obtain other attributes of the moving +object, and can not distinguish the interesting and uninteresting +moving objects. +arXiv:2301.01917v1 [cs.CV] 5 Jan 2023 + +Bird B +oBirdAIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +2 +In the early stage, methods based on deep learning were +mainly combined with traditional methods and object detection +methods: traditional methods such as frame difference method, +background difference method, principal component analysis +method, and optical flow method were combined with object +detection methods, in which the traditional method provided +time-related motion information, and the object detection +provided space-related positioning information [10], [11], [12], +[13]. These traditional methods with object detection have +made considerable progress in detecting moving objects. How- +ever, the detection performance of these methods is affected +by the motion information provided by the traditional methods +to a certain extent. At present, some researchers gradually pay +attention to the full deep learning to obtain the temporal infor- +mation and spatial information of moving objects at the same +time, such as using ConvLSTM(Convolution Long Short Term +Memory) for moving object detection [14], [15]. Or moving +object detection with input of consecutive multiple frames +merged [16], [17]. The method based on deep learning has +a certain improvement in effect and function compared with +the traditional method. It can distinguish between interested +and uninterested moving objects, and can obtain the category +and location of moving objects. However, there are still many +false detections and missed detections when detect the small +moving objects. We further analyze and find that most of the +missed detections occur because the object is small or similar +to the environment, and most of the false detections that occur +are caused by various tiny moving things or things whose +appearance is similar to the object of interest. Therefore, the +main reason for this problem is that most moving objects +account for a small proportion of pixels in the whole video +frame, and the problem of low SNR (unbalanced positive and +negative samples) is not easy to be eliminated in the training +process. +In order to solve the above problems, this paper analyzes +our human method of small moving object recognition in +complex environments. Our human approach to identifying +small moving objects is divided into two stages. In the first +stage, we will find out where the object may exist according +to the motion information. In the second stage, we will focus +on the area where the object may exist and carefully observe, +so as to remove more interference information. Therefore, we +propose a Small Moving Object Detection algorithm Based +on Motion Information (SMOD-BMI). Firstly, a moving ob- +ject detection model ConvLSTM-SCM-PAN (coarse-detection +model) is designed to fuse spatio-temporal information, which +can capture suspicious moving objects according to the mo- +tion information of moving objects. Then, the Motion Range +(MR) of suspicious moving objects is extracted by using the +object tracking algorithm. At the same time, according to +the moving speed of the suspicious moving objects, the size +of their MR is adjusted adaptively (To be specific, if the +objects move slowly, we expand their MR according their +speed to ensure the contextual environment information) to +obtain their Adaptive Candidate Motion Range (ACMR), so +as to ensure that the SNR of the moving object is improved +while the necessary context information is retained adaptively. +Finally, a lightweight moving object detection model LW- +SCM-USN (Fine detection model) based on the ACMR of the +moving object is designed to accurately classify and locate the +moving object on the basis of ensuring real-time. The main +contributions of this paper are as follows. +• The ConvLSTM-SCM-PAN model structure is designed +to capture the suspicious moving objects. Among them, +Convolution Long Short-Term Memory Network (ConvL- +STM) fuses spatio-temporal information, Selective Con- +catenation Module (SCM) to solve the problem of chan- +nel imbalance during feature fusion, and PAN locates +suspicious moving objects. +• An adaptive method of extracting ACMR based on the +amount of motion of the moving object is proposed. By +using the object tracking technology and the amount of +motion of the moving object, the ACMR of the suspected +moving object are extracted adaptively, which improves +the SNR of the moving object and retains the necessary +context information of the moving object. +• A LightWeight U-Shaped Network with SCM module +(LW-SCM-USN) model structure is designed, and the +accurate classification and location of moving objects are +realized by using the ACMR of suspected objects. +The remainder of this paper is structured as follows: Section +II is a survey of related work on moving object detection. +Section III describes the proposed SMOD-BMI in detail. +Section IV is devoted to ablation experiments and comparison +experiments of the proposed algorithm. Section V concludes +our work. +II. RELATED WORK +According to the use of different characteristics of the +object, the methods of moving object detection can be mainly +divided into three categories: methods based on appearance +information, methods based on motion information and meth- +ods based on deep learning for moving object detection. In +this section we will review these three categories. +A. Appearance-based Object Detection +From traditional methods [18], [19], [20] to deep learning- +based methods [21], [22], [23], [24], [25], [26], [27], [28], +object detection technology has now made great progress, +which can accurately determine the specific class of each +object and give the bounding box of each object. However, +since these object detection algorithms only rely on the +appearance features of the object, the detection effect is not +good for small moving objects with complex backgrounds and +unobvious appearance features [11], [29]. +B. Moving Object Detection based on Motion Information +Since the object detection algorithm based on appearance +feature can not detect small moving objects in complex +background well, researchers have proposed various moving +object detection algorithms based on motion information. The +main methods are frame difference, background subtraction, +optical flow and robust principal component analysis. + +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +3 +1) Frame Difference Method: Because the object is mov- +ing, there is a certain displacement between the position of +the historical frame and the position of the current frame. The +changed pixel, which is the pixel of the moving object, can be +extracted by subtracting the historical frame from the current +frame. When the simple frame difference method obtains the +moving object, it is easy to appear the hole or ghost phe- +nomenon [30]. Therefore, researchers have proposed various +complex frame difference methods to solve this problem [31], +[32], which have achieved certain effect improvement, but the +problem cannot be completely solved. +2) Background Subtraction Method: The environment and +moving object are regarded as background and foreground. +The background remains static, while the moving object moves +in front of the background as the foreground. The key point +of this method is the background modeling. There are many +methods of background modeling, which are widely used at +present, such as multi-frame average background modeling, +simple Gaussian modeling [33], Gaussian mixture modeling +[34], ViBe algorithm [35], etc., although the modeling effect +of these background modeling methods is getting better and +better. However, it still cannot completely overcome various +disturbances such as wind and water waves, resulting in more +interference in the extracted foreground. +3) Optical Flow Method: The moving object detection +method based on optical flow method distinguishes the back- +ground and moving object by using optical flow field according +to the feature that the brightness of adjacent points in the image +is similar [6]. The key technology of optical flow method +is to solve the estimation of optical flow. At present, the +main optical flow estimation algorithms include correlation +method, energy method, discrete optimization method and +phase method [6]. The optical flow method does not need +prior information to detect moving objects and can be used in +dynamic background. However, the calculation of optical flow +field distribution is difficult due to the change of light source, +shadow and occlusion. +4) Robust Principal Component Analysis (RPCA): The +background is considered as a low-rank matrix and the moving +objects are sparse. Therefore, this method converts the detec- +tion of moving objects into low-rank sparse decomposition +of the matrix composed of multiple frames, so as to obtain +sparse moving objects [36]. Since the original robust principal +component analysis method is time-consuming, subsequent +researchers have proposed some improved schemes, such as +Faster RPCA [37], which greatly improves the decomposition +speed. However, when the background moves or the back- +ground changes complex, the background matrix loses its low +rank property, and it is difficult to decompose the moving +object at this time. Therefore, the robust principal component +analysis method is mainly suitable for the situation that the +background is static or the background changes simple. +C. Moving Object Detection with Deep Learning +In recent years, influenced by the great progress of deep +learning technology in vision tasks, researchers have begun +to use deep learning technology to detect moving objects. +Researchers have used deep learning techniques in two differ- +ent ways to investigate how to detect moving objects, but all +related studies follow the same basic rule, that is, you need +to consider both time-based motion information and space- +based position information. The difference between these two +methods lies in how to obtain time-based motion information. +One way is to obtain the motion information by using the +traditional moving object detection method, which is called +the traditional plus deep learning method, and the other way +is to obtain the motion information directly by using deep +learning, which is called the full deep learning method. +1) Traditional plus Deep Learning Method: The traditional +and deep learning moving object detection methods are sum- +marized into two categories. 1) Firstly, the motion information +is used to extract the foreground, and then the foreground is +used for moving object detection [10], [11], [12], [13]. For +example, literature [10] introduces the Fast RPCA algorithm +to separate the foreground, and then implements Faster R-CNN +object detection on the foreground map to effectively detect +the moving small object in the panoramic video. In literature +[11], the frame difference method was used to obtain the +moving foreground, and then the CNN classification network +was used to screen the region of interest. Finally, the CNN +regression network was used to perform coordinate regression +on the region of interest to obtain the moving object. Literature +[12] uses the ViBe background modeling method to extract the +foreground, and uses this foreground as the candidate moving +object area of Fast R-CNN to set ANCHORS, so as to realize +the detection of moving objects. In reference [10], the motion +region was obtained by frame difference method, and then +the motion region was connected and expanded. Finally, Deep +CNN was used to classify and position regression the object +in the motion region. 2) Traditional methods are directly fused +with object detection to detect moving objects [38], [39]. For +example, literature [38] inputted the frame difference between +the original image and the two frames into VGG16 for fusion, +and then inputted the fused feature layer into Faster R-CNN +for object detection. Literature [39] proposed a method based +on deep learning combining RGB and optical flow to segment +moving objects. +2) Full Deep Learning Method: There are two main cat- +egories of moving object detection methods based on full +deep learning. 1) ConvLSTM is used to fuse temporal and +spatial information to segment or detect moving objects [14], +[15]. For example, reference [14] introduces the attention +Long Short-Term Memory (attention ConvLSTM) model to +simulate the change of pixels over time, and then uses a +spatial Transformer and conditional random field (CRF) to +segment moving objects. In reference [15], the Pyramid dilated +convolution (PDC) module was designed to extract multi- +scale spatial features, and then these spatial features were +concatenated and fed into the Extended deep Bidirectional +ConvLSTM (DB-ConvLSTM) to obtain spatio-temporal in- +formation. Finally, the moving objects in the video are de- +tected by using the spatio-temporal information. 2) Detecting +moving objects by merging and fusing temporal and spatial +information of consecutive frames [16], [17]. For example, the +paper [16] propose regions of objects of interest (ROOBI) by + +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +4 +using the region proposal network, which combines the spatio- +temporal information by merging the input of consecutive +frames. After getting the Propose regions, the exact position of +the object is located again by merging the input of consecutive +multiple frames. In literature [17], continuous multiple frames +are merged into CNN for background estimation, and then a +compact encoder-decoder network is used to segment moving +objects. +III. THE PROPOSED SMOD-BMI +Fig. 2 shows the overview diagram of the proposed SMOD- +BMI, which contains three parts. Firstly, ConvLSTM-SCM- +PAN model structure was designed to capture the suspicious +moving objects. Secondly, An object tracking algorithm is +used to track suspicious moving objects and calculate their +MR. At the same time, according to the moving speed of the +suspicious moving objects, the size of their MR is adjusted +adaptively to obtain their ACMR, so as to ensure that the +SNR of the moving object is improved while the necessary +context information is retained adaptively. Finally, LW-SCM- +USN based on ACMR with a SCM module is designed to clas- +sify and locate small moving objects accurately and quickly. +Section III-A describes the ConvLSTM-based suspicious mov- +ing object detection method. Section III-B ACMR extraction +method of suspicious moving object based on object tracking +technology and motion amount. Section III-C describes the +moving object detection method based on ACMR. +A. To Capture the Suspicious Moving Object +In this paper, we perform two steps to capture suspicious +moving objects (coarse-detection of moving objects) in con- +secutive video images. Firstly, the spatio-temporal information +of the moving object was fused. Secondly, the spatio-temporal +information is used to locate the suspicious moving object by +object detection. This subsection will introduce the acquisition +of spatio-temporal information of moving objects (Section +III-A1), and the localization of suspicious moving objects +(Section III-A2), respectively. +1) Fusion of Spatio-temporal Information for Small Moving +Objects: Motion is mainly reflected in time and space, that +is, at different times, according to the spatial location of +the object can show motion. Therefore, to capture the Small +moving object, it is necessary to fuse its temporal and spatial +information. +As we have introduced in Section II-C2, there are two main +ways to fuse the spatio-temporal information of the object +based on deep learning. One is based on the recurrent neural +network ConvLSTM, and the other is based on the input +merging of consecutive multiple frames. ConvLSTM(structure +shown in Fig. 3) contains three gates, namely input gate, +output gate and forget gate, which are used to control the +input and output and what information needs to be forgotten +and discarded. At the same time, the input gate and output +gate can also be understood as controlling the writing and +reading of the memory cell. Continuous multi-frame merging +input is to simply Concatenate consecutive frames of video +images together and then input into the neural network. +The coarse-detection phase captures the suspicious moving +object, and the input is the whole video, which has the char- +acteristics of many background interference and redundant in- +formation (different frames have many identical backgrounds). +According to the characteristics of ConvLSTM structure, it +can remove unimportant or redundant information while fusing +spatio-temporal information. So, in the first stage, we use Con- +vLSTM to extract and fuse the spatio-temporal information of +moving objects. Specifically, given the input n consecutive +frames of images Xt ∈ R(H×W×3)|t = (1, 2, · · · , n) (Where +H and W are the height and width of the input image, and n is +an odd number), the ConvLSTM network FConvLSTM is used to +fuse and extract the spatio-temporal features Hn ∈ R(H×W×C) +(Where C is the number of channels) of the n consecutive +frames of images, +Ht = FConvLSTM ([Xt, Ht−1] ; ΘConvLSTM) , +(1) +Where, when t = 1, H0 = 0. ΘConvLSTM is the learnable +parameter of the ConvLSTM network. The spatio-temporal +features Hn of n consecutive frames of images are input +into the subsequent classification and positioning module to +determine the category and spatial location information of the +suspicious moving object. +2) Localization of Suspicious Moving Objects: In convolu- +tional neural networks, deeper layers, which generally have +smaller size, have better global semantic information, and +can predict larger objects. The layers with shallower depth, +which generally have larger size, have more delicate spatial +information and can predict smaller objects. However, the +large feature layer often does not have a relatively high +degree of semantic information, and the small feature layer +does not have fine spatial positioning information. Therefore, +relevant researchers have proposed the structure of FPN [40] to +combine the strong semantic information of the small feature +layer and the strong spatial positioning information of the +large feature layer. However, the researchers of PANet(Path +Aggregation Network) [41] found that when FPN transmitted +information, there was information loss due to the transfer +distance when the information was transmitted to the low-level +feature layer. Therefore, path-enhanced FPN, namely PANet +structure, was proposed. The PANet structure opens up a green +channel for low-level information transmission and avoids low- +level information loss to a certain extent. At the same time, +we find that the detection performance will be improved when +Selective Concatenation Module (SCM) [42] is added to the +model (reference [42] introduces that SCM can help to better +fuse high and low layer information (refer to reference [42] for +details)). We believe that SCM can not only balance the fusion +of channel information in different layers, but also suppress +unimportant information and highlight the information that the +model needs to focus on. So, we introduce the SCM and design +the feature extraction structure of SCM-PANet (see Fig. 4). +The spatio-temporal features Hn of n consecutive frames +are input into the SCM-PANet structure to extract the features +of the suspicious moving object FMOn, +FMOn = FSCM-PAN (Hn; ΘSCM-PAN) , +(2) + +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +5 +Fig. 2: Overview of the proposed SMOD-BMI. (a) Capture the suspicious moving object, the blue box in the figure represents +the detected suspicious moving object. (b) The ACMR of the suspicious moving object is obtained. In the figure, the green +box represents the original MR of the moving object tracked by the tracking algorithm, and the red box represents the MR +adaptively adjusted according to the motion amount of the moving object. (c) Classification and localization of moving objects. +Fig. 3: Structure diagram of ConvLSTM +Where, ΘSCM-PAN is the learnable parameter of the SCM- +PANet network. +When the distance between the moving object and the +surveillance camera is different, the size of the moving object +is also different, so the moving object to be detected has the +multi-scale property. According to the multi-scale property of +the moving object, this paper uses the MultiScale Detection +Head (MS-D Head) to detect the suspicious moving object. +The objects in the middle frame of n consecutive frames +has symmetric contextual information, which can get more +accurate results in prediction. Therefore, this paper predicts +the suspicious object in the middle frame of n consecutive +frames as the detection result of the Coarse-detection stage. +Specifically, the feature FMOn of the moving object is input +into the MS-D Head to obtain the output of the model, +On = FMS-D (FMOn; ΘMS-D) , +(3) + +(a) To capture the suspicious Moving Object (MO) +MS-D +ConyLSTM +SCM-PAN +Head +Suspicious MO (Blue box) +Video Frame +Coarse Detection Model +Track and calculate the Motion Range (MR) +(b) To obtain the Adaptive Candidate MR (ACMR) +(n+4)th +MR (Green box) of the Suspicious MO +Adaptively resize and crop the MR +(c) To classify and locate the MO +BackGround +SS-D +LW-SCM-USN +Head +Bird +crop +crop +crop +ACMR (Red box) of The Suspicious MO +Fine Detection Model +BirdConvLSTM +Conv +Bias +I+X +write +read ++Bc +* +tanh +tanh +C +Xt +W:* ++B: +0 +Ct +J +Concat ++Bf +W.* +Ht-1 +Ot ++Bo +W +* +0 +0 +Ht ++ Data flow +Next iterationIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +6 +Fig. 4: Structure diagram of the SCM-PANet model +where, ΘMS-D is the learnable parameter of the MS-D Head. +Then post-processing operations such as Boxes Decoding and +non-maximum suppression were performed on the output of +the model to obtain the location of the suspicious object in +the middle frame of n consecutive frames, +{PID1, · · · , PIDk} +frame( n+1 +2 ) = FP (On) , +(4) +Where, {·} +frame( n+1 +2 ) means the locations of the moving ob- +jects in +� n+1 +2 +�th frames. PIDk indicates the predicted position +of the object with IDk (the object with IDk is taken as an +example unless otherwise specified). FP (·) denotes the post- +processing method. +B. To Obtain the ACMR +In this paper, the MR of the suspicious moving object on +n consecutive frames is extracted to improve the SNR of the +moving object. At the same time, in order to ensure the context +information of the suspicious moving object, the size of the +MR is adaptively adjusted according to the motion amount of +the suspicious moving object, so that the subsequent detection +results are more accurate. Specifically, we will divide into +two steps to obtain the ACMR of suspicious moving objects. +Respectively, the original MR of the suspicious moving object +is extracted using the object tracking technology (Section +III-B1) and the MR is adaptively adjusted using the motion +amount of the suspicious moving object to obtain the ACMR +(Section III-B2). +1) Acquisition of the Original MR of the Suspicious Moving +Object: From the +� n+1 +2 +�th frame, there are detection results +of the suspicious moving object, and we start to track the +suspicious moving object from the +� n+1 +2 ++ 1 +�th frame. In some +cases, the appearance characteristics of small moving objects +are not obvious, so we only use their motion information when +tracking them, and use a relatively simple SORT [43] object +tracking algorithm to track suspicious moving objects, +� +{PIDk}frame(i), {PIDk}frame(i+1), · · · +� += Ftrack (IDk) , +(5) +where, +� +{PIDk}frame(i), {PIDk}frame(i+1), · · · +� +represents the po- +sition on consecutive image frames of a suspicious moving +object with ID number k, and Ftrack (·) represents the SORT +object tracking method. After obtaining the position of the +suspicious moving object on consecutive image frames, we +can find the Motion Range (MR) of the suspicious moving +object on n consecutive frames. Specifically, the minimum +circumscribed rectangle RectIDk at n positions is calculated +according to the position of the same object on n consecutive +frames of images, +RectIDk = +FMinRect +�� +{PIDk}frame(i+1), · · · , {PIDk}frame(i+n) +�� +, +(6) +where, FMinRect (·) denotes the function to find the mini- +mum circumscribed rectangle of n rectangular boxes. For +example, +to +find +the +minimum +circumscribed +rectangle +[(xmin, ymin) , (xmax, ymax)] (Using the horizontal and vertical +coordinates of the top left and bottom right vertices of the rect- +angle) of {box1, · · · , boxn}, the specific calculation method is +as follows, +xmin = min +� +x1box1 , · · · , x1boxn +� +, +ymin = min +� +y1box1 , · · · , y1boxn +� +, +xmax = max +� +x2box1 , · · · , x2boxn +� +, +ymax = max +� +y2box1 , · · · , y2boxn +� +, +(7) +where, +�� +x1boxn , y1boxn +� +, +� +x2boxn , y2boxn +�� +denotes the hori- +zontal and vertical coordinates of the upper left and lower +right vertices of boxn in the image. The obtained minimum +circumscribed rectangle RectIDk is the MR of the moving +object in n consecutive frames. Fig. 5 illustrates the MR of +the moving object on five consecutive frames of images. +2) Adaptively Adjust the MR to Obtain ACMR Based on the +Amount of Motion: We crop the MR of suspicious moving +object in n consecutive frames to remove the interference of +other background and negative samples, which can improve +the SNR of the moving object. However, if the moving +object moves too slowly, the clipped MR will lack contextual +environmental information (see the Raw MR In Fig. 6), +which is not conducive to the detection of moving objects. In +order to balance the contradiction between SNR and context +information, this paper proposes an ACMR extraction method +based on the amount of motion of the moving object, which +adaptively adjusts the size of the MR of the moving object +according to the speed of the object motion. There are two +steps. +Firstly, the amount of motion of the moving object over n +consecutive frames is calculated. For an object of the same +size, if it moves fast on n consecutive frames, its MR is large; + +Backbone +SCM +SCM +FPN +PAN +MS-D HeadIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +7 +Fig. 5: MR of the moving bird over 5 consecutive frames. The blue box shows the position of the bird in each frame; The +green box represents the minimum bounding rectangle of the five blue boxes, which is the MR of the moving bird over five +consecutive frames. +Fig. 6: The left picture shows the original monitoring picture, the right picture shows the MR of the moving object on five +consecutive frames in the dashed frame, and the ACMR of the moving object in the solid frame. It is obvious that the object +in the original MR is difficult to be correctly recognized, and the object in the ACMR is easier to be recognized. +otherwise, its MR is small. Therefore, we use the ratio of the +area of the MR of the moving object on n consecutive frames +to the area of the single frame image occupied by the moving +object to define its motion amount on n consecutive frames, +σmov = S (RectIDk) +S (ObjIDk) , +(8) +where, σmov is the motion amount, S (·) represents the func- +tion to calculate the area, and ObjIDk represents the object with +ID number k. The area of MR is then the area of the minimum +circumscribed rectangle RectIDk. Since the area occupied by +a moving object in a single image frame may vary due to its +shape changes, and it is difficult to calculate accurately, we +use the rectangular area of its bounding box to approximately +represent its area in this paper. +Then, according to the amount of motion of the moving +object, the MR of the moving object is adaptively adjusted as +the Adaptive Motion Range (AMR) ARectIDk of the moving +object. Specifically, a motion hyperparameter γ is introduced. +When the amount of motion of the moving object is less than +γ, the MR of the moving object is expanded to make the +amount of motion of the moving object reach γ. Therefore, +the AMR of the moving object can be expressed as follows, +ARectIDk = +� +ARectIDk, +σmov ≥ γ +γ × ObjIDk, +otherwise. +(9) +The ARectIDk is used to crop the corresponding n consecutive +frames of video image +� +frame(1), · · · , frame(n)� +respectively, +and the n frame screenshots obtained are the Adaptive Can- +didate Moving Region (ACMR) (ACMRIDk) of the moving +object, +f(i) +ARectIDk = Fcut +� +frame(i), ARectIDk +� +, +(10) +ACMRIDk = +� +f(i) +ARectIDk|i ∈ (1, · · · , n) +� +. +(11) +C. Moving Object Detection based on ACMR +After the previous processing, we improve the SNR of the +moving object, retain its contextual environmental information, +and obtain the ACMR of the moving object. In the fine- +detection stage, we can use the ACMR of the moving object + +Raw MR +ACMR +Raw MR +ACMRIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +8 +to classify and locate the moving object. Specifically, the fine- +detection phase includes the fusion of spatio-temporal infor- +mation (III-C1) and the classification and localization(III-C2) +of moving objects. +1) Fusion of Spatio-temporal Information of Moving Ob- +jects: The input of the fine-detection model is the ACMR +of the moving object extracted earlier. The coarse-detection +model may detect multiple suspicious moving objects at one +time, so there will be multiple ACMRs, and the fine detection +model will detect each ACMR separately. So it’s possible to +run a coarse-detection model once and a fine-detection model +many times. Therefore, in order to balance accuracy and speed, +the method of fusing the spatio-temporal information of the +moving target in the fine-detection stage uses the way of +merging consecutive multiple frames. At the same time, in +order to reduce data redundancy, except the middle frame, the +rest of the frames are input in the form of grayscale image +single channel. Specifically, firstly, grayscale the screenshots +of the ACMR of the moving object except for the middle +screenshot, +f(i)′ +ARectIDk = +� +� +� +f(i) +ARectIDk, +i = int +� n +2 +� +FGray +� +f(i) +ARectIDk +� +, +otherwise. +(12) +where, FGray (·) is a function that finds the grayscale of a color +image. Then, the processed screenshots of the ACMRs are +Concatenate in the channel dimension as the input of the fine- +detection stage, +XSIDK = Fconcat +�� +f(1)′ +ARectIDk, · · · , f(n)′ +ARectIDk +� +, 2 +� +, +(13) +Where, the second argument of the Fconcat function indicates +that the concatenation operation is performed in the third input +dimension (height, width, channel). The length and width of +XSIDK is equal to the length and width of rectangle ARectIDk, +and the number of channels is n+2, which contains the motion +information and appearance information of the moving object. +It is input into the fine-detection model to accurately classify +and locate the moving object. +2) Classification and Localization of Moving objects: In +order to further improve the speed of the whole moving object +detection process, this paper uses a lightweight U-Shaped +Network (USN) (in the experiment, we use MobilenetV2 [44] +as the backbone network of the USN) as the feature extraction +network of the moving object in the fine-detection stage. At +the same time, in order to better fuse high and low layer infor- +mation, similar to the network of the coarse-detection model, +we introduce the SCM [42] module and design the lightweight +LW-SCM-USN feature extraction network structure, as shown +in Fig. 7. +The XSIDK fused with the spatio-temporal information of +the moving object is input into the LW-SCM-USN feature +extraction network to obtain the moving object feature FIDK +fused with the spatio-temporal information, +FIDK = FLW-SCM-USN (XSIDK; ΘLW-SCM-USN) , +(14) +where, ΘLW-SCM-USN is the learnable parameter of the LW- +SCM-USN. +Fig. 7: Structure diagram of LW-SCM-USN +The ACMR of moving object may contain more than one +object. And due to the interference of background and negative +samples, the detection accuracy of the coarse-detection model +is not satisfactory, there will be false detection and missed +detection. So, the ACMR may contain no object, one object +or multiple objects. Therefore, the detection model in the fine- +detection stage should still have the ability of multi-object +detection. However, since the ACMRs of moving objects are +only a small area (relative to the input image) and cannot +contain a large number of moving objects, the output of the +fine-detection model need not be designed with a complex +structure. In summary, the paper uses a relatively simple Single +Scale Detection Head (SS-D Head) structure as the output +structure of the fine-detection model (see Fig. 7). Specifically, +FIDK is fed into the SS-D Head to obtain the output of the +fine-detection model, +OIDK = FSS-D (FIDK; ΘSS-D) , +(15) +where, ΘSS-D is the learnable parameter of the SS-D Head. +Then, post-processing operations such as Boxes Decoding and +non-maximum suppression are performed on the output to +obtain the final detection result of moving object, +{ClassesIDK, BoxesIDK} = FP (OIDK) , +(16) +where, ClassesIDK represents the category of the object in +the ACMR ACMRIDk of the moving object, and BoxesIDK +(in this paper, the position of the object in the middle frame +of n consecutive frames is taken as the detection result) is the +bounding box of the corresponding object in this region. +Finally, the bounding box of the moving object in the +ACMR is mapped to the original video image, that is, the +final detection result of the moving object is obtained. +IV. EXPERIMENT +In this section, A series of experiments are conducted to +quantitatively and qualitatively evaluate the proposed SMOD- +BMI. Next, we will introduce datasets (IV-A), evaluation +metrics (IV-B), experimental platforms (IV-C), implementa- +tion details (IV-D), parameter analysis experiment (IV-F) and +comparative analysis experiment (IV-E). + +SCM +Backbone +SS-D HeadIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +9 +Fig. 8: Size distribution of the moving birds in the datasets +A. Datasets +We collected and annotated 20 videos containing moving +bird objects (the size of video images is 1280 × 720) in +an unattended traction substation. We end up with 10,381 +continuous annotated images with 11,631 objects in total. +From Fig. 8, we can see that the size of moving birds is +mainly distributed between 0 × 0 and 80 × 80 pixels, and +about more than 50% of them are below 40 × 40 pixels. So +these birds can be called moving small objects. +B. Evaluation Metrics +In this paper, the widely used measures in object detection, +precision (Prec), recall (Rec), and average precision (AP) +are adopted to evaluate the proposed SMOD-BMI. More +specifically, Prec50, Rec50, AP50 (The subscript 50 means that +the detection result is regarded as the True Positive, when +the IOU between the detection result and the ground truth is +greater than or equal to 50%. That is, the IOU threshold is set +50% ), Prec75, Rec75, AP75 (The subscript 75 has the Similar +meaning with the subscript 50) and AP (Average Precision +averaged over multiple thresholds, IOU threshold is set from +50% to 95%, in intervals of 5%) are adopted. +C. Experimental Platforms +All the experiments are implemented on a desktop computer +with an Intel Core i7-9700 CPU, 32 GB of memory, and a +single NVIDIA GeForce RTX 3090 with 24 GB GPU memory. +D. Implementation Details +We implemented the proposed method based on YOLOV4 +[28] with modifications. +Specifically, for the coarse-detection model, a ConvLSTM +module is embedded between the second and third layers of +CSPDarkNet53, the backbone network of YOLOV4 model, +and a SCM [42] is added to its PANet structure. For the input +size of the coarse-detection model, we set it to 640 × 384 to +ensure the ratio of effective input pixels as much as possible +and at the same time ensure the running speed. During training, +the input is n consecutive frames of images, the label is the +position of the object on the intermediate frame, and the loss +function of the YOLOV4 algorithm is reused. +For the fine-detection model, the lightweight MobilenetV2 +is used as the backbone network of the U-shaped network, and +the SCM [42] is added to the upsampling structure of the U- +shaped network. For the input size of the fine-detection model, +we set it to 160160. For the training data, we used the coarse- +detection model and the object tracking SORT algorithm to +collect the Motion Region (MR) containing the moving object +as the positive samples and the negative samples without the +object. During training, the input is the screenshot of the +MR of n consecutive frames, the label is the position of the +object on the intermediate screenshot, and the loss function of +YOLOV4 is reused. +In this paper, all experiments are implemented under the +Pytorch framework. All network models are trained on an +NVIDIA GeForce RTX 3090 with 24G of video memory. For +the batch size setting, it is set to 4 when training the coarse- +detection model designed in this paper and other comparison +models, and it is set to 8 when training the fine-detection +model. All the experimental models were trained from scratch, +and no pre-trained models were used. The trainable parameters +of the network were randomly initialized using a normal +distribution with mean 0 and variance 0.01. Adam was chosen +as the optimizer for the model in this paper. The initial learning +rate is set to 0.001. For each iteration, the learning rate is +multiplied by 0.95 and the model is trained for a total of +100 iterations. In the training phase, we used simple data +augmentation including random horizontal flipping, random +Gaussian noise, etc. to enhance the robustness of the model. +E. Comparative Analysis Experiments +In order to verify the advancement of the proposed moving +object detection algorithm. We design a series of comparative +experiments to compare the accuracy of different methods in +detecting moving objects. We designed and implemented some +deep learning-based methods following their main ideas. The +methods mainly compared in this paper have the following +categories. +• Object Detection method based on still images. We chose +YOLOV4 as the representative algorithm of this kind of +methods. +• Multi-frame input is used to fuse spatio-temporal features, +and then the method of object detection is used to realize +the detection or segmentation of moving objects. For +this class of methods, we use Mutlti-Input+YOLOV4 +(MI YOLOV4) to represent. +• ConvLSTM is used to fuse spatio-temporal features, +and then the object detection method is used to realize + +Distribute of Object Size +5000 +4000 +1654 +1000 +0-20 20-40 40-60 60-80 +100-120 +160-180 +220-240 +280-300 +340-360 +400-420 +460-480 +520-540 +580-600 +640-660 +700-720 +760-780 +Square Root of the Area(pixls)IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +10 +the detection or segmentation of moving objects. For +this type of methods, we use ConvLSTM+YOLOV4 +(CL YOLOV4) to represent. +By the way, the parameters of the above model are designed +as follows. The inputs are all set to 640 × 384. The number of +consecutive input frames for MI YOLOV4, CL YOLOV4 and +SMOD-MBI are set to 5 frames. For SMOD-MBI, its motion +amount parameter σmov is set to 4.0. +In the qualitative comparison experiment, we choose +YOLOV4 as the baseline for comparison. YOLOV4 algorithm +only considers the appearance features of moving objects, +while the method proposed in this paper makes full use +of the motion cues of moving objects. By comparing the +experimental results as shown in Fig. 9, it can be seen that +when the appearance characteristics of the moving object are +obvious, YOLOV4 can also achieve a certain effect. However, +when the appearance characteristics of the moving object are +not obvious, YOLOV4 will miss detection, and YOLOV4 is +also prone to false detection. However, the proposed method +can achieve good results regardless of whether the appearance +characteristics of the moving object are obvious or not. There- +fore, for the detection of moving object, its motion cues are +particularly important. +Other methods considering the motion information of the +moving object and the method proposed in this paper have +little difference in qualitative comparison, so this paper designs +a quantitative comparison experiment to compare the method +proposed in this paper with other algorithms. +The results of quantitative comparison experiments are +shown in TABLE I. For the same detection method, the AP +decreases sharply with the increase of IOU threshold. The +reason for this is that the smaller the object, the harder it is for +the detection to match the ground truth exactly, because subtle +deviations in the detection results will be more noticeable +compared to the ground truth. Compared with different detec- +tion methods, the moving object detection method YOLOV4 +based only on appearance has a poor effect on detecting +moving small objects, with AP50 of 64.34%. MI YOLOV4 +fuses the spatio-temporal information of the moving object +by merging the input of multiple frames, which can improve +the AP50 by 17.13%. Therefore, for the dataset we collected, +motion information is a more important clue for detecting +small moving targets in complex environments. CL YOLOV4 +uses ConvLSTM to merge the spatio-temporal information +of the moving object, and can obtain an AP50 increase of +1.76%, which shows that ConvLSTM is more suitable for +fusing the spatio-temporal information of the moving object +than the multi-frame merged input, because ConvLSTM has +some special structures to remove the influence of redundant +information. +On the basis of CL YOLOV4, the proposed method SMOD- +BMI uses object tracking technology and combines the motion +amount of the moving object to obtain the Adaptive Candidate +Motion Range (ACMR) of the moving object, and then finely +detects the moving object in the ACMR. We reduce the +threshold for judging the moving objects in coarse-detection +stage, which will cause some false detections but will improve +the detection rate. At the same time, we increase the threshold +that is judged as a moving object in the fine-detection stage +to reject false detections. The experimental results show that +the proposed method improves AP50 by 4.25%, and reaches +to 87.46%. +Through the qualitative and quantitative analysis of the +experimental results, it can be concluded that the small moving +object detection method proposed in this paper is advanced and +effective. +F. Parameter Analysis Experiments +1) Effect of Different Number of Consecutive Input Frames +on the Performance of the Algorithm: We design test exper- +iments with different numbers of consecutive frame inputs to +evaluate the impact on the detection accuracy and efficiency +of the proposed method. Specifically, there are 3 consecutive +frames of input, 5 consecutive frames of input, 7 consecutive +frames of input, etc. In theory, with the increase of the +number of consecutive frames, the motion information of the +moving object will be gradually enriched, and the detection +accuracy of the algorithm will be gradually improved, but its +running time will also increase accordingly. The results of +the detection performance test of the algorithm are shown in +Table II (the motion amount parameter σmov is set to 4.0). +The experimental results show that the running speed of the +algorithm is the fastest when 3 consecutive frames are input, +and the detection accuracy is the highest when 7 consecutive +frames are input. When the input is five consecutive frames, +the speed and accuracy can have a good trade-off (the AP50 +reaches to 87.46%, and the running time is 0.12s). +2) Influence of Different Amount of Motion Parameter σmov +on the Accuracy of the Algorithm: We obtain the Adaptive +Candidate Motion Ranges (ACMRs) of different sizes of the +moving object by setting different motion amount parameter +σmov. If the MR is small, the context background information +is less; if the MR is large, the SNR is large. Therefore, different +sizes of MRs of the same moving object have different effects +on the performance of the algorithm. Fig. 10 is the influence +of different motion amount parameter σmov on the accuracy +of the algorithm. +It can be seen from Fig. 10 that when the motion amount +parameter σmov is 1.0, the detection accuracy of the proposed +method is lower than that of MI YOLOV4 and CL YOLOV4 +(When the motion amount parameter σmov is set to 1.0, it +is equivalent to that the algorithm does not use the adaptive +adjustment mechanism to adjust the MR of the moving object, +because even if the moving object is still, it still satisfies +the motion amount parameter σmov of 1.0, so there is no +need to adjust the MR of the moving object according to the +motion amount of the moving object). In other words, with the +addition of the fine-detection stage, its detection accuracy is +reduced instead. This proves that when the MR of the moving +object is too small, it lacks enough context information, which +leads to the decline of detection accuracy. When we increase +the motion amount parameter σmov, the detection accuracy is +rapidly improved. However, when it is greater than 5.0, the +detection accuracy starts to slowly decrease again. +As previously analyzed, when the MR is too small, it lacks +contextual information, and when the MR is too large, it is + +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +11 +Scenario 1 +Scenario 2 +Scenario 3 +(a) YOLOV4 +(b) SMOD-BMI +Fig. 9: Detection comparisons of YOLOV4 and SMOB-BMI (green box: ground truth bounding box; red box: YOLOV4 +bounding box; blue box: proposed method bounding box). +TABLE I: Comparison with other moving object detection methods +Frame num +Prec50 +Rec50 +AP50 +Prec75 +Rec75 +AP75 +AP +YOLOV4 +0.3200 +0.7074 +0.6434 +0.0790 +0.1747 +0.0553 +0.2106 +MI YOLOV4 +0.8561 +0.8478 +0.8145 +0.4098 +0.3846 +0.2109 +0.3298 +CL YOLOV4 +0.8717 +0.8592 +0.8321 +0.4165 +0.3955 +0.2123 +0.3422 +SMOD-BMI(ours) +0.9197 +0.9118 +0.8746 +0.4827 +0.4786 +0.2482 +0.3827 +TABLE II: Effect of continuous image input with different number of frames on detection performance +Frame num +Prec50 +Rec50 +AP50 +Prec75 +Rec75 +AP75 +AP +Run Time +3 +0.9226 +0.9162 +0.8737 +0.4817 +0.4784 +0.2349 +0.3701 +0.11s +5 +0.9197 +0.9118 +0.8746 +0.4827 +0.4786 +0.2482 +0.3827 +0.12s +7 +0.9341 +0.9109 +0.8808 +0.4745 +0.4627 +0.2412 +0.3838 +0.14s +easy to introduce more noise (in an extreme case, when the +MR is already consistent with the original input image, the +previous processing will be meaningless, because the input +of the fine-detection stage is directly the original image. At +the same time, because the fine-detection model is relatively +simple and the input size is small (160 × 160), the detection +effect is bound to be poor), so whether the MR is too small or +too large, it will affect the accuracy of the algorithm. Through +experiments, we find that when the motion amount parameter +σmov is 5.0, the detection performance of the algorithm is the + +多 +快网间8888D团多 +快网间IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. XX, NO. XX, JANUARY 2023 +12 +Fig. 10: Influence of Different Amount of Motion Parameter +σmov on the Accuracy of the Algorithm +best, and its AP50 is reaching 87.85%. +Through parameter analysis experiments, we conclude that +when the number of consecutive input frames is 5, the +algorithm can get a good balance between accuracy and +speed. When the motion parameter is 5.0, the accuracy of the +algorithm reaches the highest. So we suggest the following +parameter setting scheme. The number of consecutive input +frames is set to 5, and the motion amount parameter σmov is +set to 5.0. +V. CONCLUSION +Aiming at the problem that the moving object is difficult to +detect in complex background, this paper analyzes the reason. +The reason is that the proportion of moving small object pixels +is small in complex background, which leads to low SNR. To +solve this problem, this paper proposes a Small Moving Object +Detection algorithm Based on Motion Information (SMOD- +BMI). Firstly, we use the ConvLSTM-SCM-PANet model to +coarsely detect the whole frame of a continuous video frame +and capture the suspicious moving object. Then, we used +the method of object tracking to track the suspicious moving +object to determine the MR of the suspicious moving object +on n consecutive frames. At the same time, according to the +moving speed of the suspicious moving objects, the size of +their MR is adjusted adaptively (To be specific, if the objects +move slowly, we expand their MR according their speed to +ensure the contextual environment information) to obtain their +Adaptive Candidate Motion Range (ACMR), so as to ensure +that the SNR of the moving object is improved while the +necessary context information is retained adaptively. After +that, we use LW-SCM-USN model to accurately classify and +locate the suspicious moving object by using the ACMR of the +suspicious moving object. Finally, qualitative and quantitative +experiments verify the effectiveness and advancement of the +proposed moving object detection algorithm based on motion +information. +REFERENCES +[1] K. Sehairi, F. Chouireb, and J. 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Chen, “Mo- +bilenetv2: Inverted residuals and linear bottlenecks,” in 2018 IEEE/CVF +Conference on Computer Vision and Pattern Recognition, 2018, pp. +4510–4520. + diff --git a/3NAzT4oBgHgl3EQf9P6r/content/tmp_files/load_file.txt b/3NAzT4oBgHgl3EQf9P6r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..35be8df15fd6be3459ae69296c2661fd46b4a63c --- /dev/null +++ b/3NAzT4oBgHgl3EQf9P6r/content/tmp_files/load_file.txt @@ -0,0 +1,886 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf,len=885 +page_content='IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 1 Small Moving Object Detection Algorithm Based on Motion Information Ziwei Sun, Zexi Hua, and Hengcao Li, Fellow, IEEE Abstract—A Samll Moving Object Detection algorithm Based on Motion Information (SMOD-BMI) was proposed to detect small moving objects with low Signal-to-Noise Ratio (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Firstly, To capture suspicious moving objects, a ConvLSTM- SCM-PAN model structure was designed, in which the Convo- lutional Long and Short Time Memory (ConvLSTM) network fused temporal and spatial information, the Selective Concatenate Module (SCM) was selected to solve the problem of channel unbalance during feature fusion, and the Path Aggregation Network (PAN) located the suspicious moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Then, an object tracking algorithm is used to track suspicious moving objects and calculate their Motion Range (MR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, according to the moving speed of the suspicious moving objects, the size of their MR is adjusted adaptively (To be specific, if the objects move slowly, we expand their MR according their speed to ensure the contextual environment information) to obtain their Adaptive Candidate Motion Range (ACMR), so as to ensure that the SNR of the moving object is improved while the necessary context information is retained adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Finally, a LightWeight SCM U-Shape Net (LW-SCM-USN) based on ACMR with a SCM module is designed to classify and locate small moving objects accurately and quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In this paper, the moving bird in surveillance video is used as the experimental dataset to verify the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The experimental results show that the proposed small moving object detection method based on motion information can effectively reduce the missing rate and false detection rate, and its performance is better than the existing moving small object detection method of SOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Index Terms—Object Detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Small Moving Objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Mo- tion Information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Motion Range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Low Signal-to-Noise Ratio I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' INTRODUCTION T HE intelligent video analysis technology can reduce the work intensity of the monitoring center staff and reduce the false positives and missing positives caused by manual monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' And moving object detection is one of the basic tasks of intelligent video analysis technology [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Through moving object detection technology, information such as the category, location, size and motion speed of moving objects can be obtained, which can provide basic data support for in- telligent video analysis technology such as behavior prediction and trajectory tracking of moving objects in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For the detection of small moving objects, there are two main challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The object has a low SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For the general unattended monitoring scene, the monitoring area is usually a room or an outdoor area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' If a mouse or bird intrudes into the Manuscript received January 4, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Ziwei Sun, Zexi Hua and Hengcao Li are with the School of Information Science and Technology, Southwest JiaoTong University, chengdu 611756, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' monitoring area, the number of pixels is usually small, as shown by Bird A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The moving object may blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Since most of the low-cost surveillance cameras do not have the ability of low-delay photography, the moving object captured has a certain trailing phenomenon, which may lead the moving object blur, as shown by Bird B in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 1: Small and blurred moving birds in the surveillance area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The Bird A is small but clear, the Bird B is small and blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' To solve the above problems, researchers mainly use the motion information (spatio-temporal features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Of course, like other vision tasks, the detection method of moving small objects has also experienced the development from traditional methods based on knowledge-driven to deep learning methods based on data-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At present, the knowledge-driven moving object detection algorithms mainly include frame difference method [3], back- ground difference method [4], robust principal component Analysis method [5] and optical flow method [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In the early stage, the frame difference method, background difference method, and robust principal component analysis method were only suitable for the situation that the background was static and there was no more complex interference (such as illumination change, branches and leaves swaggling, water waves and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The optical flow method was suitable for the situation of moving background, but it still could not overcome some interference such as illumination change, the object stop or slow motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, through the continuous efforts of researchers, the traditional methods can accurately extract the moving object to a certain extent [7], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, the traditional methods can only extract the pixels of the moving object at most, can not obtain other attributes of the moving object, and can not distinguish the interesting and uninteresting moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='01917v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='CV] 5 Jan 2023 Bird B oBirdAIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' JANUARY 2023 2 In the early stage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' methods based on deep learning were mainly combined with traditional methods and object detection methods: traditional methods such as frame difference method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' background difference method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' principal component analysis method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' and optical flow method were combined with object detection methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' in which the traditional method provided time-related motion information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' and the object detection provided space-related positioning information [10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' [11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' [12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' These traditional methods with object detection have made considerable progress in detecting moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' How- ever, the detection performance of these methods is affected by the motion information provided by the traditional methods to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At present, some researchers gradually pay attention to the full deep learning to obtain the temporal infor- mation and spatial information of moving objects at the same time, such as using ConvLSTM(Convolution Long Short Term Memory) for moving object detection [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Or moving object detection with input of consecutive multiple frames merged [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The method based on deep learning has a certain improvement in effect and function compared with the traditional method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' It can distinguish between interested and uninterested moving objects, and can obtain the category and location of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, there are still many false detections and missed detections when detect the small moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' We further analyze and find that most of the missed detections occur because the object is small or similar to the environment, and most of the false detections that occur are caused by various tiny moving things or things whose appearance is similar to the object of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, the main reason for this problem is that most moving objects account for a small proportion of pixels in the whole video frame, and the problem of low SNR (unbalanced positive and negative samples) is not easy to be eliminated in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In order to solve the above problems, this paper analyzes our human method of small moving object recognition in complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Our human approach to identifying small moving objects is divided into two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In the first stage, we will find out where the object may exist according to the motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In the second stage, we will focus on the area where the object may exist and carefully observe, so as to remove more interference information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, we propose a Small Moving Object Detection algorithm Based on Motion Information (SMOD-BMI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Firstly, a moving ob- ject detection model ConvLSTM-SCM-PAN (coarse-detection model) is designed to fuse spatio-temporal information, which can capture suspicious moving objects according to the mo- tion information of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Then, the Motion Range (MR) of suspicious moving objects is extracted by using the object tracking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, according to the moving speed of the suspicious moving objects, the size of their MR is adjusted adaptively (To be specific, if the objects move slowly, we expand their MR according their speed to ensure the contextual environment information) to obtain their Adaptive Candidate Motion Range (ACMR), so as to ensure that the SNR of the moving object is improved while the necessary context information is retained adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Finally, a lightweight moving object detection model LW- SCM-USN (Fine detection model) based on the ACMR of the moving object is designed to accurately classify and locate the moving object on the basis of ensuring real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The main contributions of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The ConvLSTM-SCM-PAN model structure is designed to capture the suspicious moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Among them, Convolution Long Short-Term Memory Network (ConvL- STM) fuses spatio-temporal information, Selective Con- catenation Module (SCM) to solve the problem of chan- nel imbalance during feature fusion, and PAN locates suspicious moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' An adaptive method of extracting ACMR based on the amount of motion of the moving object is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' By using the object tracking technology and the amount of motion of the moving object, the ACMR of the suspected moving object are extracted adaptively, which improves the SNR of the moving object and retains the necessary context information of the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' A LightWeight U-Shaped Network with SCM module (LW-SCM-USN) model structure is designed, and the accurate classification and location of moving objects are realized by using the ACMR of suspected objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The remainder of this paper is structured as follows: Section II is a survey of related work on moving object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Section III describes the proposed SMOD-BMI in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Section IV is devoted to ablation experiments and comparison experiments of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Section V concludes our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' RELATED WORK According to the use of different characteristics of the object, the methods of moving object detection can be mainly divided into three categories: methods based on appearance information, methods based on motion information and meth- ods based on deep learning for moving object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In this section we will review these three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Appearance-based Object Detection From traditional methods [18], [19], [20] to deep learning- based methods [21], [22], [23], [24], [25], [26], [27], [28], object detection technology has now made great progress, which can accurately determine the specific class of each object and give the bounding box of each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, since these object detection algorithms only rely on the appearance features of the object, the detection effect is not good for small moving objects with complex backgrounds and unobvious appearance features [11], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Moving Object Detection based on Motion Information Since the object detection algorithm based on appearance feature can not detect small moving objects in complex background well, researchers have proposed various moving object detection algorithms based on motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The main methods are frame difference, background subtraction, optical flow and robust principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 3 1) Frame Difference Method: Because the object is mov- ing, there is a certain displacement between the position of the historical frame and the position of the current frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The changed pixel, which is the pixel of the moving object, can be extracted by subtracting the historical frame from the current frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' When the simple frame difference method obtains the moving object, it is easy to appear the hole or ghost phe- nomenon [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, researchers have proposed various complex frame difference methods to solve this problem [31], [32], which have achieved certain effect improvement, but the problem cannot be completely solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2) Background Subtraction Method: The environment and moving object are regarded as background and foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The background remains static, while the moving object moves in front of the background as the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The key point of this method is the background modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' There are many methods of background modeling, which are widely used at present, such as multi-frame average background modeling, simple Gaussian modeling [33], Gaussian mixture modeling [34], ViBe algorithm [35], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=', although the modeling effect of these background modeling methods is getting better and better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, it still cannot completely overcome various disturbances such as wind and water waves, resulting in more interference in the extracted foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 3) Optical Flow Method: The moving object detection method based on optical flow method distinguishes the back- ground and moving object by using optical flow field according to the feature that the brightness of adjacent points in the image is similar [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The key technology of optical flow method is to solve the estimation of optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At present, the main optical flow estimation algorithms include correlation method, energy method, discrete optimization method and phase method [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The optical flow method does not need prior information to detect moving objects and can be used in dynamic background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, the calculation of optical flow field distribution is difficult due to the change of light source, shadow and occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 4) Robust Principal Component Analysis (RPCA): The background is considered as a low-rank matrix and the moving objects are sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, this method converts the detec- tion of moving objects into low-rank sparse decomposition of the matrix composed of multiple frames, so as to obtain sparse moving objects [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Since the original robust principal component analysis method is time-consuming, subsequent researchers have proposed some improved schemes, such as Faster RPCA [37], which greatly improves the decomposition speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, when the background moves or the back- ground changes complex, the background matrix loses its low rank property, and it is difficult to decompose the moving object at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, the robust principal component analysis method is mainly suitable for the situation that the background is static or the background changes simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Moving Object Detection with Deep Learning In recent years, influenced by the great progress of deep learning technology in vision tasks, researchers have begun to use deep learning technology to detect moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Researchers have used deep learning techniques in two differ- ent ways to investigate how to detect moving objects, but all related studies follow the same basic rule, that is, you need to consider both time-based motion information and space- based position information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The difference between these two methods lies in how to obtain time-based motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' One way is to obtain the motion information by using the traditional moving object detection method, which is called the traditional plus deep learning method, and the other way is to obtain the motion information directly by using deep learning, which is called the full deep learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 1) Traditional plus Deep Learning Method: The traditional and deep learning moving object detection methods are sum- marized into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 1) Firstly, the motion information is used to extract the foreground, and then the foreground is used for moving object detection [10], [11], [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For example, literature [10] introduces the Fast RPCA algorithm to separate the foreground, and then implements Faster R-CNN object detection on the foreground map to effectively detect the moving small object in the panoramic video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In literature [11], the frame difference method was used to obtain the moving foreground, and then the CNN classification network was used to screen the region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Finally, the CNN regression network was used to perform coordinate regression on the region of interest to obtain the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Literature [12] uses the ViBe background modeling method to extract the foreground, and uses this foreground as the candidate moving object area of Fast R-CNN to set ANCHORS, so as to realize the detection of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In reference [10], the motion region was obtained by frame difference method, and then the motion region was connected and expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Finally, Deep CNN was used to classify and position regression the object in the motion region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2) Traditional methods are directly fused with object detection to detect moving objects [38], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For example, literature [38] inputted the frame difference between the original image and the two frames into VGG16 for fusion, and then inputted the fused feature layer into Faster R-CNN for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Literature [39] proposed a method based on deep learning combining RGB and optical flow to segment moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2) Full Deep Learning Method: There are two main cat- egories of moving object detection methods based on full deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 1) ConvLSTM is used to fuse temporal and spatial information to segment or detect moving objects [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For example, reference [14] introduces the attention Long Short-Term Memory (attention ConvLSTM) model to simulate the change of pixels over time, and then uses a spatial Transformer and conditional random field (CRF) to segment moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In reference [15], the Pyramid dilated convolution (PDC) module was designed to extract multi- scale spatial features, and then these spatial features were concatenated and fed into the Extended deep Bidirectional ConvLSTM (DB-ConvLSTM) to obtain spatio-temporal in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Finally, the moving objects in the video are de- tected by using the spatio-temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2) Detecting moving objects by merging and fusing temporal and spatial information of consecutive frames [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For example, the paper [16] propose regions of objects of interest (ROOBI) by IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 4 using the region proposal network, which combines the spatio- temporal information by merging the input of consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' After getting the Propose regions, the exact position of the object is located again by merging the input of consecutive multiple frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In literature [17], continuous multiple frames are merged into CNN for background estimation, and then a compact encoder-decoder network is used to segment moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' THE PROPOSED SMOD-BMI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2 shows the overview diagram of the proposed SMOD- BMI, which contains three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Firstly, ConvLSTM-SCM- PAN model structure was designed to capture the suspicious moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Secondly, An object tracking algorithm is used to track suspicious moving objects and calculate their MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, according to the moving speed of the suspicious moving objects, the size of their MR is adjusted adaptively to obtain their ACMR, so as to ensure that the SNR of the moving object is improved while the necessary context information is retained adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Finally, LW-SCM- USN based on ACMR with a SCM module is designed to clas- sify and locate small moving objects accurately and quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Section III-A describes the ConvLSTM-based suspicious mov- ing object detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Section III-B ACMR extraction method of suspicious moving object based on object tracking technology and motion amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Section III-C describes the moving object detection method based on ACMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' To Capture the Suspicious Moving Object In this paper, we perform two steps to capture suspicious moving objects (coarse-detection of moving objects) in con- secutive video images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Firstly, the spatio-temporal information of the moving object was fused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Secondly, the spatio-temporal information is used to locate the suspicious moving object by object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' This subsection will introduce the acquisition of spatio-temporal information of moving objects (Section III-A1), and the localization of suspicious moving objects (Section III-A2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 1) Fusion of Spatio-temporal Information for Small Moving Objects: Motion is mainly reflected in time and space, that is, at different times, according to the spatial location of the object can show motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, to capture the Small moving object, it is necessary to fuse its temporal and spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' As we have introduced in Section II-C2, there are two main ways to fuse the spatio-temporal information of the object based on deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' One is based on the recurrent neural network ConvLSTM, and the other is based on the input merging of consecutive multiple frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ConvLSTM(structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 3) contains three gates, namely input gate, output gate and forget gate, which are used to control the input and output and what information needs to be forgotten and discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, the input gate and output gate can also be understood as controlling the writing and reading of the memory cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Continuous multi-frame merging input is to simply Concatenate consecutive frames of video images together and then input into the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The coarse-detection phase captures the suspicious moving object, and the input is the whole video, which has the char- acteristics of many background interference and redundant in- formation (different frames have many identical backgrounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' According to the characteristics of ConvLSTM structure, it can remove unimportant or redundant information while fusing spatio-temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' So, in the first stage, we use Con- vLSTM to extract and fuse the spatio-temporal information of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, given the input n consecutive frames of images Xt ∈ R(H×W×3)|t = (1, 2, · · · , n) (Where H and W are the height and width of the input image, and n is an odd number), the ConvLSTM network FConvLSTM is used to fuse and extract the spatio-temporal features Hn ∈ R(H×W×C) (Where C is the number of channels) of the n consecutive frames of images, Ht = FConvLSTM ([Xt, Ht−1] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ΘConvLSTM) , (1) Where, when t = 1, H0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ΘConvLSTM is the learnable parameter of the ConvLSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The spatio-temporal features Hn of n consecutive frames of images are input into the subsequent classification and positioning module to determine the category and spatial location information of the suspicious moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2) Localization of Suspicious Moving Objects: In convolu- tional neural networks, deeper layers, which generally have smaller size, have better global semantic information, and can predict larger objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The layers with shallower depth, which generally have larger size, have more delicate spatial information and can predict smaller objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, the large feature layer often does not have a relatively high degree of semantic information, and the small feature layer does not have fine spatial positioning information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, relevant researchers have proposed the structure of FPN [40] to combine the strong semantic information of the small feature layer and the strong spatial positioning information of the large feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, the researchers of PANet(Path Aggregation Network) [41] found that when FPN transmitted information, there was information loss due to the transfer distance when the information was transmitted to the low-level feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, path-enhanced FPN, namely PANet structure, was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The PANet structure opens up a green channel for low-level information transmission and avoids low- level information loss to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, we find that the detection performance will be improved when Selective Concatenation Module (SCM) [42] is added to the model (reference [42] introduces that SCM can help to better fuse high and low layer information (refer to reference [42] for details)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' We believe that SCM can not only balance the fusion of channel information in different layers, but also suppress unimportant information and highlight the information that the model needs to focus on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' So, we introduce the SCM and design the feature extraction structure of SCM-PANet (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The spatio-temporal features Hn of n consecutive frames are input into the SCM-PANet structure to extract the features of the suspicious moving object FMOn, FMOn = FSCM-PAN (Hn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ΘSCM-PAN) , (2) IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2: Overview of the proposed SMOD-BMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (a) Capture the suspicious moving object, the blue box in the figure represents the detected suspicious moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (b) The ACMR of the suspicious moving object is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In the figure, the green box represents the original MR of the moving object tracked by the tracking algorithm, and the red box represents the MR adaptively adjusted according to the motion amount of the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (c) Classification and localization of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 3: Structure diagram of ConvLSTM Where, ΘSCM-PAN is the learnable parameter of the SCM- PANet network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' When the distance between the moving object and the surveillance camera is different, the size of the moving object is also different, so the moving object to be detected has the multi-scale property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' According to the multi-scale property of the moving object, this paper uses the MultiScale Detection Head (MS-D Head) to detect the suspicious moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The objects in the middle frame of n consecutive frames has symmetric contextual information, which can get more accurate results in prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, this paper predicts the suspicious object in the middle frame of n consecutive frames as the detection result of the Coarse-detection stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, the feature FMOn of the moving object is input into the MS-D Head to obtain the output of the model, On = FMS-D (FMOn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ΘMS-D) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='(a) To capture the suspicious Moving Object (MO) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='MS-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='ConyLSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='SCM-PAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Suspicious MO (Blue box) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Video Frame ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Coarse Detection Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Track and calculate the Motion Range (MR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='(b) To obtain the Adaptive Candidate MR (ACMR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='(n+4)th ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='MR (Green box) of the Suspicious MO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Adaptively resize and crop the MR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='(c) To classify and locate the MO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='BackGround ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='SS-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='LW-SCM-USN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Bird ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='crop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='crop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='crop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='ACMR (Red box) of The Suspicious MO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Fine Detection Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='BirdConvLSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Bias ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='I+X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='write ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='read ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='+Bc tanh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='tanh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Xt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='W:* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='+B: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Ct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='Concat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='+Bf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='* Ht-1 Ot +Bo W 0 0 Ht + Data flow Next iterationIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 4: Structure diagram of the SCM-PANet model where, ΘMS-D is the learnable parameter of the MS-D Head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Then post-processing operations such as Boxes Decoding and non-maximum suppression were performed on the output of the model to obtain the location of the suspicious object in the middle frame of n consecutive frames, {PID1, · · · , PIDk} frame( n+1 2 ) = FP (On) , (4) Where, {·} frame( n+1 2 ) means the locations of the moving ob- jects in � n+1 2 �th frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' PIDk indicates the predicted position of the object with IDk (the object with IDk is taken as an example unless otherwise specified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' FP (·) denotes the post- processing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' To Obtain the ACMR In this paper, the MR of the suspicious moving object on n consecutive frames is extracted to improve the SNR of the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, in order to ensure the context information of the suspicious moving object, the size of the MR is adaptively adjusted according to the motion amount of the suspicious moving object, so that the subsequent detection results are more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, we will divide into two steps to obtain the ACMR of suspicious moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Respectively, the original MR of the suspicious moving object is extracted using the object tracking technology (Section III-B1) and the MR is adaptively adjusted using the motion amount of the suspicious moving object to obtain the ACMR (Section III-B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 1) Acquisition of the Original MR of the Suspicious Moving Object: From the � n+1 2 �th frame, there are detection results of the suspicious moving object, and we start to track the suspicious moving object from the � n+1 2 + 1 �th frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In some cases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' the appearance characteristics of small moving objects are not obvious,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' so we only use their motion information when tracking them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' and use a relatively simple SORT [43] object tracking algorithm to track suspicious moving objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' � {PIDk}frame(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' {PIDk}frame(i+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' · · · � = Ftrack (IDk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (5) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' � {PIDk}frame(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' {PIDk}frame(i+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' · · · � represents the po- sition on consecutive image frames of a suspicious moving object with ID number k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' and Ftrack (·) represents the SORT object tracking method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' After obtaining the position of the suspicious moving object on consecutive image frames, we can find the Motion Range (MR) of the suspicious moving object on n consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, the minimum circumscribed rectangle RectIDk at n positions is calculated according to the position of the same object on n consecutive frames of images, RectIDk = FMinRect �� {PIDk}frame(i+1), · · · , {PIDk}frame(i+n) �� , (6) where, FMinRect (·) denotes the function to find the mini- mum circumscribed rectangle of n rectangular boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' to find the minimum circumscribed rectangle [(xmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ymin) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (xmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ymax)] (Using the horizontal and vertical coordinates of the top left and bottom right vertices of the rect- angle) of {box1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' boxn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' the specific calculation method is as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' xmin = min � x1box1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' x1boxn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ymin = min � y1box1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' y1boxn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' xmax = max � x2box1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' x2boxn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ymax = max � y2box1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' y2boxn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (7) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' �� x1boxn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' y1boxn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' � x2boxn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' y2boxn �� denotes the hori- zontal and vertical coordinates of the upper left and lower right vertices of boxn in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The obtained minimum circumscribed rectangle RectIDk is the MR of the moving object in n consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 5 illustrates the MR of the moving object on five consecutive frames of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2) Adaptively Adjust the MR to Obtain ACMR Based on the Amount of Motion: We crop the MR of suspicious moving object in n consecutive frames to remove the interference of other background and negative samples, which can improve the SNR of the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, if the moving object moves too slowly, the clipped MR will lack contextual environmental information (see the Raw MR In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 6), which is not conducive to the detection of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In order to balance the contradiction between SNR and context information, this paper proposes an ACMR extraction method based on the amount of motion of the moving object, which adaptively adjusts the size of the MR of the moving object according to the speed of the object motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' There are two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Firstly, the amount of motion of the moving object over n consecutive frames is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For an object of the same size, if it moves fast on n consecutive frames, its MR is large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Backbone SCM SCM FPN PAN MS-D HeadIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 5: MR of the moving bird over 5 consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The blue box shows the position of the bird in each frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The green box represents the minimum bounding rectangle of the five blue boxes, which is the MR of the moving bird over five consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 6: The left picture shows the original monitoring picture, the right picture shows the MR of the moving object on five consecutive frames in the dashed frame, and the ACMR of the moving object in the solid frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' It is obvious that the object in the original MR is difficult to be correctly recognized, and the object in the ACMR is easier to be recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' otherwise, its MR is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, we use the ratio of the area of the MR of the moving object on n consecutive frames to the area of the single frame image occupied by the moving object to define its motion amount on n consecutive frames, σmov = S (RectIDk) S (ObjIDk) , (8) where, σmov is the motion amount, S (·) represents the func- tion to calculate the area, and ObjIDk represents the object with ID number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The area of MR is then the area of the minimum circumscribed rectangle RectIDk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Since the area occupied by a moving object in a single image frame may vary due to its shape changes, and it is difficult to calculate accurately, we use the rectangular area of its bounding box to approximately represent its area in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Then, according to the amount of motion of the moving object, the MR of the moving object is adaptively adjusted as the Adaptive Motion Range (AMR) ARectIDk of the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, a motion hyperparameter γ is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' When the amount of motion of the moving object is less than γ, the MR of the moving object is expanded to make the amount of motion of the moving object reach γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, the AMR of the moving object can be expressed as follows, ARectIDk = � ARectIDk, σmov ≥ γ γ × ObjIDk, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (9) The ARectIDk is used to crop the corresponding n consecutive frames of video image � frame(1), · · · , frame(n)� respectively, and the n frame screenshots obtained are the Adaptive Can- didate Moving Region (ACMR) (ACMRIDk) of the moving object, f(i) ARectIDk = Fcut � frame(i), ARectIDk � , (10) ACMRIDk = � f(i) ARectIDk|i ∈ (1, · · · , n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (11) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Moving Object Detection based on ACMR After the previous processing, we improve the SNR of the moving object, retain its contextual environmental information, and obtain the ACMR of the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In the fine- detection stage, we can use the ACMR of the moving object Raw MR ACMR Raw MR ACMRIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 8 to classify and locate the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, the fine- detection phase includes the fusion of spatio-temporal infor- mation (III-C1) and the classification and localization(III-C2) of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 1) Fusion of Spatio-temporal Information of Moving Ob- jects: The input of the fine-detection model is the ACMR of the moving object extracted earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The coarse-detection model may detect multiple suspicious moving objects at one time, so there will be multiple ACMRs, and the fine detection model will detect each ACMR separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' So it’s possible to run a coarse-detection model once and a fine-detection model many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, in order to balance accuracy and speed, the method of fusing the spatio-temporal information of the moving target in the fine-detection stage uses the way of merging consecutive multiple frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, in order to reduce data redundancy, except the middle frame, the rest of the frames are input in the form of grayscale image single channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, firstly, grayscale the screenshots of the ACMR of the moving object except for the middle screenshot, f(i)′ ARectIDk = � � � f(i) ARectIDk, i = int � n 2 � FGray � f(i) ARectIDk � , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (12) where, FGray (·) is a function that finds the grayscale of a color image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Then, the processed screenshots of the ACMRs are Concatenate in the channel dimension as the input of the fine- detection stage, XSIDK = Fconcat �� f(1)′ ARectIDk, · · · , f(n)′ ARectIDk � , 2 � , (13) Where, the second argument of the Fconcat function indicates that the concatenation operation is performed in the third input dimension (height, width, channel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The length and width of XSIDK is equal to the length and width of rectangle ARectIDk, and the number of channels is n+2, which contains the motion information and appearance information of the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' It is input into the fine-detection model to accurately classify and locate the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2) Classification and Localization of Moving objects: In order to further improve the speed of the whole moving object detection process, this paper uses a lightweight U-Shaped Network (USN) (in the experiment, we use MobilenetV2 [44] as the backbone network of the USN) as the feature extraction network of the moving object in the fine-detection stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, in order to better fuse high and low layer infor- mation, similar to the network of the coarse-detection model, we introduce the SCM [42] module and design the lightweight LW-SCM-USN feature extraction network structure, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The XSIDK fused with the spatio-temporal information of the moving object is input into the LW-SCM-USN feature extraction network to obtain the moving object feature FIDK fused with the spatio-temporal information, FIDK = FLW-SCM-USN (XSIDK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ΘLW-SCM-USN) , (14) where, ΘLW-SCM-USN is the learnable parameter of the LW- SCM-USN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 7: Structure diagram of LW-SCM-USN The ACMR of moving object may contain more than one object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' And due to the interference of background and negative samples, the detection accuracy of the coarse-detection model is not satisfactory, there will be false detection and missed detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' So, the ACMR may contain no object, one object or multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, the detection model in the fine- detection stage should still have the ability of multi-object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, since the ACMRs of moving objects are only a small area (relative to the input image) and cannot contain a large number of moving objects, the output of the fine-detection model need not be designed with a complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In summary, the paper uses a relatively simple Single Scale Detection Head (SS-D Head) structure as the output structure of the fine-detection model (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, FIDK is fed into the SS-D Head to obtain the output of the fine-detection model, OIDK = FSS-D (FIDK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ΘSS-D) , (15) where, ΘSS-D is the learnable parameter of the SS-D Head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' post-processing operations such as Boxes Decoding and non-maximum suppression are performed on the output to obtain the final detection result of moving object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' {ClassesIDK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' BoxesIDK} = FP (OIDK) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' (16) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ClassesIDK represents the category of the object in the ACMR ACMRIDk of the moving object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' and BoxesIDK (in this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' the position of the object in the middle frame of n consecutive frames is taken as the detection result) is the bounding box of the corresponding object in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Finally, the bounding box of the moving object in the ACMR is mapped to the original video image, that is, the final detection result of the moving object is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' EXPERIMENT In this section, A series of experiments are conducted to quantitatively and qualitatively evaluate the proposed SMOD- BMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Next, we will introduce datasets (IV-A), evaluation metrics (IV-B), experimental platforms (IV-C), implementa- tion details (IV-D), parameter analysis experiment (IV-F) and comparative analysis experiment (IV-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' SCM Backbone SS-D HeadIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 8: Size distribution of the moving birds in the datasets A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Datasets We collected and annotated 20 videos containing moving bird objects (the size of video images is 1280 × 720) in an unattended traction substation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' We end up with 10,381 continuous annotated images with 11,631 objects in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 8, we can see that the size of moving birds is mainly distributed between 0 × 0 and 80 × 80 pixels, and about more than 50% of them are below 40 × 40 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' So these birds can be called moving small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Evaluation Metrics In this paper, the widely used measures in object detection, precision (Prec), recall (Rec), and average precision (AP) are adopted to evaluate the proposed SMOD-BMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' More specifically, Prec50, Rec50, AP50 (The subscript 50 means that the detection result is regarded as the True Positive, when the IOU between the detection result and the ground truth is greater than or equal to 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' That is, the IOU threshold is set 50% ), Prec75, Rec75, AP75 (The subscript 75 has the Similar meaning with the subscript 50) and AP (Average Precision averaged over multiple thresholds, IOU threshold is set from 50% to 95%, in intervals of 5%) are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Experimental Platforms All the experiments are implemented on a desktop computer with an Intel Core i7-9700 CPU, 32 GB of memory, and a single NVIDIA GeForce RTX 3090 with 24 GB GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Implementation Details We implemented the proposed method based on YOLOV4 [28] with modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, for the coarse-detection model, a ConvLSTM module is embedded between the second and third layers of CSPDarkNet53, the backbone network of YOLOV4 model, and a SCM [42] is added to its PANet structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For the input size of the coarse-detection model, we set it to 640 × 384 to ensure the ratio of effective input pixels as much as possible and at the same time ensure the running speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' During training, the input is n consecutive frames of images, the label is the position of the object on the intermediate frame, and the loss function of the YOLOV4 algorithm is reused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For the fine-detection model, the lightweight MobilenetV2 is used as the backbone network of the U-shaped network, and the SCM [42] is added to the upsampling structure of the U- shaped network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For the input size of the fine-detection model, we set it to 160160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For the training data, we used the coarse- detection model and the object tracking SORT algorithm to collect the Motion Region (MR) containing the moving object as the positive samples and the negative samples without the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' During training, the input is the screenshot of the MR of n consecutive frames, the label is the position of the object on the intermediate screenshot, and the loss function of YOLOV4 is reused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In this paper, all experiments are implemented under the Pytorch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' All network models are trained on an NVIDIA GeForce RTX 3090 with 24G of video memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For the batch size setting, it is set to 4 when training the coarse- detection model designed in this paper and other comparison models, and it is set to 8 when training the fine-detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' All the experimental models were trained from scratch, and no pre-trained models were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The trainable parameters of the network were randomly initialized using a normal distribution with mean 0 and variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Adam was chosen as the optimizer for the model in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The initial learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For each iteration, the learning rate is multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='95 and the model is trained for a total of 100 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In the training phase, we used simple data augmentation including random horizontal flipping, random Gaussian noise, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' to enhance the robustness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Comparative Analysis Experiments In order to verify the advancement of the proposed moving object detection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' We design a series of comparative experiments to compare the accuracy of different methods in detecting moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' We designed and implemented some deep learning-based methods following their main ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The methods mainly compared in this paper have the following categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Object Detection method based on still images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' We chose YOLOV4 as the representative algorithm of this kind of methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Multi-frame input is used to fuse spatio-temporal features, and then the method of object detection is used to realize the detection or segmentation of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For this class of methods, we use Mutlti-Input+YOLOV4 (MI YOLOV4) to represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' ConvLSTM is used to fuse spatio-temporal features, and then the object detection method is used to realize Distribute of Object Size 5000 4000 1654 1000 0-20 20-40 40-60 60-80 100-120 160-180 220-240 280-300 340-360 400-420 460-480 520-540 580-600 640-660 700-720 760-780 Square Root of the Area(pixls)IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 10 the detection or segmentation of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For this type of methods, we use ConvLSTM+YOLOV4 (CL YOLOV4) to represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' By the way, the parameters of the above model are designed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The inputs are all set to 640 × 384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The number of consecutive input frames for MI YOLOV4, CL YOLOV4 and SMOD-MBI are set to 5 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For SMOD-MBI, its motion amount parameter σmov is set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In the qualitative comparison experiment, we choose YOLOV4 as the baseline for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' YOLOV4 algorithm only considers the appearance features of moving objects, while the method proposed in this paper makes full use of the motion cues of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' By comparing the experimental results as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 9, it can be seen that when the appearance characteristics of the moving object are obvious, YOLOV4 can also achieve a certain effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, when the appearance characteristics of the moving object are not obvious, YOLOV4 will miss detection, and YOLOV4 is also prone to false detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, the proposed method can achieve good results regardless of whether the appearance characteristics of the moving object are obvious or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' There- fore, for the detection of moving object, its motion cues are particularly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Other methods considering the motion information of the moving object and the method proposed in this paper have little difference in qualitative comparison, so this paper designs a quantitative comparison experiment to compare the method proposed in this paper with other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The results of quantitative comparison experiments are shown in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' For the same detection method, the AP decreases sharply with the increase of IOU threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The reason for this is that the smaller the object, the harder it is for the detection to match the ground truth exactly, because subtle deviations in the detection results will be more noticeable compared to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Compared with different detec- tion methods, the moving object detection method YOLOV4 based only on appearance has a poor effect on detecting moving small objects, with AP50 of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='34%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' MI YOLOV4 fuses the spatio-temporal information of the moving object by merging the input of multiple frames, which can improve the AP50 by 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='13%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, for the dataset we collected, motion information is a more important clue for detecting small moving targets in complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' CL YOLOV4 uses ConvLSTM to merge the spatio-temporal information of the moving object, and can obtain an AP50 increase of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='76%, which shows that ConvLSTM is more suitable for fusing the spatio-temporal information of the moving object than the multi-frame merged input, because ConvLSTM has some special structures to remove the influence of redundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' On the basis of CL YOLOV4, the proposed method SMOD- BMI uses object tracking technology and combines the motion amount of the moving object to obtain the Adaptive Candidate Motion Range (ACMR) of the moving object, and then finely detects the moving object in the ACMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' We reduce the threshold for judging the moving objects in coarse-detection stage, which will cause some false detections but will improve the detection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, we increase the threshold that is judged as a moving object in the fine-detection stage to reject false detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The experimental results show that the proposed method improves AP50 by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='25%, and reaches to 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='46%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Through the qualitative and quantitative analysis of the experimental results, it can be concluded that the small moving object detection method proposed in this paper is advanced and effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Parameter Analysis Experiments 1) Effect of Different Number of Consecutive Input Frames on the Performance of the Algorithm: We design test exper- iments with different numbers of consecutive frame inputs to evaluate the impact on the detection accuracy and efficiency of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Specifically, there are 3 consecutive frames of input, 5 consecutive frames of input, 7 consecutive frames of input, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In theory, with the increase of the number of consecutive frames, the motion information of the moving object will be gradually enriched, and the detection accuracy of the algorithm will be gradually improved, but its running time will also increase accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The results of the detection performance test of the algorithm are shown in Table II (the motion amount parameter σmov is set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The experimental results show that the running speed of the algorithm is the fastest when 3 consecutive frames are input, and the detection accuracy is the highest when 7 consecutive frames are input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' When the input is five consecutive frames, the speed and accuracy can have a good trade-off (the AP50 reaches to 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='46%, and the running time is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='12s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 2) Influence of Different Amount of Motion Parameter σmov on the Accuracy of the Algorithm: We obtain the Adaptive Candidate Motion Ranges (ACMRs) of different sizes of the moving object by setting different motion amount parameter σmov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' If the MR is small, the context background information is less;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' if the MR is large, the SNR is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Therefore, different sizes of MRs of the same moving object have different effects on the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 10 is the influence of different motion amount parameter σmov on the accuracy of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 10 that when the motion amount parameter σmov is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0, the detection accuracy of the proposed method is lower than that of MI YOLOV4 and CL YOLOV4 (When the motion amount parameter σmov is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0, it is equivalent to that the algorithm does not use the adaptive adjustment mechanism to adjust the MR of the moving object, because even if the moving object is still, it still satisfies the motion amount parameter σmov of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0, so there is no need to adjust the MR of the moving object according to the motion amount of the moving object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' In other words, with the addition of the fine-detection stage, its detection accuracy is reduced instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' This proves that when the MR of the moving object is too small, it lacks enough context information, which leads to the decline of detection accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' When we increase the motion amount parameter σmov, the detection accuracy is rapidly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' However, when it is greater than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0, the detection accuracy starts to slowly decrease again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' As previously analyzed, when the MR is too small, it lacks contextual information, and when the MR is too large, it is IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 11 Scenario 1 Scenario 2 Scenario 3 (a) YOLOV4 (b) SMOD-BMI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 9: Detection comparisons of YOLOV4 and SMOB-BMI (green box: ground truth bounding box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' red box: YOLOV4 bounding box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' blue box: proposed method bounding box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' TABLE I: Comparison with other moving object detection methods Frame num Prec50 Rec50 AP50 Prec75 Rec75 AP75 AP YOLOV4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='3200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='7074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='6434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0790 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='1747 0.' metadata={'source': 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+page_content='9109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='8808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='4745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='4627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='2412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='3838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='14s easy to introduce more noise (in an extreme case, when the MR is already consistent with the original input image, the previous processing will be meaningless, because the input of the fine-detection stage is directly the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, because the fine-detection model is relatively simple and the input size is small (160 × 160), the detection effect is bound to be poor), so whether the MR is too small or too large, it will affect the accuracy of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Through experiments, we find that when the motion amount parameter σmov is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0, the detection performance of the algorithm is the 多 快网间8888D团多 快网间IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' XX, JANUARY 2023 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' 10: Influence of Different Amount of Motion Parameter σmov on the Accuracy of the Algorithm best, and its AP50 is reaching 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Through parameter analysis experiments, we conclude that when the number of consecutive input frames is 5, the algorithm can get a good balance between accuracy and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' When the motion parameter is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0, the accuracy of the algorithm reaches the highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' So we suggest the following parameter setting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The number of consecutive input frames is set to 5, and the motion amount parameter σmov is set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' CONCLUSION Aiming at the problem that the moving object is difficult to detect in complex background, this paper analyzes the reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' The reason is that the proportion of moving small object pixels is small in complex background, which leads to low SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' To solve this problem, this paper proposes a Small Moving Object Detection algorithm Based on Motion Information (SMOD- BMI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Firstly, we use the ConvLSTM-SCM-PANet model to coarsely detect the whole frame of a continuous video frame and capture the suspicious moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Then, we used the method of object tracking to track the suspicious moving object to determine the MR of the suspicious moving object on n consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' At the same time, according to the moving speed of the suspicious moving objects, the size of their MR is adjusted adaptively (To be specific, if the objects move slowly, we expand their MR according their speed to ensure the contextual environment information) to obtain their Adaptive Candidate Motion Range (ACMR), so as to ensure that the SNR of the moving object is improved while the necessary context information is retained adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' After that, we use LW-SCM-USN model to accurately classify and locate the suspicious moving object by using the ACMR of the suspicious moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' Finally, qualitative and quantitative experiments verify the effectiveness and advancement of the proposed moving object detection algorithm based on motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAzT4oBgHgl3EQf9P6r/content/2301.01917v1.pdf'} 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Fink � � +Faculty of Informatics and Mathematics, University of Passau, Germany +Ignaz Rutter � � +Faculty of Informatics and Mathematics, University of Passau, Germany +Abstract +SPQR-trees are a central component of graph drawing and are also important in many further +areas of computer science. From their inception onwards, they have always had a strong relation +to dynamic algorithms maintaining information, e.g., on planarity and triconnectivity, under edge +insertion and, later on, also deletion. In this paper, we focus on a special kind of dynamic update, +the expansion of vertices into arbitrary biconnected graphs, while maintaining the SPQR-tree and +further information. This will also allow us to efficiently merge two SPQR-trees by identifying the +edges incident to two vertices with each other. We do this working along an axiomatic definition +lifting the SPQR-tree to a stand-alone data structure that can be modified independently from the +graph it might have been derived from. Making changes to this structure, we can now observe how +the graph represented by the SPQR-tree changes, instead of having to reason which updates to the +SPQR-tree are necessary after a change to the represented graph. +Using efficient expansions and merges allows us to improve the runtime of the Synchronized +Planarity algorithm by Bläsius et al. [8] from O(m2) to O(m · ∆), where ∆ is the maximum +pipe degree. This also reduces the time for solving several constrained planarity problems, e.g. for +Clustered Planarity from O((n + d)2) to O(n + d · ∆), where d is the total number of crossings +between cluster borders and edges and ∆ is the maximum number of edge crossings on a single +cluster border. +2012 ACM Subject Classification Mathematics of computing → Graph algorithms +Keywords and phrases SPQR-Tree, Dynamic Algorithm, Cluster Planarity +Funding Funded by DFG-grant RU-1903/3-1. +arXiv:2301.03972v1 [cs.DS] 10 Jan 2023 + +S. D. Fink and I. Rutter +1 +1 +Introduction +The SPQR-tree is a data structure that represents the decomposition of a graph at its +separation pairs, that is the pairs of vertices whose removal disconnects the graph. The +components obtained by this decomposition are called skeletons. SPQR-trees form a central +component of many graph visualization techniques and are used for, e.g., planarity testing +and variations thereof [13, 19, 29, 31, 39] and for computing embeddings and layouts [3, 7, 11, +20, 28, 42]; see [37] for a survey of graph drawing applications. Outside of graph visualization +they are used in the context of, e.g., minimum spanning trees [6, 17], triangulations [5], and +crossing optimization [28, 42]. They also have multiple applications outside of graph theory +and even computer science, e.g. for creating integrated circuits [14, 44], business processes +modelling [40], electrical engineering [24], theoretical physics [41] and genomics [22]. +Initially, SPQR-trees were devised by Di Battista and Tamassia for incremental planarity +testing [16, 19]. As such, even in their initial form, SPQR-trees already allowed dynamic +updates in the form of edge addition. Their use was quickly expanded to other on-line +problems [18, 17]. In addition to the applications mentioned above, this also sparked a series +of further papers improving the runtime of the incremental data structure [38, 39, 43] and +also extending it to be fully-dynamic, i.e., allowing insertion and deletion of vertices and +edges, in O(√n) time [21, 27], where n is the number of vertices in the graph. Recently, +Holm and Rotenberg described a fully-dynamic algorithm for maintaining planarity and +triconnectivity information in O(log3 n) time per operation [31, 32] (see also there for a short +history on dynamic SPQR-tree algorithms). +In this paper, we consider an incremental setting where we allow a single operation that +expands a vertex v into an arbitrary biconnected graph Gν. Using the approach of Holm +and Rotenberg [31], this takes O((deg(v) + |Gν|) · log3 n) time by first removing v and its +incident edges and then incrementally inserting Gν. We improve this to O(deg(v) + |Gν|) +using an algorithm that is much simpler and thus also more likely to improve performance in +practice. In addition, our approach also allows to efficiently merge two SPQR-trees as follows. +Given two biconnected graphs G1, G2 containing vertices v1, v2, respectively, together with +a bijection between their incident edges, we construct a new graph G by replacing v1 with +G2 − v2 in G1, identifying edges using the given bijection. Given the SPQR-trees of G1 and +G2, we show that the SPQR-tree of G can be found in O(deg(v1)) time. More specifically, we +present a data structure that supports the following operations: InsertGraphSPQR expands +a single vertex in time linear in the size of the expanded subgraph, MergeSPQR merges two +SPQR-trees in time linear in the degree of the replaced vertices, IsPlanar indicates whether +the currently represented graph is planar in constant time, and Rotation yields one of +the two possible planar rotations of a vertex in a triconnected skeleton in constant time. +Furthermore, our data structure can be adapted to yield consistent planar embeddings for +all triconnected skeletons and to test for the existence of three distinct paths between two +arbitrary vertices with an additional factor of α(n) for all operations, where α is the inverse +Ackermann function. +The main idea of our approach is that the subtree of the SPQR-tree affected by expanding +a vertex v has size linear in the degree of v, but may contain arbitrarily large skeletons. In a +“non-normalized” version of an SPQR-tree, the affected cycle (‘S’) skeletons can easily be +split to have a constant size, while we develop a custom splitting operation to limit the size +of triconnected ‘R’ skeletons. This limits the size of the affected structure to be linear in the +degree of v and allows us to perform the expansion efficiently. +In addition to the description of this data structure, the technical contribution of this + +2 +Maintaining Triconnected Components under Node Expansion +Problem +Running Times +before [8] +using [8] +with this paper +Atomic +Embeddability +/ +Synchronized Planarity +O(m8) [26] +O(m2) +O(m · ∆) +ClusterPlanarity +O((n + d)8) [26] +O((n + d)2) +O(n + d · ∆) +Connected SEFE +O(n16) [26] +O(n2) +O(n · ∆) +bicon: O(n2) [10] +Partially +PQ-Constrained +Planarity +bicon: O(m) [10] +O(m2) +O(m · ∆) +Row-Column +Independent +NodeTrix Planarity +bicon: O(n2) [35] +O(n2) +O(n · ∆) +Strip Planarity +O(n8) [4, 26] +O(n2) +O(n · ∆) +fixed emb: poly [4] +Table 1 The best known running times for various constrained planarity problems before Syn- +chronized Planarity [8] was published; using it as described in [8]; and using it together with +the speed-up from this paper. Running times prefixed with “bicon” only apply for certain problem +instances which expose some form of biconnectivity. The variables n and m refer to the number +of vertices and edges of the problem instance, respectively. The variable d refers to the number of +edge-cluster boundary crossings in Clustered Planarity instances, while ∆ refers to the maximum +pipe degree in the corresponding Synchronized Planarity instances. This is bounded by the +maximum number of edges crossing a single cluster border or the maximum vertex degree in the +input instance, depending on the problem. +paper is twofold: First, we develop an axiomatic definition of the decomposition at separation +pairs, putting the SPQR-tree as “mechanical” data structure into focus instead of relying on +and working along a given graph structure. As a result, we can deduce the represented graph +from the data structure instead of computing the data structure from the graph. This allows +us to make more or less arbitrary changes to the data structure (respecting its consistency +criteria) and observe how the graph changes, instead of having to reason which changes to +the graph require which updates to the data structure. +Second, we explain how our data structure can be used to improve the runtime of +the algorithm by Bläsius et al. [8] for solving Synchronized Planarity from O(m2) to +O(m · ∆), where ∆ is the maximum pipe degree (i.e. the maximum degree of a vertex with +synchronization constraints that enforce its rotation to be the same as that of another vertex). +Synchronized Planarity can be used to model and solve a vast class of different kinds of +constrained planarity, see Table 1 for an overview of problems benefiting from this speedup. +Among them is the notorious Clustered Planarity, whose complexity was open for 30 +years before Fulek and Tóth gave an algorithm with runtime O((n + d)8) in 2019 [26], where +d is the total number of crossings between cluster borders and edges. Shortly thereafter, +Bläsius et al. [8] gave a solution in O((n + d)2) time. We improve this to O(n + d · ∆), where +∆ is the maximum number of edge crossings on a single cluster border. +This work is structured as follows. Section 2 contains an overview of the definitions +used in this work. In Section 3, we describe the skeleton decomposition and show how it +relates to the SPQR-tree. Section 4 extends this data structure by the capability of splitting + +S. D. Fink and I. Rutter +3 +triconnected components. In Section 5, we exploit this feature to ensure the affected part of +the SPQR-tree is small when we replace a vertex with a new graph. Section 6 contains more +details on the background of Synchronized and Clustered Planarity and shows how +our results can be used to reduce the time required for solving them. +2 +Preliminaries +In the context of this work, G = (V, E) is a (usually biconnected and loop-free) multi-graph +with n vertices V and m (possibly parallel) edges E. For a vertex v, we denote its open +neighborhood (excluding v itself) by N(v). For a bijection or matching ϕ we call ϕ(x) the +partner of an element x. We use A ·∪ B to denote the union of two disjoint sets A, B. +A separating k-set is a set of k vertices whose removal increases the number of connected +components. Separating 1-sets are called cutvertices, while separating 2-sets are called +separation pairs. A connected graph is biconnected if it does not have a cutvertex. A +biconnected graph is triconnected if it does not have a separation pair. Maximal biconnected +subgraphs are called blocks. Each separation pair divides the graph into bridges, the maximal +subgraphs which cannot be disconnected by removing or splitting the vertices of the separation +pair. A bond is a graph that consists solely of two pole vertices connected by multiple parallel +edges, a polygon is a simple cycle, while a rigid is any simple triconnected graph. A wheel is +a cycle with an additional central vertex connected to all other vertices. +Finally, the expansion that is central to this work is formally defined as follows. Let +Gα, Gβ be two graphs where Gα contains a vertex u and Gβ contains |N(u)| marked vertices, +together with a bijection ϕ between the neighbors of u and the marked vertices in Gβ. With +Gα[u →ϕ Gβ] we denote the graph that is obtained from the disjoint union of Gα, Gβ by +identifying each neighbor x of u with its respective marked vertex ϕ(x) in Gβ and removing +u, i.e. the graph Gα where the vertex u was expanded into Gβ. +3 +Skeleton Decompositions +A skeleton structure S = (G, origV, origE, twinE) that represents a graph GS = (V, E) +consists of a set G of disjoint skeleton graphs together with three total, surjective mappings +twinE, origE, and origV that satisfy the following conditions: +Each skeleton Gµ = (Vµ, Ereal +µ +·∪ Evirt +µ +) in G is a multi-graph where each edge is either in +Ereal +µ +and thus called real or in Evirt +µ +and thus called virtual. +Bijection twinE : Evirt → Evirt matches all virtual edges Evirt = � +µ Evirt +µ +such that +twinE(e) ̸= e and twinE2 = id. +Surjection origV : � +µ Vµ → V maps all skeleton vertices to graph vertices. +Bijection origE : � +µ Ereal +µ +→ E maps all real edges to the graph edge set E. +Note that each vertex and each edge of each skeleton is in the domain of exactly one of the +three mappings. As the mappings are surjective, V and E are exactly the images of origV +and origE. For each vertex v ∈ GS, the skeletons that contain an allocation vertex v′ with +origV(v′) = v are called the allocation skeletons of v. Furthermore, let TS be the graph +where each node µ corresponds to a skeleton Gµ of G. Two nodes of TS are adjacent if their +skeletons contain a pair of virtual edges matched with each other. +We call a skeleton structure a skeleton decomposition if it satisfies the following conditions: +1 (bicon) Each skeleton is biconnected. +2 (tree) Graph TS is simple, loop-free, connected and acyclic, i.e., a tree. +3 (orig-inj) For each skeleton Gµ, the restriction origV |Vµ is injective. + +4 +Maintaining Triconnected Components under Node Expansion +u +(a) +(b) +(d) +(c) +Figure 1 Different views on the skeleton decomposition S. (a) The graph GS with a vertex u +marked in blue. (b) The skeletons of G. Virtual edges are drawn in gray with their matching twinE +being shown in orange. The allocation vertices of u are marked in blue. (c) The tree TS. The +allocation skeletons of u are marked in blue. (d) The embedding tree of vertex u as described in +Section 6.2. P-nodes are shown as white disks, Q-nodes are shown as large rectangles. The leaves of +the embedding tree correspond to the edges incident to u. +4 (orig-real) For each real edge uv, the endpoints of origE(uv) are origV(u) and origV(v). +5 (orig-virt) Let uv and u′v′ be two virtual edges with uv = twinE(u′v′). For their respective +skeletons Gµ and G′ +µ (where µ and µ′ are adjacent in TS), it is origV(Vµ) ∩ origV(Vµ′) = +origV({u, v}) = origV({u′, v′}). +6 (subgraph) The allocation skeletons of any vertex of GS form a connected subgraph of TS. +Figure 1 shows an example of S, GS, and TS. We call a skeleton decomposition with only +one skeleton Gµ trivial. Note that in this case, Gµ is isomorphic to GS, and origE and origV +are actually bijections between the edges and vertices of both graphs. +To model the decomposition into triconnected components, we define the operations +SplitSeparationPair and its converse, JoinSeparationPair, on a skeleton decomposition +S = (G, origV, origE, twinE). For SplitSeparationPair, let u, v be a separation pair of +skeleton Gµ and let (A, B) be a non-trivial bipartition of the bridges between u and v.1 +Applying SplitSeparationPair(S, (u, v), (A, B)) yields a skeleton decomposition S′ = (G′, +origV′, origE′, twinE′) as follows. In G′, we replace Gµ by two skeletons Gα, Gβ, where Gα is +obtained from Gµ[A] by adding a new virtual edge eα between u and v. The same respectively +applies to Gβ with Gµ[B] and eβ. We set twinE′(eα) = eβ and twinE′(eβ) = eα. Note that +origV maps the endpoints of eα and eβ to the same vertices. All other skeletons and the +mappings defined on them remain unchanged. +For JoinSeparationPair, consider virtual edges eα, eβ with twinE(eα) = eβ and let +Gβ ̸= Gα be their respective skeletons. +Applying JoinSeparationPair(S, eα) yields a +skeleton decomposition S′ = (G′, origV′, origE′, twinE′) as follows. In G′, we merge Gα with +Gβ to form a new skeleton Gµ by identifying the endpoints of eα and eβ that map to the +same vertex of GS. Additionally, we remove eα and eβ. All other skeletons and the mappings +defined on them remain unchanged. +The main feature of both operations is that they leave the graph represented by the +skeleton decomposition unaffected while splitting a node or contracting and edge in TS, +which can be verified by checking the individual conditions. +▶ Lemma 1. Applying SplitSeparationPair or JoinSeparationPair on a skeleton de- +composition S = (G, origV, origE, twinE) yields a skeleton decomposition S′ = (G′, origV′, +1 Note that a bridge might consist out of a single edge between u and v and that each bridge includes the +vertices u and v. + +S. D. Fink and I. Rutter +5 +origE′, twinE′) with an unchanged represented graph GS′ = GS. +Proof. We first check that all conditions still hold in the skeleton decomposition S′ returned +by SplitSeparationPair. As (A, B) is a non-trivial bipartition, each set contains at least one +bridge. Together with eα (and eβ), this bridge ensures that Gα (and Gβ) remain biconnected, +satisfying condition 1 (bicon). The operation splits a node µ of TS into two adjacent nodes +α, β, whose neighbors are defined exactly by the virtual edges in A, B, respectively. Thus, +condition 2 (tree) remains satisfied. The mappings origV′, origE′ and twinE′ obviously still +satisfy conditions 3 (orig-inj) and 4 (orig-real). We duplicated exactly two nodes, u and v of +adjacent skeletons Gα and Gβ. Because 3 (orig-inj) holds for Gµ, Gα and Gβ share no other +vertices that map to the same vertex of GS′. Thus, condition 5 (orig-virt) remains satisfied. +Condition 6 (subgraph) could only be violated if the subgraph of TS′ formed by the +allocation skeletons of some vertex z ∈ GS′ was no longer connected. This could only happen +if only one of Gα and Gβ were an allocation skeleton of z, while the other has a further +neighbor that is also an allocation skeleton of z. Assume without loss of generality that Gα +and the neighbor Gν of Gβ, but not Gβ itself, were allocation skeletons of z. Because Gν and +Gβ are adjacent in TS′ there are virtual edges xy = twinE′(x′y′) with xy ∈ Gβ and x′y′ ∈ Gν. +The same virtual edges are also present in the input instance, only with the difference that +xy ∈ Gµ and µ (instead of β) and ν are adjacent in TS. As the input instance satisfies +condition 5 (orig-virt), it is z ∈ origV(Vν) ∩ origV(Vµ) = origV({x, y}) = origV({x′, y′}). As +origV({x, y}) = origV′({x, y}), this is a contradiction to Gβ not being an allocation skeleton +of z. +Finally, the mapping origE remains unchanged and the only change to origV is to include +two new vertices mapping to already existing vertices. Due to condition 4 (orig-real) holding +for both the input and the output instance, this cannot affect the represented graph GS′. +Now consider the skeleton decomposition S′ returned by JoinSeparationPair. Identify- +ing distinct vertices of distinct connected components does not affect their biconnectivity, +thus condition 1 (bicon) remains satisfied. The operation effectively contracts and removes +an edge in TS, which does not affect TS′ being a tree satisfying condition 2 (tree). Note +that condition 2 (tree) holding for the input instance also ensures that Gα and Gβ are two +distinct skeletons. As the input instance also satisfies condition 5 (orig-virt), there are exactly +two vertices in each of the two adjacent skeletons Gα and Gβ, where origV maps to the +same vertex of GS. These two vertices must be part of the twinE pair making the two +skeletons adjacent, thus they are exactly the two pairs of vertices we identify with each other. +Thus, origV |Vµ is still injective, satisfying condition 3 (orig-inj). As we modify no real edges +and no other virtual edges, the mappings origV′ and origE′ obviously still satisfy condition +4 (orig-real). As the allocation skeletons of each graph vertex form a connected subgraph, +joining two skeletons cannot change the intersection with any of their neighbors, leaving +5 (orig-virt) satisfied. Finally, contracting a tree edge cannot lead to any of the subgraphs of +6 (subgraph) becoming disconnected, thus the condition also remains satisfied. Again, no +changes were made to origE, while condition 5 (orig-virt) makes sure that origV mapped the +two pairs of merged vertices to the same vertex of GS. Thus, the represented graph GS′ +remains unchanged. +◀ +This gives us a second way of finding the represented graph by exhaustively joining all +skeletons until there is only one left, obtaining the unique trivial skeleton decomposition: +▶ Lemma 2. Exhaustively applying JoinSeparationPair to a skeleton decomposition S = +(G, origV, origE, twinE) yields a trivial skeleton decomposition S′ = (G′, origV′, origE′, twinE′) +where origE′ and origV′ define an isomorphism between G′ +µ and GS′. + +6 +Maintaining Triconnected Components under Node Expansion +Proof. As all virtual edges are matched, and the matched virtual edge always belongs to +a different skeleton (condition 2 (tree) ensures that TS is loop-free), we can always apply +JoinSeparationPair on a virtual edge until there are none left. As TS is connected, this +means that the we always obtain a tree with a single node, that is an instance with only a +single skeleton. As a single application of JoinSeparationPair preserves the represented +graph, any chain of multiple applications also does. Note that origE′ is a bijection and the +surjective origV′ is also injective on the single remaining skeleton due to condition 3 (orig-inj), +thus it also globally is a bijection. Together with condition 4 (orig-real), this ensures that any +two vertices u and v of G′ +µ are adjacent if and only if origV′(u) and origV′(v) are adjacent +in GS′. Thus origV′ is an edge-preserving bijection, that is an isomorphism. +◀ +A key point about the skeleton decomposition and especially the operation SplitSepa- +rationPair now is that they model the decomposition of a graph at separation pairs. This +decomposition was formalized as SPQR-tree by Di Battista and Tamassia [16] and is unique +for a given graph [33, 36]; see also [28, 30]. Angelini et al. [1] describe a decomposition +tree that is conceptually equivalent to our skeleton decomposition. They also present an +alternative definition for the SPQR-tree as a decomposition tree satisfying further properties. +We adopt this definition for our skeleton decompositions as follows, not requiring planarity +of triconnected components and allowing virtual edges and real edges to appear within one +skeleton (i.e., having leaf Q-nodes merged into their parents). +▶ Definition 3. A skeleton decomposition S = (G, origV, origE, twinE) where any skeleton +in G is either a polygon, a bond, or triconnected (“rigid”), and two skeletons adjacent in TS +are never both polygons or both bonds, is the unique SPQR-tree of GS. +The main difference between the well-known ideas behind decomposition trees and our +skeleton decomposition is that the latter allow an axiomatic access to the decomposition at +separation pairs. For the skeleton decomposition, we employ a purely functional, “mechanical” +data structure instead of relying on and working along a given graph structure. In our +case, the represented graph is deduced from the data structure (i.e. SPQR-tree) instead of +computing the data structure from the graph. +4 +Extended Skeleton Decompositions +Note that most skeletons, especially polygons and bonds, can easily be decomposed into +smaller parts. The only exception to this are triconnected skeletons which cannot be split +further using the operations we defined up to now. This is a problem when modifying +a vertex that occurs in triconnected skeletons that may be much bigger than the direct +neighborhood of the vertex. To fix this, we define a further set of operations which allow +us to isolate vertices out of arbitrary triconnected components by replacing them with a +(“virtual”) placeholder vertex. This placeholder then points to a smaller component that +contains the actual vertex, see Figure 2. Modification of the edges incident to the placeholder +is disallowed, which is why we call them “occupied”. +Formally, the structures needed to keep track of the components split in this way +in an extended skeleton decomposition S = (G, origV, origE, twinE, twinV) are defined as +follows. Skeletons now have the form Gµ = (Vµ ·∪ V virt +µ +, Ereal +µ +·∪ Evirt +µ +·∪ Eocc +µ ). Bijection +twinV : V virt → V virt matches all virtual vertices V virt = � +µ V virt +µ +, such that twinV(v) ̸= v, +twinV2 = id. The edges incident to virtual vertices are contained in Eocc +µ +and thus considered +occupied; see Figure 2b. Similar to the virtual edges matched by twinE, any two virtual +vertices matched by twinV induce an edge between their skeletons in TS. Condition 2 (tree) + +S. D. Fink and I. Rutter +7 +v +Gµ +u +(a) +v +vα +vβ +Gα +Gβ +uα +uβ +(b) +Figure 2 (a) A triconnected skeleton Gµ with a highlighted vertex v incident to two gray virtual +edges. (b) The result of applying IsolateVertex to isolate v out of the skeleton. The red occupied +edges in the old skeleton Gα form a star with center vα, while the red occupied edges in Gβ connect +all neighbors of v to form a star with center vβ ̸= v. The centers vα and vβ are virtual and matched +with each other. Neighbor u of v was split into vertices uα and uβ. +also equally applies to those edges induced by twinV, which in particular ensures that there +are no parallel twinE and twinV tree edges in TS. Similarly, the connected subgraphs of +condition 6 (subgraph) can also contain tree edges induced by twinV. All other conditions +remain unchanged, but we add two further conditions to ensure that twinV is consistent: +7 (stars) For each vα, vβ with twinV(vα) = vβ, it is deg(vα) = deg(vβ). All edges incident +to vα and vβ are occupied and have distinct endpoints (except for vα and vβ). Conversely, +each occupied edge is adjacent to exactly one virtual vertex. +8 (orig-stars) Let vα and vβ again be two virtual vertices matched with each other by twinV. +For their respective skeletons Gα and Gβ (where α and β are adjacent in TS), it is +origV(Vα) ∩ origV(Vβ) = origV(N(vα)) = origV(N(vβ)). +Note that both conditions together yield a bijection γvαvβ between the neighbors of +vα and vβ, as origV is injective when restricted to a single skeleton (condition 3 (orig- +inj)) and deg(vα) = deg(vβ). Operations SplitSeparationPair and JoinSeparationPair +can also be applied to an extended skeleton decomposition, yielding an extended skeleton +decomposition without modifying twinV. To ensure that conditions 7 (stars) and 8 (orig-stars) +remain unaffected by both operations, SplitSeparationPair cannot be applied if a vertex +of the separation pair is virtual. +The operations IsolateVertex and Integrate now allow us to isolate vertices out of +triconnected components and integrate them back in, respectively. For IsolateVertex, let v +be a non-virtual vertex of skeleton Gµ, such that v has no incident occupied edges. Applying +IsolateVertex(S, v) on an extended skeleton decomposition S yields an extended skeleton +decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) as follows. Each neighbor u of v is +split into two non-adjacent vertices uα and uβ, where uβ is incident to all edges connecting u +with v, while uα keeps all other edges of u. We set origV′(uα) = origV′(uβ) = origV(u). This +creates an independent, star-shaped component with center v, which we move to skeleton +Gβ, while we rename skeleton Gµ to Gα. We connect all uα to a single new virtual vertex +vα ∈ V virt +α +using occupied edges, and all uβ to a single new virtual vertex vβ ∈ V virt +β +using +occupied edges; see Figure 2. Finally, we set twinV′(vα) = vβ, twinV′(vβ) = vα, and add Gβ +to G′. All other mappings and skeletons remain unchanged. +For Integrate, consider two virtual vertices vα, vβ with twinV(vα) = vβ and the bijec- +tion γvαvβ between the neighbors of vα and vβ. An application of Integrate(S, (vα, vβ)) +yields an extended skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) as follows. +We merge both skeletons into a skeleton Gµ (also replacing both in G′) by identifying the +neighbors of vα and vβ according to γvαvβ. Furthermore, we remove vα and vβ together with +their incident occupied edges. All other mappings and skeletons remain unchanged. + +8 +Maintaining Triconnected Components under Node Expansion +▶ Lemma 4. Applying IsolateVertex or Integrate on an extended skeleton decomposition +S = (G, origV, origE, twinE, twinV) yields an extended skeleton decomposition S′ = (G′, +origV′, origE′, twinE′, twinV′) with GS′ = GS. +Proof. We first check that all conditions still hold in the extended skeleton decomposition S′ +returned by IsolateVertex. Condition 1 (bicon) remains satisfied, as the structure of Gα +remains unchanged compared to Gµ and the skeleton Gβ is a bond. As we are again splitting +a node of TS, condition 2 (tree) also remains satisfied. Due to the neighbors of vβ and vα +mapping to the same vertices of GS′, conditions 3 (orig-inj), 4 (orig-real), and 5 (orig-virt) +remain satisfied. Conditions 7 (stars) and 8 (orig-stars) are satisfied by construction. +Lastly, condition 6 (subgraph) could only be violated if the subgraph of TS′ formed by the +allocation skeletons of some vertex z ∈ GS′ was no longer connected. This could only happen +if only one of Gα and Gβ were an allocation skeleton of z, while the other has a further +neighbor Gν that is also an allocation skeleton of z. Note that in any case, ν is adjacent +to µ in TS and µ must be an allocation skeleton of z, thus it is z ∈ origV(Gν) ∩ origV(Gµ). +Depending on the adjacency of ν, it is either origV(Gν)∩origV(Gµ) = origV′(Gν)∩origV(Gα) +or origV(Gν) ∩ origV(Gµ) = origV′(Gν) ∩ origV(Gβ), as ν is not modified by the operation +and both S and S′ satisfy 5 (orig-virt) and 8 (orig-stars). This immediately contradicts the +skeleton of {α, β}, that is adjacent to ν, not being an allocation skeleton of z. +Finally, the mapping origE remains unchanged and the only change to origV is to include +some duplicated vertices mapping to already existing vertices. Due to condition 4 (orig-real) +holding for both the input and the output instance, this cannot affect the represented graph +GS′. +Now consider the extended skeleton decomposition S′ returned by Integrate. The +merged skeleton is biconnected, as we are effectively replacing a single vertex by a connected +subgraph, satisfying 1 (bicon). The operation effectively contracts and removes an edge in +TS, which does not affect TS′ being a tree, satisfying condition 2 (tree). Note that condition +2 (tree) holding for the input instance also ensures that vα and vβ belong to two distinct +skeletons. As the input instance satisfies condition 5 (orig-virt), the vertices in each of the +two adjacent skeletons where origV maps to the same vertex of GS are exactly the neighbors +of the matched vα and vβ. Thus, origV |Vα is still injective, satisfying condition 3 (orig-inj). +As we modify no real or virtual edges, the mappings origV′, origE′ and twinE′ obviously still +satisfy conditions 4 (orig-real) and 5 (orig-virt). Finally, contracting a tree edge cannot lead to +any of the subgraphs of 6 (subgraph) becoming disconnected, thus the condition also remains +satisfied. Conditions 7 (stars) and 8 (orig-stars) also remain unaffected, as we simply remove +an entry from twinV. +Again, no changes were made to origE, while condition 8 (orig-stars) makes sure that +origV mapped each pair of merged vertices to the same vertex of GS. Thus, the represented +graph GS′ remains unchanged. +◀ +Furthermore, as Integrate is the converse of IsolateVertex and has no preconditions, +any changes made by IsolateVertex can be undone at any time to obtain a (non-extended) +skeleton decomposition, and thus possibly the SPQR-tree of the represented graph. +▶ Remark 5. Exhaustively applying Integrate to an extended skeleton decomposition +S = (G, origV, origE, twinE, twinV) yields a extended skeleton decomposition S′ = (G′, +origV′, origE′, twinE′, twinV′) where twinV′ = ∅. Thus, S′ is equivalent to a (non-extended) +skeleton decomposition S′ = (G′, origV′, origE′, twinE′). + +S. D. Fink and I. Rutter +9 +v +Gµ +(a) +Gν +(b) +(c) +Figure 3 Expanding a skeleton vertex v into a graph Gν in the SPQR-tree of Figure 4b. (a) The +single allocation skeleton Gµ of u with the single allocation vertex v of u from Figure 4b. The +neighbors of v are marked in orange. (b) The inserted graph Gν with orange marked vertices. +Note that the graph is biconnected when all marked vertices are collapsed into a single vertex. +(c) The result of applying InsertGraph(S, u, Gν, ϕ) followed by an application of Integrate on the +generated virtual vertices v and v′. +5 +Node Expansion in Extended Skeleton Decompositions +We now introduce our first dynamic operation that allows us to actually change the represented +graph by expanding a single vertex u into an arbitrary connected graph Gν. This is done +by identifying |N(u)| marked vertices in Gν with the neighbors of u via a bijection ϕ and +then removing u and its incident edges. We use the “occupied stars” from the previous +section to model the identification of these vertices, allowing us to defer the actual insertion +to an application of Integrate. We need to ensure that the inserted graph makes the same +“guarantees” to the surrounding graph in terms of connectivity as the vertex it replaces, +that is all neighbors of u (i.e. all marked vertices in Gν) need to be pairwise connected via +paths in Gν not using any other neighbor of u (i.e. any other marked vertex). Without this +requirement, a single vertex could e.g. also be split into two non-adjacent halves, which could +easily break a triconnected component apart. Thus, we require Gν to be biconnected when +all marked vertices are collapsed into a single vertex. Note that this also ensures that the +old graph can be restored by contracting the vertices of the inserted graph. For the sake of +simplicity, we require vertex u from the represented graph to have a single allocation vertex +v ∈ Gµ with origV−1(u) = {v} so that we only need to change a single allocation skeleton +Gµ in the skeleton decomposition. As we will make clear later on, this condition can be +satisfied easily. +Formally, let u ∈ GS be a vertex that only has a single allocation vertex v ∈ Gµ (and +thus only a single allocation skeleton Gµ). Let Gν be an arbitrary, new graph containing +|N(u)| marked vertices, together with a bijection ϕ between the marked vertices in Gν +and the neighbors of v in Gµ. We require Gν to be biconnected when all marked vertices +are collapsed into a single node. Operation InsertGraph(S, u, Gν, ϕ) yields an extended +skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) as follows, see also Figure 3. +We interpret Gν as skeleton and add it to G′. For each marked vertex x in Gν, we set +origV′(x) = origV(ϕ(x)). For all other vertices and edges in Gν, we set origV′ and origE′ +to point to new vertices and edges forming a copy of Gν in GS′. We connect every marked +vertex in Gν to a new virtual vertex v′ ∈ Gν using occupied edges. We also convert v to a +virtual vertex, converting its incident edges to occupied edges while removing parallel edges. +Finally, we set twinV′(v) = v′ and twinV′(v′) = v. +▶ Lemma 6. Applying InsertGraph(S, u, Gν, ϕ) on an extended skeleton decomposition +S = (G, origV, origE, twinE, twinV) yields an extended skeleton decomposition S′ = (G′, +origV′, origE′, twinE′, twinV′) with GS′ isomorphic to GS[u →ϕ Gν]. + +10 +Maintaining Triconnected Components under Node Expansion +Proof. Condition 1 (bicon) remains satisfied, as the structure of Gµ remains unchanged +and the resulting Gν is biconnected by precondition. Regarding TS, we are attaching a +degree-1 node ν to an existing node µ, thus condition 2 (tree) also remains satisfied. As +all vertices of Gν except for the vertices in N(v′) got their new, unique copy assigned by +origV′ and origV′(N(v′)) = origV(N(v)), condition 3 (orig-inj) is also satisfied for the new +Gν. As we updated origE alongside origV and Gν contains no virtual edges, conditions +4 (orig-real) and 5 (orig-virt) remain satisfied. As ν is a leaf of TS with µ being its only +neighbor, origV′(N(v′)) ⊂ origV(Vµ), and Gν is the only allocation skeleton for all vertices +in Gν \ N(v′), condition 6 (subgraph) remains satisfied. Conditions 7 (stars) and 8 (orig-stars) +are satisfied by construction. Finally, the mappings origE′ and origV′ are by construction +updated to correctly reproduce the structure of Gν in GS′. +◀ +On its own, this operation is not of much use though, as graph vertices only rarely have +a single allocation skeleton. Furthermore, our goal is to dynamically maintain SPQR-trees, +while this operation on its own will in most cases not yield an SPQR-tree. To fix this, we +introduce the full procedure InsertGraphSPQR(S, u, Gν, ϕ) that can be applied to any graph +vertex u and that, given an SPQR-tree S, yields the SPQR-tree of GS[u →ϕ Gν]. It consists +of three preparations steps, the insertion of Gν, and two further clean-up steps: +1. We apply SplitSeparationPair to each polygon allocation skeleton of u with more than +three vertices, using the neighbors of the allocation vertex of u as separation pair. +2. For each rigid allocation skeleton of u, we move the contained allocation vertex v of u to +its own skeleton by applying IsolateVertex(S, v). +3. We exhaustively apply JoinSeparationPair to any pair of allocation skeletons of u that +are adjacent in TS. Due to condition 6 (subgraph), this yields a single component Gµ that +is the sole allocation skeleton of u with the single allocation vertex v of u. Furthermore, +the size of Gµ is linear in deg(u). +4. We apply InsertGraph to insert Gν as skeleton, followed by an application of Integrate +to the virtual vertices {v, v′} introduced by the insertion, thus integrating Gν into Gµ. +5. We apply SplitSeparationPair to all separation pairs in Gµ that do not involve a +virtual vertex. These pairs can be found in linear time, e.g. by temporarily duplicating +all virtual vertices and their incident edges and then computing the SPQR-tree.2 +6. Finally, we exhaustively apply Integrate and also apply JoinSeparationPair to any +two adjacent polygons and to any two adjacent bonds to obtain the SPQR-tree of the +updated graph. +The basic idea behind the correctness of this procedure is that splitting the newly inserted +component according to its SPQR-tree in step 5 yields biconnected components that are each +either a polygon, a bond, or “almost” triconnected. The latter (and only those) might still +contain virtual vertices and all their remaining separation pairs, which were not split in step 5, +contain one of these virtual vertices. This, together with the fact that there still may be +pairs of adjacent skeletons where both are polygons or both are bonds, prevents the instance +from being an SPQR-tree. Both issues are resolved in step 6: The adjacent skeletons are +obviously fixed by the JoinSeparationPair applications. To show that the virtual vertices +are removed by the Integrate applications, making the remaining components triconnected, +we need the following lemma. +2 Note that the wheels replacing virtual vertices in the proof of Theorem 10 also ensure this. + +S. D. Fink and I. Rutter +11 +(a) +v +(b) +Figure 4 The preprocessing steps of InsertGraphSPQR being applied to the SPQR-tree of Figure 1b. +(a) The state after step 2, after all allocation skeletons of u have been split. (b) The state after +step 3, after all allocation skeletons of u have been merged into a single one. +▶ Lemma 7. Let Gα be a triconnected skeleton containing a virtual vertex vα matched with +a virtual vertex vβ of a biconnected skeleton Gβ. Furthermore, let P ⊆ +�V (Gβ) +2 +� +be the set +of all separation pairs in Gβ. An application of Integrate(S, (vα, vβ)) yields a biconnected +skeleton Gµ with separation pairs P ′ = {{u, v} ∈ P | vβ /∈ {u, v}}. +Proof. We partition the vertices of Gµ into the sets A, B, and N depending on whether the +vertex stems from Gα, Gβ, or both, respectively. The set N thus contains the neighbors of vα, +which were identified with the neighbors of vβ. We will now show by contradiction that Gµ +contains no separation pairs except for those in P ′. Thus, consider a separation pair u, v ∈ Gµ +not in P ′. First, consider the case where u, v ∈ A∪N. Observe that removing u, v in this case +leaves B connected. Thus, we can contract all vertices of B into a single vertex, reobtain Gα +and see that u, v is a separation pair in Gα. This contradicts the precondition that Gα is +triconnected. Now consider the case where u, v ∈ B ∪ N. Analogously to above, we find that +u, v is a separation pair in Gβ that does not contain vβ, a contradiction to {u, v} /∈ P ′. Finally, +consider the remaining case where, without loss of generality, u ∈ A, v ∈ B. Since {u, v} +is a separation pair, u has two neighbors x, y that lie in different connected components of +Gµ−{u, v} and therefore also in different components of (Gµ−{u, v})−B which is isomorphic +to Gα − {u, vα}. This again contradicts the precondition that Gα is triconnected. +◀ +▶ Theorem 8. Applying InsertGraphSPQR(S, u, Gν, ϕ) to an SPQR-tree S yields an SPQR- +tree S′ in O(|Gν|) time with GS′ isomorphic to GS[u →ϕ Gν]. +Proof. As all operations that are applied leave the extended skeleton decomposition valid, +the final extended skeleton decomposition S′ is also valid. Observe that the purpose of +the preprocessing steps 1–3 is solely to ensure that the preconditions of InsertGraph are +satisfied and the affected component is not too large. Note that all rigids split in step 2 +remain structurally unmodified in the sense that edges only changed their type, but the +graph and especially its triconnectedness remains unchanged. Step 4 performs the actual +insertion and yields the desired represented graph according to Lemma 6. It thus remains +to show that the clean-up steps turn the obtained extended skeleton decomposition into +an SPQR-tree. +Applying Integrate exhaustively in step 6 ensures that the extended +skeleton decomposition is equivalent to a non-extended one (Remark 5). Recall that a +non-extended skeleton decomposition is an SPQR-tree if all skeletons are either polygons, +bonds or triconnected and two adjacent skeletons are never both polygons or both bonds +(Definition 3). Step 6 ensures that the second half holds, as joining two polygons (or two +bonds) with JoinSeparationPair yields a bigger polygon (or bond, respectively). Before + +12 +Maintaining Triconnected Components under Node Expansion +step 6, all skeletons that are not an allocation skeleton of u are still unmodified and thus +already have a suitable structure, i.e., they are either polygons, bonds or triconnected. +Furthermore, the allocation skeletons of u not containing virtual vertices also have a suitable +structure, as their splits were made according to the SPQR-tree in step 5. It remains to +show that the remaining skeletons, that is those resulting from the Integrate applications +in step 6, are triconnected. Note that in these skeletons, step 5 ensures that every separation +pair consists of at least one virtual vertex, as otherwise the computed SPQR-tree would +have split the skeleton further. Further note that, for each of these virtual vertices, the +matched partner vertex is part of a structurally unmodified triconnected skeleton that was +split in step 2. Lemma 7 shows that applying Integrate does not introduce new separation +pairs while removing two virtual vertices if one of the two sides is triconnected. We can +thus exhaustively apply Integrate and thereby remove all virtual vertices and thus also all +separation pairs, obtaining triconnected components. This shows that the criteria for being +an SPQR-tree are satisfied and, as InsertGraph expanded u to Gν in the represented graph, +we now have the unique SPQR-tree of GS[u →ϕ Gν]. +Note that all operations we used can be performed in time linear in the degree of the +vertices they are applied on. For the bipartition of bridges input to SplitSeparationPair, +it is sufficient to describe each bridge via its edges incident to the separation pair instead of +explicitly enumerating all in vertices in the bridge. Thus, the applications of SplitSepara- +tionPair and IsolateVertex in steps 1 and 2 touch every edge incident to u at most once +and thus take O(deg(u)) time. Furthermore, they yield skeletons that have a size linear in +the degree of their respective allocation vertex of u. As the subtree of u’s allocation skeletons +has size at most deg(u), the JoinSeparationPair applications of step 3 also take at most +O(deg(u)) time. It also follows that the resulting single allocation skeleton of u has size +O(deg(u)). The applications of InsertGraph and Integrate in step 4 can be done in time +linear in the number of identified neighbors, which is O(deg(u)). Generating the SPQR-tree +of the inserted graph in step 5 (where all virtual vertices where replaced by wheels) can +be done in time linear in the size of the inserted graph [30, 33], that is O(|Gν|). Applying +SplitSeparationPair according to all separation pairs identified by this SPQR-tree can +also be done in O(|Gν|) time in total. Note that there are at most deg(u) edges between +the skeletons that existed before step 4 and those that were created or modified in steps 4 +and 5, and these are the only edges that might now connect two polygons or two bonds. As +these tree edges have one endpoint in the single allocation skeleton of u, the applications of +Integrate and JoinSeparationPair in step 6 run in O(deg(u)) time in total. Furthermore, +they remove all pairs of adjacent polygons and all pairs of adjacent bonds. This shows that +all steps take O(deg(u)) time, except for step 5, which takes O(|Gν|) time. As the inserted +graph contains at least one vertex for each neighbor of u, the total runtime is in O(|Gν|). +◀ +▶ Corollary 9. Let S1, S2 be two SPQR-trees together with vertices u1 ∈ GS1, u2 ∈ GS2, and +let ϕ be a bijection between the edges incident to u1 and the edges incident to u2. Operation +MergeSPQR(S1, S2, u1, u2, ϕ) yields the SPQR-tree of the graph GS1[u1 →ϕ GS2 − u2], i.e. the +union of both graphs where the edges incident to u1, u2 were identified according to ϕ and +u1, u2 removed, in time O(deg(u1)) = O(deg(u2)). +Proof. Operation MergeSPQR works similar to the more general InsertGraphSPQR, although +the running time is better because we already know the SPQR-tree for the graph being +inserted. We apply the preprocessing steps 1–3 to ensure that both u1 and u2 have sole +allocation vertices v1 and v2, respectively. To properly handle parallel edges, we subdivide +all edges incident to u1, u2 (and thus also the corresponding real edges incident to v1, v2) and + +S. D. Fink and I. Rutter +13 +then identify the subdivision vertices of each pair of edges matched by ϕ. By deleting vertices +v1 and v2 and suppressing the subdivision vertices (that is, removing them and identifying +each pair of incident edges) we obtain a skeleton Gµ that has size O(deg(u1)) = O(deg(u2)). +Finally, we apply the clean-up steps 5 and 6 to Gµ to obtain the final SPQR-tree. Again, +as the partner vertex of every virtual vertex in the allocation skeletons of u is part of a +triconnected skeleton, applying Integrate exhaustively in step 6 yields triconnected skeletons. +As previously discussed, the preprocessing and clean-up steps run in time linear in degree of +the affected vertices, thus the overall runtime is O(deg(u1)) = O(deg(u2)) in this case. +◀ +5.1 +Maintaining Planarity and Vertex Rotations +Note that expanding a vertex of a planar graph using another planar graph using Insert- +GraphSPQR (or merging two SPQR-trees of planar graphs using Corollary 9) might actually +yield a non-planar graph. This is, e.g., because the rigids of both graphs might require +incompatible orders for the neighbors of the replaced vertex. The aim of this section is to +efficiently detect this case, that is a planar graph turning non-planar. To check a general +graph for planarity, it suffices to check the rigids in its SPQR-tree for planarity and each rigid +allows exactly two planar embeddings, where one is the reverse of the other [19]. Thus, if a +graph becomes non-planar through an application of InsertGraphSPQR, this will be noticeable +from the triconnected allocation skeletons of the replaced vertex. To be able to immediately +report if the instance became non-planar, we need to maintain a rotation, that is a cyclic +order of all incident edges, for each vertex in any triconnected skeleton. Note that we do not +track the direction of the orders, that is we only store the order up to reversal. As discussed +later, the exact orders can also be maintained with a slight overhead. +▶ Theorem 10. SPQR-trees support the following operations: +InsertGraphSPQR(S, u, Gν, ϕ): expansion of a single vertex u in time O(|Gν|), +MergeSPQR(S1, S2, u1, u2, ϕ): merging of two SPQR-trees in time O(deg(u1)), +IsPlanar: queries whether the represented graph is planar in time O(1), and +Rotation(u): queries for one of the two possible rotations of vertices u in planar tricon- +nected skeletons in time O(1). +Proof. Note that the boolean flag IsPlanar together with the Rotation information can +be computed in linear time when creating a new SPQR-tree and that expanding a vertex or +merging two SPQR-trees cannot turn a non-planar graph planar. We make the following +changes to the operations InsertGraphSPQR and MergeSPQR described in Theorem 8 and +Corollary 9 to maintain the new information. After a triconnected component is split in +step 2 we now introduce further structure to ensure that the embedding is maintained on both +sides. The occupied edges generated around the split-off vertex v (and those around its copy +v′) are subdivided and connected cyclically according to Rotation(v). Instead of “stars”, we +thus now generate occupied “wheels” that encode the edge ordering in the embedding of the +triconnected component. When generating the SPQR-tree of the modified subgraph in step 5, +now containing occupied wheels instead of only stars, we also generate a planar embedding for +all its triconnected skeletons. If no planar embedding can be found for at least one skeleton, +we report that the resulting instance is non-planar by setting IsPlanar to false. Otherwise, +after performing all splits indicated by the SPQR-tree, we assign Rotation by generating +embeddings for all new rigids. Note that for all skeletons with virtual vertices, the generated +embedding will be compatible with the one of the neighboring triconnected component, that +is, the rotation of each virtual vertex will line up with that of its matched partner vertex, +thanks to the inserted wheel. Finally, before applying Integrate in step 6, we contract each + +14 +Maintaining Triconnected Components under Node Expansion +occupied wheel into a single vertex to re-obtain occupied stars. The creation and contraction +of wheels adds an overhead that is at most linear in the degree of the expanded vertex and +the generation of embeddings for the rigids can be done in time linear in the size of the rigid. +Thus, this does not affect the asymptotic runtime of both operations. +◀ +▶ Corollary 11. The data structure from Theorem 10 can be adapted to also provide the exact +rotations with matching direction for every vertex in a rigid. Furthermore, it can support +queries whether two vertices v1, v2 are connected by at least 3 different vertex-disjoint paths +via 3Paths(v1, v2) in O((deg(v1)+deg(v2))·α(n)) time. These adaptions change the runtime +of InsertGraphSPQR to O(deg(u) · α(n) + |Gν|), that of MergeSPQR to O(deg(u1) · α(n)), and +that of Rotation(u) to O(α(n)). +Proof. The exact rotation information for Rotation can be maintained by using union-find +to keep track of the rigid a vertex belongs to and synchronizing the reversal of all vertices +within one rigid when two rigids are merged by Integrate as follows. We create a union-find +set for every vertex in a triconnected component and apply Union to all vertices in the same +rigid. Next to the pointer indicating the representative in the union-find structure, we store +a boolean flag indicating whether the rotation information for the current vertex is reversed +with regard to rotation of its direct representative. To find whether a Rotation needs to +be flipped, we accumulate all flags along the path to the actual representative of a vertex +by using an exclusive-or. As Rotation(u) thus relies on the Find operation, its amortized +runtime is O(α(n)). When merging two rigids with Integrate, we also perform a Union on +their respective representatives (which we need to Find first), making Integrate(S, (vα, vβ)) +run in O(deg(vα) + α(n)). We also compare the Rotation of the replaced vertices and flip +the flag stored with the vertex that does not end up as the representative if they do not +match. In total, this makes InsertGraphSPQR run in O(deg(u) · α(n) + |Gν|) time as there +can be up to deg(u) split rigids. Furthermore, MergeSPQR now runs in O(deg(u1) · α(n)) time. +Maintaining the information in which rigid a skeleton vertex is contained in can then +also be used to answer queries whether two arbitrary vertices are connected by three disjoint +paths. This is exactly the case if they are part of the same rigid, appear as poles of the same +bond or are connected by a virtual edge in a polygon. This can be checked by enumerating +all allocation skeletons of both vertices, which can be done in time linear in their degree. +As finding each of the skeletons may require a Find call, the total runtime for this is in +O((deg(v1) + deg(v2)) · α(n)). +◀ +6 +Application to Synchronized Planarity +In this section, we will give some background on the historical development of and further +details on the problems Clustered Planarity and Synchronized Planarity together +with summary of the algorithm of Bläsius et al. for solving both problems. Furthermore, +we will show how our and also previous work on dynamic SPQR-trees can be used in the +context of both problems. +6.1 +Background and Discussion +Lengauer [34] first discussed Clustered Planarity under a different name in 1989, which is +why it was later independently rediscovered by Feng et al. [23] in 1995. Both gave polynomial- +time algorithms for the case where the subgraph induced by any cluster is connected. In +contrast, the question whether the general problem with disconnected clusters allows an + +S. D. Fink and I. Rutter +15 +Figure 5 Schematic representation of the three operations used by Bläsius et al. [8] for solving +Synchronized Planarity. Matched vertices are shown as bigger disks, the matching is indicated +by the orange dotted lines. Top: Two cut-vertices matched with each other (left), the result of +encapsulating their incident blocks (middle) and the bipartite graph resulting from joining both +cut-vertices (right). Middle: A matched non-cut-vertex with a non-trivial embedding tree (left) +that is propagated to replace both the vertex and its partner (right). Bottom: Three different +cases of matched vertices with trivial embedding trees (blue) and how their pipes can be removed or +replaced (red). +efficient solution remained open for 30 years. In that time, polynomial-time algorithms were +found for many special-cases [2, 15, 25, 29] before Fulek and Tóth [26] found an O((n + d)8) +solution in 2019. +Shortly thereafter, Bläsius et al. [8] gave a solution with runtime in +O((n + d)2) that also exposes the main concepts needed to solve Clustered Planarity. +The solution works via a linear-time reduction to the problem Synchronized Planarity, +for which Bläsius et al. gave a quadratic algorithm. We improve the runtime of the latter +algorithm. As Synchronized Planarity can be used as a modeling tool for several other +constrained planarity problems next to Clustered Planarity [8], this also improves the +time needed for solving any constrained planarity problem that can be solved via a linear-time +reduction to Synchronized Planarity; see Table 1. +In Clustered Planarity, the embedding has to respect a laminar family of clusters [9, +34], that is every vertex is part of some (hierarchically nested) cluster and an edge may +only cross a cluster boundary if it connects a vertex from the inside with one from the +outside. In Synchronized Planarity, we are given a matching on some of the vertices in +the graph and seek an embedding such that the rotations matched vertices line up under a +given bijection [8]. The synchronization constraints imposed by matching two vertices are +also called pipe. The reduction from the former problem to the latter employs the CD-tree +representation of Clustered Planarity [9], where each cluster is represented as individual +skeleton in which adjacent clusters were collapsed into single “virtual vertices”. The order +of the edges “leaving” one cluster via a virtual vertex now needs to line up with the order +in which they “enter” an adjacent cluster via its corresponding virtual vertex (see also [8, +Figure 6]). + +16 +Maintaining Triconnected Components under Node Expansion +The algorithm for solving Synchronized Planarity works by removing an arbitrary +pipe each step, using one of three operations depending on the graphs around the matched +vertices, see Figure 5. +EncapsulateAndJoin If both vertices of the pipe are cut-vertices, they are “encapsulated” +by taking a copy of their respective components and then collapsing each incident block +to a single vertex to obtain stars with matched centers that have multiple parallel edges +connecting them to their ray vertices. The original cut-vertices are split up so that each +incident block gets its own copy and these copies are synchronized with the respective +vertex representing a collapsed block. Now the cut-vertices can be removed by “joining” +both stars, that is identifying their incident edges according to the bijection that is given +alongside the matching. +PropagatePQ If one of the vertices is not a cut-vertex and has an embedding tree that +not only consists of a single P-node, two copies of this embedding tree are inserted +(“propagated”) in place of both matched vertices, respectively. The inner nodes of the +embedding trees are synchronized by matching corresponding vertices. +SimplifyMatching In the remaining case, one of the vertices is not a cut-vertex but has a +trivial embedding tree, i.e., only appears in a single parallel skeleton and no rigid skeleton +in the SPQR-tree. If the vertex (or, more precisely, the parallel that completely defines +it rotation) can respect arbitrary rotations, we can simply remove the pipe. The only +exception to this is when the other pole of the parallel is also matched, in which case we +can “short-circuit” the matching across the parallel. +To summarize, every operation removes a pipe from the matching, while potentially +introducing new pipes with vertices that have a smaller degree. Using a potential function, +it can be shown that the progress made by the removal always dominates overhead of the +newly-introduced pipes, and that the operations needed to remove all pipes is limited by the +total degree of all matched vertices. Furthermore, the resulting instance without pipes can +be solved in linear time. All of the three operations run in time linear in the degree of the +un-matched vertices if the embedding trees they depend on are available. The contribution of +this paper is to efficiently provide the embedding trees, which would require processing entire +connected components at each step when done naïvely. Using the fully-dynamic SPQR-tree +by Holm and Rotenberg [31, 32], this can be achieved with a poly-log cost of O(∆ · log3 n) +leading to an overall runtime of O(m · ∆ · log3 n). Using the node expansion from this paper, +we can improve the runtime from spending time linear in the size of the input instance (O(m)) +for each of the linearly many operations, to only spending time linear in the maximum degree +(O(∆)) on each operation. The reduction from Clustered Planarity creates an instance +of size O(n+d) in which the total degree of matched vertices is in O(d), corresponding to the +total number of times an edge crosses a cluster boundary. Note that, while this means that +O(d) operations are sufficient to reach a reduced instance, the number of crossings between +edges and cluster boundaries can be quadratic in the number of vertices in a planar graph. +We also note that while the improvement over using the Holm and Rotenberg approach is +only poly-logarithmic, our datastructure has the additional benefit of being conceptually +simpler and thus also more likely to improve performance in practice. +6.2 +Using Node Expansion for Solving Synchronized Planarity +We show how extended skeleton decompositions and their dynamic operation InsertGraphSPQR +can be used to improve the runtime of the algorithm for solving Synchronized Planarity +by Bläsius et al. [8] from O(m2) to O(m · ∆), where ∆ is the maximum pipe degree. As + +S. D. Fink and I. Rutter +17 +already explained in the previous section, the algorithm spends a major part of its runtime on +computing so-called embedding trees, which describe all possible rotations of a single vertex +in a planar graph and are used to communicate embedding restrictions between vertices with +synchronized rotation. Once the embedding trees are available, the at most O(m) executed +operations run in time linear in the degree of the pipe/vertex they are applied on, that is +in O(∆) [8]. Thus, being able to generate these embedding trees efficiently by maintaining +the SPQR-trees they are derived from is our main contribution towards the speedup of the +Synchronized Planarity algorithm. +An embedding tree Tv for a vertex v of a biconnected graph G describes the possible +cyclic orderings or rotations of the edges incident to v in all planar embeddings of G [12]. +The leaves of Tv are the edges incident to v, while its inner nodes are partitioned into two +categories: Q-nodes define an up-to-reversal fixed rotation of their incident tree edges, while +P-nodes allow arbitrary rotation; see Figure 1d. To generate the embedding tree we use +the observation about the relationship of SPQR-trees and embedding trees described by +Bläsius and Rutter [10, Section 2.5]: there is a bijection between the P- and Q-nodes in the +embedding tree of v and the bond and triconnected allocation skeletons of v in the SPQR-tree +of G, respectively. +▶ Lemma 12. Let S be an SPQR-tree with a planar represented graph GS. The embedding +tree for a vertex v ∈ GS can be found in time O(deg(v)). +Proof. We use the rotation information from Theorem 10 and furthermore maintain an +(arbitrary) allocation vertex for each vertex in GS. To compute the embedding tree of a +vertex v starting at the allocation vertex u of v, we will explore the SPQR-tree by using +twinE on one of the edges incident to u and then finding the next allocation vertex of v +as one endpoint of the obtained edge. If u has degree 2, it is part of a polygon skeleton +that does not induce a node in the embedding tree. We thus move on to its neighboring +allocation skeletons and will also similarly skip over any other polygon skeleton we encounter. +If u has degree 3 or greater, we inspect two arbitrary incident edges: if they lead to the +same vertex, u is the pole of a bond, and we generate a P-node. Otherwise it is part of a +triconnected component, and we generate a Q-node. We now iterate over the edges incident +to u, in the case of a triconnected component using the order given by the rotation of u. For +each real edge, we attach a corresponding leaf to the newly generated node. The graph edge +corresponding to the leaf can be obtained from origE. For each virtual edge, we recurse on +the respective neighboring skeleton and attach the recursively generated node to the current +node. As u can only be part of deg(u) many skeletons, which form a subtree of TS, and the +allocation vertices of u in total only have O(deg(u)) many virtual and real edges incident, +this procedure yields the embedding tree of u in time linear in its degree. +◀ +Our data structure can now be used to reduce the runtime of solving Synchronized +Planarity by generating an SPQR-tree upfront, maintaining it throughout all applied +operations, and deriving any needed embedding tree from the SPQR-tree. +▶ Theorem 13. Synchronized Planarity can be solved in time in O(m · ∆), where m is +the number of edges and ∆ is the maximum degree of a pipe. +Proof. The algorithm works by splitting the pipes representing synchronization constraints +until they are small enough to be trivial. It does so by exhaustively applying the three +operations EncapsulateAndJoin, PropagatePQ and SimplifyMatching depending on the +graph structure around the pairs of synchronized vertices. As mentioned by Bläsius et al., +all operations run in time linear in the degree of the pipe they are applied on if the used + +18 +Maintaining Triconnected Components under Node Expansion +embedding trees are known, and O(m) operations are sufficient to solve a given instance [8]. +Our modification is that we maintain an SPQR-tree for each biconnected component and +then generate the needed embedding trees on-demand in linear time using Lemma 12. See +Section 6.1 for more background on the Synchronized Planarity operations modified in +the following. +Operation SimplifyMatching can be applied if the graph around a synchronized vertex +v allows arbitrary rotations of v, that is the embedding tree of v is trivial. In this case, the +pipe can be removed without modifying the graph structure. Thus, we can now easily check +the preconditions of this operations without making any changes to the SPQR-tree. +PropagatePQ takes the non-trivial embedding tree of one synchronized vertex v and inserts +copies of the tree in place of v and its partner, respectively. Synchronization constraints on +the inner vertices of the inserted trees are used to ensure that they are embedded in the +same way. We use InsertGraphSPQR to also insert the embedding tree into the respective +SPQR trees, representing Q-nodes using wheels. When propagating into a cutvertex we also +need to check whether two or more incident blocks merge. We form equivalence classes on +the incident blocks, where two blocks are in the same class if 1) the two subtrees induced by +their respective edges share at least two nodes 2) both induced subtrees share a C-node that +has degree at least 2 in both subtrees. Blocks in the same equivalence class will end up in the +same biconnected component as follows: We construct the subtree induced by all edges in +the equivalence class and add a single further node for each block in the class, connecting all +leaves to the node of the block the edges they represent lead to. We calculate the SPQR-tree +for this biconnected graph and then merge the SPQR-trees of the individual blocks into it by +applying Corollary 9. As InsertGraphSPQR (and similarly all MergeSPQR applications) runs in +time linear in the size of the inserted PQ-tree, which is limited by the degree of the vertex it +represents, this does not negatively impact the running time of the operation. +Operation EncapsulateAndJoin generates a new bipartite component representing how +the edges of the blocks incident to two synchronized cutvertices are matched with each other. +The size of this component is linear in the degree of the synchronized vertices. Thus, we can +freshly compute the SPQR-tree for the generated component in linear time, which also does +not negatively impact the running time. +Furthermore, as we now no longer need to iterate over whole connected components to +generate the embedding trees, we are also no longer required to ensure those components do +not grow to big. We can thus also directly contract pipes between two distinct biconnected +components using Corollary 9 instead of having to insert PQ-trees using PropagatePQ. This +may improve the practical runtime, as PropagatePQ might require further operations to +clean up the generated pipes, while the direct contraction entirely removes a pipe without +generating new ones. +◀ +▶ Corollary 14. Clustered Planarity can be solved in time in O(n + d · ∆), where d +is the total number of crossings between cluster borders and edges and ∆ is the maximum +number of edge crossings on a single cluster border. +Proof. Note that for a graph not containing parallel edges to be planar, the number of +edges has to be linear in the number of vertices. We apply the reduction from Clustered +Planarity to Synchronized Planarity as described by Bläsius et al. [8]. Ignoring the +parallel edges generated by the CD-tree, we can generate an SPQR-tree for every component +of the resulting instance in O(n) time in total. 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IEEE, 2013. +doi:10.1109/ICCAD.2013.6691115. + diff --git a/4dE2T4oBgHgl3EQfjwe5/content/tmp_files/load_file.txt b/4dE2T4oBgHgl3EQfjwe5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b888a63d0cbdcbb8f96cfcb0b2e6844b658bda0d --- /dev/null +++ b/4dE2T4oBgHgl3EQfjwe5/content/tmp_files/load_file.txt @@ -0,0 +1,1073 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf,len=1072 +page_content='Maintaining Triconnected Components under Node Expansion Simon D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink � � Faculty of Informatics and Mathematics, University of Passau, Germany Ignaz Rutter � � Faculty of Informatics and Mathematics, University of Passau, Germany Abstract SPQR-trees are a central component of graph drawing and are also important in many further areas of computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' From their inception onwards, they have always had a strong relation to dynamic algorithms maintaining information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', on planarity and triconnectivity, under edge insertion and, later on, also deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In this paper, we focus on a special kind of dynamic update, the expansion of vertices into arbitrary biconnected graphs, while maintaining the SPQR-tree and further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This will also allow us to efficiently merge two SPQR-trees by identifying the edges incident to two vertices with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We do this working along an axiomatic definition lifting the SPQR-tree to a stand-alone data structure that can be modified independently from the graph it might have been derived from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Making changes to this structure, we can now observe how the graph represented by the SPQR-tree changes, instead of having to reason which updates to the SPQR-tree are necessary after a change to the represented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Using efficient expansions and merges allows us to improve the runtime of the Synchronized Planarity algorithm by Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [8] from O(m2) to O(m · ∆), where ∆ is the maximum pipe degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This also reduces the time for solving several constrained planarity problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' for Clustered Planarity from O((n + d)2) to O(n + d · ∆), where d is the total number of crossings between cluster borders and edges and ∆ is the maximum number of edge crossings on a single cluster border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 2012 ACM Subject Classification Mathematics of computing → Graph algorithms Keywords and phrases SPQR-Tree, Dynamic Algorithm, Cluster Planarity Funding Funded by DFG-grant RU-1903/3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='03972v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='DS] 10 Jan 2023 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 1 1 Introduction The SPQR-tree is a data structure that represents the decomposition of a graph at its separation pairs, that is the pairs of vertices whose removal disconnects the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The components obtained by this decomposition are called skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' SPQR-trees form a central component of many graph visualization techniques and are used for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', planarity testing and variations thereof [13, 19, 29, 31, 39] and for computing embeddings and layouts [3, 7, 11, 20, 28, 42];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' see [37] for a survey of graph drawing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Outside of graph visualization they are used in the context of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', minimum spanning trees [6, 17], triangulations [5], and crossing optimization [28, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' They also have multiple applications outside of graph theory and even computer science, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' for creating integrated circuits [14, 44], business processes modelling [40], electrical engineering [24], theoretical physics [41] and genomics [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Initially, SPQR-trees were devised by Di Battista and Tamassia for incremental planarity testing [16, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As such, even in their initial form, SPQR-trees already allowed dynamic updates in the form of edge addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Their use was quickly expanded to other on-line problems [18, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In addition to the applications mentioned above, this also sparked a series of further papers improving the runtime of the incremental data structure [38, 39, 43] and also extending it to be fully-dynamic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', allowing insertion and deletion of vertices and edges, in O(√n) time [21, 27], where n is the number of vertices in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Recently, Holm and Rotenberg described a fully-dynamic algorithm for maintaining planarity and triconnectivity information in O(log3 n) time per operation [31, 32] (see also there for a short history on dynamic SPQR-tree algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In this paper, we consider an incremental setting where we allow a single operation that expands a vertex v into an arbitrary biconnected graph Gν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Using the approach of Holm and Rotenberg [31], this takes O((deg(v) + |Gν|) · log3 n) time by first removing v and its incident edges and then incrementally inserting Gν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We improve this to O(deg(v) + |Gν|) using an algorithm that is much simpler and thus also more likely to improve performance in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In addition, our approach also allows to efficiently merge two SPQR-trees as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Given two biconnected graphs G1, G2 containing vertices v1, v2, respectively, together with a bijection between their incident edges, we construct a new graph G by replacing v1 with G2 − v2 in G1, identifying edges using the given bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Given the SPQR-trees of G1 and G2, we show that the SPQR-tree of G can be found in O(deg(v1)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' More specifically, we present a data structure that supports the following operations: InsertGraphSPQR expands a single vertex in time linear in the size of the expanded subgraph, MergeSPQR merges two SPQR-trees in time linear in the degree of the replaced vertices, IsPlanar indicates whether the currently represented graph is planar in constant time, and Rotation yields one of the two possible planar rotations of a vertex in a triconnected skeleton in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, our data structure can be adapted to yield consistent planar embeddings for all triconnected skeletons and to test for the existence of three distinct paths between two arbitrary vertices with an additional factor of α(n) for all operations, where α is the inverse Ackermann function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The main idea of our approach is that the subtree of the SPQR-tree affected by expanding a vertex v has size linear in the degree of v, but may contain arbitrarily large skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In a “non-normalized” version of an SPQR-tree, the affected cycle (‘S’) skeletons can easily be split to have a constant size, while we develop a custom splitting operation to limit the size of triconnected ‘R’ skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This limits the size of the affected structure to be linear in the degree of v and allows us to perform the expansion efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In addition to the description of this data structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' the technical contribution of this ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Maintaining Triconnected Components under Node Expansion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Running Times ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='before [8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='using [8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='with this paper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Atomic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Embeddability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Synchronized Planarity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(m8) [26] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(m2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(m · ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='ClusterPlanarity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O((n + d)8) [26] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O((n + d)2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(n + d · ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Connected SEFE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(n16) [26] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(n2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(n · ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='bicon: O(n2) [10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Partially ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='PQ-Constrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Planarity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='bicon: O(m) [10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(m2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(m · ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Row-Column ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Independent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='NodeTrix Planarity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='bicon: O(n2) [35] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(n2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(n · ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='Strip Planarity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='O(n8) [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 26] O(n2) O(n · ∆) fixed emb: poly [4] Table 1 The best known running times for various constrained planarity problems before Syn- chronized Planarity [8] was published;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' using it as described in [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' and using it together with the speed-up from this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Running times prefixed with “bicon” only apply for certain problem instances which expose some form of biconnectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The variables n and m refer to the number of vertices and edges of the problem instance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The variable d refers to the number of edge-cluster boundary crossings in Clustered Planarity instances, while ∆ refers to the maximum pipe degree in the corresponding Synchronized Planarity instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This is bounded by the maximum number of edges crossing a single cluster border or the maximum vertex degree in the input instance, depending on the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' paper is twofold: First, we develop an axiomatic definition of the decomposition at separation pairs, putting the SPQR-tree as “mechanical” data structure into focus instead of relying on and working along a given graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As a result, we can deduce the represented graph from the data structure instead of computing the data structure from the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This allows us to make more or less arbitrary changes to the data structure (respecting its consistency criteria) and observe how the graph changes, instead of having to reason which changes to the graph require which updates to the data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Second, we explain how our data structure can be used to improve the runtime of the algorithm by Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [8] for solving Synchronized Planarity from O(m2) to O(m · ∆), where ∆ is the maximum pipe degree (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' the maximum degree of a vertex with synchronization constraints that enforce its rotation to be the same as that of another vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Synchronized Planarity can be used to model and solve a vast class of different kinds of constrained planarity, see Table 1 for an overview of problems benefiting from this speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Among them is the notorious Clustered Planarity, whose complexity was open for 30 years before Fulek and Tóth gave an algorithm with runtime O((n + d)8) in 2019 [26], where d is the total number of crossings between cluster borders and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Shortly thereafter, Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [8] gave a solution in O((n + d)2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We improve this to O(n + d · ∆), where ∆ is the maximum number of edge crossings on a single cluster border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This work is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Section 2 contains an overview of the definitions used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In Section 3, we describe the skeleton decomposition and show how it relates to the SPQR-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Section 4 extends this data structure by the capability of splitting S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 3 triconnected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In Section 5, we exploit this feature to ensure the affected part of the SPQR-tree is small when we replace a vertex with a new graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Section 6 contains more details on the background of Synchronized and Clustered Planarity and shows how our results can be used to reduce the time required for solving them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 2 Preliminaries In the context of this work, G = (V, E) is a (usually biconnected and loop-free) multi-graph with n vertices V and m (possibly parallel) edges E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For a vertex v, we denote its open neighborhood (excluding v itself) by N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For a bijection or matching ϕ we call ϕ(x) the partner of an element x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We use A ·∪ B to denote the union of two disjoint sets A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' A separating k-set is a set of k vertices whose removal increases the number of connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Separating 1-sets are called cutvertices, while separating 2-sets are called separation pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' A connected graph is biconnected if it does not have a cutvertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' A biconnected graph is triconnected if it does not have a separation pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Maximal biconnected subgraphs are called blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Each separation pair divides the graph into bridges, the maximal subgraphs which cannot be disconnected by removing or splitting the vertices of the separation pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' A bond is a graph that consists solely of two pole vertices connected by multiple parallel edges, a polygon is a simple cycle, while a rigid is any simple triconnected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' A wheel is a cycle with an additional central vertex connected to all other vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, the expansion that is central to this work is formally defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Let Gα, Gβ be two graphs where Gα contains a vertex u and Gβ contains |N(u)| marked vertices, together with a bijection ϕ between the neighbors of u and the marked vertices in Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' With Gα[u →ϕ Gβ] we denote the graph that is obtained from the disjoint union of Gα, Gβ by identifying each neighbor x of u with its respective marked vertex ϕ(x) in Gβ and removing u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' the graph Gα where the vertex u was expanded into Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 3 Skeleton Decompositions A skeleton structure S = (G, origV, origE, twinE) that represents a graph GS = (V, E) consists of a set G of disjoint skeleton graphs together with three total, surjective mappings twinE, origE, and origV that satisfy the following conditions: Each skeleton Gµ = (Vµ, Ereal µ ∪ Evirt µ ) in G is a multi-graph where each edge is either in Ereal µ and thus called real or in Evirt µ and thus called virtual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Bijection twinE : Evirt → Evirt matches all virtual edges Evirt = � µ Evirt µ such that twinE(e) ̸= e and twinE2 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Surjection origV : � µ Vµ → V maps all skeleton vertices to graph vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Bijection origE : � µ Ereal µ → E maps all real edges to the graph edge set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that each vertex and each edge of each skeleton is in the domain of exactly one of the three mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As the mappings are surjective, V and E are exactly the images of origV and origE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For each vertex v ∈ GS, the skeletons that contain an allocation vertex v′ with origV(v′) = v are called the allocation skeletons of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, let TS be the graph where each node µ corresponds to a skeleton Gµ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Two nodes of TS are adjacent if their skeletons contain a pair of virtual edges matched with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We call a skeleton structure a skeleton decomposition if it satisfies the following conditions: 1 (bicon) Each skeleton is biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 2 (tree) Graph TS is simple, loop-free, connected and acyclic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 3 (orig-inj) For each skeleton Gµ, the restriction origV |Vµ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 4 Maintaining Triconnected Components under Node Expansion u (a) (b) (d) (c) Figure 1 Different views on the skeleton decomposition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (a) The graph GS with a vertex u marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (b) The skeletons of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Virtual edges are drawn in gray with their matching twinE being shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The allocation vertices of u are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (c) The tree TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The allocation skeletons of u are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (d) The embedding tree of vertex u as described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' P-nodes are shown as white disks, Q-nodes are shown as large rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The leaves of the embedding tree correspond to the edges incident to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 4 (orig-real) For each real edge uv, the endpoints of origE(uv) are origV(u) and origV(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 5 (orig-virt) Let uv and u′v′ be two virtual edges with uv = twinE(u′v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For their respective skeletons Gµ and G′ µ (where µ and µ′ are adjacent in TS), it is origV(Vµ) ∩ origV(Vµ′) = origV({u, v}) = origV({u′, v′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 6 (subgraph) The allocation skeletons of any vertex of GS form a connected subgraph of TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Figure 1 shows an example of S, GS, and TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We call a skeleton decomposition with only one skeleton Gµ trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that in this case, Gµ is isomorphic to GS, and origE and origV are actually bijections between the edges and vertices of both graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To model the decomposition into triconnected components, we define the operations SplitSeparationPair and its converse, JoinSeparationPair, on a skeleton decomposition S = (G, origV, origE, twinE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For SplitSeparationPair, let u, v be a separation pair of skeleton Gµ and let (A, B) be a non-trivial bipartition of the bridges between u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='1 Applying SplitSeparationPair(S, (u, v), (A, B)) yields a skeleton decomposition S′ = (G′, origV′, origE′, twinE′) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In G′, we replace Gµ by two skeletons Gα, Gβ, where Gα is obtained from Gµ[A] by adding a new virtual edge eα between u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The same respectively applies to Gβ with Gµ[B] and eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We set twinE′(eα) = eβ and twinE′(eβ) = eα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that origV maps the endpoints of eα and eβ to the same vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' All other skeletons and the mappings defined on them remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For JoinSeparationPair, consider virtual edges eα, eβ with twinE(eα) = eβ and let Gβ ̸= Gα be their respective skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Applying JoinSeparationPair(S, eα) yields a skeleton decomposition S′ = (G′, origV′, origE′, twinE′) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In G′, we merge Gα with Gβ to form a new skeleton Gµ by identifying the endpoints of eα and eβ that map to the same vertex of GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Additionally, we remove eα and eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' All other skeletons and the mappings defined on them remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The main feature of both operations is that they leave the graph represented by the skeleton decomposition unaffected while splitting a node or contracting and edge in TS, which can be verified by checking the individual conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ▶ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Applying SplitSeparationPair or JoinSeparationPair on a skeleton de- composition S = (G, origV, origE, twinE) yields a skeleton decomposition S′ = (G′, origV′, 1 Note that a bridge might consist out of a single edge between u and v and that each bridge includes the vertices u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 5 origE′, twinE′) with an unchanged represented graph GS′ = GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We first check that all conditions still hold in the skeleton decomposition S′ returned by SplitSeparationPair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As (A, B) is a non-trivial bipartition, each set contains at least one bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Together with eα (and eβ), this bridge ensures that Gα (and Gβ) remain biconnected, satisfying condition 1 (bicon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The operation splits a node µ of TS into two adjacent nodes α, β, whose neighbors are defined exactly by the virtual edges in A, B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, condition 2 (tree) remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The mappings origV′, origE′ and twinE′ obviously still satisfy conditions 3 (orig-inj) and 4 (orig-real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We duplicated exactly two nodes, u and v of adjacent skeletons Gα and Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Because 3 (orig-inj) holds for Gµ, Gα and Gβ share no other vertices that map to the same vertex of GS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, condition 5 (orig-virt) remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Condition 6 (subgraph) could only be violated if the subgraph of TS′ formed by the allocation skeletons of some vertex z ∈ GS′ was no longer connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This could only happen if only one of Gα and Gβ were an allocation skeleton of z, while the other has a further neighbor that is also an allocation skeleton of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Assume without loss of generality that Gα and the neighbor Gν of Gβ, but not Gβ itself, were allocation skeletons of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Because Gν and Gβ are adjacent in TS′ there are virtual edges xy = twinE′(x′y′) with xy ∈ Gβ and x′y′ ∈ Gν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The same virtual edges are also present in the input instance, only with the difference that xy ∈ Gµ and µ (instead of β) and ν are adjacent in TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As the input instance satisfies condition 5 (orig-virt), it is z ∈ origV(Vν) ∩ origV(Vµ) = origV({x, y}) = origV({x′, y′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As origV({x, y}) = origV′({x, y}), this is a contradiction to Gβ not being an allocation skeleton of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, the mapping origE remains unchanged and the only change to origV is to include two new vertices mapping to already existing vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Due to condition 4 (orig-real) holding for both the input and the output instance, this cannot affect the represented graph GS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Now consider the skeleton decomposition S′ returned by JoinSeparationPair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Identify- ing distinct vertices of distinct connected components does not affect their biconnectivity, thus condition 1 (bicon) remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The operation effectively contracts and removes an edge in TS, which does not affect TS′ being a tree satisfying condition 2 (tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that condition 2 (tree) holding for the input instance also ensures that Gα and Gβ are two distinct skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As the input instance also satisfies condition 5 (orig-virt), there are exactly two vertices in each of the two adjacent skeletons Gα and Gβ, where origV maps to the same vertex of GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' These two vertices must be part of the twinE pair making the two skeletons adjacent, thus they are exactly the two pairs of vertices we identify with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, origV |Vµ is still injective, satisfying condition 3 (orig-inj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As we modify no real edges and no other virtual edges, the mappings origV′ and origE′ obviously still satisfy condition 4 (orig-real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As the allocation skeletons of each graph vertex form a connected subgraph, joining two skeletons cannot change the intersection with any of their neighbors, leaving 5 (orig-virt) satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, contracting a tree edge cannot lead to any of the subgraphs of 6 (subgraph) becoming disconnected, thus the condition also remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Again, no changes were made to origE, while condition 5 (orig-virt) makes sure that origV mapped the two pairs of merged vertices to the same vertex of GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, the represented graph GS′ remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ This gives us a second way of finding the represented graph by exhaustively joining all skeletons until there is only one left, obtaining the unique trivial skeleton decomposition: ▶ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Exhaustively applying JoinSeparationPair to a skeleton decomposition S = (G, origV, origE, twinE) yields a trivial skeleton decomposition S′ = (G′, origV′, origE′, twinE′) where origE′ and origV′ define an isomorphism between G′ µ and GS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 6 Maintaining Triconnected Components under Node Expansion Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As all virtual edges are matched, and the matched virtual edge always belongs to a different skeleton (condition 2 (tree) ensures that TS is loop-free), we can always apply JoinSeparationPair on a virtual edge until there are none left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As TS is connected, this means that the we always obtain a tree with a single node, that is an instance with only a single skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As a single application of JoinSeparationPair preserves the represented graph, any chain of multiple applications also does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that origE′ is a bijection and the surjective origV′ is also injective on the single remaining skeleton due to condition 3 (orig-inj), thus it also globally is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Together with condition 4 (orig-real), this ensures that any two vertices u and v of G′ µ are adjacent if and only if origV′(u) and origV′(v) are adjacent in GS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus origV′ is an edge-preserving bijection, that is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ A key point about the skeleton decomposition and especially the operation SplitSepa- rationPair now is that they model the decomposition of a graph at separation pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This decomposition was formalized as SPQR-tree by Di Battista and Tamassia [16] and is unique for a given graph [33, 36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' see also [28, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Angelini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [1] describe a decomposition tree that is conceptually equivalent to our skeleton decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' They also present an alternative definition for the SPQR-tree as a decomposition tree satisfying further properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We adopt this definition for our skeleton decompositions as follows, not requiring planarity of triconnected components and allowing virtual edges and real edges to appear within one skeleton (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', having leaf Q-nodes merged into their parents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ▶ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' A skeleton decomposition S = (G, origV, origE, twinE) where any skeleton in G is either a polygon, a bond, or triconnected (“rigid”), and two skeletons adjacent in TS are never both polygons or both bonds, is the unique SPQR-tree of GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The main difference between the well-known ideas behind decomposition trees and our skeleton decomposition is that the latter allow an axiomatic access to the decomposition at separation pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For the skeleton decomposition, we employ a purely functional, “mechanical” data structure instead of relying on and working along a given graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In our case, the represented graph is deduced from the data structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' SPQR-tree) instead of computing the data structure from the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 4 Extended Skeleton Decompositions Note that most skeletons, especially polygons and bonds, can easily be decomposed into smaller parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The only exception to this are triconnected skeletons which cannot be split further using the operations we defined up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This is a problem when modifying a vertex that occurs in triconnected skeletons that may be much bigger than the direct neighborhood of the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To fix this, we define a further set of operations which allow us to isolate vertices out of arbitrary triconnected components by replacing them with a (“virtual”) placeholder vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This placeholder then points to a smaller component that contains the actual vertex, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Modification of the edges incident to the placeholder is disallowed, which is why we call them “occupied”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Formally, the structures needed to keep track of the components split in this way in an extended skeleton decomposition S = (G, origV, origE, twinE, twinV) are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Skeletons now have the form Gµ = (Vµ ·∪ V virt µ , Ereal µ ∪ Evirt µ ∪ Eocc µ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Bijection twinV : V virt → V virt matches all virtual vertices V virt = � µ V virt µ , such that twinV(v) ̸= v, twinV2 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The edges incident to virtual vertices are contained in Eocc µ and thus considered occupied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' see Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Similar to the virtual edges matched by twinE, any two virtual vertices matched by twinV induce an edge between their skeletons in TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Condition 2 (tree) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 7 v Gµ u (a) v vα vβ Gα Gβ uα uβ (b) Figure 2 (a) A triconnected skeleton Gµ with a highlighted vertex v incident to two gray virtual edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (b) The result of applying IsolateVertex to isolate v out of the skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The red occupied edges in the old skeleton Gα form a star with center vα, while the red occupied edges in Gβ connect all neighbors of v to form a star with center vβ ̸= v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The centers vα and vβ are virtual and matched with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Neighbor u of v was split into vertices uα and uβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' also equally applies to those edges induced by twinV, which in particular ensures that there are no parallel twinE and twinV tree edges in TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Similarly, the connected subgraphs of condition 6 (subgraph) can also contain tree edges induced by twinV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' All other conditions remain unchanged, but we add two further conditions to ensure that twinV is consistent: 7 (stars) For each vα, vβ with twinV(vα) = vβ, it is deg(vα) = deg(vβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' All edges incident to vα and vβ are occupied and have distinct endpoints (except for vα and vβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Conversely, each occupied edge is adjacent to exactly one virtual vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 8 (orig-stars) Let vα and vβ again be two virtual vertices matched with each other by twinV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For their respective skeletons Gα and Gβ (where α and β are adjacent in TS), it is origV(Vα) ∩ origV(Vβ) = origV(N(vα)) = origV(N(vβ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that both conditions together yield a bijection γvαvβ between the neighbors of vα and vβ, as origV is injective when restricted to a single skeleton (condition 3 (orig- inj)) and deg(vα) = deg(vβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Operations SplitSeparationPair and JoinSeparationPair can also be applied to an extended skeleton decomposition, yielding an extended skeleton decomposition without modifying twinV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To ensure that conditions 7 (stars) and 8 (orig-stars) remain unaffected by both operations, SplitSeparationPair cannot be applied if a vertex of the separation pair is virtual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The operations IsolateVertex and Integrate now allow us to isolate vertices out of triconnected components and integrate them back in, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For IsolateVertex, let v be a non-virtual vertex of skeleton Gµ, such that v has no incident occupied edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Applying IsolateVertex(S, v) on an extended skeleton decomposition S yields an extended skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Each neighbor u of v is split into two non-adjacent vertices uα and uβ, where uβ is incident to all edges connecting u with v, while uα keeps all other edges of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We set origV′(uα) = origV′(uβ) = origV(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This creates an independent, star-shaped component with center v, which we move to skeleton Gβ, while we rename skeleton Gµ to Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We connect all uα to a single new virtual vertex vα ∈ V virt α using occupied edges, and all uβ to a single new virtual vertex vβ ∈ V virt β using occupied edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, we set twinV′(vα) = vβ, twinV′(vβ) = vα, and add Gβ to G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' All other mappings and skeletons remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For Integrate, consider two virtual vertices vα, vβ with twinV(vα) = vβ and the bijec- tion γvαvβ between the neighbors of vα and vβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' An application of Integrate(S, (vα, vβ)) yields an extended skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We merge both skeletons into a skeleton Gµ (also replacing both in G′) by identifying the neighbors of vα and vβ according to γvαvβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, we remove vα and vβ together with their incident occupied edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' All other mappings and skeletons remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 8 Maintaining Triconnected Components under Node Expansion ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Applying IsolateVertex or Integrate on an extended skeleton decomposition S = (G, origV, origE, twinE, twinV) yields an extended skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) with GS′ = GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We first check that all conditions still hold in the extended skeleton decomposition S′ returned by IsolateVertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Condition 1 (bicon) remains satisfied, as the structure of Gα remains unchanged compared to Gµ and the skeleton Gβ is a bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As we are again splitting a node of TS, condition 2 (tree) also remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Due to the neighbors of vβ and vα mapping to the same vertices of GS′, conditions 3 (orig-inj), 4 (orig-real), and 5 (orig-virt) remain satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Conditions 7 (stars) and 8 (orig-stars) are satisfied by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Lastly, condition 6 (subgraph) could only be violated if the subgraph of TS′ formed by the allocation skeletons of some vertex z ∈ GS′ was no longer connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This could only happen if only one of Gα and Gβ were an allocation skeleton of z, while the other has a further neighbor Gν that is also an allocation skeleton of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that in any case, ν is adjacent to µ in TS and µ must be an allocation skeleton of z, thus it is z ∈ origV(Gν) ∩ origV(Gµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Depending on the adjacency of ν, it is either origV(Gν)∩origV(Gµ) = origV′(Gν)∩origV(Gα) or origV(Gν) ∩ origV(Gµ) = origV′(Gν) ∩ origV(Gβ), as ν is not modified by the operation and both S and S′ satisfy 5 (orig-virt) and 8 (orig-stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This immediately contradicts the skeleton of {α, β}, that is adjacent to ν, not being an allocation skeleton of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, the mapping origE remains unchanged and the only change to origV is to include some duplicated vertices mapping to already existing vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Due to condition 4 (orig-real) holding for both the input and the output instance, this cannot affect the represented graph GS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Now consider the extended skeleton decomposition S′ returned by Integrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The merged skeleton is biconnected, as we are effectively replacing a single vertex by a connected subgraph, satisfying 1 (bicon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The operation effectively contracts and removes an edge in TS, which does not affect TS′ being a tree, satisfying condition 2 (tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that condition 2 (tree) holding for the input instance also ensures that vα and vβ belong to two distinct skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As the input instance satisfies condition 5 (orig-virt), the vertices in each of the two adjacent skeletons where origV maps to the same vertex of GS are exactly the neighbors of the matched vα and vβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, origV |Vα is still injective, satisfying condition 3 (orig-inj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As we modify no real or virtual edges, the mappings origV′, origE′ and twinE′ obviously still satisfy conditions 4 (orig-real) and 5 (orig-virt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, contracting a tree edge cannot lead to any of the subgraphs of 6 (subgraph) becoming disconnected, thus the condition also remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Conditions 7 (stars) and 8 (orig-stars) also remain unaffected, as we simply remove an entry from twinV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Again, no changes were made to origE, while condition 8 (orig-stars) makes sure that origV mapped each pair of merged vertices to the same vertex of GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, the represented graph GS′ remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ Furthermore, as Integrate is the converse of IsolateVertex and has no preconditions, any changes made by IsolateVertex can be undone at any time to obtain a (non-extended) skeleton decomposition, and thus possibly the SPQR-tree of the represented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ▶ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Exhaustively applying Integrate to an extended skeleton decomposition S = (G, origV, origE, twinE, twinV) yields a extended skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) where twinV′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, S′ is equivalent to a (non-extended) skeleton decomposition S′ = (G′, origV′, origE′, twinE′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 9 v Gµ (a) Gν (b) (c) Figure 3 Expanding a skeleton vertex v into a graph Gν in the SPQR-tree of Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (a) The single allocation skeleton Gµ of u with the single allocation vertex v of u from Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The neighbors of v are marked in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (b) The inserted graph Gν with orange marked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that the graph is biconnected when all marked vertices are collapsed into a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (c) The result of applying InsertGraph(S, u, Gν, ϕ) followed by an application of Integrate on the generated virtual vertices v and v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 5 Node Expansion in Extended Skeleton Decompositions We now introduce our first dynamic operation that allows us to actually change the represented graph by expanding a single vertex u into an arbitrary connected graph Gν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This is done by identifying |N(u)| marked vertices in Gν with the neighbors of u via a bijection ϕ and then removing u and its incident edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We use the “occupied stars” from the previous section to model the identification of these vertices, allowing us to defer the actual insertion to an application of Integrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We need to ensure that the inserted graph makes the same “guarantees” to the surrounding graph in terms of connectivity as the vertex it replaces, that is all neighbors of u (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' all marked vertices in Gν) need to be pairwise connected via paths in Gν not using any other neighbor of u (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' any other marked vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Without this requirement, a single vertex could e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' also be split into two non-adjacent halves, which could easily break a triconnected component apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, we require Gν to be biconnected when all marked vertices are collapsed into a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that this also ensures that the old graph can be restored by contracting the vertices of the inserted graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For the sake of simplicity, we require vertex u from the represented graph to have a single allocation vertex v ∈ Gµ with origV−1(u) = {v} so that we only need to change a single allocation skeleton Gµ in the skeleton decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As we will make clear later on, this condition can be satisfied easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Formally, let u ∈ GS be a vertex that only has a single allocation vertex v ∈ Gµ (and thus only a single allocation skeleton Gµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Let Gν be an arbitrary, new graph containing |N(u)| marked vertices, together with a bijection ϕ between the marked vertices in Gν and the neighbors of v in Gµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We require Gν to be biconnected when all marked vertices are collapsed into a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Operation InsertGraph(S, u, Gν, ϕ) yields an extended skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) as follows, see also Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We interpret Gν as skeleton and add it to G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For each marked vertex x in Gν, we set origV′(x) = origV(ϕ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For all other vertices and edges in Gν, we set origV′ and origE′ to point to new vertices and edges forming a copy of Gν in GS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We connect every marked vertex in Gν to a new virtual vertex v′ ∈ Gν using occupied edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We also convert v to a virtual vertex, converting its incident edges to occupied edges while removing parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, we set twinV′(v) = v′ and twinV′(v′) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ▶ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Applying InsertGraph(S, u, Gν, ϕ) on an extended skeleton decomposition S = (G, origV, origE, twinE, twinV) yields an extended skeleton decomposition S′ = (G′, origV′, origE′, twinE′, twinV′) with GS′ isomorphic to GS[u →ϕ Gν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 10 Maintaining Triconnected Components under Node Expansion Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Condition 1 (bicon) remains satisfied, as the structure of Gµ remains unchanged and the resulting Gν is biconnected by precondition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Regarding TS, we are attaching a degree-1 node ν to an existing node µ, thus condition 2 (tree) also remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As all vertices of Gν except for the vertices in N(v′) got their new, unique copy assigned by origV′ and origV′(N(v′)) = origV(N(v)), condition 3 (orig-inj) is also satisfied for the new Gν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As we updated origE alongside origV and Gν contains no virtual edges, conditions 4 (orig-real) and 5 (orig-virt) remain satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As ν is a leaf of TS with µ being its only neighbor, origV′(N(v′)) ⊂ origV(Vµ), and Gν is the only allocation skeleton for all vertices in Gν \\ N(v′), condition 6 (subgraph) remains satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Conditions 7 (stars) and 8 (orig-stars) are satisfied by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, the mappings origE′ and origV′ are by construction updated to correctly reproduce the structure of Gν in GS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ On its own, this operation is not of much use though, as graph vertices only rarely have a single allocation skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, our goal is to dynamically maintain SPQR-trees, while this operation on its own will in most cases not yield an SPQR-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To fix this, we introduce the full procedure InsertGraphSPQR(S, u, Gν, ϕ) that can be applied to any graph vertex u and that, given an SPQR-tree S, yields the SPQR-tree of GS[u →ϕ Gν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' It consists of three preparations steps, the insertion of Gν, and two further clean-up steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We apply SplitSeparationPair to each polygon allocation skeleton of u with more than three vertices, using the neighbors of the allocation vertex of u as separation pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For each rigid allocation skeleton of u, we move the contained allocation vertex v of u to its own skeleton by applying IsolateVertex(S, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We exhaustively apply JoinSeparationPair to any pair of allocation skeletons of u that are adjacent in TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Due to condition 6 (subgraph), this yields a single component Gµ that is the sole allocation skeleton of u with the single allocation vertex v of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, the size of Gµ is linear in deg(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We apply InsertGraph to insert Gν as skeleton, followed by an application of Integrate to the virtual vertices {v, v′} introduced by the insertion, thus integrating Gν into Gµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We apply SplitSeparationPair to all separation pairs in Gµ that do not involve a virtual vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' These pairs can be found in linear time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' by temporarily duplicating all virtual vertices and their incident edges and then computing the SPQR-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, we exhaustively apply Integrate and also apply JoinSeparationPair to any two adjacent polygons and to any two adjacent bonds to obtain the SPQR-tree of the updated graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The basic idea behind the correctness of this procedure is that splitting the newly inserted component according to its SPQR-tree in step 5 yields biconnected components that are each either a polygon, a bond, or “almost” triconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The latter (and only those) might still contain virtual vertices and all their remaining separation pairs, which were not split in step 5, contain one of these virtual vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This, together with the fact that there still may be pairs of adjacent skeletons where both are polygons or both are bonds, prevents the instance from being an SPQR-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Both issues are resolved in step 6: The adjacent skeletons are obviously fixed by the JoinSeparationPair applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To show that the virtual vertices are removed by the Integrate applications, making the remaining components triconnected, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 2 Note that the wheels replacing virtual vertices in the proof of Theorem 10 also ensure this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 11 (a) v (b) Figure 4 The preprocessing steps of InsertGraphSPQR being applied to the SPQR-tree of Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (a) The state after step 2, after all allocation skeletons of u have been split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' (b) The state after step 3, after all allocation skeletons of u have been merged into a single one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ▶ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Let Gα be a triconnected skeleton containing a virtual vertex vα matched with a virtual vertex vβ of a biconnected skeleton Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, let P ⊆ �V (Gβ) 2 � be the set of all separation pairs in Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' An application of Integrate(S, (vα, vβ)) yields a biconnected skeleton Gµ with separation pairs P ′ = {{u, v} ∈ P | vβ /∈ {u, v}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We partition the vertices of Gµ into the sets A, B, and N depending on whether the vertex stems from Gα, Gβ, or both, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The set N thus contains the neighbors of vα, which were identified with the neighbors of vβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We will now show by contradiction that Gµ contains no separation pairs except for those in P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, consider a separation pair u, v ∈ Gµ not in P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' First, consider the case where u, v ∈ A∪N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Observe that removing u, v in this case leaves B connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, we can contract all vertices of B into a single vertex, reobtain Gα and see that u, v is a separation pair in Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This contradicts the precondition that Gα is triconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Now consider the case where u, v ∈ B ∪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Analogously to above, we find that u, v is a separation pair in Gβ that does not contain vβ, a contradiction to {u, v} /∈ P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, consider the remaining case where, without loss of generality, u ∈ A, v ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Since {u, v} is a separation pair, u has two neighbors x, y that lie in different connected components of Gµ−{u, v} and therefore also in different components of (Gµ−{u, v})−B which is isomorphic to Gα − {u, vα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This again contradicts the precondition that Gα is triconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ ▶ Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Applying InsertGraphSPQR(S, u, Gν, ϕ) to an SPQR-tree S yields an SPQR- tree S′ in O(|Gν|) time with GS′ isomorphic to GS[u →ϕ Gν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As all operations that are applied leave the extended skeleton decomposition valid, the final extended skeleton decomposition S′ is also valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Observe that the purpose of the preprocessing steps 1–3 is solely to ensure that the preconditions of InsertGraph are satisfied and the affected component is not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that all rigids split in step 2 remain structurally unmodified in the sense that edges only changed their type, but the graph and especially its triconnectedness remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Step 4 performs the actual insertion and yields the desired represented graph according to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' It thus remains to show that the clean-up steps turn the obtained extended skeleton decomposition into an SPQR-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Applying Integrate exhaustively in step 6 ensures that the extended skeleton decomposition is equivalent to a non-extended one (Remark 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Recall that a non-extended skeleton decomposition is an SPQR-tree if all skeletons are either polygons, bonds or triconnected and two adjacent skeletons are never both polygons or both bonds (Definition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Step 6 ensures that the second half holds, as joining two polygons (or two bonds) with JoinSeparationPair yields a bigger polygon (or bond, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Before 12 Maintaining Triconnected Components under Node Expansion step 6, all skeletons that are not an allocation skeleton of u are still unmodified and thus already have a suitable structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', they are either polygons, bonds or triconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, the allocation skeletons of u not containing virtual vertices also have a suitable structure, as their splits were made according to the SPQR-tree in step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' It remains to show that the remaining skeletons, that is those resulting from the Integrate applications in step 6, are triconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that in these skeletons, step 5 ensures that every separation pair consists of at least one virtual vertex, as otherwise the computed SPQR-tree would have split the skeleton further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Further note that, for each of these virtual vertices, the matched partner vertex is part of a structurally unmodified triconnected skeleton that was split in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Lemma 7 shows that applying Integrate does not introduce new separation pairs while removing two virtual vertices if one of the two sides is triconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We can thus exhaustively apply Integrate and thereby remove all virtual vertices and thus also all separation pairs, obtaining triconnected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This shows that the criteria for being an SPQR-tree are satisfied and, as InsertGraph expanded u to Gν in the represented graph, we now have the unique SPQR-tree of GS[u →ϕ Gν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that all operations we used can be performed in time linear in the degree of the vertices they are applied on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For the bipartition of bridges input to SplitSeparationPair, it is sufficient to describe each bridge via its edges incident to the separation pair instead of explicitly enumerating all in vertices in the bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, the applications of SplitSepara- tionPair and IsolateVertex in steps 1 and 2 touch every edge incident to u at most once and thus take O(deg(u)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, they yield skeletons that have a size linear in the degree of their respective allocation vertex of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As the subtree of u’s allocation skeletons has size at most deg(u), the JoinSeparationPair applications of step 3 also take at most O(deg(u)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' It also follows that the resulting single allocation skeleton of u has size O(deg(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The applications of InsertGraph and Integrate in step 4 can be done in time linear in the number of identified neighbors, which is O(deg(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Generating the SPQR-tree of the inserted graph in step 5 (where all virtual vertices where replaced by wheels) can be done in time linear in the size of the inserted graph [30, 33], that is O(|Gν|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Applying SplitSeparationPair according to all separation pairs identified by this SPQR-tree can also be done in O(|Gν|) time in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that there are at most deg(u) edges between the skeletons that existed before step 4 and those that were created or modified in steps 4 and 5, and these are the only edges that might now connect two polygons or two bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As these tree edges have one endpoint in the single allocation skeleton of u, the applications of Integrate and JoinSeparationPair in step 6 run in O(deg(u)) time in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, they remove all pairs of adjacent polygons and all pairs of adjacent bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This shows that all steps take O(deg(u)) time, except for step 5, which takes O(|Gν|) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As the inserted graph contains at least one vertex for each neighbor of u, the total runtime is in O(|Gν|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ ▶ Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Let S1, S2 be two SPQR-trees together with vertices u1 ∈ GS1, u2 ∈ GS2, and let ϕ be a bijection between the edges incident to u1 and the edges incident to u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Operation MergeSPQR(S1, S2, u1, u2, ϕ) yields the SPQR-tree of the graph GS1[u1 →ϕ GS2 − u2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' the union of both graphs where the edges incident to u1, u2 were identified according to ϕ and u1, u2 removed, in time O(deg(u1)) = O(deg(u2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Operation MergeSPQR works similar to the more general InsertGraphSPQR, although the running time is better because we already know the SPQR-tree for the graph being inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We apply the preprocessing steps 1–3 to ensure that both u1 and u2 have sole allocation vertices v1 and v2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To properly handle parallel edges, we subdivide all edges incident to u1, u2 (and thus also the corresponding real edges incident to v1, v2) and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 13 then identify the subdivision vertices of each pair of edges matched by ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' By deleting vertices v1 and v2 and suppressing the subdivision vertices (that is, removing them and identifying each pair of incident edges) we obtain a skeleton Gµ that has size O(deg(u1)) = O(deg(u2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, we apply the clean-up steps 5 and 6 to Gµ to obtain the final SPQR-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Again, as the partner vertex of every virtual vertex in the allocation skeletons of u is part of a triconnected skeleton, applying Integrate exhaustively in step 6 yields triconnected skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As previously discussed, the preprocessing and clean-up steps run in time linear in degree of the affected vertices, thus the overall runtime is O(deg(u1)) = O(deg(u2)) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='1 Maintaining Planarity and Vertex Rotations Note that expanding a vertex of a planar graph using another planar graph using Insert- GraphSPQR (or merging two SPQR-trees of planar graphs using Corollary 9) might actually yield a non-planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', because the rigids of both graphs might require incompatible orders for the neighbors of the replaced vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The aim of this section is to efficiently detect this case, that is a planar graph turning non-planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To check a general graph for planarity, it suffices to check the rigids in its SPQR-tree for planarity and each rigid allows exactly two planar embeddings, where one is the reverse of the other [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, if a graph becomes non-planar through an application of InsertGraphSPQR, this will be noticeable from the triconnected allocation skeletons of the replaced vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To be able to immediately report if the instance became non-planar, we need to maintain a rotation, that is a cyclic order of all incident edges, for each vertex in any triconnected skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that we do not track the direction of the orders, that is we only store the order up to reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As discussed later, the exact orders can also be maintained with a slight overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ▶ Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' SPQR-trees support the following operations: InsertGraphSPQR(S, u, Gν, ϕ): expansion of a single vertex u in time O(|Gν|), MergeSPQR(S1, S2, u1, u2, ϕ): merging of two SPQR-trees in time O(deg(u1)), IsPlanar: queries whether the represented graph is planar in time O(1), and Rotation(u): queries for one of the two possible rotations of vertices u in planar tricon- nected skeletons in time O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that the boolean flag IsPlanar together with the Rotation information can be computed in linear time when creating a new SPQR-tree and that expanding a vertex or merging two SPQR-trees cannot turn a non-planar graph planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We make the following changes to the operations InsertGraphSPQR and MergeSPQR described in Theorem 8 and Corollary 9 to maintain the new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' After a triconnected component is split in step 2 we now introduce further structure to ensure that the embedding is maintained on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The occupied edges generated around the split-off vertex v (and those around its copy v′) are subdivided and connected cyclically according to Rotation(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Instead of “stars”, we thus now generate occupied “wheels” that encode the edge ordering in the embedding of the triconnected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' When generating the SPQR-tree of the modified subgraph in step 5, now containing occupied wheels instead of only stars, we also generate a planar embedding for all its triconnected skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' If no planar embedding can be found for at least one skeleton, we report that the resulting instance is non-planar by setting IsPlanar to false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Otherwise, after performing all splits indicated by the SPQR-tree, we assign Rotation by generating embeddings for all new rigids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that for all skeletons with virtual vertices, the generated embedding will be compatible with the one of the neighboring triconnected component, that is, the rotation of each virtual vertex will line up with that of its matched partner vertex, thanks to the inserted wheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Finally, before applying Integrate in step 6, we contract each 14 Maintaining Triconnected Components under Node Expansion occupied wheel into a single vertex to re-obtain occupied stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The creation and contraction of wheels adds an overhead that is at most linear in the degree of the expanded vertex and the generation of embeddings for the rigids can be done in time linear in the size of the rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, this does not affect the asymptotic runtime of both operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ ▶ Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The data structure from Theorem 10 can be adapted to also provide the exact rotations with matching direction for every vertex in a rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, it can support queries whether two vertices v1, v2 are connected by at least 3 different vertex-disjoint paths via 3Paths(v1, v2) in O((deg(v1)+deg(v2))·α(n)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' These adaptions change the runtime of InsertGraphSPQR to O(deg(u) · α(n) + |Gν|), that of MergeSPQR to O(deg(u1) · α(n)), and that of Rotation(u) to O(α(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The exact rotation information for Rotation can be maintained by using union-find to keep track of the rigid a vertex belongs to and synchronizing the reversal of all vertices within one rigid when two rigids are merged by Integrate as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We create a union-find set for every vertex in a triconnected component and apply Union to all vertices in the same rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Next to the pointer indicating the representative in the union-find structure, we store a boolean flag indicating whether the rotation information for the current vertex is reversed with regard to rotation of its direct representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To find whether a Rotation needs to be flipped, we accumulate all flags along the path to the actual representative of a vertex by using an exclusive-or.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As Rotation(u) thus relies on the Find operation, its amortized runtime is O(α(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' When merging two rigids with Integrate, we also perform a Union on their respective representatives (which we need to Find first), making Integrate(S, (vα, vβ)) run in O(deg(vα) + α(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We also compare the Rotation of the replaced vertices and flip the flag stored with the vertex that does not end up as the representative if they do not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In total, this makes InsertGraphSPQR run in O(deg(u) · α(n) + |Gν|) time as there can be up to deg(u) split rigids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, MergeSPQR now runs in O(deg(u1) · α(n)) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Maintaining the information in which rigid a skeleton vertex is contained in can then also be used to answer queries whether two arbitrary vertices are connected by three disjoint paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This is exactly the case if they are part of the same rigid, appear as poles of the same bond or are connected by a virtual edge in a polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This can be checked by enumerating all allocation skeletons of both vertices, which can be done in time linear in their degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As finding each of the skeletons may require a Find call, the total runtime for this is in O((deg(v1) + deg(v2)) · α(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ 6 Application to Synchronized Planarity In this section, we will give some background on the historical development of and further details on the problems Clustered Planarity and Synchronized Planarity together with summary of the algorithm of Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' for solving both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, we will show how our and also previous work on dynamic SPQR-trees can be used in the context of both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='1 Background and Discussion Lengauer [34] first discussed Clustered Planarity under a different name in 1989, which is why it was later independently rediscovered by Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [23] in 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Both gave polynomial- time algorithms for the case where the subgraph induced by any cluster is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In contrast, the question whether the general problem with disconnected clusters allows an S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 15 Figure 5 Schematic representation of the three operations used by Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [8] for solving Synchronized Planarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Matched vertices are shown as bigger disks, the matching is indicated by the orange dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Top: Two cut-vertices matched with each other (left), the result of encapsulating their incident blocks (middle) and the bipartite graph resulting from joining both cut-vertices (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Middle: A matched non-cut-vertex with a non-trivial embedding tree (left) that is propagated to replace both the vertex and its partner (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Bottom: Three different cases of matched vertices with trivial embedding trees (blue) and how their pipes can be removed or replaced (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' efficient solution remained open for 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In that time, polynomial-time algorithms were found for many special-cases [2, 15, 25, 29] before Fulek and Tóth [26] found an O((n + d)8) solution in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Shortly thereafter, Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [8] gave a solution with runtime in O((n + d)2) that also exposes the main concepts needed to solve Clustered Planarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The solution works via a linear-time reduction to the problem Synchronized Planarity, for which Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' gave a quadratic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We improve the runtime of the latter algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As Synchronized Planarity can be used as a modeling tool for several other constrained planarity problems next to Clustered Planarity [8], this also improves the time needed for solving any constrained planarity problem that can be solved via a linear-time reduction to Synchronized Planarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In Clustered Planarity, the embedding has to respect a laminar family of clusters [9, 34], that is every vertex is part of some (hierarchically nested) cluster and an edge may only cross a cluster boundary if it connects a vertex from the inside with one from the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In Synchronized Planarity, we are given a matching on some of the vertices in the graph and seek an embedding such that the rotations matched vertices line up under a given bijection [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The synchronization constraints imposed by matching two vertices are also called pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The reduction from the former problem to the latter employs the CD-tree representation of Clustered Planarity [9], where each cluster is represented as individual skeleton in which adjacent clusters were collapsed into single “virtual vertices”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The order of the edges “leaving” one cluster via a virtual vertex now needs to line up with the order in which they “enter” an adjacent cluster via its corresponding virtual vertex (see also [8, Figure 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 16 Maintaining Triconnected Components under Node Expansion The algorithm for solving Synchronized Planarity works by removing an arbitrary pipe each step, using one of three operations depending on the graphs around the matched vertices, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' EncapsulateAndJoin If both vertices of the pipe are cut-vertices, they are “encapsulated” by taking a copy of their respective components and then collapsing each incident block to a single vertex to obtain stars with matched centers that have multiple parallel edges connecting them to their ray vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The original cut-vertices are split up so that each incident block gets its own copy and these copies are synchronized with the respective vertex representing a collapsed block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Now the cut-vertices can be removed by “joining” both stars, that is identifying their incident edges according to the bijection that is given alongside the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' PropagatePQ If one of the vertices is not a cut-vertex and has an embedding tree that not only consists of a single P-node, two copies of this embedding tree are inserted (“propagated”) in place of both matched vertices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The inner nodes of the embedding trees are synchronized by matching corresponding vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' SimplifyMatching In the remaining case, one of the vertices is not a cut-vertex but has a trivial embedding tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', only appears in a single parallel skeleton and no rigid skeleton in the SPQR-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' If the vertex (or, more precisely, the parallel that completely defines it rotation) can respect arbitrary rotations, we can simply remove the pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The only exception to this is when the other pole of the parallel is also matched, in which case we can “short-circuit” the matching across the parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To summarize, every operation removes a pipe from the matching, while potentially introducing new pipes with vertices that have a smaller degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Using a potential function, it can be shown that the progress made by the removal always dominates overhead of the newly-introduced pipes, and that the operations needed to remove all pipes is limited by the total degree of all matched vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, the resulting instance without pipes can be solved in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' All of the three operations run in time linear in the degree of the un-matched vertices if the embedding trees they depend on are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The contribution of this paper is to efficiently provide the embedding trees, which would require processing entire connected components at each step when done naïvely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Using the fully-dynamic SPQR-tree by Holm and Rotenberg [31, 32], this can be achieved with a poly-log cost of O(∆ · log3 n) leading to an overall runtime of O(m · ∆ · log3 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Using the node expansion from this paper, we can improve the runtime from spending time linear in the size of the input instance (O(m)) for each of the linearly many operations, to only spending time linear in the maximum degree (O(∆)) on each operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The reduction from Clustered Planarity creates an instance of size O(n+d) in which the total degree of matched vertices is in O(d), corresponding to the total number of times an edge crosses a cluster boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that, while this means that O(d) operations are sufficient to reach a reduced instance, the number of crossings between edges and cluster boundaries can be quadratic in the number of vertices in a planar graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We also note that while the improvement over using the Holm and Rotenberg approach is only poly-logarithmic, our datastructure has the additional benefit of being conceptually simpler and thus also more likely to improve performance in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='2 Using Node Expansion for Solving Synchronized Planarity We show how extended skeleton decompositions and their dynamic operation InsertGraphSPQR can be used to improve the runtime of the algorithm for solving Synchronized Planarity by Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [8] from O(m2) to O(m · ∆), where ∆ is the maximum pipe degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 17 already explained in the previous section, the algorithm spends a major part of its runtime on computing so-called embedding trees, which describe all possible rotations of a single vertex in a planar graph and are used to communicate embedding restrictions between vertices with synchronized rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Once the embedding trees are available, the at most O(m) executed operations run in time linear in the degree of the pipe/vertex they are applied on, that is in O(∆) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, being able to generate these embedding trees efficiently by maintaining the SPQR-trees they are derived from is our main contribution towards the speedup of the Synchronized Planarity algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' An embedding tree Tv for a vertex v of a biconnected graph G describes the possible cyclic orderings or rotations of the edges incident to v in all planar embeddings of G [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The leaves of Tv are the edges incident to v, while its inner nodes are partitioned into two categories: Q-nodes define an up-to-reversal fixed rotation of their incident tree edges, while P-nodes allow arbitrary rotation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' see Figure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To generate the embedding tree we use the observation about the relationship of SPQR-trees and embedding trees described by Bläsius and Rutter [10, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='5]: there is a bijection between the P- and Q-nodes in the embedding tree of v and the bond and triconnected allocation skeletons of v in the SPQR-tree of G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ▶ Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Let S be an SPQR-tree with a planar represented graph GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The embedding tree for a vertex v ∈ GS can be found in time O(deg(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We use the rotation information from Theorem 10 and furthermore maintain an (arbitrary) allocation vertex for each vertex in GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' To compute the embedding tree of a vertex v starting at the allocation vertex u of v, we will explore the SPQR-tree by using twinE on one of the edges incident to u and then finding the next allocation vertex of v as one endpoint of the obtained edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' If u has degree 2, it is part of a polygon skeleton that does not induce a node in the embedding tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We thus move on to its neighboring allocation skeletons and will also similarly skip over any other polygon skeleton we encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' If u has degree 3 or greater, we inspect two arbitrary incident edges: if they lead to the same vertex, u is the pole of a bond, and we generate a P-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Otherwise it is part of a triconnected component, and we generate a Q-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We now iterate over the edges incident to u, in the case of a triconnected component using the order given by the rotation of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For each real edge, we attach a corresponding leaf to the newly generated node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The graph edge corresponding to the leaf can be obtained from origE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' For each virtual edge, we recurse on the respective neighboring skeleton and attach the recursively generated node to the current node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As u can only be part of deg(u) many skeletons, which form a subtree of TS, and the allocation vertices of u in total only have O(deg(u)) many virtual and real edges incident, this procedure yields the embedding tree of u in time linear in its degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ Our data structure can now be used to reduce the runtime of solving Synchronized Planarity by generating an SPQR-tree upfront, maintaining it throughout all applied operations, and deriving any needed embedding tree from the SPQR-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ▶ Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Synchronized Planarity can be solved in time in O(m · ∆), where m is the number of edges and ∆ is the maximum degree of a pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The algorithm works by splitting the pipes representing synchronization constraints until they are small enough to be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' It does so by exhaustively applying the three operations EncapsulateAndJoin, PropagatePQ and SimplifyMatching depending on the graph structure around the pairs of synchronized vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As mentioned by Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=', all operations run in time linear in the degree of the pipe they are applied on if the used 18 Maintaining Triconnected Components under Node Expansion embedding trees are known, and O(m) operations are sufficient to solve a given instance [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Our modification is that we maintain an SPQR-tree for each biconnected component and then generate the needed embedding trees on-demand in linear time using Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' See Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content='1 for more background on the Synchronized Planarity operations modified in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Operation SimplifyMatching can be applied if the graph around a synchronized vertex v allows arbitrary rotations of v, that is the embedding tree of v is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' In this case, the pipe can be removed without modifying the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, we can now easily check the preconditions of this operations without making any changes to the SPQR-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' PropagatePQ takes the non-trivial embedding tree of one synchronized vertex v and inserts copies of the tree in place of v and its partner, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Synchronization constraints on the inner vertices of the inserted trees are used to ensure that they are embedded in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We use InsertGraphSPQR to also insert the embedding tree into the respective SPQR trees, representing Q-nodes using wheels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' When propagating into a cutvertex we also need to check whether two or more incident blocks merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We form equivalence classes on the incident blocks, where two blocks are in the same class if 1) the two subtrees induced by their respective edges share at least two nodes 2) both induced subtrees share a C-node that has degree at least 2 in both subtrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Blocks in the same equivalence class will end up in the same biconnected component as follows: We construct the subtree induced by all edges in the equivalence class and add a single further node for each block in the class, connecting all leaves to the node of the block the edges they represent lead to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We calculate the SPQR-tree for this biconnected graph and then merge the SPQR-trees of the individual blocks into it by applying Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' As InsertGraphSPQR (and similarly all MergeSPQR applications) runs in time linear in the size of the inserted PQ-tree, which is limited by the degree of the vertex it represents, this does not negatively impact the running time of the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Operation EncapsulateAndJoin generates a new bipartite component representing how the edges of the blocks incident to two synchronized cutvertices are matched with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The size of this component is linear in the degree of the synchronized vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, we can freshly compute the SPQR-tree for the generated component in linear time, which also does not negatively impact the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Furthermore, as we now no longer need to iterate over whole connected components to generate the embedding trees, we are also no longer required to ensure those components do not grow to big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We can thus also directly contract pipes between two distinct biconnected components using Corollary 9 instead of having to insert PQ-trees using PropagatePQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' This may improve the practical runtime, as PropagatePQ might require further operations to clean up the generated pipes, while the direct contraction entirely removes a pipe without generating new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ ▶ Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Clustered Planarity can be solved in time in O(n + d · ∆), where d is the total number of crossings between cluster borders and edges and ∆ is the maximum number of edge crossings on a single cluster border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Note that for a graph not containing parallel edges to be planar, the number of edges has to be linear in the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' We apply the reduction from Clustered Planarity to Synchronized Planarity as described by Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Ignoring the parallel edges generated by the CD-tree, we can generate an SPQR-tree for every component of the resulting instance in O(n) time in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' The instance contains one pipe for every cluster boundary, where the degree of a pipe corresponds to the number of edges crossing the respective cluster boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, the potential described by Bläsius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' [8], which sums S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Fink and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Rutter 19 up the degrees of all pipes with a constant factor depending on the endpoints of each pipe, is in O(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Each operation applied when solving the Synchronized Planarity instance runs in time O(∆) (the maximum degree of a pipe) and reduces the potential by at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Thus, a reduced instance without pipes, which can be solved in linear time, can be reached in O(d · ∆) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' ◀ References 1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Angelini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfjwe5/content/2301.03972v1.pdf'} +page_content=' Bläsius, and I.' metadata={'source': 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sha256:52b4fa4e17472600823f19817e2ebeed62c8122b3267fc48d4eaca36fc2c5ea8 +size 31741 diff --git a/7NE4T4oBgHgl3EQfcgzB/content/tmp_files/2301.05084v1.pdf.txt b/7NE4T4oBgHgl3EQfcgzB/content/tmp_files/2301.05084v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..df634cd91231b4d8e5d46ba9d97d9fee80467f14 --- /dev/null +++ b/7NE4T4oBgHgl3EQfcgzB/content/tmp_files/2301.05084v1.pdf.txt @@ -0,0 +1,2526 @@ +LOCAL CONSISTENCY AS A REDUCTION BETWEEN +CONSTRAINT SATISFACTION PROBLEMS +Victor Dalmau +Universitat Pompeu Fabra, Barcelona, Spain +victor.dalmau@upf.edu +Jakub Opršal +Institute of Science and Technology Austria, Klosterneuburg, Austria +jakub.oprsal@ist.ac.at +Abstract. We study the use of local consistency methods as reductions between constraint satisfaction +problems (CSPs), and promise version thereof, with the aim to classify these reductions in similar way as +the algebraic approach classifes gadget reductions between CSPs. We classify a use of arc-consistency in +this way, provide frst steps into classifcation of general 𝑘-consistency, and ask whether every tractable +fnite template CSP is reducible by such a reduction to solving systems of afne Diophantine equations. +Key words and phrases. constraint satisfaction problem, Datalog, bounded width, reduction. +1. INTRODUCTION +Is it possible to fnd a mathematical invariant that characterises when one com- +putational problem reduces to another by a polynomial-time (Karp) reduction? Any +such characterisation would give a characterisation of when a problem is solvable in +polynomial time, possibly resolving the P vs. NP problem which makes this question +far beyond our current understanding. Nevertheless, a classifcation of a certain +subclass of polynomial-time reductions between certain well-structured problems is +the core of the theory referred to as the algebraic approach to constraint satisfaction +problem. These reductions have a very precise strict structure that allows such +classifcation. The goal of the present paper is to introduce a larger framework +of reductions extending the scope of the algebraic approach. Our setup includes +reductions that are commonly used to prove NP-hardness of a promise variant +of constraint satisfaction problems, and can be also used to provide new efcient +algorithms by reducing to a tractable problem. +The constraint satisfaction problem (CSP) is a decision problem whose input +consists of a list of variables, with each variable allowed to attain values from a fnite +domain, and a list of constraints each involving a tuple of variables. The goal is to +decide if there is an assignment of values to variables that simultaneously satisfes +all the constraints. Many problems can be directly expressed in this framework, e.g., +SAT, graph 3-colouring, solving systems of linear equations. These problems are +studied for numerous reasons from many diferent research directions. Let us focus +on the direction of the so-called algebraic approach. The main motivation of this +This project has received funding from the European Research Council (ERC) under the European +Union’s Horizon 2020 research and innovation programme (grant agreement No 714532). This project has +received funding from the European Union’s Horizon 2020 research and innovation programme under +the Marie Skłodowska-Curie Grant Agreement No 101034413. Jakub Opršal was also supported by the +UK EPSRC grant EP/R034516/1. +1 +arXiv:2301.05084v1 [cs.LO] 12 Jan 2023 + +2 +VICTOR DALMAU AND JAKUB OPRŠAL +direction is to fnd out what inherent property makes a computational problem hard +and which properties can lead to efcient algorithms. The constraint satisfaction +problem, where we study the complexity depending on the shape of the constraints +allowed, is an ideal scope for rigorous investigations of this question since the +CSPs are well-structured from the mathematical perspective. On top of that, the +fnite-domain CSP have been identifed by Feder and Vardi [FV98] as one of the +largest natural subclasses of the class NP that could exhibit a P vs. NP-complete +dichotomy. This was confrmed by the algebraic approach after 20 years of research +[Bul17, Zhu20]. +The essence of the algebraic approach, whose development started with [JCG97, +BJK05], and was further refned in many subsequent papers including [BOP18], is a +characterisation of gadget reductions between constraint satisfaction problems in +terms of certain mathematical invariants of the problem. Loosely speaking a gadget +reduction replaces each variable of the input with a tuple of variables of a fxed +length, and each constraint on the input with a gadget of constraints on the newly +introduced variables (possibly introducing more new variables). The main theorem +of the approach then asserts that these gadget reductions are characterised in terms +of polymorphisms of the CSP template (see Theorem 4.7 below for a formal statement). +A remarkable fact is that, in the realm of fnite template CSPs, gadget reductions +are enough to provide all necessary NP-hardness, i.e., a fnite template CSP is NP- +complete if all other fnite template CSPs reduce to it via a gadget reduction, and in +P otherwise. The tractability side of the CSP dichotomy is given by providing an +algorithm for all other CSPs. These algorithms of [Bul17, Zhu20] are involved and +rely on a structural analysis of the template and its polymorphisms. +Why do we need new reductions? +Gadget reductions and the algebraic theory show that it is possible to characterise +well-structured reductions between structured problems, and such a characteri- +sation can yield interesting results. In this paper, we introduce a wider class of +reductions that could be possibly prone to a similar characterisation. We have +several motivations to do that. +Firstly, such a characterisation would provide a more refned view on the CSP +dichotomy. Though gadget reductions are enough to provide NP-hardness in the +scope of fnite domain CSPs, there are infnitely many classes of CSPs up to such +reductions, and their order (the class a problem Γ is below the class of problem Δ if +Γ reduces to Δ by a gadget reduction) is incredibly complicated.1 For example, the +order of Boolean CSPs, i.e., when each variable is allowed to attain one of two values, +w.r.t. gadget reductions is infnite (the precise shape was described in [BV20], and is +shown in Figure 1a) even though only two polynomial-time algorithms sufce to +solve all tractable Boolean CSPs. +Secondly, extending gadget reductions with other reductions could also improve +understanding of the dichotomy from the perspective of descriptive complexity: It +is known that gadget reductions between CSPs can be also achieved using Datalog +with parameters [ABD09], but eforts to translate Bulatov’s and Zhuk’s algorithms +to expressibility in some logic computable in polynomial time (e.g., fxed point logic +with linear algebraic operators) have not yet been successful. Extending reductions +with well-behaved logical reductions would allow us to focus on a fewer tractable +CSPs to provide such a characterisation. +1Likely, it is as complicated as a countable partial order can get [Bar19]. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +3 +(a) gadget reductions +(b) consistency reductions +Figure 1. Boolean CSPs ordered by two classes of reductions. +Thirdly, a lot of recent research is focused on a more general version of CSPs, +promise constraint satisfaction problems. In this promise version of the problem, +each instance comes with two versions of each constraint, a stronger and a weaker +one. The goal is then decide between two (disjoint, but not complementary) cases: +all stronger constraints can be satisfed, or not even the weaker constraints can +be satisfed. A prominent problem described in this scope is approximate graph +colouring which asks, e.g., to decide between graphs that are 3-colourable and those +that are not 6-colourable. A systematic study of promise CSPs has been started in +[AGH17] which studied a certain promise version of SAT, and continued in a series +of papers including generalisations of the algebraic approach to promise CSPs in +[BG21, BBKO21]. +Unlike the case of CSPs, there are many NP-hardness results of promise CSPs +that are not explainable by gadget reductions, e.g., [DRS05, Hua13, AGH17, KO19, +FKOS19, WŽ20, BWŽ21]. These hardness results usually rely on some version of +the PCP theorem [AS98] which is better suited for a diferent, more analytical, +approximation version of CSPs. This is one of the main reasons that new reductions +have been called for in [BBKO21] and [BK22]. We note that the frst of these papers +is the aforementioned classifcation of gadget reductions, while the second paper +gives a sufcient condition for a more general reduction that replaces the necessity +of using the PCP theorem in almost all of the above NP-hardness results on promise +CSPs. +Our contribution +In the present paper, we identify a well-behaved class of reductions between +(promise) CSPs that is closed under composition, extends gadget reductions described +by the algebraic approach, covers the reduction used in [BK22], and can express +Sherali-Adams hierarchy [SA90] through reductions to linear programming. We +present two descriptions of such reductions: one using the language of logic, namely +monotone Datalog interpretations, and one combinatorial which is based on the local +consistency algorithm for CSPs. We show that these two descriptions are equivalent in +the context of promise CSPs. In short, we call these reductions consistency reductions. + +4 +VICTOR DALMAU AND JAKUB OPRŠAL +Some form of local consistency checking is often run as a preprocessing step in +many CSP algorithms (including SAT-solvers, and Zhuk’s polynomial time algorithm +[Zhu20]). The local consistency checking that we will use is a procedure that +considers at most 𝑘 variables at a time, and keeps track of what are the possible +tuples of values that could be assigned to these variables. This list of tuples is then +iteratively updated by removing tuples that cannot appear in a solution, and the +algorithm stops when no more tuples can be ruled out. We can view this procedure +as a reduction from the CSP to a CSP with binary constraints. The full 𝑘-consistency +reduction is then obtained by joining this procedure with a standard gadget reduction. +We start the quest of characterisation of consistency reductions by providing sev- +eral preliminary observations that suggest that a characterisation might be possible. +First, we prove that the logical and combinatorial descriptions are equivalent in the +scope of promise CSPs. Second, we characterise a special case of consistency reduc- +tions, which can be described by replacing the 𝑘-consistency with arc-consistency, +by the means of polymorphisms and a certain transformation 𝜔 on the classes of +polymorphisms of a given template. We describe these constructions in detail in Sec- +tion 4. This result hints towards what a more general characterisation of consistency +reductions might look like. +In the last section, we investigate several classes of (promise) CSPs that are defned +as those that reduce to a fxed CSP by a consistency reduction. A few examples +of such classes are: promise CSPs with bounded width which correspond to those +that reduce to Horn-3SAT, promise CSPs solvable by some level of Sherali-Adams +hierarchy which correspond to those that reduce to linear programming, and promise +CSPs solvable by some level of Lasserre hierarchy. We show that all such classes are +closed under gadget reductions, and as an immediate consequence we obtain several +generalisations of [LZ07] and [BBKO21, Lemma 7.5]. In particular, we show the class +of promise CSPs that are solvable by some level of Sherali-Adams is closed under +gadget reductions, and hence could be theoretically characterised by the means of +polymorphisms. +The class of bounded width promise CSPs corresponds to the bottom element +of the order of promise CSPs up to consistency reductions. Since the majority of +(non-promise) Boolean CSPs have bounded width, we get immediately that the order +of Boolean CSPs up to consistency reductions consists only of three classes: the +class of bounded width CSPs, the class of XOR-3Sat, and the class consisting of +NP-complete CSPs (see Figure 1b). This observation suggests that the order of all +CSPs up to consistency reductions might be substantially simpler than the order of +CSPs up to gadget reductions. +Finally, one class of problems that has not been studied before is of particular +interest: the class of all CSPs that reduce to solving systems of afne equations over +integers by a consistency reduction. This family contains both CSPs solvable by +local consistency, and systems of linear equations over all fnite felds. This leads us +to the question: which fnite-domain CSPs are contained in this class? Could it be +that the class contains all fnite-domain CSPs that do not allow a gadget reduction +from 3-colouring? If that was the case, it would provide a new polynomial time +algorithm for those CSPs, and hence a new proof of the CSP dichotomy theorem. +In the last section, we discuss several observations that lead us to believe that the +answer to these questions could be afrmative. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +5 +Related work +We mention two papers which provide results in a similar direction as the present +paper, though they have diferent motivations from ours. +First, a recent paper of Barto and Kozik [BK22] describes sufcient condition for +the existence of polynomial-time (and possibly a log-space) reduction between two +promise CSPs more general that the one given in [BBKO21, Theorem 3.1]. They aim +to avoid dependence on the PCP theorem in current hardness results for PCSPs. The +reduction they use to provide their result falls into our framework of reductions, and +hence [BK22] provide a sufcient condition for a consistency reduction between +two promise CSPs though this condition is not necessary. +Second, Ciardo and Živný [CŽ23] defned a general framework for hierarchies +of algorithms for promise CSPs generalising the Sherali-Adams hierarchy. Again, +their hierarchies fall within our framework in the sense that a problem is solved by +some level of the Ciardo-Živný hierarchy if and only if it reduces to a corresponding +(promise) CSP by a consistency reduction. The motivation of Ciardo and Živný is +to provide classes of algorithms for promise CSPs that could be investigated by a +uniform approach, and provide better understanding of those algorithms. +Organisation of the paper +Section 2 provides basic defnitions of the problems considered, and each of the +three following sections contains one of our main results together with all the prelim- +inaries that are necessary for that particular result. Section 3 provides equivalence +of the logical and combinatorial descriptions, Section 4 the characterisation of the +arc-consistency reduction, and Section 5 our observations on classes of promise +CSPs that are locally reducible to a fxed CSP. +2. PRELIMINARIES +In the present paper, every object has a given ‘type’ in the sense that every symbol +is either a set, a function, a structure, an integer, etc. Sets are generally denoted by +capital letters, and integers by lower-case letters. We often do not explicitly specify +that such symbols represent sets or integers when we are introducing a new symbol +and the ‘type’ is clear from the context, e.g., when the symbol appears in an index. +Many of constructions described in this paper can be defned using several dif- +ferent languages, e.g., an instance of a CSP can be viewed as a list of constraints, a +logical formula, or a relational structure. Often one of this languages allows for a +more elegant introduction of a certain notion, or a more elegant proof. The same as +true about other concepts that we introduce, and we often switch between several +languages to avoid technical and obscuring constructions in the proofs. We try to +pick the language that is ‘locally best’ for the current argument. The cost for this is +that we often have to reinterpret a result of one construction as a diferent object. +Technically, all the proofs could be written in any one language of your choice, +though, we believe, it would result in unwieldy long proofs. +We denote by [𝑛] the set of the frst 𝑛 positive integers, i.e., [𝑛] = {1, . . . ,𝑛}. +We denote the set of all functions 𝑓 : 𝑋 → 𝐴 by 𝐴𝑋, and for such a function 𝑓 and +𝑌 ⊆ 𝑋, we denote its restriction to 𝑌 by 𝑓 |𝑌 : 𝑌 → 𝐴. For a function 𝜋 : 𝑋 → 𝑌 +and 𝑦 ∈ 𝑌, we denote by 𝜋−1(𝑦) the set of preimages of 𝑦 under 𝜋, i.e., the set +{𝑥 ∈ 𝑋 | 𝜋(𝑥) = 𝑦}. We will denote entries in a tuple 𝑎 by lower indices, i.e., +𝑎 = (𝑎1, . . . ,𝑎𝑘), and occasionally view a tuple 𝑎 ∈ 𝐴𝑘 as a function 𝑎: [𝑘] → 𝐴, + +6 +VICTOR DALMAU AND JAKUB OPRŠAL +and hence 𝑎(𝑖) is an alternative notation for 𝑎𝑖. Finally, we will write 𝑓 ◦ 𝑔 for the +composition of 𝑓 and 𝑔 and 𝑓 𝑔(𝑥) for (𝑓 ◦ 𝑔)(𝑥). +2.1. Constraint satisfaction problems +To settle on the playing feld, we formally defne the constraint satisfaction +problem, and its promise variant. We refer to [BKW17] for a deeper exposition of +the algebraic theory of CSPs, and to [KO22] for more background and examples +of promise CSPs. Since it will bring further simplifcation below, we defne a non- +homogeneous CSP where each variable is given its own domain. Before we get to +defne a fxed-template CSP, let us start with a defnition of the uniform CSP. +Definition 2.1. The constraint satisfaction problem gets on input a list of variables +with each variable 𝑣 assigned a fnite domain 𝐷𝑣, and a list of constraints each of +the form (𝑣1, . . . , 𝑣𝑘) ∈ 𝑅 where 𝑅 ⊆ 𝐷𝑣1 × · · · × 𝐷𝑣𝑘 is given as a list of tuples (the +number 𝑘 is called the arity of the constraint and the tuple (𝑣1, . . . , 𝑣𝑘) is called +the scope of the constraint). The goal is to decide if there is an assignment 𝑠 with +𝑠(𝑣) ∈ 𝐷𝑣 for each variable 𝑣 that simultaneously satisfes all of the constraints, i.e., +(𝑠(𝑣1), . . . ,𝑠(𝑣𝑘)) ∈ 𝑅 for each constraint as above. +Commonly, instances are restricted in some way, e.g., we could insist that all +constraints are of arity at most 𝑚 (i.e., 𝑘 ≤ 𝑚 in the defnition above) for a fxed 𝑚 +to get 𝑚-CSP. A 2-CSP where each constraint is a graph of a function, i.e., of the +form 𝜋(𝑣1) = 𝑣2 for some 𝜋 : 𝐷𝑣1 → 𝐷𝑣2 is called label cover. Another reasonable +restriction is to limit the domains; we could fx a set 𝐷 and require that 𝐷𝑣 = 𝐷 for +each variable 𝑣 to get the usual defnition of the CSP. +Let us move to the fxed-template CSP. A template consists of a sequence of +allowed domains 𝐷𝑣, and relations 𝑅 appearing in the constraints, and it is encoded +in a single structure defned as follows. +Definition 2.2. A non-homogeneous relational structure is a tuple A = (𝐴1, . . . ,𝐴𝑛; +𝑅A +1 , . . . , 𝑅A +𝑚) where 𝐴𝑖 is a set for each 𝑖 ∈ [𝑛] called the 𝑖-th domain of A, and, +for each 𝑖 ∈ [𝑚], 𝑅A +𝑖 +⊆ 𝐴𝑖1 × · · · × 𝐴𝑖𝑘 for some 𝑘 ≥ 0 and 𝑖1, . . . ,𝑖𝑘 ∈ [𝑛]. The +word 𝑖1 . . .𝑖𝑘 is called the arity of 𝑅𝑖 and it is denoted by ar𝑅𝑖. We use the notation +ar𝑅𝑖 (𝑗) = 𝑖𝑗, sometimes view ar𝑅𝑖 as a function, and for simplifcation just refer to +the number 𝑘 (the length of the arity word) as the arity of 𝑅𝑖. We assume that empty +relations have a fxed arity. +The signature of A is consists of: the number of domains 𝑛, and relational symbols +𝑅1, . . . , 𝑅𝑚 together with their arities ar𝑅1, . . . , ar𝑅𝑚. We also call any such collection +𝜎 a relational signature. An element 𝑖 ∈ [𝑛] is called a 𝜎-type, and the symbols 𝑅𝑗 +are called 𝜎-symbols. +A structure A is fnite if 𝑛, 𝑚, and all 𝐴𝑖’s are fnite. We denote the domains +of some structure by the same letter, e.g., a structure X has domains 𝑋1, . . . and +relations 𝑅X, etc. +One way to think about such structures is simply imagine them as a single-sorted +structures whose domain is the disjoint union of the domains 𝐴𝑖, and each element +has a given type (which is formally just a number in [𝑛]). The relations then only +relate elements of given types. In concordance with this interpretation we will call +any 𝑎 ∈ 𝐴𝑖 an element of A of type 𝑖, and we will simply write 𝑎 ∈ A for an element +with understanding that such 𝑎 is assigned a fxed domain 𝐴𝑖. We will also use the +symbol 𝐴 for the disjoint union of 𝐴𝑖’s, hence 𝑎 ∈ 𝐴 and 𝑎 ∈ A are synonymous. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +7 +Throughout this paper we will often say structure or 𝜎-structure instead of non- +homogeneous relational structure, and we will silently assume that every element +and variable has a given type, and that the domains of A are disjoint (this can be +always achieved by picking a suitable isomorphic structure). +We silently assume that every object in this paper is properly typed, e.g., every +variable has a type that defnes in which domain it should belong even when the +type is not mentioned. We will often assume those types are given implicitly, e.g., +if 𝜙 is a transformation on structures of certain signature, and A is a structure to +which 𝜙 is applied, it is assumed that the signature of A agrees with the signature +required on the input. +Loosely speaking, a homomorphism between two structures of the same signature +is a mapping that maps elements of one structure to elements of the second structure +that preserves types and all relations. Formally, we defne a homomorphism as +follows. +Definition 2.3. Given two structures A and B of the same signature, a homomorphism +from A to B is a collection of maps 𝑓𝑖 : 𝐴𝑖 → 𝐵𝑖, one for each type 𝑖, such that for +each relational symbol 𝑅 and (𝑎1, . . . ,𝑎𝑘) ∈ 𝑅A, we have +(𝑓ar𝑅 (1) (𝑎1), . . . , 𝑓ar𝑅 (𝑘) (𝑎𝑘)) ∈ 𝑅B. +Below, we will simply write 𝑓 (𝑎) instead of 𝑓𝑖 (𝑎) if 𝑓 is a collection of mappings +as above, and the type 𝑖 of 𝑎 is clear from the context. Similarly, we will use the +symbol 𝑓𝑅(𝑟) or simply 𝑓 (𝑟) for the component-wise application of 𝑓 to 𝑟 ∈ 𝑅. We +write 𝑓 : A → B if 𝑓 is a homomorphism, and A → B if such a homomorphism +exists. Such a homomorphism can be also viewed as a function 𝑓 : 𝐴 → 𝐵 that +preserves types and relations. Note that a homomorphism also implies that the two +structures have the same signature. +Having settled on these defnitions, we are ready to defne a fxed-template CSP. +Definition 2.4. +Let D be a non-homogeneous relational structure, the constraint +satisfaction problem CSP(D) is a decision problem whose goal is: given X with the +same signature as D, decide whether X → D. +Given A and A′ such that A → A′, we defne the promise constraint satisfaction +problem PCSP(A, A′) as a promise problem whose goal is, on an input X which is a +structure with the same signature as both A and A′, output yes if X → A, and no if +X ̸→ A′. The pair A, A′ with A → A′ is called a promise template. We assume that +A in such a promise template is fnite. +Note that in the promise version, the requirement that A → B ensures that +the yes- and no-instances are disjoint. We could defne promise CSP as a promise +search problem, the goal would be, given X that is promised to map to A via a +homomorphism, fnd a homomorphism X → B. It is not clear, and currently not +known, whether these two versions of promise CSPs are of equivalent complexity. +This paper focuses on the decision version of PCSP though a few results (mostly +with some caveats) apply also to the search version. Finally, note that CSP(A) is +PCSP(A, A), and therefore every result about promise CSPs is applicable to (fnite +template) CSPs. +Every instance X of CSP(D) can be interpreted as an instance of a general CSP +in the following way: the variables are elements of X, where each variable 𝑣 ∈ 𝑋𝑖 +is assigned the domain 𝐷𝑖, and constraints are of the form (𝑣1, . . . , 𝑣𝑘) ∈ 𝑆 where +(𝑣1, . . . , 𝑣𝑘) is a tuple in 𝑅X and 𝑆 = 𝑅A. Following this translation, we introduce the + +8 +VICTOR DALMAU AND JAKUB OPRŠAL +following notions for an instance of a fxed-template (promise) CSP. A constraint +of an instance X of PCSP(A, A′) is an expression of the form (𝑣1, . . . , 𝑣𝑘) ∈ 𝑅X for +some symbol 𝑅 which is formally seen as a pair ((𝑣1, . . . , 𝑣𝑘), 𝑅) where (𝑣1, . . . , 𝑣𝑘) +is called the scope of the constraint. We will often work with such an instance of +promise CSP as an instance X of CSP(A) since most constructions in this paper only +depend on the frst part of a promise template. +We formally defne a notion of a reduction between two (promise) CSPs. A +reduction between decision problems is a (usually efciently computable) function +that maps instances of one problem to instances of the other problem in such a way +that the answer is preserved. +Definition 2.5. We say that a mapping 𝜓 from instances of PCSP(A, A′) to instances +of PCSP(B, B′) is a reduction between these two problems if for all X, we have +• if X → A then 𝜓 (X) → B, and +• if X ̸→ A′ then 𝜓 (X) ̸→ B′. +The frst item, preserving the yes instances, is usually referred to as completeness, +and the second, preserving the no instances, as soundness. We will usually use and +prove the soundness in its converse form: if 𝜓 (X) → B′ then X → A′. Note that +if we are interested in the complexity of search version of (promise) CSP, for a +reduction between search versions, we need that this converse is witnessed by an +efciently computable function that given a homomorphism 𝜓 (X) → B′ outputs a +homomorphism X → A′. +Reductions are more general than algorithms: every decision algorithm can be +viewed as a reduction to a problem with two admissible instances yes and no where +yes is the positive instance. In our setting, we use a specifc CSP instead of this trivial +problem, so that our reductions do not leave the scope of (promise) CSPs. There are +a few sensible ways we could defne such a trivial CSP. We pick the template with +no satisfable constraints, i.e., such that all non-trivial instances are negative. +Definition 2.6. The trivial CSP is the CSP with template T = (𝐷; ⊥T) where 𝐷 is a +1-element set, and ⊥T is the nullary empty relation (i.e., a constraint that is always +false). +There is an obvious algorithm which decides the trivial CSP which is, depending +on the encoding of the input, either constant or linear time. We will also implicitly +assume that all (promise) CSPs considered in this paper have negative instances — +this can be always ensured by adding a nullary constraint ⊥ that cannot be satisfed. +2.2. Label cover and projective CSPs +Many constructions in this paper hinge on seeing one concept in diferent per- +spectives, and we often switch between these perspectives. Here we present one +such concept with two formal defnitions and show how they relate to each other. +First perspective is an instance of the label cover problem mentioned above. +Definition 2.7. Label cover is the following decision problem: given an instance of a +non-homogeneous CSP such that each constraint is binary and of the form 𝑣 = 𝜋(𝑢) +for some function 𝜋 : 𝐷𝑢 → 𝐷𝑣 (i.e., defned by the relation 𝑃𝜋 = {(𝑎, 𝜋(𝑎)) | 𝑎 ∈ +𝐷𝑢}), decide whether it is solvable. +The second perspective is a fxed-template version of label cover, which gives a +restriction on the relations of the template in the following way. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +9 +Definition 2.8. +A projective structure is a non-homogeneous structure A whose +every relation 𝑅A is a graph of a function 𝜋𝑅 : 𝐴𝑖 → 𝐴𝑗, i.e., 𝑅 is binary and 𝑅A = +{(𝑎, 𝜋𝑅(𝑎)) | 𝑎 ∈ 𝐴𝑖}. We will call such a binary relation projective. +In a similar way as an instance of a fxed-template CSP can be interpreted as an +instance of general CSP, an instance X of CSP(A), where A is projective, can be +interpreted as a label cover instance I in such a way that X → A if and only if I +is solvable. Since we will often use this interpretation, we present it as a formal +defnition. +Definition 2.9. Let X be an instance of CSP(A) where A is a projective structure +whose relations are defned by maps 𝜋𝑅 as in Defnition 2.8. We defne the corre- +sponding label cover instance I in the following way: The variables of I are 𝑣𝑥 for +𝑥 ∈ 𝑋 with domain 𝐴𝑖 where 𝑖 is the type of 𝑥. The constraints of I are of the form +𝑣𝑥 = 𝜋𝑅(𝑣𝑦), where (𝑥,𝑦) ∈ 𝑅X. +A mapping 𝑠 : 𝑉 → 𝐴, where 𝑉 is the set of variables of I satisfying 𝑠(𝑣𝑥) ∈ 𝐷𝑥 +is a solution of I if and only if the mapping 𝑓 : 𝑋 → 𝐴 defned by 𝑓 (𝑥) = 𝑠(𝑣𝑥) is a +homomorphism from X to A. +This interpretation of an instance of projective CSP as a label cover instance can be +also reversed though with subtle caveats. Namely, that the process of Defnition 2.9 +can loose some information about the template A. More importantly, if we defne +a construction 𝜙 (e.g., a reduction between two promise problems) that on input +gets an instance X of some fxed-template CSP, say CSP(A) and produces a label +cover instance I = 𝜙(X, A), which in turn is interpreted as an instance Y of CSP(B) +with a projective B. To stay in the scope of fxed-template CSP, we would like to +insist that B does not depend on X, but depends only on A and 𝜙. This can be done +by defning B to have one domain for each possible domain of I, and one relation +𝑅B for each possible 𝜋 appearing in a constraint of I. For all constructions in this +paper, there is an implicit bound on the sizes of domains of B in terms of A and 𝜙 (or +parameters thereof) that can be obtained in a straight-forward way. Consequently, +we can ensure that the template B is fnite. We will not comment in much detail on +these bounds and templates B — we will work directly with the instance I bearing in +mind that when needed it can be interpreted as an instance of CSP(B) for a suitable +projective B. +We also note that label cover instances can be interpreted as minor conditions +which were extensively used in [BBKO21]. This will be important in Section 4, +where we return to this issue. +3. LOCAL CONSISTENCY REDUCTIONS +We will defne several classes of efciently computable functions on relational +structures, Datalog interpretations, gadget replacements, and (local) consistency re- +ductions. Also, we will show that arbitrary composition of Datalog interpretations +and gadget replacements are equivalent to consistency reductions in the sense that +one reduction can be used in the place of the other when reducing between two +(promise) CSPs; this statement is formalised in Theorem 3.27 below. While Datalog +interpretations and gadgets have many parameters that could be adjusted, consis- +tency has only one parameter, a width 𝑘, in this sense we can view local consistency +as a canonical normal form of these reductions. + +10 +VICTOR DALMAU AND JAKUB OPRŠAL +3.1. Gadgets +Gadget reductions are the more traditional reductions in the realm of fnite- +template CSPs and promise CSPs. They have been classifed by the algebraic ap- +proach. We outline this classifcation briefy in Section 4.1 below, and refer to +[BBKO21], [BKW17], or [KO22] for a detailed exposition. We formally defne gadget +reductions as a two-step process since this decomposition will be useful below, and it +is not hard to see that the resulting procedure is equivalent with the more traditional +one, e.g., as described in [KOWŽ22, Section 4.1]. The frst step is reifcation. +Definition 3.1. Assume A = (𝐴1, . . . ,𝐴𝑛;𝑅A +1 , . . . , 𝑅A +𝑚) is a structure, we denote by +A∗ the structure with domains (𝐴1, . . . ,𝐴𝑛, 𝑅A +1 , . . . , 𝑅A +𝑚) and binary relations 𝑃A∗ +𝑅,𝑖 ⊆ +𝑅A × 𝐴ar𝑅 (𝑖), for each 𝑅 and each 𝑖 ∈ [𝑘] where 𝑘 is the arity of 𝑅, defned as +((𝑎1, . . . ,𝑎𝑘),𝑏) ∈ 𝑃A∗ +𝑅,𝑖 if and only if 𝑏 = 𝑎𝑖 and (𝑎1, . . . ,𝑎𝑘) ∈ 𝑅A. The structure A∗ +is called the reifcation of A. +The result of a reifcation is a projective structure in the sense of Defnition 2.8. +Hence, if I is an instance of CSP(A), then I∗ is an instance of a binary CSP with +template A∗ with projective relations. As mentioned before, in Section 2.2, instances +of such a binary CSPs can be viewed as instances of label cover. In fact, reifcation is +a common way to reduce from any CSP to label cover, and if a label cover instance +is interpreted as a minor condition, it is also equivalent to the construction Σ(A, I) +from [BBKO21, Section 3.1] — we will talk about this interpretation in more detail +later, in Section 4. +The second step of a gadget reduction has the freedom to choose a gadget. We +defne a bit restrictive version of gadgets which are enough for reductions from +CSPs with a projective template. +Definition 3.2. Let 𝜏 and 𝜎 be relational signatures, and assume that 𝜏 consists of +𝑛 types and binary relational symbols 𝑅1, . . . , 𝑅𝑚. A strict gadget 𝜸 is an 𝑛-tuple +(D1, . . . , D𝑛) of 𝜎-structures along with a homomorphism 𝑝𝑅 : D𝑗 → D𝑖 for each +𝜏-symbol 𝑅 where 𝑖𝑗 = ar𝑅. +This gadget can be then applied on a 𝜏-structure A to produce a 𝜎-structure 𝜸 (A) +in the following way: +(1) For each 𝑎 ∈ 𝐴𝑖 introduce to 𝜸 (A) a copy of D𝑖, whose elements will be +denoted as (𝑎;𝑑) for 𝑑 ∈ D𝑖. +(2) For each 𝑅 of arity 𝑖𝑗, (𝑎,𝑏) ∈ 𝑅A, and every 𝑑 ∈ 𝐷𝑗, add an equality +constraint (𝑎;𝑝𝑅(𝑑)) = (𝑏;𝑑). +(3) Collapse all equality constraints (i.e., identify all pairs of elements involved +in one of the constraints introduced in the previous step). We denote by +[𝑎;𝑑] the class of an element (𝑎;𝑑) after collapsing. +We will also call this operation a strict gadget replacement. +Example 3.3. Throughout this section, we will give several connected examples +of relatively trivial cases of our constructions. Let us start with a strict gadget +replacement 𝜸 that produces from a graph another graph. This gadget is defned by +a graph D1 = K2 = ({0, 1}; ≠), and a map 𝑝𝐸 : K2 → K2 that switches 0 and 1 (the +non-trivial automorphism of K2). +The gadget replacement 𝜸 then produces from a graph G another graph H in the +following way: +• Replace each vertex 𝑣 of G with a pair of vertices 𝑣0, 𝑣1 connected by an edge, +i.e., replace each vertex with the gadget K2. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +11 +• Replace each edge (𝑢, 𝑣) of G with the constraints 𝑢0 = 𝑣1 and 𝑢1 = 𝑣0, i.e., +for each edge, we identify the two gadgets introduced by replacing the two +vertices in the reverse orientation. +• Collapse all equality constraints as in Defnition 3.2. +Note that this gadget produces for each connected component of the input G +either a loop, if the component contains an odd cycle, or an edge, if the component +is bipartite. Hence, 𝜸 is a reduction from CSP(K2) to CSP(K∞) where K∞ is the +countable clique.2 +A gadget replacement is a composition of the reifcation and a strict gadget +replacement. It is well-known that such a gadget replacement can be computed +in log-space (the fact that collapsing equality constraints is in log-space is due to +[Rei08]). The classifcation of gadget reductions of the algebraic approach also gives +a universal gadget to reduce from a (projective) structure A to any other structure. +These gadgets are constructed using direct powers. +Definition 3.4. Let 𝑋 be a set, and B a structure. The 𝑋-fold direct power of B, denoted +by B𝑋, is the structure +(𝐵𝑋 +1 , . . . , 𝐵𝑋 +𝑛 ;𝑅B𝑋 , . . . ) +where for each 𝑅, 𝑅B𝑋 contains all tuples (𝑏1, . . . ,𝑏𝑘) of functions 𝑏𝑖 : 𝑋 → 𝐵ar𝑅 (𝑖) +such that (𝑏1(𝑥), . . . ,𝑏𝑘 (𝑥)) ∈ 𝑅B for all 𝑥 ∈ 𝑋. +Definition 3.5. The universal gadget from a projective A to another structure B is +then defned as D𝑖 = B𝐴𝑖 for each 𝑖, and 𝑝𝑅 : B𝐴𝑗 → B𝐴𝑖 is defned as 𝑝𝑅(𝑏) = 𝑏 ◦ 𝜋𝑅 +where 𝜋𝑅 : 𝐴𝑖 → 𝐴𝑗 is the mapping defning 𝑅 in A. +The universality of these gadgets is given by the following lemma which we +phrase without a proof. The proof is implicit in [BBKO21, Section 9]. +Lemma 3.6. If there is a gadget reduction from PCSP(A, A′) to PCSP(B, B′), then the +composition of reifcation with the universal gadgets for A∗ and B is also a reduction +between these two PCSPs. +Let us also note that the universal gadget can be applied directly on a label cover +instance I as follows: (1) Replace each variable 𝑣 with domain 𝐷𝑣 with a copy +B𝐷𝑣, (2) for each constraint 𝜋(𝑢) = 𝑣, and 𝑏 ∈ B𝐷𝑣, add an equality constraint +(𝑢;𝑏 ◦ 𝜋) = (𝑣;𝑏), and (3) collapse the equality constraints. +Finally, let us defne a notion that plays a major role in the characterisation of +gadget reduction, pp-powers. Though we will not use this fact directly, it is a special +case of a Datalog interpretation defned below, and it relates the current work with +other existing literature. In brief, a pp-power is a transformation of the template +that is adjoint to a gadget replacement; a precise construction is described below +after the defnition of pp-formulae. +Definition 3.7. Let 𝜏 be a signature. A primitive positive 𝜏-formula (pp-formula) is a +logical formula using only existential quantifcation, conjunctions, and equality, i.e., +a formula that can be rewritten as +𝜙(𝑥1, . . . ,𝑥𝑛) ≡ ∃𝑥𝑛+1, . . . ,𝑥𝑘 . 𝑡1 ∧ 𝑡2 ∧ · · · ∧ 𝑡𝑚 +where each 𝑡𝑖 is an atomic 𝜏-formula, i.e., a formula of the form 𝑅(𝑥𝑖1, . . . ,𝑥𝑖𝑙 ) or +𝑥𝑖1 = 𝑥𝑖2 where 𝑖𝑗 ∈ [𝑘]. The arity of 𝜙 is the word 𝑖1 . . .𝑖𝑛 where 𝑖𝑗 is the type of 𝑥𝑗. +(An atomic formula 𝑅(𝑥1, . . . ,𝑥𝑙) is satisfed by 𝑎1, . . . ,𝑎𝑙 in A if (𝑎1, . . . ,𝑎𝑙) ∈ 𝑅A.) +2In this and connected examples, we are deviating from the convention that requires that the template +is fnite. + +12 +VICTOR DALMAU AND JAKUB OPRŠAL +We note here that we are defning pp-formulae over typed (non-homogeneous) +signatures. This means that each variable occurring in the formula must have one +of the types of the signature and that, in addition, the atomic formulae respect the +type, i.e., the arity (or type) of 𝑅 is the concatenation of the types of 𝑥𝑖1, . . . ,𝑥𝑖𝑙 in +each atomic formula 𝑅(𝑥𝑖1, . . . ,𝑥𝑖𝑙 ). Same applies to the other logical formalisms +introduced in this paper such as Datalog programs. +We now defne logical interpretations in the special case of primitive positive +logic, which are closely connected to gadget reductions. +Definition 3.8. Fix two relational signatures 𝜎 and 𝜏, let 𝑛 be the number of domains +in 𝜎, and let 𝑅1, . . . , 𝑅𝑚 be the list of all relational symbols in 𝜎. A primitive positive +interpretation (pp-interpretation) is an (𝑛 + 𝑚)-tuple +𝝓 = (𝜙1, . . . ,𝜙𝑛;𝜙𝑅1, . . . ,𝜙𝑅𝑚) +of primitive positive 𝜏-formulae such that, for each relational symbol 𝑅 in 𝜎, the +arity of 𝜙𝑅 is the concatenation of the arities of 𝜙𝑖1, ..., 𝜙𝑖𝑘 where ar𝑅 = 𝑖1 . . .𝑖𝑘. +The application of such a pp-interpretation 𝝓 to a 𝜏-structure is the 𝜎-structure +𝝓(A) = (𝜙A +1 , . . . ,𝜙A +𝑛 ;𝜙A +𝑅1, . . . ,𝜙A +𝑅𝑚) +where 𝜙A +𝑖 denotes the set of all tuples in A that satisfy 𝜙𝑖, and 𝜙A +𝑅 is interpreted as +a relation on 𝜙ar𝑅 (1) (A) × · · · × 𝜙ar𝑅 (𝑘) (A) in the natural way, i.e., it consists of all +tuples +((𝑎11, . . . ,𝑎1𝑖1), . . . , (𝑎𝑘1, . . . ,𝑎𝑘𝑖𝑘)) ∈ 𝜙A +ar𝑅 (1) × · · · × 𝜙A +ar𝑅 (𝑘) +such that +A |= 𝜙𝑅(𝑎11, . . . ,𝑎1𝑖1, . . . ,𝑎𝑘1, . . . ,𝑎𝑘𝑖𝑘). +We note that our defnition of an interpretation slightly difers from common +defnitions, in particular, we do not allow factoring the newly defned domains by +defnable equivalence relations.3 +These pp-interpretations can describe gadget reductions in the following way: +there is a correspondence between pp-interpretations and gadgets, s.t., if 𝝓 is a pp- +interpretation corresponding to a gadget 𝜸, then 𝜸 (A) → B if and only if A → 𝝓(B). +This relation is called adjunction and it is enough to prove that such 𝜸 gives a +reduction from CSP(𝝓(B)) to CSP(B) for each structure B. For more details, see +[KOWŽ22, Section 4.1] and [BKW17, Section 3]. +3.2. Logical reductions +We frst described the class of reductions we consider in this paper in terms of +logic, or more precisely Datalog. We start with introducing Datalog and some of its +known properties. +3Our pp-interpretations are equivalent to pp-powers defned in [BOP18, Defnition 3.6] in the following +sense: every pp-interpretation of A is homomorphically equivalent to a pp-power of A, and conversely, +every pp-power of A is homomorphically equivalent to a pp-interpretation of A. Consequently, a structure +is pp-constructible from A in the sense of [BOP18, Defnition 3.4] if and only if it is homomorphically +equivalent to a pp-interpretation of A (see [BOP18, Corollary 3.10]). + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +13 +3.2.1. Datalog. Datalog is a language of logic programs without functional symbols. +Let 𝜏 be a relational signature. A Datalog program 𝜙 with input signature 𝜏 is +constituted by a relational signature 𝛿 satisfying 𝜏 ⊆ 𝛿 (meaning that it has the same +types as 𝜏 and each symbol of 𝜏 appears in 𝛿 with the same arity) along with a fnite +collection of rules that are traditionally written in the form +𝑡0 ← 𝑡1, . . . ,𝑡𝑟 +where 𝑡0, ..., 𝑡𝑟 are atomic 𝛿-formulae, i.e., formulae of the form 𝑅(𝑥1, . . . ,𝑥𝑛) where +𝑅 is a relational symbol in 𝛿 of arity 𝑛 and variables 𝑥1, . . . ,𝑥𝑛, or 𝑥1 = 𝑥2 for +variables 𝑥1 and 𝑥2. In such a rule 𝑡0 is called the head and 𝑡1, . . . ,𝑡𝑟 the body of the +rule. Moreover, we require that neither the symbols in 𝜏 nor the equality appear in +the head of any rule. The symbols in 𝜏 are called input symbols, or EDBs (standing for +extensional database predicates, while all the other symbols would be IDBs, intensional +database predicates). Furthermore, one of the 𝛿-symbols is designed as output.4 A +Datalog program receives as input a 𝜏-structure A and produces a relation with the +same arity of the output predicate in the following way: Let B be the 𝛿-structure +computed in the following way. +• Start with setting 𝑅B = 𝑅A if 𝑅 ∈ 𝜏 and 𝑅B = ∅ otherwise. +• If there is a rule 𝑡0 ← 𝑡1, . . . ,𝑡𝑟 and some assignment ℎ: 𝑋 → 𝐴, where 𝑋 are +the variables occurring in the rule, such that all the atomic predicates hold +in B then include (ℎ(𝑥1), . . . ,ℎ(𝑥𝑘)) in 𝑅B where 𝑡0 = 𝑅(𝑥1, . . . ,𝑥𝑘). Repeat +until we get to a fxed point. +• Output the relation 𝑅B ⊆ 𝐴ar𝑅 (1) × · · · × 𝐴ar𝑅 (𝑘) where 𝑅 is the output +predicate. +We will denote the output of such a Datalog program by 𝜙A. Commonly in the +context of CSPs, Datalog is restricted so that the output is a nullary predicate, so +that such a program outputs either true or false — we call such Datalog programs +Datalog sentences. The arity of a Datalog program is the arity of the output predicate, +and the width of a Datalog program is the maximal number of variables in a rule. +We defne Datalog interpretations in a similar way as pp-interpretations. This +hinges on interpretation of Datalog programs as formulae in a fragment of the +infnitary logic L∞,𝜔 (for defnition see, e.g., [ABD09]). More precisely it is included +in the existential positive fnite variable fragment of the said logic, and also in the +fxed point logic. A Datalog program 𝜙 whose output is a relation with arity 𝑖1 . . .𝑖𝑘 +is then seen as a formula of the same arity, so that A |= 𝜙(𝑎1, . . . ,𝑎𝑘) if and only if +(𝑎1, . . . ,𝑎𝑘) ∈ 𝜙A. +Definition 3.9. Fix two relational signatures 𝜎 and 𝜏, let 𝑛 be the number of types in +𝜎, and let 𝑅1, . . . , 𝑅𝑚 be the list of all relational symbols in 𝜎. A Datalog interpretation +is an (𝑛 + 𝑚)-tuple +𝝓 = (𝜙1, . . . ,𝜙𝑛;𝜙𝑅1, . . . ,𝜙𝑅𝑚) +of Datalog programs with input signature 𝜏 such that, for each relational symbol 𝑅 in +𝜎, the arity of 𝜙𝑅 is the concatenation of the arities of 𝜙𝑖1, ..., 𝜙𝑖𝑘 where ar𝑅 = 𝑖1 . . .𝑖𝑘. +Such an interpretation is said to be of width 𝑘 if all 𝜙𝑖 and 𝜙𝑅𝑗 are of width 𝑘. +The application of such a Datalog interpretation 𝝓 to a 𝜏-structure A is then +defned in the same way as in Defnition 3.8, i.e., it is the 𝜎-structure +𝝓(A) = (𝜙A +1 , . . . ,𝜙A +𝑛 ;𝜙A +𝑅, . . . ,𝜙A +𝑅𝑚) +4In database theory, Datalog programs very often have several output predicates (and defne structures +instead of relations). Such program can be viewed as a special case of a Datalog interpretation which we +defne later. + +14 +VICTOR DALMAU AND JAKUB OPRŠAL +where 𝜙A +𝑅 is interpreted as a relation on 𝜙ar𝑅 (1) (A) × · · · × 𝜙ar𝑅 (𝑘) (A) in the natural +way (i.e., as in Defnition 3.8). +Example 3.10. Let us describe an easy Datalog interpretation (𝜙1,𝜙⊥) from graphs to +structures of the same signature as the trivial template T. That is the input signature +consists of single type and a binary relation 𝐸 and the output signature consists +also of one type and one nullary relation ⊥. Hence, the input of both 𝜙1 and 𝜙⊥ is a +graph. We let 𝜙1 be the program with a single rule 𝑉 (𝑥) ← 𝑥 = 𝑥 and output 𝑉 , and +𝜙⊥ be the program with rule ⊥ ← 𝐸(𝑥,𝑥) with output ⊥. +The interpretation 𝝓 then produces from a graph G = (𝐺; 𝐸G), the structure +𝝓(G) = (𝐺; ⊥𝝓(G)) where ⊥𝝓(G) is true if G contains a loop, and false otherwise. +Hence, 𝝓(G) → T if and only if G does not have a loop, and consequently, 𝝓 is a +reduction from CSP(K∞) → CSP(T). +We again note a diference to Datalog interpretations defned in [ABD09, Section +2.3]: in the above defnition we do not allow parameters. We do this to ensure that +Datalog interpretations are monotone in the following sense. +Lemma 3.11. Let 𝝓 be a Datalog interpretation, and A and B structures. If A → B, +then 𝝓(A) → 𝝓(B). +Proof sketch. The claim follows from monotonicity of Datalog programs: assuming +a homomorphism ℎ: A → B, the component-wise application of ℎ gives a well- +defned mapping 𝜙A +𝑖 → 𝜙B +𝑖 for each 𝑖. It is then straightforward to check that the +collection of these mappings is a homomorphism 𝝓(A) → 𝝓(B). +□ +Let us briefy mention that a CSP is said to be solved by Datalog if its complement +is defnable by a Datalog sentence. Formalized in the following defnition. The +apparent negation in the defnition will become clearer when we discuss using +Datalog as reductions (see Lemma 5.1). +Definition 3.12. A PCSP(A, A′) is said to be solvable by Datalog if there is a Datalog +sentence 𝜓 such that, for all X with X → A, 𝜙X is false, and for all X with X ̸→ A′, +𝜙X is true. +3.2.2. Datalog∪ reductions. Datalog interpretations are powerful reductions; they +can reduce a substantial class of CSPs, including 2SAT and Horn-3SAT, to the trivial +problem. Also another particular example, the arc digraph construction, was used to +improve the state-of-the-art hardness of approximate graph colouring [KOWŽ22, +Section 4.3]. Nevertheless, the monotone version of the interpretation that we +defned above cannot fully emulate gadget reductions, in particular, they cannot +express taking disjoint unions of domains and relations. We note that [ABD09] use +parameters in their Datalog interpretations only to express these disjoint unions +up to a fnite number of exceptions — we do not have that liberty here since in the +multisorted setting, we would have to deal with an infnite number of exceptions. +Instead, we deal with this by extending Datalog interpretation with this operation. +We introduce a new operator called union gadget, that loosely speaking constructs +a new structure by taking disjoint unions of domains of an input structure, and in a +similar fashion defnes new relations as unions of relations of the original structure. +In the formal defnition, we need to take care to preserve types. +Definition 3.13. Assume that 𝜏 and 𝜎 are two relational signatures, and let 𝑛 and +𝑚 be the number of types in 𝜏 and 𝜎 respectively. A union gadget 𝝊 is defned by a +pair of mappings (𝑑,𝑟), where 𝑑 maps 𝜏-types to 𝜎-types and 𝑟 maps 𝜏-symbols to + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +15 +𝜎-symbols, such that for each 𝜏-symbol 𝑅 of arity 𝑖1 · · ·𝑖𝑘, the 𝜎-symbol 𝑟 (𝑅) is of +arity 𝑑(𝑖1) · · ·𝑑(𝑖𝑘). +Such a union gadget can be then applied on a structure A to produce a 𝜎-structure +𝝊(A) in the following way. The 𝑖-th domain of 𝝊(A) is the disjoint union of 𝐴𝑗’s +where 𝑑(𝑗) = 𝑖, i.e., the set {(𝑎; 𝑗) | 𝑑(𝑗) = 𝑖 and 𝑎 ∈ 𝐴𝑖}, and similarly, for a +𝜎-symbol 𝑆, 𝑆𝝊 (A) is the disjoint union of 𝑅A for 𝑅 ∈ 𝑟 −1(𝑆), more precisely, +𝑆𝝊 (A) = {((𝑎1;𝑑(𝑖1)), . . . , (𝑎𝑘;𝑑(𝑖𝑘))) | 𝑟 (𝑅) = 𝑆,𝑎 ∈ 𝑅,𝑖1 . . .𝑖𝑘 = ar𝑅}. +Example 3.14. Let us describe a union gadget that produces a graph from a multisorted +structure with two types, 0 and 1, and four binary relations 𝐸𝑖,𝑗, for 𝑖, 𝑗 ∈ {0, 1}, each +𝐸𝑖,𝑗 of arity 𝑖𝑗. There is only one choice for the maps 𝑑 and 𝑟 in this case. Given a +structure V in this signature, the union gadget 𝝊 produces a graph G with vertex set +𝑉0 ∪ 𝑉1 (assuming 𝑉0 and 𝑉1 are disjoint) and edges 𝐸 = 𝐸V +0,0 ∪ 𝐸V +0,1 ∪ 𝐸V +1,0 ∪ 𝐸V +1,1. +Note that this would not be possible by a Datalog interpretation: Consider the +structure V where 𝑉0 = {0}, 𝑉1 = {1}, and 𝐸𝑖,𝑗 = {(𝑖, 𝑗)} if 𝑖 ≠ 𝑗 and 𝐸𝑖,𝑖 = ∅. The +graph G = 𝝊(V) is then isomorphic to K2. Nevertheless, it can be verifed that every +Datalog defnable relation on V is either empty or a singleton, and hence a Datalog +interpretation 𝝓 can only produce graphs with a single vertex. None of such graphs +is homomorphically equivalent to K2. +As we will show below, it is enough to consider compositions of Datalog inter- +pretations with union gadgets in this specifc order, so we defne our extension of +Datalog as follows. +Definition 3.15. A Datalog∪ reduction is a composition 𝝊 ◦ 𝝓 where 𝝊 is a union +gadget and 𝝓 a Datalog interpretation. We write PCSP(A, A′) ≤Datalog PCSP(B, B′) if +there exists a Datalog∪ reduction that is a valid reduction between the two problems. +Clearly, every Datalog interpretation is a Datalog∪ reduction since we take the +trivial union gadget (both 𝑑 and 𝑟 being the identity map). We will show that +every gadget replacement can be also expressed as a Datalog∪ reduction up to +homomorphic equivalence (i.e., they produce homomorphically equivalent output +on the same input). We also note that a union gadget can be expressed as a gadget +though not necessarily strict gadget. We start by showing that Datalog∪ reductions +compose, i.e., the following. +Theorem 3.16. Assume that 𝝍1 and 𝝍2 are two Datalog∪ reductions such that the +output signature of 𝝍1 coincides with the input signature of 𝝍2, then there is a Datalog∪ +reduction 𝝍 such that, for all structures A, 𝝍(A) and 𝝍2𝝍1(A) are isomorphic. +The obvious consequence of the above theorem is that if PCSP(A, A′) ≤Datalog +PCSP(B, B′) ≤Datalog PCSP(C, C′), then PCSP(A, A′) ≤Datalog PCSP(C, C′), justify- +ing the notation. +We prove the theorem by showing that both Datalog interpretations and union +gadgets compose, and that a union gadget and a Datalog interpretation can be +permuted in the following sense. +Lemma 3.17. +(1) Let 𝝊 and 𝝊′ be union gadgets such that the output signature +of 𝝊 and the input signature of 𝝊′ coincide. There is a union gadget 𝝂 such that, +for all structures A, 𝝂(A) and 𝝊′𝝊(A) are isomorphic. +(2) Let 𝝓 and 𝝓′ be Datalog interpretations such that the output signature of 𝝓 +and the input signature of 𝝓′ coincide. There is a Datalog interpretation 𝝍 such +that, for all structures A, 𝝓(A) and 𝝓′𝝓(A) are isomorphic. + +16 +VICTOR DALMAU AND JAKUB OPRŠAL +(3) Let 𝝊 be a union gadget and 𝝓 be a Datalog interpretation such that the output +signature of 𝝊 and the input signature of 𝝓 coincide. There exist a union gadget +𝝊′ and a Datalog interpretation 𝝓′ such that, for all structures A, 𝝊′𝝓′(A) and +𝝓𝝊(A) are isomorphic. +Proof. +(1) Let 𝝊 be defned by (𝑑,𝑟) and 𝝊′ by (𝑑′,𝑟 ′). It is immediate that the +union gadget defned by (𝑑′ ◦ 𝑑,𝑟 ′ ◦ 𝑟) gives isomorphic outputs to 𝝊′ ◦ 𝝊. +(2) It is well-known that Datalog programs are closed under composition. We +sketch how to extend it to Datalog interpretations. We construct a new +Datalog interpretation 𝝍 as follows. For every Datalog program 𝜙 ′ +𝑗 in 𝝓′ we +include a Datalog program 𝜓𝑗 in 𝝍 obtained from 𝜙 ′ +𝑗 and 𝝓 as follows: Start +with the union of all programs in 𝝓. For each symbol 𝑅 of arity 𝑖1 . . .𝑖𝑘 in 𝜙 ′ +𝑗, +include in 𝜓𝑗 a new symbol 𝑅′ whose arity is the concatenation of arities of +𝜙𝑖1, ..., 𝜙𝑖𝑘. Then, for each rule 𝑟 of 𝜙 ′ +𝑗, include in 𝜓𝑗 a rule 𝑟 ′ obtained from +𝑟 as follows. First, in every atomic formula in 𝑟 we replace its predicate 𝑅 +by 𝑅′ and replace every variable 𝑥 occurring in it by a tuple (𝑥𝑗1, . . . ,𝑥𝑗𝑘) of +fresh variables where 𝑖 is the type of 𝑥, 𝑗1 . . . 𝑗𝑘 = ar𝑆𝑖, and 𝑆𝑖 is the output +symbol of 𝜙𝑖. It is immediate to verify that 𝝍 and 𝝓′ ◦ 𝝓 give isomorphic +outputs. +(3) Let us denote by 𝜏, 𝜎, and 𝜌 the input signature of 𝝊, the output signature of +𝝊 which coincides with the input signature of 𝝓, and the output signature of +𝝓, and let 𝝊 be defned by (𝑑,𝑟). +The proof is a simple typing exercise. Generally, we would want to adapt +𝝓 into 𝝓′ that runs directly on A by adding a rule +𝑟 (𝑅)(𝑥1, . . . ,𝑥𝑘) ← 𝑅(𝑥1, . . . ,𝑥𝑘) +for each 𝜏-symbol 𝑅. These rules obviously compute the union of 𝑅’s with +𝑟 (𝑅) = 𝑆. Nevertheless, there is a formal problem with them: the variables +𝑥𝑖 present in the rule are not properly typed since the union contains tuples +of various arities. We circumvent this by adding copies of each relational +symbol. +Let 𝜙 be a Datalog program with input signature 𝜎. We defne a new +Datalog program, or more precisely, a collection of Datalog rules without +an output predicate, which will be chosen later. Let us denote this ‘program’ +by 𝜙 ′. Its input signature is 𝜏, and it has an IDB 𝑅𝑗1...𝑗𝑘 of arity 𝑗1 · · · 𝑗𝑘 for +each symbol 𝑅 (IDB or EDB) in 𝜙 of arity 𝑑(𝑗1) · · ·𝑑(𝑗𝑘). We include, for +each 𝜎-symbol 𝑅 of arity 𝑗1 · · · 𝑗𝑘 and 𝑆 = 𝑟 (𝑅), the following rule into 𝜙 ′: +𝑆 𝑗1...𝑗𝑘 (𝑥1, . . . ,𝑥𝑘) ← 𝑅(𝑥1, . . . ,𝑥𝑘) +where each 𝑥𝑖 is of type 𝑗𝑖. Further, for each function 𝑓 from 𝜎-types to +𝜏-types such that 𝑑𝑓 is the identity, and each rule +𝑡0 ← 𝑡1, . . . ,𝑡𝑚 +of 𝜙, we introduce to 𝜙 ′ a rule +𝑡 ′ +0 ← 𝑡 ′ +1, . . . ,𝑡 ′ +𝑚 +where 𝑡 ′ +𝑖 is obtained from 𝑡𝑖 by replacing the symbol 𝑆 with 𝑆 𝑓 (𝑖1)...𝑓 (𝑖𝑘) +where 𝑖1 · · ·𝑖𝑘 is the arity of 𝑆. Note that this rule is properly typed since if +a variable 𝑥 in the original rule was of type 𝑖, it becomes a variable of type +𝑓 (𝑖). Now, for every relation symbol 𝑆 𝑗 where 𝑆 is the output predicate of 𝜙, + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +17 +we shall denote by 𝜙 𝑗 the Datalog program obtained from 𝜙 ′ by choosing +𝑆 𝑗 as the output predicate. +We construct 𝝓′ by including all possible choices of outputs in the above +construction, that is the output signature will have a type 𝑖 𝑗 for each 𝜌-type +𝑖, and 𝑗 = 𝑗1 . . . 𝑗𝑘 where 𝜙𝑖 is of arity 𝑑(𝑗1) . . .𝑑(𝑗𝑘) defned by 𝜙 𝑗 +𝑖 , and +similarly, a symbol 𝑆 𝑗 for each 𝜌-symbol 𝑆 and 𝑗 = 𝑗1 . . . 𝑗𝑚 where 𝜙𝑆 has +arity 𝑑(𝑗1) . . .𝑑(𝑗𝑚) of corresponding arity defned by 𝜙 𝑗 +𝑆. The union gadget +𝝊′ is defned by (𝑑′,𝑟 ′) where 𝑑′(𝑖 𝑗) = 𝑖 and 𝑟 ′(𝑆 𝑗) = 𝑆. It is immediate to +verify that 𝝊′𝝓′ and 𝝓𝝊 give isomorphic outputs. +□ +Example 3.18. +To bring a little light to the proof of case (3), let us describe the +construction of 𝝓′ and 𝝊′ in the particular case of the Datalog interpretation 𝝓 from +Example 3.10 and the union gadget 𝝊 from Example 3.14. +Let us start with describing the component 𝜙 ′ +⊥ of 𝝓′. Recall that 𝜙⊥ is the Datalog +sentence given by the rule ⊥ ← 𝐸(𝑥,𝑥), and that the input signature 𝜎 of 𝝊 has two +types 0, 1 and four binary relations 𝐸𝑖,𝑗, for 𝑖, 𝑗 ∈ {0, 1}, of arity 𝑖𝑗. The union gadget +𝝊 then produces digraphs in the only possible way by taking the disjoint union of all +domains and of all relations. We defne a new program 𝜙 ′ +⊥ with this input signature +by including two rules: +⊥ ← 𝐸0,0(𝑥,𝑥) +and +⊥ ← 𝐸1,1(𝑦,𝑦) +and outputting ⊥. Intuitively, instead of looking for a loop in 𝐸 which is a disjoint +union of four relations, 𝜙 ′ +⊥ looks for a loop in each of the relations separately. Note +that 𝐸0,1 and 𝐸1,0 cannot contain a loop, and hence can be skipped. +To fnish the defnition of 𝝓′, we need to defne two more Datalog programs 𝜙 ′ +0 +and 𝜙 ′ +1. Recall that the component 𝜙1 of 𝝓 has a single rule 𝑉 (𝑥) ← 𝑥 = 𝑥 and +outputs 𝑉 – each 𝜙 ′ +𝑖 therefore has a single rule 𝑉𝑖 (𝑥) ← 𝑥 = 𝑥 where 𝑥 is of type 𝑖 +and output 𝑉𝑖. +Finally, we take the disjoint union gadget 𝝊′ defned by 𝑑(𝑖) = 1 and 𝑟 (⊥) = ⊥. +It is easy to check that, indeed, for each 𝜎-structure V, 𝝊′𝝓′(V) and 𝝓𝝊(V) are +isomorphic. +Let us now prove that Datalog∪ reductions compose. +Proof of Theorem 3.16. Assume that either of 𝝍1 and 𝝍2 is decomposed as 𝝍𝑖 = 𝝊𝑖◦𝝓𝑖, +hence the composition is𝝊2◦𝝓2◦𝝊1◦𝝓1. By Lemma 3.17(3) this produces isomorphic +outputs as 𝝊2 ◦ 𝝊′ +1 ◦ 𝝓′ +2 ◦ 𝝓1 for some Datalog interpretation 𝝓′ +2 and a union gadget +𝝊′ +1. Finally, by Lemma 3.17(1–2), we can compose the two union gadgets and two +Datalog interpretations while producing isomorphic outputs. +□ +We continue with showing that a gadget can be expressed as a Datalog∪ reduction. +The precise statement is as follows. +Theorem 3.19. For every gadget 𝜸 there is a Datalog∪ reduction 𝝍 such that, for all +A, 𝜸 (A) and 𝝍(A) are homomorphically equivalent. +We prove this by frst producing a Datalog program that almost emulates a single +relation in 𝜸 as per the following lemma. We slightly abuse the terminology and +say that a tuple 𝑔 ∈ G1 × · · · × G𝑘 has arity 𝑖1 · · ·𝑖𝑘 where 𝑖𝑗 is the type of 𝑔𝑗 (recall +that by convention 𝑔 = (𝑔1, . . . ,𝑔𝑘)) even when the components of the tuple do not +belong to the same structure, and we will say that such a tuple has the same arity as +a symbol 𝑅 if ar𝑅 = 𝑖1 · · ·𝑖𝑘. + +18 +VICTOR DALMAU AND JAKUB OPRŠAL +Lemma 3.20. Let 𝜸 be a strict gadget with input signature 𝜏 and output signature 𝜎 +composed of 𝜎-structures G𝑖, let 𝑅 be a 𝜎-symbol of arity 𝑘 and let 𝑔 ∈ G𝑖1 × · · · × G𝑖𝑘 +be a tuple of the same arity as 𝑅. There is a Datalog program 𝜙𝑅𝑔 of arity 𝑖1 · · ·𝑖𝑘 with +the following property: For every 𝜏-structure A and every tuple 𝑎 ∈ 𝐴𝑖1 × · · · × 𝐴𝑖𝑘, +𝑎 ∈ (𝜙𝑅𝑔)A if and only if ([𝑎1;𝑔1], . . . , [𝑎𝑘;𝑔𝑘]) ∈ 𝑅𝜸 (A). +Proof. Let us defne program 𝜙𝑔. In addition to all input predicates in 𝜎, 𝜙𝑔 has an +IDB 𝐸ℎ1,ℎ2 of arity 𝑗1𝑗2 for each ℎ1 ∈ G𝑗1 and ℎ2 ∈ G𝑗2, and an IDB 𝑅𝑔 of arity 𝑖1 · · ·𝑖𝑘 +which is designated as output. Now, let us describe the rules of 𝜙𝑔. +(1) For every 𝜏 symbol 𝑆 of arity 𝑖𝑗, ℎ ∈ G𝑗, include the rule: +𝐸𝑝𝑆 (ℎ),ℎ(𝑥,𝑦) ← 𝑆(𝑥,𝑦) +(2) Include the rules: +𝐸ℎ1,ℎ1 (𝑥,𝑥) ← 𝑥 = 𝑥 +𝐸ℎ1,ℎ2(𝑥,𝑦) ← 𝐸ℎ2,ℎ1 (𝑦,𝑥) +𝐸ℎ1,ℎ3 (𝑥,𝑧) ← 𝐸ℎ1,ℎ2 (𝑥,𝑦), 𝐸ℎ2,ℎ3(𝑦,𝑧) +for all ℎ1,ℎ2,ℎ3 in the disjoint union of all the domains of G𝑖. +(3) Finally, add the rule: +𝑅𝑔(𝑥1, . . . ,𝑥𝑘) ← 𝐸𝑔1,ℎ1(𝑥1,𝑦), . . . , 𝐸𝑔𝑘,ℎ𝑘 (𝑥𝑘,𝑦) +for each 𝜎-type 𝑗 all ℎ ∈ 𝑅G𝑗 . +Let us show that 𝜙𝑔 has the required property. We start by observing that a +relation 𝐸ℎ1,ℎ2(𝑥,𝑦) is derived in A if and only if [𝑥;ℎ1] = [𝑦;ℎ2] — which follows +since rules introduces in item (1) introduce into 𝐸 the equality constraints of the +gadget, and the rules in item (2) then compute the transitive symmetric refexive +closure of 𝐸. Observing this, it is straightforward to see that +([𝑎1;𝑔1], . . . , [𝑎𝑘;𝑔𝑘]) ∈ 𝑅𝜸 (A) +if and only if there exists 𝑎 ∈ 𝐴𝑗 and ℎ ∈ 𝑅G𝑗 such that [𝑎;ℎ𝑖] = [𝑎𝑖;𝑔𝑖] for all 𝑖, +which equivalent to triggering the rule (3). +□ +Example 3.21. Let us explain in detail, how to obtain programs 𝜓𝐸𝑖,𝑗 (where 𝑖, 𝑗 ∈ +{0, 1}) satisfying the claim of the above lemma for the relational symbol 𝐸 and the +gadget replacement 𝜸 described in Example 3.3. We use the symbol 𝐼 instead of 𝐸 +as in the above proof so that our notation does not clash. The programs 𝜓𝐸𝑖,𝑗 are +composed of the same rules, with diferent outputs: +𝐼 0,1(𝑥,𝑦) ← 𝐸(𝑥,𝑦) +𝐼 1,0(𝑥,𝑦) ← 𝐸(𝑥,𝑦) +𝐼𝑖,𝑖 (𝑥,𝑥) ← 𝑥 = 𝑥 +𝐼𝑖,𝑗 (𝑥,𝑦) ← 𝐼 𝑗,𝑖 (𝑦,𝑥) +𝐼𝑖,𝑘 (𝑥,𝑧) ← 𝐼𝑖,𝑗 (𝑥,𝑦), 𝐼 𝑗,𝑘 (𝑦,𝑧) +𝐸𝑖,𝑗 (𝑥,𝑦) ← 𝐼𝑖,0(𝑥,𝑧), 𝐼 𝑗,1(𝑦,𝑧) +𝐸𝑖,𝑗 (𝑥,𝑦) ← 𝐼𝑖,1(𝑥,𝑧), 𝐼 𝑗,0(𝑦,𝑧) +where 𝑖, 𝑗, and 𝑘 above ranges over 0 and 1. To recall the intuition, 𝐼𝑖,𝑗 (𝑥,𝑦) is +derived if 𝑥𝑖 and 𝑦𝑗 were identifed in the third step of application of 𝜸. The output +of each 𝜓𝐸𝑖,𝑗 is then 𝐸𝑖,𝑗. +Observe that 𝐼 0,1(𝑥,𝑦) and 𝐼 1,0(𝑥,𝑦) are derived in G if and only if 𝑥 and 𝑦 are +connected by a path of odd length in G, and 𝐼 0,0(𝑥,𝑦) and 𝐼 1,1(𝑥,𝑦) are derived if +and only if 𝑥 and 𝑦 are connected by a path of even length. Consequently, we get +that, if 𝑖 ≠ 𝑗, 𝐸𝑖,𝑗 (𝑥,𝑦) is derived if and only if 𝑥 and 𝑦 are connected by a path of + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +19 +even length, and 𝐸𝑖,𝑖 is derived if and only if 𝑥 and 𝑦 are connected by a path of odd +length. This exactly corresponds to when 𝑥𝑖 and 𝑦𝑗 are identifed in the third step of +application of 𝜸. +We are ready to prove that Datalog∪ reductions can emulate gadget reductions. +Proof of Theorem 3.19. Let us assume that 𝜸 is strict which can be done without +loss of generality due to Lemma 3.17 and the fact that reifcation is a Datalog +interpretation. We now construct a Datalog interpretation 𝝓 with output 𝜎′ (to be +defned later) and a union gadget 𝝊. We assume that 𝜸 is composed of 𝜎-structures +G1, ..., G𝑚, and that these structures are disjoint. +Signature 𝜎′ contains a type𝑔 for each𝑔 ∈ G1∪· · ·∪G𝑚. Let 𝑗 be such that𝑔 ∈ G𝑗. +We defne 𝜙𝑔 to be the Datalog program composed of a single rule 𝐷𝑔(𝑥) ← 𝑥 = 𝑥, +where 𝑥 is of type 𝑗, and the output symbol of 𝜙𝑔 is 𝐷𝑔. The equality 𝑥 = 𝑥 in +the body is just an artefact to satisfy the formal requirement that the body is non +empty. Further, 𝜎′ contains a relational symbol 𝑅𝑔 for each 𝜎-symbol 𝑅 of arity 𝑘 +and each tuple 𝑔 = (𝑔1, . . . ,𝑔𝑘) of the same arity as 𝑅 where each 𝑔𝑖 is an element of +G1 ∪ · · · ∪ G𝑚. The arity of 𝑅𝑔 is 𝑔1 · · ·𝑔𝑘 and it is defned by the program 𝜙𝑅𝑔 as in +Lemma 3.20. This concludes the defnition of the interpretation 𝝓. The consecutive +union gadget 𝝊 is defned by maps (𝑑,𝑟) such that 𝑑(𝑔) is the 𝜎-type of 𝑔, and +𝑟 (𝑅𝑔) = 𝑅. +Let us now prove that𝝊𝝓(A) and𝜸 (A) are homomorphically equivalent. Observe +that we have the following chain of equivalences for 𝑘-tuples 𝑎 and 𝑔: +([𝑎1;𝑔1], . . . , [𝑎𝑘,𝑔𝑘]) ∈ 𝑅𝜸 (A) ⇔ 𝑎 ∈ 𝜙A +𝑅𝑔 ⇔ +𝑎 ∈ (𝑅𝑔)𝝓(A) ⇔ ((𝑎1;𝑔1), . . . , (𝑎𝑘,𝑔𝑘)) ∈ 𝑅𝝊𝝓(A) +where the frst equivalence is by Lemma 3.20 and the last equivalence is by the +defnition of 𝝊. Hence, the mapping 𝑓 : 𝝊𝝓(A) → 𝝓(A) defned by 𝑓 (𝑎;𝑔) = [𝑎;𝑔] is +a homomorphism, and any mapping 𝑔 such that 𝑔(𝑏) ∈ 𝑓 −1(𝑏) is a homomorphism +in the opposite direction. +□ +Example 3.22. Let us describe a Datalog∪ reduction obtained from the gadget replace- +ment 𝜸 from Example 3.3. The reduction is composed of a Datalog interpretation +𝝍 and a union gadget 𝝂. The interpretation 𝝍 outputs structures with signature 𝜎 +composed of two types 0 and 1 and four binary relations of the same arities as in +Example 3.14; to unify the signatures, we identify the symbols 𝐸𝑖,𝑗 with 𝐸𝑖,𝑗. The +interpretation 𝝍 is (𝜓0,𝜓1;𝜓𝐸0,0, . . . ) where𝜓𝐸𝑖,𝑗 are the programs from Example 3.21 +and 𝜓𝑖 for 𝑖 ∈ {0, 1} are programs with a single rule 𝐷𝑖 (𝑥) ← 𝑥 = 𝑥 and output 𝐷𝑖. +We claim that connecting this 𝝍 with the union gadget 𝝊 from Example 3.14 then +produces homomorphically equivalent outputs to 𝜸. To show that, let us assume +for simplicity that the input is a connected unoriented graph G. We distinguish two +cases: +(1) G is bipartite, and 𝜸 (G) ≃ K2. Since G is connected it splits to two parts +𝐴 and 𝐵 uniquely. Observe that 𝑥 and 𝑦 are in the same path if and only +if 𝑥 and 𝑦 are connected by a path of even length, and they are in distinct +parts if and only if 𝑥 and 𝑦 are connected by a path of odd length. Hence +𝝊𝝓(G) is the complete bipartite graph with parts (𝐴 × {0}) ∪ (𝐵 × {1}) and +(𝐴 × {1}) ∪ (𝐵 × {0}). Clearly, it is homomorphically equivalent to K2. +(2) G contains an odd cycle, and 𝜸 (G) is a loop. In this case there is an odd path +from 𝑥 to 𝑥 for each 𝑥. Hence all pairs of elements 𝑥 and 𝑦 are connected by + +20 +VICTOR DALMAU AND JAKUB OPRŠAL +both odd and even path. This implies that 𝝊𝝓(G) is a clique with all loops, +and hence clearly homomorphically equivalent to a loop. +Consequently, we have that 𝝊 ◦ 𝝍 is a reduction from CSP(K2) to CSP(K∞). +Let us fnish this section with a fnal example. +Example 3.23. Recall Examples 3.22 and 3.10 which produce reductions +CSP(K2) ≤Datalog CSP(K∞) ≤Datalog CSP(T). +Theorem 3.16 then asserts that CSP(K2) ≤Datalog CSP(K∞). This reduction uses the +composition 𝝍′ of interpretations 𝝓′ from Example 3.18 and 𝝍 from Example 3.22: +the program 𝜓 ′ +⊥ is obtained from the Datalog rules described in the above example +by adding two rules of 𝜙 ′ +⊥: +⊥ ← 𝐸0,0(𝑥,𝑥) +and +⊥ ← 𝐸1,1(𝑥,𝑥) +Note that 𝜓 ′ +⊥ in fact decides whether the input has an odd cycle or not. The program +that we implicitly constructed by joining our proofs is a more verbose version of +much simpler Datalog program solving CSP(K2) described in [KV00, Section 4.1]. +To complete the Datalog interpretation 𝝍′, we extend this Datalog program 𝜓 ′ +⊥ +with programs𝜓 ′ +0 and𝜓 ′ +1 that are identical to𝜓0 and𝜓1. Finally, we compose 𝝍′ with +the disjoint union 𝝊′ defned by (𝑑,𝑟) with 𝑑(𝑖) = 1 for 𝑖 ∈ {0, 1} and 𝑟 (⊥) = ⊥. The +composition 𝝊′ ◦ 𝝍′ is then a valid Datalog∪ reduction from CSP(K2) to CSP(T). +Finally, let us remark that although we started with a strict gadget replacement 𝜸 +and a very trivial Datalog interpretation 𝝓 of width 1 that does not uses recursion +in any of its Datalog programs (i.e., 𝝓 is actually a pp-interpretation), the program +𝝍′ has width 3 and uses recursion. It can be shown, using [LLT07] and [KV00], +that this cannot be avoided: there is no Datalog interpretation of width 2 or a +pp-interpretation that could be used instead of 𝝓′. +3.3. Combinatorial reductions +In this subsection, we describe the combinatorial counterpart of Datalog∪ reduc- +tions — called consistency reductions. This reduction is based on the local consistency +algorithm for CSPs and the uniform gadget reduction. We also prove that, in the +scope of promise CSPs, consistency reductions have the same power as Datalog∪ +reductions. +Consistency reductions are defned by the two templates and a single parameter +𝑘. Let us assume that we are trying to reduce PCSP(A, ∗) to PCSP(B, ∗); the second +part of the templates are irrelevant for the defnition of the reduction, but play +an important role for the soundness which we do not characterise here. The 𝑘- +consistency reduction is composed of two steps: (1) establishing 𝑘-consistency. This +step is possibly the most intuitive approach to solving CSPs, and it is used in many +CSPs algorithms (including, e.g., Zhuk’s polynomial algorithm). We describe it as a +procedure that inputs a CSP instance, i.e., a pair of structures X and A, and outputs +a label cover instance. (2) the universal gadget replacement. The second step is the +standard reduction from label cover to CSP(B) which we described in Defnition 3.5. +We start with describing the frst step. +Let X and A be structures of the same signature and 𝐾 ⊆ 𝑋, a partial homomor- +phism from 𝐾 to A is a mapping 𝑓 : 𝐾 → A that preserves types and relations, i.e., it +satisfes the defnition of a homomorphism when all variables are quantifed in 𝐾 +instead of 𝑋. Intuitively, a partial homomorphism is simply a partial solution to the +instance defned on the given set 𝐾. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +21 +Definition 3.24 (𝑘-consistency step). +Fix an integer 𝑘 > 1. The input to the 𝑘- +consistency procedure is a pair (X, D) where X is an instance of CSP(D) (it is useful +to assume that𝑘 is bigger than the maximal arity of relations of X since the procedure +below ignores all constraints of arity bigger than 𝑘). Also we denote by � 𝑋 +≤𝑘 +� the set +of all at most 𝑘-element subsets of 𝑋. +(1) For each 𝐾 ∈ � 𝑋 +≤𝑘 +�, let F𝐾 be the set of all partial homomorphisms from 𝐾 +to D. +(2) Ensure that for each 𝐿 ⊂ 𝐾 ∈ � 𝑋 +≤𝑘 +�, the sets F𝐾 and F𝐿 are consistent, i.e., +• remove from F𝐿 all ℎ: 𝐿 → 𝐷 that do not extend to a 𝑔: 𝐾 → 𝐷, +𝑔 ∈ F𝐾, +• remove from F𝐾 all 𝑔: 𝐾 → 𝐷 whose restriction 𝑔|𝐿 is not in F𝐿. +(3) Repeat step (2), while anything changes. +(4) Output the label cover instance with a variable 𝑣𝐾 with domain F𝐾 for each +𝐾 ∈ � 𝑋 +≤𝑘 +�, and a constraint between 𝑣𝐾 and 𝑣𝐿 for each 𝐿 ⊂ 𝐾 of the form +𝜋(𝑣𝐾) = 𝑣𝐿 where 𝜋(𝑔) = 𝑔|𝐿. +We denote the output of this step by 𝜅𝑘 (X, D). +The 𝑘-consistency step is turned into a decision algorithm for CSPs by outputting +no if one of the sets F𝐾 (and consequently each of them) is empty, and outputting yes +if all F𝐾’s are non-empty. The resulting algorithm coincides with [BK14, Algorithm +1 on p.5] though we are using a single parameter 𝑘 instead of two parameters 𝑘 and +𝑙. (Promise) CSPs solved by this algorithm are said to have bounded width.5 +Definition 3.25 (bounded width). We say that a PCSP(A, A′) has width at most 𝑘 if +the 𝑘-consistency algorithm solves this problem, i.e., if every X, such that F𝐾 ≠ ∅ +for all 𝐾 ∈ � 𝑋 +≤𝑘 +� where F𝐾 denotes the domains in 𝜅𝑘 (X, A), maps homomorphically +to A′. Such a problem is said to be of bounded width if there exists 𝑘 such that the +problem has width at most 𝑘. +Let us now describe how to turn the 𝑘-consistency step into a reduction. The +resulting reduction depends on the two structures A and B, appearing in the tem- +plates of the promise CSPs involved, and the parameter 𝑘. The higher 𝑘 we chose, +the better reduction we obtain, i.e., it will be a valid reduction for more choices of +the second part of the templates. The running time depends exponentially on 𝑘, +hence 𝑘 has to be fxed in order to obtain a polynomial-time algorithm. +Definition 3.26 (𝑘-consistency reduction). Fix two relational structures A and B, and +an integer 𝑘 > 1. The 𝑘-consistency reduction applied to an instance X of CSP(A) is +the following construction: +(K1) First run 𝑘-consistency step on input X, A as described above, i.e., compute +the label cover instance 𝜅𝑘 (X, A). +(K2) Encode the label cover instance into an instance of CSP(B), i.e., a structure +with the same signature as B, using the universal gadget for B in the following +way: +• Replace each variable 𝑣𝐾 of the label cover with domain F𝐾 by a copy +of BF𝐾 whose elements are denoted by (𝐾;𝑏) where 𝑏 : F𝐾 → 𝐵. +• For each constraint 𝐿 ⊂ 𝐾 of the label cover, identify the elements (𝐿;𝑏) +with (𝐾;𝑏′) where 𝑏′(𝑓 ) = 𝑏(𝑓 |𝐿), for all 𝑏 : F𝐿 → 𝐵. +5The name ‘bounded width’ is derived from the fact that the template of such CSP has bounded +treewidth duality [FV98, Theorem 23]. + +22 +VICTOR DALMAU AND JAKUB OPRŠAL +Similarly as for gadget replacements, we will denote by [𝐾;𝑏] the element +obtained from (𝐾;𝑏) after identifcation. +We denote the resulting structure by 𝜿A,B +𝑘 +(X). +For 𝑘 > 1, we say that PCSP(A, A′) reduces PCSP(B, B′) by the 𝑘-consistency +reduction, and write PCSP(A, A′) ≤𝑘-cons PCSP(B, B′) if 𝜿A,B +𝑘 +is a reduction between +these promise CSPs. If such a 𝑘 exists, we say that that PCSP(A, A′) reduces to +PCSP(B, B′) by a consistency reduction. +The 𝑘-consistency reduction produces from a structure of the same type as A, a +structure 𝜿A,B +𝑘 +(X) of the same signature as B through producing an intermediate +instance of label cover 𝜅𝑘 (X, A). +The main result of this section is that Datalog∪ reductions and consistency +reductions have the same power in the scope of promise CSPs in the following sense. +Theorem 3.27. PCSP(A, A′) ≤Datalog PCSP(B, B′) if and only if there exists 𝑘 > 1 +such that PCSP(A, A′) ≤𝑘-cons PCSP(B, B′). +We prove this theorem using the following special case which also relates the +width of a Datalog interpretation with the parameter 𝑘 of the 𝑘-consistency reduc- +tion. +Theorem 3.28. +Fix 𝑘 > 1. PCSP(A, A′) ≤𝑘-cons PCSP(B, B′) if and only if there +exists a Datalog interpretation 𝝓 of width 𝑘 and a strict gadget 𝜸 such that 𝜸 ◦ 𝝓 is a +valid reduction between these promise CSPs. +Let us briefy outline how Theorem 3.27 follows from the above. +Proof. Proof of Theorem 3.27 given Theorem 3.28 Assume that PCSP(A, A′) reduces +to PCSP(B, B′) by a Datalog∪ reduction. Since a union gadget can be expressed as a +gadget replacement and each gadget replacement decomposes as a reifcation and a +strict gadget replacement, we can assume that this reduction is of the form 𝜸 ◦ 𝝓 +for some Datalog interpretation 𝝓 and a strict gadget replacement 𝜸. Therefore, +PCSP(A, A′) reduces to PCSP(B, B′) via the 𝑘-consistency reduction where 𝑘 is the +width of 𝝓 by Theorem 3.28. Conversely, we get that if PCSP(A, A′) reduces to +PCSP(B, B′) by a 𝑘-consistency reduction then it reduces by a Datalog∪ reduction +by the other implication of Theorem 3.28 and Theorems 3.19 and 3.16. +□ +In order to prove Theorem 3.28, we will frst show that the𝑘-consistency reduction +can be equivalently expressed as a composition of a Datalog interpretation of width +𝑘 and a gadget replacement — namely, it is essentially a composition of a canonical +Datalog interpretation of width 𝑘 and the universal gadget. Formally, we will prove +the following. +Lemma 3.29. Let A and B be two structures, and 𝑘 > 1. There is a Datalog interpre- +tation 𝝓 of width 𝑘, and a strict gadget replacement 𝜸 such that, for all structures X of +the same signature as A, 𝜸𝝓(X) and 𝜿A,B +𝑘 +(X) are isomorphic. +In the proof of this lemma, we will use some basic facts about the connection +between Datalog and local consistency. This connection is well-studied (see, e.g., +[KV95, FV98]), and the above lemma is a consequence of known results through a +new perspective which involves a rather technical notation. +Let us frst outline a few notational simplifcations. We adopt functional notation +for tuples. Let 𝑤 ∈ 𝑋𝑛 be a tuple, we denote by im𝑤 the set of all its entries, i.e., +im𝑤 = {𝑤1, . . . ,𝑤𝑛}, and we will view𝑤 as a function [𝑛] → im𝑤. If 𝜋 : [𝑚] → [𝑛], + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +23 +we denote by 𝑤 ◦ 𝜋 the tuple (𝑤𝜋 (1), . . . ,𝑤𝜋 (𝑚)), and if 𝑤 is an injective tuple, +𝑤−1 : im𝑤 → [𝑛] is the function returning the index of appearance of a component +in 𝑤, i.e., 𝑤−1(𝑤𝑖) = 𝑖. We will allow to compose functions 𝑎: 𝐾 → 𝐿 and 𝑎′: 𝐾 ′ → +𝐿′ if 𝐿 ⊆ 𝐾 ′ resulting with a function 𝑎′ ◦ 𝑎: 𝐾 → 𝐿′. If 𝑅 is a 𝑛-ary relation, and +𝜋 : [𝑚] → [𝑛], we write 𝑅 ◦ 𝜋 for the 𝑚-ary relation {𝑎 ◦ 𝜋 | 𝑎 ∈ 𝑅}. Assuming F𝐾 +for 𝐾 ∈ � 𝑋 +≤𝑘 +� is a system output by the 𝑘-consistency procedure, and 𝑤 ∈ 𝑋𝑛 is such +that im𝑤 has at most 𝑘 elements, we let F𝑤 = Fim 𝑤 ◦ 𝑤. These sets satisfy +F𝑤 = F𝐾 ◦ 𝑤 = {(𝑓 (𝑤1), . . . , 𝑓 (𝑤𝑛)) | 𝑓 ∈ F𝐾} +for all 𝑤 ∈ 𝑋𝑛 and 𝐾 such that im𝑤 ⊂ 𝐾, which is a direct consequence of the +defnition and the consistency of F𝐾’s. Similarly, we can show that +F𝑤◦𝜋 = F𝑤 ◦ 𝜋 +for all 𝑤 and 𝜋 which will be useful later. We write 𝑋 ≤𝑘 for � +𝑚≤𝑘 𝑋𝑚, and note +that if 𝑤 ∈ 𝑋 ≤𝑘 then F𝑤 is defned (the converse is not true since there are tuples +with higher arity and repeated entries with im𝑤 of size at most 𝑘). In the rest +of this section we will work with 𝑘-consistency using this notation. Finally, we +will work with functions 𝑑 : 𝑅 → 𝐵 where 𝑅 ⊆ 𝐴𝐾 for some 𝐾. We will use the +following notation, if 𝜋 : 𝐾 → 𝐿, 𝑆 ⊆ 𝐴𝐿, and 𝑆 ◦ 𝜋 ⊆ 𝑅, then 𝑑𝜋 : 𝑆 → 𝐵 is defned +as 𝑑𝜋 (𝑎) = 𝑑(𝑎 ◦ 𝜋). This is in agreement with an established notation for minors of +a function of arity 𝐾 (see Section 4.1). +One of the keys to showing that 𝑘-consistency procedure can replace a Datalog +interpretation of width𝑘 is the following statement, which has been proved implicitly +in [KV95] (see, e.g., Theorem 4.8). +Lemma 3.30. Let A and X be structures, 𝑘 > 1, and F𝐾 denote the sets output by +𝜅𝑘 (X, A) of the 𝑘-consistency procedure. For each Datalog program 𝜙 of width 𝑘, we +have that if 𝑤 ∈ 𝜙X then F𝑤 ⊆ 𝜙A. +Note that in the statement above, 𝑤 ∈ 𝜙X immediately gives that im𝑤 has at +most 𝑘 elements even when the tuple might have larger arity since 𝜙 is of width +𝑘, and hence it can only introduce tuples with at most 𝑘 diferent entries into the +relation 𝜙X. +Next, we construct the canonical Datalog interpretation. This interpretation is +based on so-called canonical Datalog program 𝜙 of width 𝑘 for a 𝜏-structure A defned +as follows: Its signature is the set of all relations that are defnable in A by a Datalog +program of width 𝑘, i.e., all 𝑅 ⊆ 𝐴≤𝑘 for which there is a Datalog program 𝜓 of +width 𝑘 with 𝑅 = 𝜓 A. We treat each such relation as an abstract symbol. The rules +of the canonical Datalog are all the rules that are satisfed in A when each rule is +interpreted as an implication 𝑡0 ← 𝑡1 ∧ · · · ∧ 𝑡𝑟 where 𝑡0 is the head and 𝑡1, . . . ,𝑡𝑟 is +the body. With this program at hand, we can construct the canonical interpretation +by choosing diferent output predicates, namely, we let 𝜙𝑖 to be the Datalog formula +obtained from the canonical Datalog program by designating 𝑅𝑖 as an output, where +𝑅1, . . . , 𝑅𝑚 is a list of all symbols of 𝜙. Further, we add a binary relation 𝑃𝑖,𝑗,𝜋 of +arity 𝑖𝑗 for each 𝑖, 𝑗 and 𝜋 : [ar𝑅𝑗 ] → [ar𝑅𝑖] such that 𝑅𝑖 ◦ 𝜋 ⊆ 𝑅𝑗. The relation 𝑃𝑖,𝑗,𝜋 +contains all pairs of the form (𝑎,𝑎 ◦ 𝜋) where 𝑎 ∈ 𝑅𝑖. Clearly, such a relation is +defnable by the canonical Datalog program extended with the rule +𝑃𝑖,𝑗,𝜋 (𝑥1, . . . ,𝑥𝑛,𝑥𝜋 (1), . . . ,𝑥𝜋 (𝑚)) ← 𝑅𝑖 (𝑥1, . . . ,𝑥𝑛) +where𝑛 is the arity of 𝑅𝑖 and𝑚 is the arity of 𝑅𝑗. Note that this Datalog interpretation +satisfes 𝜙𝑖 (A) = 𝑅𝑖 for all 𝑖 where 𝑅𝑖 on the right-hand side is interpreted as the +actual relation on 𝐴, i.e., each 𝑅𝑖 is actually defned by 𝜙𝑖 in A. + +24 +VICTOR DALMAU AND JAKUB OPRŠAL +The key property of the canonical Datalog program is that it is in a sense most +general Datalog program of width 𝑘 with respect to A. In particular, we will use the +following. +Lemma 3.31. Let 𝝓 be the canonical Datalog interpretation of width 𝑘 for A, X a +structure, F𝐾 denote the sets output by 𝜅𝑘 (X, A), and let 𝜏 denote the output signature +of 𝝓. If 𝑤 ∈ 𝑋 ≤𝑘, then there exists a 𝜏-type 𝑖 such that F𝑤 = 𝜙A +𝑖 and 𝑤 ∈ 𝜙X +𝑖 . +Proof sketch. The lemma follows from, e.g., [KV95]. We briefy sketch a direct +argument. We show that F𝑤 is Datalog defnable for each 𝑤 ∈ 𝑋 ≤𝑘, which can be +argued by induction through the evaluation of the 𝑘-consistency algorithm. In the +frst step, F𝑤 is initiated as 𝐴𝑙 where 𝑙 is the length of 𝑤, which is clearly Datalog +defnable in A, and is satisfed by 𝑤 in X. Then in each of the iterated steps, we alter +F𝑤 by removing certain values, which is equivalently expressed by, e.g., a Datalog +rule of the form +F ′ +𝑤(𝑥1, . . . ,𝑥𝑙) ← F𝑤(𝑥1, . . . ,𝑥𝑙), F𝑤′(𝑥1, . . . ,𝑥𝑙′). +where F ′ +𝑤 denotes the new value for F𝑤, which is a rule in 𝑘 variables valid in A +and hence a rule of the canonical Datalog. Again, it is easy to observe that the +corresponding relation is derived for 𝑤 on X. +□ +This observation, together with Lemma 3.30 describes the close relationship +between the 𝑘-consistency procedure and the canonical Datalog interpretation: F𝑤 +is the minimal 𝜙A +𝑖 (w.r.t. inclusion) such that 𝑤 ∈ 𝜙X +𝑖 . The diference between the +𝑘-consistency step and the canonical Datalog interpretation (apart from the output +of 𝑘-consistency step being a label cover instance and the output of the canonical +Datalog a binary projective structure) is that in 𝝓(X), each 𝑤 might appear in several +domains of 𝝓(X), hence it represents several elements of diferent types: one of +each type 𝑖 where 𝑤 ∈ 𝜙X +𝑖 . The domain of the corresponding variable 𝑤 of type 𝑖 +is 𝜙A +𝑖 . In the output of 𝑘-consistency, 𝑤 has only one copy with domain F𝑤 = 𝜙A +𝑖 +for a suitable 𝑖. This diference is then smoothed by the use of the universal gadget. +Let us for future reference note that the universal gadget for 𝝓(A) and B introduces +an equality constraint between (𝑣;𝑑) and (𝑤;𝑒) where 𝑣 ∈ 𝜙X +𝑖 , 𝑤 ∈ 𝜙X +𝑗 , 𝑑 : 𝜙A +𝑖 → 𝐵, +and 𝑒 : 𝜙A +𝑗 → 𝐵, if (𝑣,𝑤) ∈ 𝑃𝝓(X) +𝑖,𝑗,𝜋 (which is equivalent to 𝜙A +𝑖 ◦ 𝜋 ⊆ 𝜙A +𝑗 and 𝑣 ◦ 𝜋 = 𝑤) +and 𝑑 = 𝑒𝜋 for some 𝜋. +Proof of Lemma 3.29. Let 𝜸 be the universal gadget for 𝝓(A) and B. We claim that +𝜸𝝓(X) is isomorphic to 𝜿A,B +𝑘 +(X) for each X. We show this by constructing two +mutually inverse homomorphisms. In both cases, we defne the homomorphisms +on the disjoint union of the gadgets before introducing and collapsing equality +constraint, and subsequently argue that the value is not changed after collapsing +said constraints. +First, we construct ℎ: 𝜿A,B +𝑘 +(X) → 𝜸𝝓(X). We defne ℎ on the disjoint union of +BF𝐾 ’s before collapsing the equality constraints, and then show that it preserves +equality constraints introduced by the gadget replacement, hence inducing the +required homomorphism. For each 𝐾, fx a bijection 𝑤𝐾 : [𝑙] → 𝐾 and let 𝑖𝐾 be +such that F𝑤𝐾 = 𝜙A +𝑖𝐾 and 𝑤𝐾 ∈ 𝜙X +𝑖𝐾 , which exists by Lemma 3.31. Henceforth, we +interpret the symbol 𝑤𝐾 as the element of type 𝑖𝑘 in 𝝓(X). Let +ℎ(𝐾;𝑑) = [𝑤𝐾;𝑑𝑤−1 +𝐾 ] +where 𝑑𝑤−1 +𝐾 : 𝜙A +𝑖𝐾 → 𝐵 is defned to map 𝑎 to 𝑑(𝑎 ◦ 𝑤−1). We claim that ℎ preserves +the relations. The proof of the claim is straight-forward: Each tuple in a relation + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +25 +𝑅 of the disjoint union of BF𝐾 ’s is of the form (𝑑1, . . . ,𝑑𝑙) ∈ 𝑅BF𝐾 for some 𝐾, and +𝑑 ↦→ 𝑑𝑤−1 +𝐾 is a homomorphism BF𝐾 → B𝜙𝑖𝑘 (A). +We check that ℎ preserves the equality constraints, i.e., that ℎ(𝐾;𝑑) = ℎ(𝐿;𝑒) +whenever the universal gadget introduces an equality between (𝐾;𝑑) and (𝐿;𝑒), +i.e., when 𝐿 ⊂ 𝐾 and 𝑑 = 𝑒1𝐿 where 1𝐿 : 𝐿 → 𝐾 is the inclusion. Let 𝜋 = 𝑤−1 +𝐾 ◦ 𝑤𝐿, +which means that 𝑤𝐾 ◦ 𝜋 = 𝑤𝐿, and hence (𝑤𝐿,𝑤𝐾) ∈ 𝑃𝝓(X) +𝑖𝐿,𝑖𝐾,𝜋. Consequently, the +constraint (𝑤𝐿;𝑏) = (𝑤𝐾;𝑏𝜋) is introduced to 𝜸𝝓(X) for each 𝑏. By substituting +𝑏 = 𝑒𝑤−1 +𝐿 , we get that +ℎ(𝐿;𝑒) = [𝑤𝐿;𝑒𝑤−1 +𝐿 ] = [𝑤𝐾;𝑒𝜋◦𝑤−1 +𝐿 ] = [𝑤𝐾;𝑒𝑤−1 +𝐾 ◦1𝐿] = [𝑤𝐾;𝑑𝑤−1 +𝐾 ] = ℎ(𝐾;𝑑) +as we wanted to show. +Second, we construct 𝑔: 𝜸𝝓(X) → 𝜿A,B +𝑘 +(X). Again, we defne 𝑔 on the disjoint +union of B𝜙𝑖 (A)’s introduced to𝜸𝝓(X) by replacing 𝑤 ∈ 𝜙𝑖 (X) for all 𝑖. The mapping +𝑔 is defned as +𝑔(𝑤;𝑑) = [im𝑤;𝑑𝑤] +where 𝑤 ∈ 𝜙X +𝑖 , 𝑑 : 𝜙A +𝑖 → 𝐵, and 𝑑𝑤 : F𝐾 → 𝐵 is defned by 𝑑𝑤(𝑓 ) = 𝑑(𝑓 ◦ 𝑤). Since +𝑤 ∈ 𝜙X +𝑖 we have F𝑤 ⊆ 𝜙A +𝑖 by Lemma 3.30, and hence 𝑑𝑤 is a well-defned function +with domain F𝐾. Checking that 𝑔 is well-defned is straightforward and analogous +to the argument that ℎ is well-defned. +Finally, we argue that ℎ and 𝑔 are mutually inverse. The fact that 𝑔ℎ is the identity +is immediate from the defnition since im𝑤𝐾 = 𝐾 and 𝑑𝑤𝐾 ◦𝑤−1 +𝐾 = 𝑑. Let us show that +ℎ𝑔 is the identity as well. We have +ℎ𝑔(𝑤;𝑑) = ℎ(im𝑤;𝑑𝑤) = [𝑣;𝑑𝑣−1◦𝑤] +for 𝑤 ∈ 𝜙X +𝑖 where 𝑣 = 𝑤im 𝑤 is an element of type 𝑗 = 𝑖im 𝑤 and 𝜙A +𝑗 = F𝑣. Let +𝜋 = 𝑣−1 ◦ 𝑤, hence 𝑣 ◦ 𝜋 = 𝑤. Moreover, 𝜙A +𝑗 ◦ 𝜋 = F𝑣 ◦ 𝜋 = F𝑤 ⊆ 𝜙A +𝑖 where +the last inclusion is given by Lemma 3.31, hence the required equality constraint +(𝑣;𝑑𝜋) = (𝑣 ◦𝜋;𝑑) is ensured by replacing 𝑃𝑗,𝑖,𝜋. Consequently, ℎ𝑔(𝑤;𝑑) = [𝑣;𝑑𝜋] = +[𝑣 ◦ 𝜋;𝑑] = [𝑤;𝑑] as we wanted to show. +□ +An important consequence of the above lemma is that the𝑘-consistency reduction +is expressible as a Datalog∪ reduction up to homomorphic equivalence. This fact +has a few consequences, e.g., the 𝑘-consistency reduction is monotone. +Let us now move to the proof of the other implication. We will use the following +lemma to prove the completeness of the 𝑘-consistency reduction. +Lemma 3.32. +Let A, B be relational structures, and 𝑘 > 1. If X → A for some +structure X, then 𝜿A,B +𝑘 +(X) → B. +Proof. Let 𝑔: X → A be a homomorphism. First, we observe that, for each 𝐾 ∈ � 𝑋 +≤𝑘 +�, +the restriction 𝑔|𝐾, satisfes 𝑔|𝐾 ∈ F𝐾: this is since the restrictions are locally +consistent partial homomorphisms and will never be removed from F𝐾. We use this +observation to defne a homomorphism ℎ: 𝜿A,B +𝑘 +(A) → B. Again, we defne ℎ on the +disjoint union of F𝐾’s for 𝐾 ∈ � 𝑋 +≤𝑘 +� frst. Let ℎ(𝐾;𝑏) = 𝑏(𝑔|𝐾), 𝑏 ∈ 𝐵F𝐾 . Clearly ℎ +is a homomorphism. We need to show that for every 𝐾, 𝐿 ∈ � 𝑋 +≤𝑘 +�, ℎ gives the same +value on the elements glued in step (K2). This is straightforward since if 𝐿 ⊂ 𝐾, and +𝑏(𝑓 ) = 𝑏′(𝑓 |𝐿), then ℎ(𝐾;𝑏) = 𝑏(𝑔|𝐾) = 𝑏′((𝑔|𝐾)𝐿) = 𝑏′(𝑔|𝐿) = ℎ(𝐿;𝑏′). +□ +The soundness of the 𝑘-consistency is achieved by the following lemma, which +is a combination of two statements: frst, that 𝑘-consistency can be used instead + +26 +VICTOR DALMAU AND JAKUB OPRŠAL +of any other Datalog interpretation of width 𝑘 in a similar way as the canonical +Datalog interpretation, and that the universal gadgets can be used instead of any +other gadget replacement. We present the proof as a common generalisation of these +two statements instead of presenting the proofs of two cases separately. +Lemma 3.33. Let A, B be a pair of structure 𝑘 > 1, 𝝓 be a Datalog interpretation of +width 𝑘, 𝜸 be a strict gadget replacement, and assume 𝜸𝝓(A) → B. Then for all X, +𝜸𝝓(X) → 𝜿A,B +𝑘 +(X). +Proof. Assume that 𝜸 is composed of structures D1, ..., D𝑛 and homomorphisms +𝑝𝑅, and let 𝑏 : 𝜸𝝓(A) → B. We defne a homomorphism ℎ: 𝜸𝝓(X) → 𝜿A,B +𝑘 +(X) as +follows. Assume that [𝑤;𝑔] ∈ 𝜸𝝓(X), i.e., 𝑤 ∈ 𝜙X +𝑖 for some type 𝑖 and 𝑔 ∈ D𝑖. By +Lemma 3.30, we have that F𝑤 ⊆ 𝜙A +𝑖 . We let +ℎ(𝑤;𝑔) = [im𝑤;𝑑] +where 𝑑 : Fim 𝑤 → B is defned by 𝑑(𝑎) = 𝑏([𝑎 ◦ 𝑤;𝑔]); note that 𝑎 ◦ 𝑤 ∈ 𝜙A +𝑖 and +hence (𝑎 ◦ 𝑤;𝑔) ∈ 𝜸𝝓(A). We have to argue that ℎ is a homomorphism, and that +it is well-defned, i.e., it preserves equality constraints introduced by the gadget +replacement. +First, we argue that the mapping ℎ is well-defned. Let (𝑤;𝑔) and (𝑤 ′;𝑔′) be +related by an equality constraint introduced to𝜸𝝓(X) by replacing a pair of elements +of 𝝓(X) related by 𝑅, i.e., (𝑤,𝑤 ′) ∈ 𝑅𝝓(X) and 𝑔′ = 𝑝𝑅(𝑔) for some relational symbol +𝑅 of arity 𝑖𝑖′. More precisely, we have 𝑤 ∈ 𝜙X +𝑖 , 𝑤 ′ ∈ 𝜙X +𝑖′ and 𝑤𝑤 ′ ∈ 𝜙X +𝑅 where 𝑤𝑤 ′ +denotes the concatenation of 𝑤 and 𝑤 ′. Let 𝐾 = im𝑤 ∪ im𝑤 ′ and observe that +(𝑓 ◦𝑤, 𝑓 ◦𝑤 ′) ∈ 𝑅𝝓(A) for each 𝑓 ∈ F𝐾 since F𝑤𝑤′ ⊆ 𝜙A +𝑅 (Lemma 3.31). We get that +ℎ(𝑤;𝑔) = [im𝑤;𝑑] = [𝐾;𝑒] +where 𝑑(𝑎) = 𝑏([𝑎 ◦ 𝑤;𝑔]) and 𝑒(𝑓 ) = 𝑑(𝑓 |im 𝑤); the second equality is given +by an equality constraint introduced to 𝜅A,B +𝑘 +(X). Note that 𝑒(𝑓 ) = 𝑏([𝑓 ◦ 𝑤;𝑔]) +since the restriction of 𝑓 to im𝑤 is implicit in this expression. Similarly, we have +ℎ(𝑤 ′;𝑝𝑅(𝑔)) = [𝐾;𝑒′] where 𝑒′(𝑓 ) = 𝑏([𝑎◦𝑤 ′;𝑝𝑅(𝑔)]). Since (𝑓 ◦𝑤, 𝑓 ◦𝑤 ′) ∈ 𝑅𝝓(A), +the equality constraint (𝑓 ◦𝑤;𝑔) = (𝑓 ◦𝑤 ′;𝑝𝑅(𝑔)) is introduced to𝜸𝝓(A) by replacing +this pair. Consequently, 𝑏([𝑓 ◦𝑤;𝑔]) = 𝑏([𝑓 ◦𝑤 ′;𝑝𝑅(𝑔)]) for all 𝑓 ∈ F𝐾, and hence +𝑒 = 𝑒′ concluding ℎ(𝑤;𝑔) = [𝐾;𝑒] = [𝐾;𝑒′] = ℎ(𝑤 ′;𝑝𝑅(𝑔)). +Second, we argue that ℎ is a homomorphism. Tuples in 𝑆𝜸𝝓(X) are of the form +([𝑤;𝑔1], . . . , [𝑤;𝑔𝑚]) where (𝑔1, . . . ,𝑔𝑚) ∈ 𝑆D𝑗 and 𝑗 is the type of 𝑤. We get that +ℎ(𝑤;𝑔𝑖) = [im𝑤;𝑑𝑖] where 𝑑𝑖 (𝑎) = 𝑏([𝑎 ◦ 𝑤;𝑔𝑖]). Observe that +([𝑎 ◦ 𝑤;𝑔1], . . . , [𝑎 ◦ 𝑤;𝑔𝑚]) ∈ 𝑆𝜸𝝓(A). +Therefore, we get the claim from the defnition of power and the fact that 𝑏 is a +homomorphism. +□ +Finally, we can fnish the proof that 𝑘-consistency and Datalog∪ reductions have +the same power. +Proof of Theorem 3.28. The implication (2)→(1) follows directly from Lemma 3.29. +Let us prove (1)→(2). We assume that 𝜸𝝓 is a reduction from PCSP(A, A′) to +PCSP(B, B′), and claim that so is 𝜅A,B +𝑘 +. First, 𝜅A,B +𝑘 +preserves positive instances by +Lemma 3.32. We show soundness by the contrapositive, assuming 𝜅A,B +𝑘 +(X) → B′. +Note that 𝜸𝝓(A) → B by completeness of 𝜸𝝓. Consequently, we can invoke +Lemma 3.33 to get that 𝜸𝝓(X) → B′, and conclude that X → A′ by the sound- +ness of 𝜸𝝓. +□ + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +27 +Theorem 3.28 has a few consequences, most prominently, since Datalog∪ re- +ductions compose, we get that if one (promise) CSP reduced by a 𝑘-consistency +reduction to a second (promise) CSP which in turn reduces to a third (promise) +CSP by an 𝑙-consistency reduction, then the frst problem reduces to the third by +a 𝑚-consistency reduction for some 𝑚 that depends on 𝑘, 𝑙, and the signatures +involved. +4. ARC-CONSISTENCY REDUCTION +In this section we characterise the applicability of a special case of a Datalog∪ +reduction. Namely, a reduction that is obtained from the 𝑘-consistency reduction out- +lined in the previous section by replacing the 𝑘-consistency step with arc-consistency +step. We describe how is this procedure run on projective instances (instances of +label cover) — in the full reduction it will be preceded by a reifcation. +Definition 4.1 (arc-consistency step). Assume I is a label cover instance with each +variable 𝑣 having the domain 𝐷𝑣. +(1) For each variable 𝑣, set F𝑣 = 𝐷𝑣, +(2) Ensure that for each constraint 𝜋(𝑣) = 𝑤, the sets F𝑣 and F𝑤 are consistent: +• remove from F𝑤 all 𝑑 that are not in the image of F𝑣 under 𝜋, +• remove from F𝑣 all 𝑑 s.t. 𝜋(𝑑) ∉ F𝑤. +(3) Repeat step (2), while anything changes. +(4) Output the label cover instance with the same variables as X where each +variable 𝑣 is assigned domain F𝑣 (instead of 𝐷𝑣), and with constraints of +the form 𝜋|F𝑣 (𝑣) = 𝑤 for each original constraint 𝜋(𝑣) = 𝑤 (note that 𝜋 +restricts to a function F𝑣 → F𝑤). +We denote the resulting label cover instance by 𝜅arc(I). +Compare the arc-consistency step with the 𝑘-consistency step described in Sec- +tion 3.3, the only signifcant diference is the initialisation: the local consistency can +be also described as running the above arc-consistency on a label cover instance +with a variable 𝑣𝐾 for each 𝐾 ∈ � 𝑋 +≤𝑘 +� with the domain F𝐾 as defned by item (1) in the +local consistency procedure, and a constraint 𝑅𝜋 for each 𝐿 ⊂ 𝐾 where 𝜋 : F𝐾 → F𝐿 +is defned by 𝜋(ℎ) = ℎ|𝐿. +Definition 4.2 (arc-consistency reduction). The arc-consistency reduction applied an +instance X of CSP(A) to obtain an instance of CSP(B) is then the following 3-step +construction: +(A1.1) create a label cover instance I from X and A by reifcation and Defnition 2.9; +(A1.2) apply the arc-consistency step as described above to get an instance 𝜅arc(I) +of label cover; and +(A2) turn the resulting instance of label cover into an instance of CSP(B) using +the universal gadget, i.e., in the same way as in Step (K2) of 𝑘-consistency +(see Defnition 3.26). +We denote the resulting structure by 𝜅A,B +arc (X). +We say that PCSP(A, A′) reduces to PCSP(B, B′) by the arc-consistency reduction, +and write PCSP(A, A′) ≤arc-cons PCSP(B, B′), if 𝜅A,B +arc is a proper reduction between +these two problems. +We characterise when arc-consistency reduction gives a proper reduction between +two promise CSPs using the notions of polymorphisms, minion homomorphisms, + +28 +VICTOR DALMAU AND JAKUB OPRŠAL +and a certain transformation on the polymorphism minion of the frst template. We +defne the necessary notions in the following subsection. +4.1. Polymorphisms and minions +We recall the known characterisation of gadget reductions, and the notions +necessary for that. These notions will be also used in the characterisation of the +arc-consistency reduction. +Definition 4.3. Let A, A′ be a promise template, and 𝑋 a fnite set. A polymorphism +from A to A′ of arity 𝑋 is a homomorphism 𝑓 : A𝑋 → A′. The set of all 𝑋-ary +polymorphisms from A to A′ is denoted by Pol(𝑋) (A, A′), and the collection of all +these sets is denoted by Pol(A, A′). If A = A′, we write Pol(A) instead of Pol(A, A). +To spell out what being a polymorphism means, let us repeat the preservation +property without referring to the direct product. For each relational symbol 𝑅 of +arity 𝑖1 · · ·𝑖𝑘, and a tuple (𝑎1, . . . ,𝑎𝑘) ∈ 𝐴𝑋 +𝑖1 × · · · × 𝐴𝑋 +𝑖𝑘 of functions from 𝑋, write +the elements of these tuples into a 𝑘 × |𝑋 | matrix by placing each 𝑎𝑖 onto its own +row. The condition on 𝑓 being a polymorphism reads: if every column of this matrix +is in the relation 𝑅A, then the 𝑘-tuple obtained by applying 𝑓 on each row, is in 𝑅A′. +Polymorphisms form an object that is called a minion. Minions, or more precisely +functions minions are defned in [BBKO21, Defnition 2.20]. Here we defne an +abstraction of this notion which is better suited for the non-homogeneous setting +and is necessary for a construction that we introduce below to characterise the +arc-consistency reduction. +Definition 4.4. +An abstract minion is a functor from the category of fnite sets +to the category of sets, i.e., a mapping M that assigns to each fnite set 𝑋 a set +M (𝑋), and each function 𝜋 : 𝑋 → 𝑌 between two fnite sets 𝑋, 𝑌, a function +M 𝜋 : M (𝑋) → M (𝑌), such that +• M 1𝑋 = 1M (𝑋 ) where 1𝑋 denotes the identity function on 𝑋, and +• M 𝜋 ◦ M 𝜎 = M 𝜋◦𝜎. +For 𝑓 ∈ M (𝑋) and 𝜋 : 𝑋 → 𝑌, we write 𝑓 𝜋 for M 𝜋 (𝑓 ) ∈ M (𝑌), and M (𝑛) for +M ([𝑛]). This is done to unite the notation for an abstract minion and function +minion. We further require that every minion M is non-empty and maps the empty +set to itself, i.e., that M (𝑋) = ∅ if and only if 𝑋 = ∅. +The polymorphisms of a template A, A′ form a minion, which we denote by +the same symbol Pol(A, A′). It is the minion A where A (𝑋) = Pol(𝑋) (A, A′), and +A 𝜋 for 𝜋 : 𝑋 → 𝑌 is defned to map 𝑓 ∈ Pol(𝑋) (A, B) to 𝑓 𝜋 ∈ Pol(𝑌) (A, B) where +𝑓 𝜋 +𝑖 (𝑎) = 𝑓𝑖 (𝑎 ◦ 𝜋) for each type 𝑖. The polymorphism 𝑓 𝜋 is said to be a minor of 𝑓 +defned by 𝜋. +Note that since A → A′ for each promise template, we have that 𝑋 ≠ ∅ implies +Pol(𝑋) (A, A′) ≠ ∅. The converse follows from the assumption that every promise +CSP in this paper has a negative instance: observe that X → A∅ for all structures X, +and hence if X ̸→ A′ for some X, there is no homomorphism A∅ → A′. +Definition 4.5. A minion homomorphism is a mapping between the two minions which +preserves the minor taking operations. More precisely, a minion homomorphism +from M to N is a natural transformation 𝜉 : M → N , i.e., a collection of maps +𝜉𝑋 : M (𝑋) → N (𝑋), one for each fnite set 𝑋, such that for all 𝜋 : 𝑋 → 𝑌, we have +N 𝜋 ◦ 𝜉𝑋 = 𝜉𝑌 ◦ M 𝜋, i.e., 𝜉𝑋 (𝑓 )𝜋 = 𝜉𝑌 (𝑓 𝜋) for all 𝑓 ∈ M (𝑋). + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +29 +Example 4.6. Defne an abstract minion H to be the non-empty powerset functor, +i.e., H (𝑋) = {𝑈 ⊆ 𝑋 | 𝑈 ≠ ∅}. The minor taking operation is then defned as +𝑈 𝜋 = 𝜋(𝑈 ) = {𝜋(𝑢) | 𝑢 ∈ 𝑈 }. It is straightforward to check that this defnition +satisfes the properties in Defnition 4.4. Also it is not hard to observe that H is in +fact isomorphic to the minion of polymorphisms of Horn-3SAT using a well-known +classifcation of polymorphisms of Horn-3SAT (see, e.g., [BKW17, Example 5]). +Finally, we are ready to formulate the fundamental theorem of the algebraic +approach. We give a modern formulation that is essentially [BBKO21, Theorem 3.1], +though that theorem does not explicitly say that the log-space reduction that exists +is a gadget reduction, and does not show the converse (the converse did not receive +much attention until [KOWŽ22] though it was essentially folklore in the algebraic +community). The theorem builds on a long line of refnements of these reductions +between CSPs and promise CSPs [JCG97, BJK05, BOP18, BG21]. Finally, none of the +mentioned papers works with non-homogeneous structures, thought the proofs, +in particular those of [BBKO21], apply essentially verbatim to this case. See also +[BJ03] for an algebraic treatment of non-homogeneous CSP including a description +of encoding such a CSP in a single-sorted one. +Theorem 4.7. The following are equivalent for any two PCSPs with templates A, A′, +and B, B′. +(1) There is a minion homomorphism Pol(B, B′) → Pol(A, A′); +(2) PCSP(A, A′) is reducible to PCSP(B, B′) via a gadget reduction. +Our characterisation of arc-consistency reductions relies heavily on the above the- +orem and its proof. In fact, the gadget reduction provided in [BBKO21, Section 3] in +presence of a minion homomorphism from Pol(B, B′) to Pol(A, A′) can equivalently +be described as the arc-consistency replacement with omitted step (A1.2) above. +Naturally, this means that we can relate our constructions to [BBKO21]. The above +theorem is proved by a two step reduction through an intermediate problem denoted +by PMC(M ) in [BBKO21]; the frst step corresponds to the step (A1.1) above and +formally produces a minor condition, and the second step which is equivalent to +(A2) then transforms this minor condition to an instance of CSP(A). Naturally, if we +want to draw a parallel with this proof, we will need to interpret the step (A1.2) as a +transformation on those minor conditions which we will do in the next subsection. +Let us now explain what a minor condition is, and describe how the other two +steps relate to the theory described in [BBKO21]. Minor conditions form the third +perspective on label cover instances, though written in a slightly diferent language. +Definition 4.8 (Minor conditions). An algebraic language is a collection of function +symbols denoted by 𝑓 , 𝑔, etc., together with their arities which are fnite sets 𝐷𝑓 , +𝐷𝑔, etc. We view a function symbol 𝑓 with arity 𝐷𝑓 as a placeholder for function +of arity 𝐷𝑓 , i.e., as 𝑓 : 𝑋 𝐷𝑓 → 𝑌 for some (at this point irrelevant) sets 𝑋 as 𝑌. A +minor condition is a fnite set of identities of the form 𝑓 = 𝑔𝜋 for some 𝜋 : 𝐷𝑔 → 𝐷𝑓 +in some algebraic language, i.e., each identity can be written as a formal identity +𝑓 (𝑥1, . . . ,𝑥𝑚) ≈ 𝑔(𝑥𝜋 (1), . . . ,𝑥𝜋 (𝑛)) +assuming 𝐷𝑓 = {1, . . . ,𝑚} and 𝐷𝑔 = {1, . . . ,𝑛}; the symbol ≈ is used to distinguish +formal identity from an equality. +Such a condition is then said to be satisfed in a minion M if there is an assignment +𝜉 that assigns to each 𝑓 of arity𝑋 an element 𝜉(𝑓 ) ∈ M (𝑋) such that for each identity + +30 +VICTOR DALMAU AND JAKUB OPRŠAL +𝑓 = 𝑔𝜋, we have 𝜉(𝑓 ) = 𝜉(𝑔)𝜋. And it is called trivial if it is satisfed in the minion +P of projections, i.e., the minion with P (𝑛) = [𝑛], and 𝑖𝜋 = 𝜋(𝑖). +We note that a condition is trivial if and only if it is satisfed in every minion +since P maps to any other minion by a minion homomorphism, and minion ho- +momorphisms preserve satisfaction of minor conditions; for details see [BBKO21, +Section 3.1]. +The translation between minor conditions and label cover instances uses the +following dictionary: function symbol ∼ variable, arity ∼ domain, identity ∼ constraint. +More precisely, each identity 𝑓 = 𝑔𝜋 between 𝑓 of arity 𝐷𝑓 and 𝑔 of arity 𝐷𝑔 +corresponds to a constraint 𝑓 = 𝜋(𝑔) where 𝑓 has domain 𝐷𝑓 and 𝑔 has domain +𝐷𝑔. It is also straightforward to reverse this process. Note that the resulting label +cover instance is solvable if and only if the condition is trivial; the key diference +is that we can talk about ‘satisfying a minor condition in a minion’ while ‘solving +label cover instance in a minion’ does not make much sense. With this distinction in +mind, we will not distinguish between minor conditions and label cover instances. +As we mentioned before, the frst step of the reduction in [BBKO21, Section 3.2] +transforms an instance I of a CSP(A) into a minor condition denoted by Σ(A, I) +— this condition then directly translates to the label cover instance obtained by +reifcation, i.e., the label cover instance I obtained from I∗ and A∗ via Defnition 2.9. +Below, we will adopt the notation Σ(A, I) and we interpret this label cover instance +as a minor condition. We will use the following key property of this construction. +Lemma 4.9 ([BBKO21, Lemma 3.14(2)]). If Pol(A, A′) satisfes Σ(A, X) then X → +A′. +The step (A2) corresponds to the indicator structure defned in [BBKO21, Section +3.3]. This construction takes as input a minor condition (a label cover instance) Σ +and a relational structure B, and produces a new relational structure that we will +denote BΣ since it is obtained from Σ by the universal (power) gadget replacement +for B.6 We will use the following properties of BΣ. +Lemma 4.10 ([BBKO21, Lemma 3.16 & Remark 3.17]). The following are equivalent +for each minor condition Σ and every promise template B, B′. +• BΣ → B′, +• Σ is satisfed in Pol(B, B′). +The fnal piece of the puzzle is a certain construction called the free structure +of a minion M generated by A. This is the last of four fundamental constructions +that were used in [BBKO21] that we have not described yet. We will use it as a +black box, only noting that the free structure of M generated by a fnite structure A +is fnite as long as M (𝑛) is fnite for all 𝑛 [BBKO21, Defnition 4.1], and using the +following fundamental lemma for the construction. This lemma can be taken as an +alternative defnition of the free structure since it characterises such structure up to +isomorphism. +Lemma 4.11 ([BBKO21, Lemma 4.4]). Let M be a minion, let A, A′ be a promise +template, and let F be the free structure of M generated by A. Then there is a 1-to-1 cor- +respondence between minion homomorphisms M → Pol(A, A′) and homomorphisms +F → A′. +We will also use the following observation. +6[BBKO21] used the notation IΣ (B). + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +31 +Lemma 4.12 ([BBKO21, Lemma 4.3]). Let F be the free structure of a minion M +generated by a structure A. Then M satisfes Σ(A, F). +4.2. The characterisation of arc-consistency reduction +The characterisation of the arc-consistency as a reduction uses the following +construction on minions which from a minion produces another abstract minion. +Definition 4.13. Let M be an abstract minion. We say that an element 𝑓 ∈ M (𝑁 ) +only depends on variables in 𝑋 ⊆ 𝑁 if there is 𝑔 ∈ M (𝑋) such that 𝑓 = 𝑔1𝑋 where +1𝑋 : 𝑋 → 𝑁 is the inclusion map.7 If this is the case we write 𝑓 ≺ 𝑋.8 +Given a minion M , we defne a minion 𝜔(M ) as follows. +𝜔(M )(𝑛) = {(𝑓 ,𝑋) | 𝑓 ∈ M (𝑛), 𝑓 ≺ 𝑋 } +and for 𝜋 : [𝑛] → [𝑚], we let (𝑓 ,𝑋)𝜋 = (𝑓 𝜋, 𝜋(𝑋)) where 𝜋(𝑋) = {𝜋(𝑥) | 𝑥 ∈ 𝑋 }. +The construction 𝜔 is closely related to the minion H of polymorphisms of +Horn-SAT (see Example 4.6): 𝜔(M ) is a subminion of the product M × H of +the two minions. Note that the frst projection (𝑓 ,𝑋) ↦→ 𝑓 is always a minion +homomorphism from 𝜔(M ) to M and the second projection (𝑓 ,𝑋) ↦→ 𝑋 is a +minion homomorphism from 𝜔(M ) to H . We also note that the 𝜔-image of the +polymorphism minion of the trivial CSP, 𝜔 Pol(T), is isomorphic to H : Pol(T) has +only one function of each arity which does not depend on any of its variables, let us +call it 𝑡, then (𝑡,𝑋) ↦→ 𝑋 is an isomorphism. Finally, we are ready to state the main +theorem. +Theorem 4.14. Let A, A′ and B, B′ be two promise templates. Then the following are +equivalent. +(1) there is a minion homomorphism 𝜔 Pol(B, B′) → Pol(A, A′); +(2) PCSP(A, A′) ≤arc-cons PCSP(B, B′). +The theorem is proved by closely following the proof of Theorem 4.7 as it is +presented in [BBKO21, Sections 3] with inserting an additional reduction in the +middle. Namely, we replace the middle reduction which is trivial in [BBKO21] with +the arc-consistency replacement, and we prove that the reduction is sound if (and +only if) 𝜔 Pol(B, B′) → Pol(A, A′). The novel ingredient is then Lemma 4.15 below. +In order to use the strategy of [BBKO21] we describe the arc-consistency pro- +cedure as a transformation of a minor condition: it iteratively decreases arity of +function symbols in the given minor condition, until each identity contains the same +variables on either of the sides. Following our notation above, we will also denote +by 𝜅arc(Σ) the minor condition obtained by applying arc-consistency to Σ, and for +a symbol 𝑓 of Σ, we will denote by F𝑓 its updated arity in 𝜅arc(Σ). The following +lemma then explains the relation of the transformation 𝜅arc on minor conditions and +the transformation 𝜔 on minions. We will also use the symbol 1𝑋 for the inclusion +map 1𝑋 : 𝑋 → 𝑌 where 𝑋 ⊆ 𝑌. +Lemma 4.15. For all minor conditions Σ and minions M , 𝜅arc(Σ) is satisfed in M +if and only if Σ is satisfed in 𝜔(M ) +7If M is a function minion on 𝐷, this is expressed as 𝑓 (𝑥) = 𝑔(𝑥 |𝑋 ) for all 𝑥 : 𝑁 → 𝐷. +8The set of essential variables of 𝑓 would be the smallest 𝑋 w.r.t. inclusion such that 𝑓 ≺ 𝑋. It is not +hard to prove that such a set must exist, but it is not necessary for our use. + +32 +VICTOR DALMAU AND JAKUB OPRŠAL +Proof. Assume M satisfes 𝜅arc(Σ). This implies that for each symbol 𝑓 in Σ, there +is 𝑓 ′ ∈ M such that ar 𝑓 ′ = F𝑓 , and these 𝑓 ′ satisfy all identities of 𝜅arc(Σ). Conse- +quently, by putting +𝜇(𝑓 ) = (𝑓 ′)1F𝑓 +we get that M satisfes the condition Σ as witnessed by 𝜇. Moreover, observe +that 𝑓 ↦→ F𝑓 satisfes Σ in H , hence 𝑓 ↦→ (𝜇(𝑓 ), F𝑓 ) satisfes Σ in 𝜔(M ) — it is +immediate from the defnition of 𝜇(𝑓 )’s that these elements indeed belong to 𝜔(M ). +For the other direction, assume Σ is satisfed in 𝜔(M ) by elements (𝜇(𝑓 ),𝑋𝑓 ). +We note that for each minor identity 𝑓 = 𝑔𝜋 in Σ, since 𝜋(𝑋𝑓 ) = 𝑋𝑔, it follows that all +variables in 𝑋𝑓 and 𝑋𝑔 are not removed in any of the iterations of the arc-consistency +procedure. This implies that 𝑋𝑓 ⊆ F𝑓 , and that we can defne 𝑓 ′ of arity F𝑓 by +𝑓 ′(𝑥) = 𝑔1𝑋𝑓 +where 𝑔 is a witness for 𝜇(𝑓 ) ≺ 𝑋𝑓 . It is easy to check that the collection of 𝑓 ′’s +satisfy 𝜅arc(Σ). +□ +We get back to the proof of the main theorem. Again, let us stress that with the +exception of the above lemma, we are simply following the proof of Theorem 4.7 as +described in [BBKO21, Sections 3 & 4]. +Proof of Theorem 4.14. Let A = Pol(A, A′) and B = Pol(B, B′). First, we show +that if there is a minion homomorphism 𝜉 : 𝜔(B) → A then PCSP(A, A′) reduces +to PCSP(B, B′) via an arc-consistency reduction. Starting with an instance X of +CSP(A), the arc-consistency reduction produces frst a minor condition Σ(A, X) in +step (A1.1), then the condition 𝜅arcΣ(A, X) in step (A1.2), and fnally the instance +B𝜅arcΣ(A,X) in step (A2). The completeness, i.e., that the resulting instance maps to B +if X → A is straightforward, let us focus on the soundness. Assume B𝜅arcΣ(A,X) → B′, +then we get that: B satisfes 𝜅arc(Σ(A, X)) by Lemma 4.10, 𝜔(B) satisfes Σ(A, X) +by Lemma 4.15, A satisfes Σ(A, X) since a minion homomorphism preserves satis- +faction of minor conditions, and fnally X → A′ by Lemma 4.9. This concludes the +proof of the converse implication. +To show the other implication, assume that PCSP(A, A′) reduces to PCSP(B, B′) +by the arc-consistency reduction. Let F be the free structure of 𝜔(B) generated by +A. In order to obtain a minion homomorphism from 𝜔(B) to A , we use Lemma 4.11 +stating that such a minion homomorphism exists if and only if F → A′. We prove this +by using the soundness of arc-consistency applied on F. Observe that: 𝜔(B) satisfes +Σ(A, F) by Lemma 4.12, B satisfes 𝜅arcΣ(A, F) by Lemma 4.15, and B𝜅arcΣ(A,F) → B′ +by Lemma 4.10. Since the last homomorphism witnesses that𝜅A,B +arc (F) is not a negative +instance of PCSP(B, B′), we get that F → A′ by the soundness of the arc-consistency +reduction as we wanted to show. +□ +We briefy note that the above proof can be adapted for the cases when F is not +fnite (e.g., when B, B′ and B(𝑛) are not fnite) using the standard compactness +argument (see, e.g., [BBKO21, Remark 7.13]). +Theorem 4.14 together with [BK22, Theorem 5.1] provides an immediate corollary +which gives a new sufcient condition for the existence of a local reduction. Namely, +we can connect the notion of minion (𝑑,𝑟)-homomorphisms defned in [BK22, Def- +nition 5.1] with the 𝜔 construction; we refer to the cited paper for the defnition of +this notion. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +33 +Corollary 4.16. Assume A, A′ and B, B′ are two promise templates. If there exist 𝑑,𝑟 +and a minion (𝑑,𝑟)-homomorphism from 𝜔 Pol(B, B′) to Pol(A, A′), then +PCSP(A, A′) ≤Datalog PCSP(B, B′). +While (𝑑,𝑟)-homomorphisms between polymorphism minions are provably not +necessary for existence of a Datalog∪ reduction — this would contradict Example ??. +It is not known if existence of a (𝑑,𝑟)-homomorphisms from the 𝜔-image is also a +necessary for such a reduction. +Let us mention an example of a CSP (and a minion) for which Theorem 4.14 does +not provide any new leverage. +Example 4.17. The following minion Qconv, that was defned in [BBKO21, Section 7.2] +to describe the power of basic linear programming relaxation (BLP) for promise CSPs, +has a remarkable property that it is homomorphically equivalent to its 𝜔-image. This +is underlined by the intuition that BLP is stronger than arc-consistency, and thus +running arc-consistency before BLP does not increase the power of the algorithm. +The minion Qconv is defned as follows: Q(𝑋) +conv consists of all rational probability +distributions 𝜆 on 𝑋, i.e., 𝜆: 𝑋 → Q with � +𝑥 ∈𝑋 𝜆(𝑥) = 1 and 𝜆(𝑥) ≥ 0 for all +𝑥 ∈ 𝑋, and for 𝜋 : 𝑋 → 𝑌, we let 𝜆𝜋 be the probability distribution of 𝜋(𝑥) when +𝑥 is sampled according to 𝜆, i.e., 𝜆𝜋 (𝑥) = � +𝑦∈𝜋−1(𝑥) 𝜆(𝑦). We construct a minion +homomorphism 𝜉 : Qconv → 𝜔(Qconv), a homomorphism in the other direction is +immediate. Let +𝜉(𝜆) = (𝜆, supp(𝜆)) +where supp(𝜆) is the set of all values 𝑥 ∈ 𝑋 that have a non-zero probability in 𝜆. +It is not hard to check that indeed 𝜆 ≺ supp(𝜆). To show that 𝜉 is indeed a minion +homomorphism, we need to show that supp(𝜆𝜋) = 𝜋(supp(𝑓 )) for all 𝜋 : 𝑋 → 𝑌 +which is rather easy: if 𝑥 has non-zero probability according to 𝜆 then 𝜋(𝑥) has a +non-zero probability according to 𝜆𝜋, and conversely 𝑦 has a non-zero probability +according to 𝜆𝜋 only if there is 𝑥 with 𝜋(𝑥) = 𝑦 and a non-zero probability in 𝜆. This +concludes that 𝜉 is a minion homomorphism. +As a direct consequence, we get that if a promise CSP is reducible by arc- +consistency reduction to another promise CSP that is solved by the basic linear +programming relaxation (BLP), then it is itself solvable by BLP. We will also use this +fact in the next section to derive some properties of Sherali-Adams hierarchy. +4.2.1. Conic minions. Another group of examples of minions M that satisfy M → +𝜔(M ) are conic minions introduced in [CŽ23] to describe several algorithms includ- +ing Sherali-Adams, Lasserre hierarchy of semi-defnite programming, and hierarchies +of CLAP [CŽ22b] and a combination of linear programming and afne integer pro- +gramming [BG20, BGWŽ20]. Conic minions are a special case of so-called linear +minions. For more clarity, we present this notion in a restricted setting of matrices +over felds, see [CŽ23, Defnitions 16 & 20] for the general case.9 +Definition 4.18. Let 𝑑 ≥ 1 be a possibly infnite cardinal (a dimension of a vector +space) and fx a feld F. A linear minion of depth 𝑑 is a minion M , such that, for all +𝑛, M (𝑛) consists of some 𝑛 × 𝑑-matrices over F. The minor taking operation are +defned as 𝑀𝜋 = 𝑃𝜋𝑀 for 𝜋 : [𝑛] → [𝑚] where 𝑃𝜋 is the incidence matrix of the +graph of 𝜋, i.e., the (𝜋(𝑖),𝑖)-th entry of 𝑃𝜋 is 1 for all 𝑖 and all other entries are 0.10 +9Most examples of linear minions fall under our defnition with the exception of H . +10This minor taking operations coincide with the identifcation minors of functions if we interpret an +element 𝑀 ∈ M (𝑛) as linear map 𝑀𝑇 : F𝑛 → F𝑑. + +34 +VICTOR DALMAU AND JAKUB OPRŠAL +Such a linear minion M is said to be conic if 𝑀 ≠ 0 for all 𝑀 ∈ M and, for all +𝑀 ∈ M (𝑛) and 𝑋 ⊆ [𝑛], whenever � +𝑖 ∈𝑋 𝑀𝑇 e𝑖 = 0 we get 𝑀𝑇 e𝑖 = 0 for all 𝑖 ∈ 𝑋 +where e𝑖 denotes the 𝑖-th vector of the canonical basis. +See [CŽ23, Proposition 21] for examples of conic (and non-conic) minions. Let +us now prove that every conic minion admits a homomorphism to its 𝜔-image. +Naturally, this also raises the question whether a linear minion is conic if and only +if it allows a homomorphism from its 𝜔-image? +Lemma 4.19. For every conic minion M , there is a minion homomorphism M → +𝜔(M ). +Proof. We defne a homomorphism 𝜉 : M → 𝜔(M ) by +𝜉(𝑀) = (𝑀; {𝑖 | 𝑀𝑇 e𝑖 ≠ 0}). +The mapping 𝜉 is well-defned since if 𝑀 does not depend on 𝑖 then 𝑀𝑇 e𝑖 = 0. We +claim that the fact that 𝜉 preserves minors follows from the conicity of 𝑀. Indeed +if 𝑀 = 𝑁 𝜋, then 𝑀𝑇 e𝑗 = � +𝜋 (𝑖)=𝑗 𝑁𝑇 e𝑖, and hence 𝑀𝑇 e𝑗 = 0 if and only if 𝑁𝑇 e𝑖 = 0 +for all 𝑖 such that 𝜋(𝑖) = 𝑗 since M is conic. Consequently, we get that +{𝜋(𝑖) | 𝑁𝑇 e𝑖 ≠ 0} = {𝑗 | (𝑁 𝜋)𝑇 e𝑗 ≠ 0} +Since we only need to check preserving minors in the second coordinate (it is trivial +in the frst), this equality gives the claim. +□ +We believe that this general property of conic minions is the key to why hierar- +chies of conic minions introduced in [CŽ23] have particularly nice properties. +4.3. Arc-consistency reductions are transitive: comonads and Kleisli arrows +We argue that arc-consistency reductions compose by using the language of cate- +gory theory that is incredibly elegant for this purpose. Essentially, this claim follows +from an observation that 𝜔 is a comonad, and Theorem 4.14 which characterises +the arc-consistency reduction in terms of co-Kleisli arrows of this comonad. Let us +briefy outline the defnitions. Although it is not the traditional way, we defne these +notions together to highlight their connection. +Definition 4.20. Assume 𝜂 is an endofunctor of some category, and 𝐴, 𝐵 are two +objects. A co-Kleisli arrow is a morphism 𝑓 : 𝜂(𝐴) → 𝐵.11 A functor 𝜂 is a comonad +if its co-Kleisli arrows form a category with the same objects as the underlying +category, i.e., +• there is an associative binary operator ◦𝜂 that assigns to every pair of +co-Kleisli arrows 𝑓 : 𝜂(𝐴) → 𝐵 and 𝑔: 𝜂(𝐵) → 𝐶 a co-Kleisli arrow 𝑔 ◦𝜂 +𝑓 : 𝜂(𝐴) → 𝐶; +• for each object 𝐴, there is a co-Kleisli arrow 𝑒𝐴 : 𝜂(𝐴) → 𝐴 that acts as the +identity w.r.t. ◦𝜂. +Lemma 4.21. 𝜔 is a comonad. +Proof. We start with defning the units 𝑒M where M is a minion; this unit 𝑒M +is simply the projection to the frst coordinate, i.e., 𝑒M (𝑓 ,𝑋) = 𝑓 . To defne the +composition, it is easier to show how to get from a co-Kleisli arrow 𝜉 : 𝜔(M ) → N +11Usually, a morphism 𝜂(𝐴) → 𝐵 is called a co-Kleisli arrow only if 𝜂 is a comonad, and in that case +the ‘co-’ prefx is often dropped since its implied. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +35 +a minion homomorphism 𝜉♯ : 𝜔(M ) → 𝜔(N ); this 𝜉♯ is defned by 𝜉♯(𝑓 ,𝑋) = +(𝜉(𝑓 ,𝑋),𝑋). The composition is then defned as +𝜉 ◦𝜔 𝜁 = 𝜉 ◦ 𝜁 ♯. +It is not hard to check that ◦𝜔 is associative. Note that 𝑒♯ +M it the identity on 𝜔(M ) +which immediately gives that 𝜉 ◦𝜔 𝑒M = 𝜉. The other identity 𝜉 = 𝑒M ◦ 𝜉♯ is +straightforward. +□ +Note that in the above proof, we showed that 𝜔(M ) → 𝜔(N ) if and only if +𝜔(M ) → N . Finally, as an easy corollary of the above lemma and the fact that +co-Kleisli arrows of a comonad compose, we get that arc-consistency reductions are +transitive. +Corollary 4.22. +If PCSP(A, A′) ≤arc-cons PCSP(B, B′) ≤arc-cons PCSP(C, C′), then +PCSP(A, A′) ≤arc-cons PCSP(C, C′). +5. HIERARCHIES AND CSP ALGORITHMS +Several hierarchies of algorithms are studied for the tractability of CSPs and +promise CSPs. The most prominent two are arguably the local consistency hierarchy +and the Sherali-Adams hierarchy [SA90]. Both of these hierarchies can be viewed +as a reduction to a certain polynomial-time solvable CSP; Horn-SAT and linear +programming, respectively. We will describe this in a bigger detail below, and +explain that this reduction is actually a special case of a Datalog∪ reduction. In +fact for these two hierarchies we show that a promise CSP is reducible by the 𝑘- +consistency reduction to a CSP that defnes the hierarchy if and only if the promise +CSP is solved by the 𝑘-th level of the hierarchy. A similar result can be obtained also +for a few other hierarchies introduced in [CŽ23]. As a direct consequence we obtain +that the class of promise CSPs solvable by some level of such a hierarchy is closed +under consistency reductions, and in particular under minion homomorphisms +between their templates. +We also suggest a new way to defne a hierarchy of any fxed promise CSP +by simply defning it as all the problems reducible to the fxed problem via the +𝑘-consistency reduction. Such a hierarchy has a natural grading that corresponds to +the parameter 𝑘. A slightly diferent general approach to hierarchies has been also +suggested in [CŽ23], which defnes a hierarchy of ‘minion tests’ for a fxed minion +M , generalising Sherali-Adams, Lasserre, and others. While we defne the hierarchy +in a diferent way, under a certain condition, namely that M → 𝜔(M ), where M +is the polymorphism minion of the template of the base problem of the hierarchy, +the two hierarchies coincide. It is likely that if the above condition is not satisfed, +the two hierarchies difer. In that case the algorithm provided through our hierarchy +is always stronger. +Often, it is useful that the problem we reduce to is an infnite template CSP — +this is the case for, e.g., linear programming, and for afne integer programming. +This creates a practical problem with the 𝑘-consistency reduction since it produces +infnite instances. Nevertheless, the templates we consider have a certain property +that allows us to always produce an equivalent, but fnite, instance. + +36 +VICTOR DALMAU AND JAKUB OPRŠAL +5.1. Problems solvable by Datalog +The bounded width hierarchy corresponds to problems solvable by Datalog. The +equivalence of the local consistency algorithm and Datalog is well-known [FV98]. +We will show that the same coincides as a hierarchy of Horn-SAT, or the hierarchy +of the trivial CSP. +Recall that a (promise) CSP is said to have width 𝑘 if it is solvable by the 𝑘- +consistency algorithm, and that a promise CSP is said to be solvable by Datalog +if there is a Datalog sentence which is false on all positive instances, and true +on all negative instances (see Defnitions 3.25 and 3.12). Note that both the 𝑘- +consistency reduction and the 𝑘-consistency decision algorithm use as a subroutine +the𝑘-consistency step, and recall that the𝑘-consistency algorithm rejects an instance +if and only if the 𝑘-consistency step derives that some variable has an empty domain. +CSPs that have bounded width were characterised in [BK14] as exactly those CSPs +which do not allow a gadget reduction from solving systems of linear equations over +a fnite feld. The situation is more complicated for promise CSPs: 1-in-3- vs NAE- +SAT does not have bounded width [AD22] and it does not allow a gadget reduction +from solving systems linear equations (which can be shown using Theorem 4.7). +Let us frst observe that promise CSPs are defnable by Datalog if and only if +they reduce to the trivial CSP by a Datalog∪ reduction. This is achieved by a +straightforward translation between the two Datalog programs. +Lemma 5.1. Let 𝑘 > 1. A promise CSP is defnable by a Datalog program if and only +if there is a Datalog∪ reduction that is a reduction from the promise CSP to the trivial +CSP. +Proof. Assuming 𝜙⊥ is a Datalog sentence, we defne a Datalog interpretation 𝝓 = +(𝜙1, . . . ,𝜙𝑛;𝜙⊥) where 𝑛 is the number of types in the input signature of 𝜙⊥. Each +𝜙𝑖 is defned by the rule 𝜙𝑖 (𝑥) ← 𝑥 = 𝑥 where 𝑥 of type 𝑖. Finally, in order to turn 𝝓 +into a reduction to CSP(T), we compose it with the obvious disjoint union; defned +by (𝑑,𝑟) where 𝑑(𝑖) = 1 for all 𝑖 ∈ [𝑛], and 𝑟 (⊥) = ⊥ for the single relation. It is +straightforward to check that 𝜙⊥ solves PCSP(A, A′) if and only if this Datalog∪ +reduction is a valid reduction from PCSP(A, A′) to CSP(T). +□ +Observe that the width of 𝝓 in the above proof coincides with the width of 𝜙⊥, +and that the disjoint union used above can be expressed as a strict gadget reduction. +The following theorem shows that the hierarchies of Horn-SAT and the trivial +CSP coincide, and they also coincides with CSPs of bounded width. The proof is +an exercise of applying the results of previous sections using the fact that running +arc-consistency on the output of 𝑘-consistency has no efect, and the fact that +the polymorphisms of the trivial CSP and Horn-SAT are connected through the +construction 𝜔 described in the previous section. We note that if we would not insist +on keeping the parameter 𝑘 constant, the theorem below would follow relatively +easily from Theorems 3.27 and 4.14. +Theorem 5.2. The following are equivalent for every promise template A, A′ and +every 𝑘 ≥ 2. +(1) PCSP(A, A′) is has width at most 𝑘; +(2) PCSP(A, A′) is solvable by Datalog with width 𝑘; +(3) PCSP(A, A′) reduces to the trivial CSP by the 𝑘-consistency reduction; +(4) PCSP(A, A′) reduces to Horn-3SAT by the 𝑘-consistency reduction. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +37 +As we mentioned above, the equivalence of (1) and (2) is well-known for CSPs, +see, e.g., [FV98, BK14], the same argument works for promise CSPs as well. We also +note that this could be argued in a similar way as Theorem 3.28. We will frst show +that (3) and (4) are equivalent, for which we use the following lemma. +Lemma 5.3. If 𝜔 Pol(B, B′) → Pol(A, A′) then every promise CSP that reduces to +PCSP(A, A′) by the𝑘-consistency reduction reduces to PCSP(B, B′) by the𝑘-consistency +reduction. +Proof. Assume that PCSP(C, C′) reduces to PCSP(A, A′) by the 𝑘-consistency reduc- +tion, i.e., if A𝜅𝑘 (X,C) → A′ then X → C′. We will show that 𝑘-consistency is also a +sound reduction to PCSP(B, B′). Assume that B𝜅𝑘 (X,C) → B′, which is equivalent to +𝜅𝑘 (X, C) being satisfed in Pol(B, B′) by Lemma 4.10. By Lemma 4.15, using the obser- +vation that 𝜅arc𝜅𝑘 (X, C) = 𝜅𝑘 (X, C), we get that 𝜅𝑘 (X, C) is satisfed in 𝜔 Pol(B, B′). +The minion homomorphism then implies that 𝜅𝑘 (X, C) is satisfed in Pol(A, A′), and +the converse implication of Lemma 4.10 that A𝜅𝑘 (X,C) → A′. Therefore, we obtain +X → C′ from the soundness of the reduction to PCSP(A, A′). +□ +Note that a direct consequence of the above lemma and Theorem 4.14 is that +if PCSP(C, C′) ≤𝑘-cons PCSP(A, A′) ≤arc-cons PCSP(B, B′) then PCSP(C, C′) ≤𝑘-cons +PCSP(B, B′). We are now ready to return the proof of the theorem. +Proof of Theorem 5.2. As we mentioned above, the equivalence of (1) and (2) is +known. The equivalence of (3) and (4) is a direct consequence of Lemma 5.3, the fact +that 𝜔(Pol(T)) is isomorphic to the polymorphism minion H of Horn-3SAT, and +the trivial homomorphism 𝜔(H ) → Pol(T). +The implication (2)→(3) is a direct consequence of Lemma 5.1 and Theorem 3.28 +observing that the Datalog∪ reduction provided by the lemma is a composition of a +Datalog interpretation of width 𝑘 and a particular disjoint union which is expressible +strict gadget replacement. +Finally, we argue that (3) implies (2). The key to proving this implication is +observing that a minor condition Σ is satisfed in Pol(T) if and only if it has no +nullary symbols: Observe that T𝑋 is isomorphic to T if X ≠ ∅, and T∅ ̸→ T since +the relation ⊥ of T∅ is true. The claim follows immediately from this observation. +Assume that PCSP(A, A′) reduces by the 𝑘-consistency reduction to CSP(T), and +let X be an instance of CSP(A). The 𝑘-consistency algorithm is defned to output +yes on the input X of if and only if F𝐾 is non-empty in 𝜅𝑘 (X, A) for all 𝐾. This +condition exactly correspond with 𝜅𝑘 (X, A) being satisfed in Pol(T), and hence +with T𝜅𝑘 (X,A) → T (Lemma 4.10). Hence, if the 𝑘-consistency algorithm outputs yes, +then X → A′. Conversely, if the 𝑘-consistency algorithm outputs no, then X ̸→ A +by Lemma 3.32. +□ +A direct corollary of Theorem 5.2 is that PCSPs solvable by local consistency are +closed under gadget reductions which was frst shown in [BBKO21, Lemma 7.5], +and in the non-promise setting in [LZ07]. +Corollary 5.4. Let A, A′ and B, B′ be two promise templates such that Pol(A, A′) → +Pol(B, B′). If PCSP(A, A′) is solvable by Datalog, then so is PCSP(B, B′). +5.2. Sherali-Adams +There are several slightly diferent ways to defne Sherali-Adams relaxation for +CSPs and promise CSPs, e.g., [TŽ17, CŽ23, BD21, BB22] — the last of these references +introduces Sherali-Adams in the most general setting, namely promise valued CSPs. + +38 +VICTOR DALMAU AND JAKUB OPRŠAL +Most of these defnitions difer in minor technical details, and all of them are based +in some way on a paper of Sherali and Adams [SA90] which describes the hierarchy +as a reduction from 0-1 integer programming to linear programming. All of the +defnitions have a few things in common: They produce, from an instance of CSP, an +instance of linear programming with 0-1 coefcients. This instance can be interpreted +as an instance of CSP(Qconv) for some suitable template Qconv, which we describe +below. +Our defnition below agrees with the (𝑘−1,𝑘)-SA system in [TŽ17] assuming that +𝑘 is at least the maximal arity of the constraints. We make a choice of not projecting +larger arity constraints not to increase the width of the Datalog interpretation above +𝑘. Otherwise, e.g., when we would use the system in [TŽ17] directly, it would +require a Datalog interpretation of width 𝑚, where 𝑚 is the maximal arity of the +constraints. This is related to the fact that interpreting the identity map as a Datalog +interpretation still requires width 𝑚. Though it is not strictly necessary, we assume +𝑘 > 1, since smaller values of 𝑘 lead to rather trivial systems. +In plain words, the goal of the 𝑘-th level of Sherali-Adams relaxation is to fnd a +collection of probability distributions on partial solutions on each of the subsets of +variables of size at most 𝑘 that have consistent marginals. +Definition 5.5 (𝑘-th level of Sherali-Adams hierarchy). Fix 𝑘 > 1. The 𝑘-th level of +SA hierarchy is the following reduction from an instance X of CSP(A) to linear +programming: +(SA1) Create the following label cover instance: +• for each 𝐾 ∈ � 𝑄 +≤𝑘 +�, introduce a variable 𝑣𝑘 with domain F𝐾 which is the +set of all partial homomorphisms from 𝐾 to A; +• for each 𝐿 ⊂ 𝐾 ∈ � 𝑄 +≤𝑘 +�, introduce a constraint between 𝑣𝐾 and 𝑣𝐿 defned +by the map 𝑔 = 𝑔|𝐿 from F𝐾 to F𝐿. +(SA2) Encode the label cover instance into linear program in a similar way as for +basic linear programming relaxation with variables 𝜆𝐾,𝑓 ∈ [0, 1] for each +𝑓 ∈ F𝐾: +∑︁ +𝑓 ∈F𝐾 +𝜆𝐾,𝑓 = 1 +for all 𝐾 ∈ � 𝑄 +≤𝑘 +�, +(1) +∑︁ +𝑓 ∈F𝐾,𝑓 |𝐿=𝑔 +𝜆𝐾,𝑓 = 𝜆𝐿,𝑔 +for all 𝐿 ⊂ 𝐾 ∈ � 𝑄 +≤𝑘 +� and 𝑔 ∈ F𝐿. +(2) +Observe that the above system has a 0-1 solution if there is a homomorphism +ℎ: X → A: simply assign 𝜆𝑓 ,𝐾 = 1 if 𝑓 = ℎ|𝐾 and 𝜆𝑓 ,𝐾 = 0 otherwise. +Linear programming can be viewed as a fxed template CSP though with infnite +domain and infnitely many relations: Namely, the domain is Q, and the relations +are all relations defned by afne inequalities, e.g., of the form +𝑎1𝑥1 + · · · + 𝑎𝑛𝑥𝑛 ≤ 𝑏 +for some 𝑎1, . . . ,𝑎𝑛,𝑏 ∈ Q. We have to address technicalities that arise due to the fact +that the template of linear programming is infnite and has infnitely many relations. +In particular, in an instance of linear programming, the relations are usually given +as the tuples of their coefcients. We note that all of these relations are in fact +pp-defnable using the following three 𝑥 ≤ 𝑦, 𝑥1 + 𝑥2 = 𝑦, and 𝑦 = 1, and with some +care, the length of these pp-defnitions is proportionate to the binary encoding of +the coefcients. We will henceforth ignore this issue with having infnitely many +relations, and defne Qconv as the structure with the above relations. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +39 +Another formal problem with the 𝑘-consistency reduction to linear programming +when viewed as CSP(Qconv) is that the resulting instance is infnite. This issue is +caused by the fact that the universal gadgets for Qconv are infnite. Nevertheless, +linear programming and several other CSPs including afne integer programming, +systems of linear equations over fnite felds, and semi-defnite programming, have a +curious property that allows us to use fnite gadgets instead of the universal gadget +with equivalent properties. We informally call this property short code property. +These ‘smaller gadgets’ are exactly the gadgets used in the step (SA2) above, i.e., the +following. +Definition 5.6. Let Σ be a label cover instance. We defne a system of linear equations +and inequalities 𝜖LP(Σ) in the following way. We replace a variable 𝑣 with domain +𝐷 with variables 𝑥𝑣,𝑑 for 𝑑 ∈ 𝐷, an equation � +𝑑 ∈𝐷 𝑥𝑣,𝑑 = 1, and inequalities 𝑥𝑣,𝑑 ≥ 0 +for all 𝑑 ∈ 𝐷, and we replace a label cover constraint 𝜋(𝑢) = 𝑣 defned by a map +𝜋 : 𝐷 → 𝐷′ is the collection of equations +∑︁ +𝑑 ∈𝜋−1(𝑑′) +𝑥𝑢,𝑑 = 𝑥𝑣,𝑑′, +one for each 𝑑′ ∈ 𝐷′. +The goal of the linear program is then to solve this system in Q. Note that 𝜖LP(Σ) +can be expressed as a structure of the same type as Qconv, i.e., a linear program. It is +nevertheless easier to work with it as a system of linear equations and inequalities +with 0-1 coefcients. +We note that Qconv, defned in Example 4.17, is isomorphic to the polymorphism +minion of Qconv; a minion isomorphism 𝜉 : Qconv → Pol(Qconv) can be defned by +𝜉(𝜆)(𝑥1, . . . ,𝑥𝑛) = +𝑛 +∑︁ +𝑖=1 +𝜆(𝑖)𝑥𝑖, +i.e., 𝜉(𝜆) is the convex combination with coefcients given by 𝜆. It is straightfor- +ward to check that 𝜉(𝜆) preserves the relations of Qconv. On the other hand, each +polymorphism of Qconv is a convex combination, and hence its coefcients are a +rational probability distribution which implies that 𝜉 is onto. It is clearly 1-to-1, and +checking that 𝜉 preserves minors is straightforward. +The purpose of the lemma below is to show that 𝜖LP is as good as the universal +gadget for Qconv, and we will use it to freely exchange them. In particular, replacing +Σ ↦→ QΣ +conv with 𝜖LP in the 𝑘-consistency reduction to Qconv we obtain actually +a polynomial time reduction to linear programming that is equivalent to the 𝑘- +consistency reduction but does not produce infnite instances. +Lemma 5.7. The following are equivalent for each minor condition Σ. +(1) the system 𝜖LP(Σ) is solvable in Q; +(2) Qconv satisfes Σ; +(3) QΣ +conv → Qconv. +Proof. The equivalence of (2) and (3) follows from Lemma 4.10 and the fact that Qconv +is isomorphic to Pol(Qconv) discussed above. We prove that (1) and (2) are equivalent. +Assume that 𝑥𝑓 ,𝑑 ∈ Q ∩ [0, 1] for 𝑓 and 𝑑 ∈ 𝐷𝑓 is a solution of 𝜖LP(Σ). For each 𝑓 , +we defne a probability distribution 𝜆𝑓 on 𝐷𝑓 by taking 𝑑 with probability 𝑥𝑓 ,𝑑; 𝜆𝑓 +is a probability distribution since 𝑥𝑓 ,𝑑 ≥ 0 and � +𝑑 ∈𝐷𝑓 𝑥𝑓 ,𝑑 = 1. The fact that each +identity of Σ is satisfed is straightforward to check using that � +𝑑 ∈𝜋−1(𝑑′) 𝑥𝑓 ,𝑑 = 𝑥𝑓 ′,𝑑′ +for 𝑓 𝜋 = 𝑓 ′ and 𝑑′ ∈ 𝐷𝑓 ′. This concludes that (1) implies (2). The converse is given + +40 +VICTOR DALMAU AND JAKUB OPRŠAL +by reversing this argument, i.e., checking that 𝑥𝑖,𝑓 = 𝜆𝑓 (𝑖) is a solution to 𝜖LP(Σ) if +𝜆𝑓 are probability distributions witnessing satisfaction of Σ in Qconv. +□ +Once we have addressed the formal problems with the 𝑘-consistency reduction +to linear programming, we can formulate the main result of this subsection. +Theorem 5.8. Let A, A′ be a promise template. Then the following are equivalent. +(1) PCSP(A, A′) is solvable by the 𝑘-th level of Sherali-Adams. +(2) PCSP(A, A′) reduces to linear programming by the 𝑘-consistency reduction. +The proof is a comparison of Sherali-Adams, as a reduction to linear programming, +and the 𝑘-consistency reduction to linear programming. First, note that the 𝑘- +consistency procedure can be decomposed using the frst step (SA1) of Sherali-Adams +as follows. Let 𝜃𝑘 (X, A) denote the label cover instance obtained by the step (SA1) +on input X as an instance of CSP(A). Then the output of 𝑘-consistency procedure +𝜅𝑘 (X, A) is the same as frst applying𝜃𝑘 and then running the arc consistency𝜅arc, i.e., +we have that 𝜅arc𝜃𝑘 (X, A) = 𝜅𝑘 (X, A) for all X and A. Second, we compare the last +step, which is 𝜖LP for Sherali-Adams and the universal gadget for the 𝑘-consistency +reduction, using the short-code property (Lemma 5.7). Intuitively, the proof claims +that the two gadgets, 𝜖LP and the universal gadget for Qconv, are equivalent, and that +the arc-consistency step can be omitted since linear programming can emulate arc- +consistency. The latter is a general property of promise CSPs whose polymorphism +minion M admits a homomorphism M → 𝜔(M ); note that Qconv → 𝜔(Qconv) is +discussed in Example 4.17. +Lemma 5.9. Fix a promise template B, B′, and let B be the polymorphism minion of +its template. If B → 𝜔(B), then the following are equivalent for any other promise +template A, A′. +(1) X ↦→ B𝜃𝑘 (X,A) is a reduction from PCSP(A, A′) to PCSP(B, B′); +(2) PCSP(A, A′) ≤𝑘-cons PCSP(B, B′). +Proof. The implication (1)→(2) follows by a similar argument as Theorem 3.28. Let +us focus on the converse, and in particular on the soundness of the reductions since +completeness is again straightforward. +As we discussed above, the 𝑘-consistency reduction maps X to B𝜅arc𝜃𝑘 (X,A). As- +suming that the result maps to B′, we get by Lemma 4.10 that B satisfes𝜅arc𝜃𝑘 (X, A), +and consequently, by Lemma 4.15, that 𝜔(B) satisfes 𝜃𝑘 (X, A). Since 𝜔(B) → B +and minion homomorphisms preserve satisfaction of minor conditions, also B satis- +fes 𝜃𝑘 (X, A), and consequently by Lemma 4.10, B𝜃𝑘 (X,A) → B′. Finally, we get that +X → A′ by invoking the soundness of X ↦→ B𝜃𝑘 (X,A). +□ +We are ready to prove Theorem 5.8 as a consequence of Lemmas 5.7 and 5.9. +Proof of Theorem 5.8. For sanity, let us write Q instead of Qconv. The theorem claims +that the 𝑘-consistency reduction to linear programming, i.e., X ↦→ Q𝜅arc𝜃𝑘 (X,A), has +the same power as the Sherali-Adams reduction, i.e., the mapping X ↦→ 𝜖LP𝜃𝑘 (X, A). +Using Lemma 5.9, for B = B′ = Q and B = Qconv, we get that PCSP(A, A′) +reduces to CSP(Q) by the 𝑘-consistency reduction if and only if X ↦→ Q𝜃𝑘 (X,A) is a +reduction between these problems as well. Finally, we have that Q𝜃𝑘 (X,A) → Q if and +only if 𝜖LP𝜃𝑘 (X, A) is a system of equations solvable over Q ∩ [0, 1] by Lemma 5.7. +Consequently, Sherali-Adams solves PCSP(A, A′) if and only if X → Q𝜃𝑘 (X,A) is a +reduction from PCSP(A, A′) to linear programming, which concludes the proof. +□ + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +41 +We also obtain an immediate corollary of Theorems 5.8 and 3.27, which we believe +has not been shown before in the scope of promise CSPs, though the analogous +statement in the non-promise setting follows from the characterisation of Sherali- +Adams for CSPs [TŽ17]. +Corollary 5.10. Let A, A′ and B, B′ be two promise templates such that Pol(A, A′) → +Pol(B, B′). If PCSP(A, A′) is solvable by some level of Sherali-Adams hierarchy, then +so is PCSP(B, B′). +Finally, we note that the key to the above arguments is the property that Qconv → +𝜔(Qconv), and the short code property of linear programming (Lemma 5.7). This +in particular means that the arguments of this section generalise to many of the +hierarchies of conic minions introduced in [CŽ23]; In particular, we showed that +a conic minion M allows a minion homomorphism M → 𝜔(M ) in Lemma 4.19. +With some extra efort it can be shown that semi-defnite programming has also a +short code property similar to the linear programming. It follows that there is an +analogue of Corollary 5.10 for the Lasserre hierarchy of semi-defnite programs for +PCSPs. Naturally, the same would apply to hierarchies of CLAP and the combination +of linear programming and afne integer programming. +5.3. Hierarchies of groups and obstructions to consistency reductions +Solving equations over a (fnite) Abelian group is a well-known CSP that is not +solved by local consistency algorithm. In this section, we argue that at least some +of these problems may serve as obstruction for our consistency reductions. We +will also describe some properties of hierarchies of these problems, i.e., the classes +of problems that reduce by a consistency reduction to solving systems of linear +equations. One particularly interesting class is the class of all problems that reduce +to solving systems of equations over integers. +Definition 5.11. +Let G be an Abelian group, by CSP(G) we mean the following +problem: given a systems of equations of the form +(3) +𝑎1𝑥1 + · · · + 𝑎𝑛𝑥𝑛 = 𝑏 +where 𝑎1, . . . ,𝑎𝑛 ∈ Z and 𝑏 ∈ G. Decide whether the system is solvable. Equivalently, +it is formulated as the CSP with the template G, where the domain of G is 𝐺 and +G has a relation 𝑅𝑎1,...,𝑎𝑛,𝑏 for each 𝑛, 𝑎𝑖’s, and 𝑏 as above. Similarly as above, it is +not hard to check that all these relations are generated by fnitely many relations, +namely 𝑥1 + 𝑥2 = 𝑦 and 𝑦 = 𝑏 for each 𝑏 ∈ 𝐺, or equivalently, for each 𝑏 from some +set of generators of G. +Note that CSP(G) for a non-Abelian group is NP-complete [GR02]. We restrict to +Abelian groups, and in particular, to cyclic groups: Every fnite Abelian group is a +product of cyclic groups which allows us to ‘decompose’ CSP(G) for a fnite group +G to several CSPs of Abelian groups. Also note that the structure G assigned to an +Abelian group G has an alternating polymorphism, and hence is solvable by afne +integer programming relaxation, i.e., reducible via a gadget reduction to CSP(Z) +which is a CSP of a cyclic group; for details see [BBKO21, Section 7.3]. In the rest of +the section, we assume that G is a cyclic group generated by an element denoted by +1, i.e., it is either isomorphic to the group Z𝑛 of addition modulo 𝑛 ∈ Z, or Z itself. +Similarly to linear programming, all the problems of the form CSP(G) have the +short code property. The ‘short universal gadget’ reduction from a label cover +instance to an instance of CSP(G) is similar to the systems of linear inequalities + +42 +VICTOR DALMAU AND JAKUB OPRŠAL +we used for the linear programming. More precisely, let us denote by 𝜖(Σ) the +system of equations obtained from 𝜖LP(Σ) defned in Defnition 5.6 by dropping the +inequalities 𝑥𝑣,𝑑 ≥ 0. The key diference here is that we are asking for a solution of +this system in G instead of Q ∩ [0, 1]. +The basic afne integer programming relaxation is an example of the reduction +of the above form. It can be described by a composition of the standard reduction to +label cover, Σ(A, X), and 𝜖. The applicability of this basic level has been characterised +in [BBKO21, Theorem 7.19] using a minion Zaf. Following our exposition of Sherali- +Adams, we can see this minion in two equivalent ways, either as polymorphism +minion of Z (the structure corresponding to Z), or as an abstract minion Zaf defned +below in a slightly more general setting of cyclic groups. +Definition 5.12. Let G be a cyclic group generated by an element denoted by 1. The +minion Gaf is defned as follows: +G (𝑛) +af = {𝑎 ∈ 𝐺𝑛 | �𝑛 +𝑖=1 𝑎𝑖 = 1} +with the minor-taking operations 𝑎 ↦→ 𝑎𝜋 defned as 𝑎𝜋 (𝑖) = � +𝑗 ∈𝜋−1(𝑖) 𝑎𝑗. +Similarly as for linear programming, it is straight-forward to check that Pol(G) +is isomorphic to Gaf. We also present the analogue of Lemma 5.7. +Lemma 5.13. Let G be a cyclic group. The following are equivalent for each minor +condition Σ. +(1) 𝜖(Σ) is solvable in G; +(2) Gaf satisfes Σ; +(3) GΣ → G. +Proof. The proof is analogous to the proof of Lemma 5.7. The equivalence of (2) +and (3) is given by Lemma 4.10 and the fact that Gaf is isomorphic to Pol(G). The +equivalence of (1) and (2) is proved by following the proof of Lemma 5.7 only replac- +ing ‘probability distribution 𝜆 on 𝐷’ with ‘a tuple 𝜆 ∈ 𝐺𝐷 such that � +𝑔∈𝐺 𝜆(𝑔) = 1’ +which can intuitively be understood as “G-valued probability distribution”. +□ +Using the same argumentation as in the previous subsection, we can again argue +that a 𝑘-consistency reduction to CSP(G) is always equivalent to the following +procedure, which we describe as a decision algorithm. Since solving equations over +G is in P, and the reduction is polynomial-time computable, this algorithm runs in +polynomial time. +Definition 5.14. (𝑘-consistency reduction to group equations) Fix 𝑘 > 1 and a cyclic +group G generated by 1. Given an instance X of CSP(A). +(1) Run the 𝑘-consistency step as described in Defnition 3.24 with the output +being a label cover instance with variables 𝑣𝐾 for 𝐾 ∈ � 𝑄 +≤𝑘 +� each with the +domain F𝐾. +(2) Solve the following system of linear equations over G with variables 𝑥𝐾,𝑓 +for each 𝑓 ∈ F𝐾: +∑︁ +𝑓 ∈F𝐾 +𝑥𝐾,𝑓 = 1 +for all 𝐾 ∈ � 𝑄 +≤𝑘 +�, +∑︁ +𝑓 ∈F𝐾,𝑓 |𝐿=𝑔 +𝑥𝐾,𝑓 = 𝑥𝐿,𝑔 +for all 𝐿 ⊂ 𝐾 ∈ � 𝑄 +≤𝑘 +� and 𝑔 ∈ F𝐿. +(3) Output yes if the system is solvable, and no otherwise. + +LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS +43 +It is easy to check that if X → A then the above algorithm answers yes: if +ℎ: X → A is a homomorphism, then the assignment 𝑥𝐾,ℎ|𝐾 = 1 extends to a +solution of the system by assigning 0 to all other variables. Hence, in order to prove +correctness of this algorithm, soundness needs to be analysed. The equivalence is +formally given by the following. +Lemma 5.15. PCSP(A, A′) ≤𝑘-cons CSP(G) if and only if the 𝑘-consistency reduction +to G-equations solves PCSP(A, A′). +□ +Using our framework, we get immediately that the above algorithm with 𝑘 > 2 +and G = Z solves correctly all promise CSPs solvable by the 𝑘-consistency algorithm +as well as systems of linear equations over a fnite Abelian group, i.e., it solves +the two prime examples of tractable CSPs. We conjecture that it can be used as a +replacement for Bulatov’s and Zhuk’s algorithms. +Conjecture 5.16. For every fnite structure A, either 3-colouring reduces to CSP(A) +by a gadget reduction, or there exists 𝑘 > 1 such that CSP(A) reduces to afne integer +programming by the 𝑘-consistency reduction. +An intuition behind the above conjecture is an observation that solving systems +of equations over a group is somewhat typical obstruction for an existence of a +𝑘-consistency reduction. More precisely, we can prove the following from which +we can derive further non-existence of a Datalog∪ reduction between certain CSPs. +The core of the proof of the proposition below is an adaptation of the proof that +CSP(Z𝑝) is not solved by any level of Sherali-Adams hierarchy with a twist. +Proposition 5.17. CSP(Z𝑝) does not reduce to CSP(Z𝑞) by a consistency reduction +for any distinct primes 𝑝 and 𝑞. +Proof. We start with a 𝑘-consistent but unsolvable instance of CSP(Z𝑝), i.e., with +a system of equations modulo 𝑝 which is not solvable, but is accepted by the 𝑘- +consistency algorithm. Such a system exists due to, e.g., [FV98, ABD09]. Observe +that, after each step in the 𝑘-consistency procedure, the sets F𝐾 are always afne +subspaces of Z𝐾 +𝑝 — since to begin with they are defned by some equations, and +in each step, we intersect with a projection of another afne subspace. Since the +system is consistent, the subspaces are non-empty. +Further observe that the maps 𝜋 : F𝐾 → F𝐿 defned by restriction to 𝐿 are afne, +and hence the size of the preimage of an element 𝑓 ∈ F𝐿 under 𝜋 does not depend +on 𝑓 — it only depends on the dimension of the kernel of 𝜋. Hence the uniform +probability on F𝐾, i.e., 𝜆: F𝐾 → Q defned as 𝜆(𝑓 ) = 1/𝑝𝑑 where 𝑑 is the dimension +of F𝐾, solves the corresponding Sherali-Adams linear program. Since 𝑝 and 𝑞 are +coprime, we can interpret the expression 1/𝑝𝑑 as an element of Z𝑞. Consequently, +this assignment solves the system in the second step of Algorithm 5.14 with G = Z𝑞, +and hence the algorithm accepts a non-solvable instance. +□ +We note that the above proposition could be also proved by refning [ABD09, +Theorem 3] (to adapt for composition with another Datalog reduction) and using +the fact that CSP(Z𝑝) is not expressible in fxed-point logic with modulo 𝑞 rank +operators (see, e.g., [Hol11]). This alternative approach has been communicated to +us by Anuj Dawar [Daw22] and precedes the above proof. +As a consequence of the above proposition, we get that, e.g., CSP(K3) does not +reduce to CSP(Z2) by a consistency reduction, since CSP(Z3) reduces to CSP(K3) +by a gadget reduction and consistency reductions are transitive by Theorem 3.27. +This concludes that the upper of the two inequalities in Figure 1b is strict. More + +44 +VICTOR DALMAU AND JAKUB OPRŠAL +generally, it can be shown that CSP(K3) does not reduce to CSP(Z) by a consistency +reduction — this statement follows from the main result of [CŽ22a]. +Further, it is known that CSP(Z𝑝)’s are a complete set of obstructions for a +consistency reduction from a CSP to the trivial CSP, in the sense that CSP(A) +reduces by 𝑘-consistency reduction to the trivial CSP if and only if CSP(Z𝑝) does +not reduce to CSP(A) by a gadget reduction for any prime 𝑝 — this follows from +the characterisation of CSPs solvable by local consistency [BK14] and Theorem 5.2. +Could it be that equations over a group form a complete set of obstructions for +consistency reductions? Assuming, it is the case and it is also true for the infnite +template Z of solving equations over integers, we can explain the CSP dichotomy +in the following way: Either CSP(A) allows a gadget reduction from CSP(G) for +a non-Abelian group G, and hence it is NP-complete, or all groups G, such that +CSP(A) allows a gadget reduction from CSP(G), are Abelian, and hence CSP(A) +reduces to CSP(Z) by a consistency reduction and it is in P. Finally, it is also possible +that for each tractable CSP(A), there exist 𝑛 and 𝑘, that depend on A, such that +CSP(A) reduces to CSP(Z𝑛) by the 𝑘-consistency reduction. Such a result would be +a considerable step in providing a descriptive complexity of tractable CSPs. +Finally, there are several algorithms related to Algorithm 5.14 whose power is +not fully described for fnite template CSPs, and are actually stronger than the one +we suggest. A few of them are hierarchies of problems that can express CSP(Z): the +question whether higher levels of LP+AIP solves all tractable CSPs was asked in +[BGWŽ20], and higher levels of CLAP was suggested in [CŽ22b] (note that [CŽ22b] +also asks whether CLAP itself could solve all fnite tractable CSPs which is not +immediately implied by our conjecture). Another algorithm, called cohomological +𝑘-consistency was suggested in [ÓCo22] — the cohomological 𝑘-consistency is es- +sentially a stronger version of our algorithm: the diference is that it does not only +ask whether the above system of linear equations has a solution over Z, but also +attempts to fnd a solution, for each 𝑓 ∈ F𝐾, that satisfes 𝑥𝐾,𝑓 = 1, and iteratively +removes values for which there is none. In short, our conjecture implies that all +CSPs tractable by [Bul17, Zhu20] can be solved by any of these algorithms, and +the comparison of power of, e.g., cohomological 𝑘-consistency and higher levels of +LP+AIP is not clear. +Acknowledgements +Parts of this paper are based on part of an unpublished note by Marcin Wrochna +which in particular contains the characterisation of the arc-consistency reduction, +and several observations and statements that served as a ground for research pre- +sented in this paper. We are grateful to Marcin for allowing us to publish his results +among ours. +References +[ABD09] +Albert Atserias, Andrei A. Bulatov, and Anuj Dawar. Afne systems of equations and counting +infnitary logic. Theor. Comput. Sci., 410(18):1666–1683, 2009. doi:10.1016/j.tcs.2008.12.049. +[AD22] +Albert Atserias and Víctor Dalmau. Promise Constraint Satisfaction and Width, pages 1129– +1153. 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ACM, 67(5):30:1–30:78, August +2020. doi:10.1145/3402029. + diff --git a/7NE4T4oBgHgl3EQfcgzB/content/tmp_files/load_file.txt b/7NE4T4oBgHgl3EQfcgzB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..004b22844183d7eb3a23e745202b412acd378951 --- /dev/null +++ b/7NE4T4oBgHgl3EQfcgzB/content/tmp_files/load_file.txt @@ -0,0 +1,2453 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf,len=2452 +page_content='LOCAL CONSISTENCY AS A REDUCTION BETWEEN CONSTRAINT SATISFACTION PROBLEMS Victor Dalmau Universitat Pompeu Fabra, Barcelona, Spain victor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='dalmau@upf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='edu Jakub Opršal Institute of Science and Technology Austria, Klosterneuburg, Austria jakub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='oprsal@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='at Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in similar way as the algebraic approach classifes gadget reductions between CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We classify a use of arc-consistency in this way, provide frst steps into classifcation of general 𝑘-consistency, and ask whether every tractable fnite template CSP is reducible by such a reduction to solving systems of afne Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' constraint satisfaction problem, Datalog, bounded width, reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' INTRODUCTION Is it possible to fnd a mathematical invariant that characterises when one com- putational problem reduces to another by a polynomial-time (Karp) reduction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Any such characterisation would give a characterisation of when a problem is solvable in polynomial time, possibly resolving the P vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' NP problem which makes this question far beyond our current understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Nevertheless, a classifcation of a certain subclass of polynomial-time reductions between certain well-structured problems is the core of the theory referred to as the algebraic approach to constraint satisfaction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' These reductions have a very precise strict structure that allows such classifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The goal of the present paper is to introduce a larger framework of reductions extending the scope of the algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Our setup includes reductions that are commonly used to prove NP-hardness of a promise variant of constraint satisfaction problems, and can be also used to provide new efcient algorithms by reducing to a tractable problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The constraint satisfaction problem (CSP) is a decision problem whose input consists of a list of variables, with each variable allowed to attain values from a fnite domain, and a list of constraints each involving a tuple of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The goal is to decide if there is an assignment of values to variables that simultaneously satisfes all the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Many problems can be directly expressed in this framework, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', SAT, graph 3-colouring, solving systems of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' These problems are studied for numerous reasons from many diferent research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us focus on the direction of the so-called algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The main motivation of this This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714532).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No 101034413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Jakub Opršal was also supported by the UK EPSRC grant EP/R034516/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='05084v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='LO] 12 Jan 2023 2 VICTOR DALMAU AND JAKUB OPRŠAL direction is to fnd out what inherent property makes a computational problem hard and which properties can lead to efcient algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The constraint satisfaction problem, where we study the complexity depending on the shape of the constraints allowed, is an ideal scope for rigorous investigations of this question since the CSPs are well-structured from the mathematical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' On top of that, the fnite-domain CSP have been identifed by Feder and Vardi [FV98] as one of the largest natural subclasses of the class NP that could exhibit a P vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' NP-complete dichotomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This was confrmed by the algebraic approach after 20 years of research [Bul17, Zhu20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The essence of the algebraic approach, whose development started with [JCG97, BJK05], and was further refned in many subsequent papers including [BOP18], is a characterisation of gadget reductions between constraint satisfaction problems in terms of certain mathematical invariants of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Loosely speaking a gadget reduction replaces each variable of the input with a tuple of variables of a fxed length, and each constraint on the input with a gadget of constraints on the newly introduced variables (possibly introducing more new variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The main theorem of the approach then asserts that these gadget reductions are characterised in terms of polymorphisms of the CSP template (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7 below for a formal statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A remarkable fact is that, in the realm of fnite template CSPs, gadget reductions are enough to provide all necessary NP-hardness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a fnite template CSP is NP- complete if all other fnite template CSPs reduce to it via a gadget reduction, and in P otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The tractability side of the CSP dichotomy is given by providing an algorithm for all other CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' These algorithms of [Bul17, Zhu20] are involved and rely on a structural analysis of the template and its polymorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Why do we need new reductions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Gadget reductions and the algebraic theory show that it is possible to characterise well-structured reductions between structured problems, and such a characteri- sation can yield interesting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In this paper, we introduce a wider class of reductions that could be possibly prone to a similar characterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We have several motivations to do that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Firstly, such a characterisation would provide a more refned view on the CSP dichotomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Though gadget reductions are enough to provide NP-hardness in the scope of fnite domain CSPs, there are infnitely many classes of CSPs up to such reductions, and their order (the class a problem Γ is below the class of problem Δ if Γ reduces to Δ by a gadget reduction) is incredibly complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1 For example, the order of Boolean CSPs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', when each variable is allowed to attain one of two values, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' gadget reductions is infnite (the precise shape was described in [BV20], and is shown in Figure 1a) even though only two polynomial-time algorithms sufce to solve all tractable Boolean CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Secondly, extending gadget reductions with other reductions could also improve understanding of the dichotomy from the perspective of descriptive complexity: It is known that gadget reductions between CSPs can be also achieved using Datalog with parameters [ABD09], but eforts to translate Bulatov’s and Zhuk’s algorithms to expressibility in some logic computable in polynomial time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', fxed point logic with linear algebraic operators) have not yet been successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Extending reductions with well-behaved logical reductions would allow us to focus on a fewer tractable CSPs to provide such a characterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 1Likely, it is as complicated as a countable partial order can get [Bar19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 3 (a) gadget reductions (b) consistency reductions Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Boolean CSPs ordered by two classes of reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Thirdly, a lot of recent research is focused on a more general version of CSPs, promise constraint satisfaction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In this promise version of the problem, each instance comes with two versions of each constraint, a stronger and a weaker one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The goal is then decide between two (disjoint, but not complementary) cases: all stronger constraints can be satisfed, or not even the weaker constraints can be satisfed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A prominent problem described in this scope is approximate graph colouring which asks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', to decide between graphs that are 3-colourable and those that are not 6-colourable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A systematic study of promise CSPs has been started in [AGH17] which studied a certain promise version of SAT, and continued in a series of papers including generalisations of the algebraic approach to promise CSPs in [BG21, BBKO21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Unlike the case of CSPs, there are many NP-hardness results of promise CSPs that are not explainable by gadget reductions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [DRS05, Hua13, AGH17, KO19, FKOS19, WŽ20, BWŽ21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' These hardness results usually rely on some version of the PCP theorem [AS98] which is better suited for a diferent, more analytical, approximation version of CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This is one of the main reasons that new reductions have been called for in [BBKO21] and [BK22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We note that the frst of these papers is the aforementioned classifcation of gadget reductions, while the second paper gives a sufcient condition for a more general reduction that replaces the necessity of using the PCP theorem in almost all of the above NP-hardness results on promise CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Our contribution In the present paper, we identify a well-behaved class of reductions between (promise) CSPs that is closed under composition, extends gadget reductions described by the algebraic approach, covers the reduction used in [BK22], and can express Sherali-Adams hierarchy [SA90] through reductions to linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We present two descriptions of such reductions: one using the language of logic, namely monotone Datalog interpretations, and one combinatorial which is based on the local consistency algorithm for CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We show that these two descriptions are equivalent in the context of promise CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In short, we call these reductions consistency reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 4 VICTOR DALMAU AND JAKUB OPRŠAL Some form of local consistency checking is often run as a preprocessing step in many CSP algorithms (including SAT-solvers, and Zhuk’s polynomial time algorithm [Zhu20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The local consistency checking that we will use is a procedure that considers at most 𝑘 variables at a time, and keeps track of what are the possible tuples of values that could be assigned to these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This list of tuples is then iteratively updated by removing tuples that cannot appear in a solution, and the algorithm stops when no more tuples can be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We can view this procedure as a reduction from the CSP to a CSP with binary constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The full 𝑘-consistency reduction is then obtained by joining this procedure with a standard gadget reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We start the quest of characterisation of consistency reductions by providing sev- eral preliminary observations that suggest that a characterisation might be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First, we prove that the logical and combinatorial descriptions are equivalent in the scope of promise CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Second, we characterise a special case of consistency reduc- tions, which can be described by replacing the 𝑘-consistency with arc-consistency, by the means of polymorphisms and a certain transformation 𝜔 on the classes of polymorphisms of a given template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We describe these constructions in detail in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This result hints towards what a more general characterisation of consistency reductions might look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In the last section, we investigate several classes of (promise) CSPs that are defned as those that reduce to a fxed CSP by a consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A few examples of such classes are: promise CSPs with bounded width which correspond to those that reduce to Horn-3SAT, promise CSPs solvable by some level of Sherali-Adams hierarchy which correspond to those that reduce to linear programming, and promise CSPs solvable by some level of Lasserre hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We show that all such classes are closed under gadget reductions, and as an immediate consequence we obtain several generalisations of [LZ07] and [BBKO21, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In particular, we show the class of promise CSPs that are solvable by some level of Sherali-Adams is closed under gadget reductions, and hence could be theoretically characterised by the means of polymorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The class of bounded width promise CSPs corresponds to the bottom element of the order of promise CSPs up to consistency reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since the majority of (non-promise) Boolean CSPs have bounded width, we get immediately that the order of Boolean CSPs up to consistency reductions consists only of three classes: the class of bounded width CSPs, the class of XOR-3Sat, and the class consisting of NP-complete CSPs (see Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This observation suggests that the order of all CSPs up to consistency reductions might be substantially simpler than the order of CSPs up to gadget reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, one class of problems that has not been studied before is of particular interest: the class of all CSPs that reduce to solving systems of afne equations over integers by a consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This family contains both CSPs solvable by local consistency, and systems of linear equations over all fnite felds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This leads us to the question: which fnite-domain CSPs are contained in this class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Could it be that the class contains all fnite-domain CSPs that do not allow a gadget reduction from 3-colouring?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If that was the case, it would provide a new polynomial time algorithm for those CSPs, and hence a new proof of the CSP dichotomy theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In the last section, we discuss several observations that lead us to believe that the answer to these questions could be afrmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 5 Related work We mention two papers which provide results in a similar direction as the present paper, though they have diferent motivations from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First, a recent paper of Barto and Kozik [BK22] describes sufcient condition for the existence of polynomial-time (and possibly a log-space) reduction between two promise CSPs more general that the one given in [BBKO21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' They aim to avoid dependence on the PCP theorem in current hardness results for PCSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The reduction they use to provide their result falls into our framework of reductions, and hence [BK22] provide a sufcient condition for a consistency reduction between two promise CSPs though this condition is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Second, Ciardo and Živný [CŽ23] defned a general framework for hierarchies of algorithms for promise CSPs generalising the Sherali-Adams hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Again, their hierarchies fall within our framework in the sense that a problem is solved by some level of the Ciardo-Živný hierarchy if and only if it reduces to a corresponding (promise) CSP by a consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The motivation of Ciardo and Živný is to provide classes of algorithms for promise CSPs that could be investigated by a uniform approach, and provide better understanding of those algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Organisation of the paper Section 2 provides basic defnitions of the problems considered, and each of the three following sections contains one of our main results together with all the prelim- inaries that are necessary for that particular result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Section 3 provides equivalence of the logical and combinatorial descriptions, Section 4 the characterisation of the arc-consistency reduction, and Section 5 our observations on classes of promise CSPs that are locally reducible to a fxed CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' PRELIMINARIES In the present paper, every object has a given ‘type’ in the sense that every symbol is either a set, a function, a structure, an integer, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Sets are generally denoted by capital letters, and integers by lower-case letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We often do not explicitly specify that such symbols represent sets or integers when we are introducing a new symbol and the ‘type’ is clear from the context, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', when the symbol appears in an index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Many of constructions described in this paper can be defned using several dif- ferent languages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', an instance of a CSP can be viewed as a list of constraints, a logical formula, or a relational structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Often one of this languages allows for a more elegant introduction of a certain notion, or a more elegant proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The same as true about other concepts that we introduce, and we often switch between several languages to avoid technical and obscuring constructions in the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We try to pick the language that is ‘locally best’ for the current argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The cost for this is that we often have to reinterpret a result of one construction as a diferent object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Technically, all the proofs could be written in any one language of your choice, though, we believe, it would result in unwieldy long proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We denote by [𝑛] the set of the frst 𝑛 positive integers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [𝑛] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We denote the set of all functions 𝑓 : 𝑋 → 𝐴 by 𝐴𝑋, and for such a function 𝑓 and 𝑌 ⊆ 𝑋, we denote its restriction to 𝑌 by 𝑓 |𝑌 : 𝑌 → 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For a function 𝜋 : 𝑋 → 𝑌 and 𝑦 ∈ 𝑌, we denote by 𝜋−1(𝑦) the set of preimages of 𝑦 under 𝜋, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the set {𝑥 ∈ 𝑋 | 𝜋(𝑥) = 𝑦}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will denote entries in a tuple 𝑎 by lower indices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝑎 = (𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘), and occasionally view a tuple 𝑎 ∈ 𝐴𝑘 as a function 𝑎: [𝑘] → 𝐴, 6 VICTOR DALMAU AND JAKUB OPRŠAL and hence 𝑎(𝑖) is an alternative notation for 𝑎𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we will write 𝑓 ◦ 𝑔 for the composition of 𝑓 and 𝑔 and 𝑓 𝑔(𝑥) for (𝑓 ◦ 𝑔)(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Constraint satisfaction problems To settle on the playing feld, we formally defne the constraint satisfaction problem, and its promise variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We refer to [BKW17] for a deeper exposition of the algebraic theory of CSPs, and to [KO22] for more background and examples of promise CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since it will bring further simplifcation below, we defne a non- homogeneous CSP where each variable is given its own domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Before we get to defne a fxed-template CSP, let us start with a defnition of the uniform CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The constraint satisfaction problem gets on input a list of variables with each variable 𝑣 assigned a fnite domain 𝐷𝑣, and a list of constraints each of the form (𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑣𝑘) ∈ 𝑅 where 𝑅 ⊆ 𝐷𝑣1 × · · · × 𝐷𝑣𝑘 is given as a list of tuples (the number 𝑘 is called the arity of the constraint and the tuple (𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑣𝑘) is called the scope of the constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The goal is to decide if there is an assignment 𝑠 with 𝑠(𝑣) ∈ 𝐷𝑣 for each variable 𝑣 that simultaneously satisfes all of the constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', (𝑠(𝑣1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑠(𝑣𝑘)) ∈ 𝑅 for each constraint as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Commonly, instances are restricted in some way, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', we could insist that all constraints are of arity at most 𝑚 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝑘 ≤ 𝑚 in the defnition above) for a fxed 𝑚 to get 𝑚-CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A 2-CSP where each constraint is a graph of a function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', of the form 𝜋(𝑣1) = 𝑣2 for some 𝜋 : 𝐷𝑣1 → 𝐷𝑣2 is called label cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Another reasonable restriction is to limit the domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' we could fx a set 𝐷 and require that 𝐷𝑣 = 𝐷 for each variable 𝑣 to get the usual defnition of the CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us move to the fxed-template CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A template consists of a sequence of allowed domains 𝐷𝑣, and relations 𝑅 appearing in the constraints, and it is encoded in a single structure defned as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A non-homogeneous relational structure is a tuple A = (𝐴1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝐴𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝑅A 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑅A 𝑚) where 𝐴𝑖 is a set for each 𝑖 ∈ [𝑛] called the 𝑖-th domain of A, and, for each 𝑖 ∈ [𝑚], 𝑅A 𝑖 ⊆ 𝐴𝑖1 × · · · × 𝐴𝑖𝑘 for some 𝑘 ≥ 0 and 𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑖𝑘 ∈ [𝑛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The word 𝑖1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑖𝑘 is called the arity of 𝑅𝑖 and it is denoted by ar𝑅𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We use the notation ar𝑅𝑖 (𝑗) = 𝑖𝑗, sometimes view ar𝑅𝑖 as a function, and for simplifcation just refer to the number 𝑘 (the length of the arity word) as the arity of 𝑅𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We assume that empty relations have a fxed arity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The signature of A is consists of: the number of domains 𝑛, and relational symbols 𝑅1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑅𝑚 together with their arities ar𝑅1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , ar𝑅𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We also call any such collection 𝜎 a relational signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' An element 𝑖 ∈ [𝑛] is called a 𝜎-type, and the symbols 𝑅𝑗 are called 𝜎-symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A structure A is fnite if 𝑛, 𝑚, and all 𝐴𝑖’s are fnite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We denote the domains of some structure by the same letter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a structure X has domains 𝑋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' and relations 𝑅X, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' One way to think about such structures is simply imagine them as a single-sorted structures whose domain is the disjoint union of the domains 𝐴𝑖, and each element has a given type (which is formally just a number in [𝑛]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The relations then only relate elements of given types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In concordance with this interpretation we will call any 𝑎 ∈ 𝐴𝑖 an element of A of type 𝑖, and we will simply write 𝑎 ∈ A for an element with understanding that such 𝑎 is assigned a fxed domain 𝐴𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will also use the symbol 𝐴 for the disjoint union of 𝐴𝑖’s, hence 𝑎 ∈ 𝐴 and 𝑎 ∈ A are synonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 7 Throughout this paper we will often say structure or 𝜎-structure instead of non- homogeneous relational structure, and we will silently assume that every element and variable has a given type, and that the domains of A are disjoint (this can be always achieved by picking a suitable isomorphic structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We silently assume that every object in this paper is properly typed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', every variable has a type that defnes in which domain it should belong even when the type is not mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will often assume those types are given implicitly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', if 𝜙 is a transformation on structures of certain signature, and A is a structure to which 𝜙 is applied, it is assumed that the signature of A agrees with the signature required on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Loosely speaking, a homomorphism between two structures of the same signature is a mapping that maps elements of one structure to elements of the second structure that preserves types and all relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Formally, we defne a homomorphism as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Given two structures A and B of the same signature, a homomorphism from A to B is a collection of maps 𝑓𝑖 : 𝐴𝑖 → 𝐵𝑖, one for each type 𝑖, such that for each relational symbol 𝑅 and (𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘) ∈ 𝑅A, we have (𝑓ar𝑅 (1) (𝑎1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑓ar𝑅 (𝑘) (𝑎𝑘)) ∈ 𝑅B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Below, we will simply write 𝑓 (𝑎) instead of 𝑓𝑖 (𝑎) if 𝑓 is a collection of mappings as above, and the type 𝑖 of 𝑎 is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Similarly, we will use the symbol 𝑓𝑅(𝑟) or simply 𝑓 (𝑟) for the component-wise application of 𝑓 to 𝑟 ∈ 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We write 𝑓 : A → B if 𝑓 is a homomorphism, and A → B if such a homomorphism exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Such a homomorphism can be also viewed as a function 𝑓 : 𝐴 → 𝐵 that preserves types and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that a homomorphism also implies that the two structures have the same signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Having settled on these defnitions, we are ready to defne a fxed-template CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let D be a non-homogeneous relational structure, the constraint satisfaction problem CSP(D) is a decision problem whose goal is: given X with the same signature as D, decide whether X → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Given A and A′ such that A → A′, we defne the promise constraint satisfaction problem PCSP(A, A′) as a promise problem whose goal is, on an input X which is a structure with the same signature as both A and A′, output yes if X → A, and no if X ̸→ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The pair A, A′ with A → A′ is called a promise template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We assume that A in such a promise template is fnite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that in the promise version, the requirement that A → B ensures that the yes- and no-instances are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We could defne promise CSP as a promise search problem, the goal would be, given X that is promised to map to A via a homomorphism, fnd a homomorphism X → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is not clear, and currently not known, whether these two versions of promise CSPs are of equivalent complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This paper focuses on the decision version of PCSP though a few results (mostly with some caveats) apply also to the search version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, note that CSP(A) is PCSP(A, A), and therefore every result about promise CSPs is applicable to (fnite template) CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Every instance X of CSP(D) can be interpreted as an instance of a general CSP in the following way: the variables are elements of X, where each variable 𝑣 ∈ 𝑋𝑖 is assigned the domain 𝐷𝑖, and constraints are of the form (𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑣𝑘) ∈ 𝑆 where (𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑣𝑘) is a tuple in 𝑅X and 𝑆 = 𝑅A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Following this translation, we introduce the 8 VICTOR DALMAU AND JAKUB OPRŠAL following notions for an instance of a fxed-template (promise) CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A constraint of an instance X of PCSP(A, A′) is an expression of the form (𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑣𝑘) ∈ 𝑅X for some symbol 𝑅 which is formally seen as a pair ((𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑣𝑘), 𝑅) where (𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑣𝑘) is called the scope of the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will often work with such an instance of promise CSP as an instance X of CSP(A) since most constructions in this paper only depend on the frst part of a promise template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We formally defne a notion of a reduction between two (promise) CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A reduction between decision problems is a (usually efciently computable) function that maps instances of one problem to instances of the other problem in such a way that the answer is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We say that a mapping 𝜓 from instances of PCSP(A, A′) to instances of PCSP(B, B′) is a reduction between these two problems if for all X, we have if X → A then 𝜓 (X) → B, and if X ̸→ A′ then 𝜓 (X) ̸→ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The frst item, preserving the yes instances, is usually referred to as completeness, and the second, preserving the no instances, as soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will usually use and prove the soundness in its converse form: if 𝜓 (X) → B′ then X → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that if we are interested in the complexity of search version of (promise) CSP, for a reduction between search versions, we need that this converse is witnessed by an efciently computable function that given a homomorphism 𝜓 (X) → B′ outputs a homomorphism X → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Reductions are more general than algorithms: every decision algorithm can be viewed as a reduction to a problem with two admissible instances yes and no where yes is the positive instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In our setting, we use a specifc CSP instead of this trivial problem, so that our reductions do not leave the scope of (promise) CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' There are a few sensible ways we could defne such a trivial CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We pick the template with no satisfable constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', such that all non-trivial instances are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The trivial CSP is the CSP with template T = (𝐷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ⊥T) where 𝐷 is a 1-element set, and ⊥T is the nullary empty relation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a constraint that is always false).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' There is an obvious algorithm which decides the trivial CSP which is, depending on the encoding of the input, either constant or linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will also implicitly assume that all (promise) CSPs considered in this paper have negative instances — this can be always ensured by adding a nullary constraint ⊥ that cannot be satisfed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Label cover and projective CSPs Many constructions in this paper hinge on seeing one concept in diferent per- spectives, and we often switch between these perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Here we present one such concept with two formal defnitions and show how they relate to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First perspective is an instance of the label cover problem mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Label cover is the following decision problem: given an instance of a non-homogeneous CSP such that each constraint is binary and of the form 𝑣 = 𝜋(𝑢) for some function 𝜋 : 𝐷𝑢 → 𝐷𝑣 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', defned by the relation 𝑃𝜋 = {(𝑎, 𝜋(𝑎)) | 𝑎 ∈ 𝐷𝑢}), decide whether it is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The second perspective is a fxed-template version of label cover, which gives a restriction on the relations of the template in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 9 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A projective structure is a non-homogeneous structure A whose every relation 𝑅A is a graph of a function 𝜋𝑅 : 𝐴𝑖 → 𝐴𝑗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝑅 is binary and 𝑅A = {(𝑎, 𝜋𝑅(𝑎)) | 𝑎 ∈ 𝐴𝑖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will call such a binary relation projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In a similar way as an instance of a fxed-template CSP can be interpreted as an instance of general CSP, an instance X of CSP(A), where A is projective, can be interpreted as a label cover instance I in such a way that X → A if and only if I is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since we will often use this interpretation, we present it as a formal defnition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let X be an instance of CSP(A) where A is a projective structure whose relations are defned by maps 𝜋𝑅 as in Defnition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne the corre- sponding label cover instance I in the following way: The variables of I are 𝑣𝑥 for 𝑥 ∈ 𝑋 with domain 𝐴𝑖 where 𝑖 is the type of 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The constraints of I are of the form 𝑣𝑥 = 𝜋𝑅(𝑣𝑦), where (𝑥,𝑦) ∈ 𝑅X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A mapping 𝑠 : 𝑉 → 𝐴, where 𝑉 is the set of variables of I satisfying 𝑠(𝑣𝑥) ∈ 𝐷𝑥 is a solution of I if and only if the mapping 𝑓 : 𝑋 → 𝐴 defned by 𝑓 (𝑥) = 𝑠(𝑣𝑥) is a homomorphism from X to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This interpretation of an instance of projective CSP as a label cover instance can be also reversed though with subtle caveats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Namely, that the process of Defnition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9 can loose some information about the template A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' More importantly, if we defne a construction 𝜙 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a reduction between two promise problems) that on input gets an instance X of some fxed-template CSP, say CSP(A) and produces a label cover instance I = 𝜙(X, A), which in turn is interpreted as an instance Y of CSP(B) with a projective B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To stay in the scope of fxed-template CSP, we would like to insist that B does not depend on X, but depends only on A and 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This can be done by defning B to have one domain for each possible domain of I, and one relation 𝑅B for each possible 𝜋 appearing in a constraint of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For all constructions in this paper, there is an implicit bound on the sizes of domains of B in terms of A and 𝜙 (or parameters thereof) that can be obtained in a straight-forward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, we can ensure that the template B is fnite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will not comment in much detail on these bounds and templates B — we will work directly with the instance I bearing in mind that when needed it can be interpreted as an instance of CSP(B) for a suitable projective B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We also note that label cover instances can be interpreted as minor conditions which were extensively used in [BBKO21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This will be important in Section 4, where we return to this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY REDUCTIONS We will defne several classes of efciently computable functions on relational structures, Datalog interpretations, gadget replacements, and (local) consistency re- ductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Also, we will show that arbitrary composition of Datalog interpretations and gadget replacements are equivalent to consistency reductions in the sense that one reduction can be used in the place of the other when reducing between two (promise) CSPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' this statement is formalised in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='27 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' While Datalog interpretations and gadgets have many parameters that could be adjusted, consis- tency has only one parameter, a width 𝑘, in this sense we can view local consistency as a canonical normal form of these reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 10 VICTOR DALMAU AND JAKUB OPRŠAL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Gadgets Gadget reductions are the more traditional reductions in the realm of fnite- template CSPs and promise CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' They have been classifed by the algebraic ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We outline this classifcation briefy in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1 below, and refer to [BBKO21], [BKW17], or [KO22] for a detailed exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We formally defne gadget reductions as a two-step process since this decomposition will be useful below, and it is not hard to see that the resulting procedure is equivalent with the more traditional one, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', as described in [KOWŽ22, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The frst step is reifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume A = (𝐴1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝐴𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑅A 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑅A 𝑚) is a structure, we denote by A∗ the structure with domains (𝐴1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝐴𝑛, 𝑅A 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑅A 𝑚) and binary relations 𝑃A∗ 𝑅,𝑖 ⊆ 𝑅A × 𝐴ar𝑅 (𝑖), for each 𝑅 and each 𝑖 ∈ [𝑘] where 𝑘 is the arity of 𝑅, defned as ((𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘),𝑏) ∈ 𝑃A∗ 𝑅,𝑖 if and only if 𝑏 = 𝑎𝑖 and (𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘) ∈ 𝑅A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The structure A∗ is called the reifcation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The result of a reifcation is a projective structure in the sense of Defnition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence, if I is an instance of CSP(A), then I∗ is an instance of a binary CSP with template A∗ with projective relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' As mentioned before, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2, instances of such a binary CSPs can be viewed as instances of label cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In fact, reifcation is a common way to reduce from any CSP to label cover, and if a label cover instance is interpreted as a minor condition, it is also equivalent to the construction Σ(A, I) from [BBKO21, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1] — we will talk about this interpretation in more detail later, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The second step of a gadget reduction has the freedom to choose a gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne a bit restrictive version of gadgets which are enough for reductions from CSPs with a projective template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜏 and 𝜎 be relational signatures, and assume that 𝜏 consists of 𝑛 types and binary relational symbols 𝑅1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑅𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A strict gadget 𝜸 is an 𝑛-tuple (D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , D𝑛) of 𝜎-structures along with a homomorphism 𝑝𝑅 : D𝑗 → D𝑖 for each 𝜏-symbol 𝑅 where 𝑖𝑗 = ar𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This gadget can be then applied on a 𝜏-structure A to produce a 𝜎-structure 𝜸 (A) in the following way: (1) For each 𝑎 ∈ 𝐴𝑖 introduce to 𝜸 (A) a copy of D𝑖, whose elements will be denoted as (𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) for 𝑑 ∈ D𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) For each 𝑅 of arity 𝑖𝑗, (𝑎,𝑏) ∈ 𝑅A, and every 𝑑 ∈ 𝐷𝑗, add an equality constraint (𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑝𝑅(𝑑)) = (𝑏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (3) Collapse all equality constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', identify all pairs of elements involved in one of the constraints introduced in the previous step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We denote by [𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑] the class of an element (𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) after collapsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will also call this operation a strict gadget replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Throughout this section, we will give several connected examples of relatively trivial cases of our constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us start with a strict gadget replacement 𝜸 that produces from a graph another graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This gadget is defned by a graph D1 = K2 = ({0, 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ≠), and a map 𝑝𝐸 : K2 → K2 that switches 0 and 1 (the non-trivial automorphism of K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The gadget replacement 𝜸 then produces from a graph G another graph H in the following way: Replace each vertex 𝑣 of G with a pair of vertices 𝑣0, 𝑣1 connected by an edge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', replace each vertex with the gadget K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 11 Replace each edge (𝑢, 𝑣) of G with the constraints 𝑢0 = 𝑣1 and 𝑢1 = 𝑣0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', for each edge, we identify the two gadgets introduced by replacing the two vertices in the reverse orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Collapse all equality constraints as in Defnition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that this gadget produces for each connected component of the input G either a loop, if the component contains an odd cycle, or an edge, if the component is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence, 𝜸 is a reduction from CSP(K2) to CSP(K∞) where K∞ is the countable clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2 A gadget replacement is a composition of the reifcation and a strict gadget replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is well-known that such a gadget replacement can be computed in log-space (the fact that collapsing equality constraints is in log-space is due to [Rei08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The classifcation of gadget reductions of the algebraic approach also gives a universal gadget to reduce from a (projective) structure A to any other structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' These gadgets are constructed using direct powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝑋 be a set, and B a structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The 𝑋-fold direct power of B, denoted by B𝑋, is the structure (𝐵𝑋 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝐵𝑋 𝑛 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑅B𝑋 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ) where for each 𝑅, 𝑅B𝑋 contains all tuples (𝑏1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑏𝑘) of functions 𝑏𝑖 : 𝑋 → 𝐵ar𝑅 (𝑖) such that (𝑏1(𝑥), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑏𝑘 (𝑥)) ∈ 𝑅B for all 𝑥 ∈ 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The universal gadget from a projective A to another structure B is then defned as D𝑖 = B𝐴𝑖 for each 𝑖, and 𝑝𝑅 : B𝐴𝑗 → B𝐴𝑖 is defned as 𝑝𝑅(𝑏) = 𝑏 ◦ 𝜋𝑅 where 𝜋𝑅 : 𝐴𝑖 → 𝐴𝑗 is the mapping defning 𝑅 in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The universality of these gadgets is given by the following lemma which we phrase without a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The proof is implicit in [BBKO21, Section 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If there is a gadget reduction from PCSP(A, A′) to PCSP(B, B′), then the composition of reifcation with the universal gadgets for A∗ and B is also a reduction between these two PCSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us also note that the universal gadget can be applied directly on a label cover instance I as follows: (1) Replace each variable 𝑣 with domain 𝐷𝑣 with a copy B𝐷𝑣, (2) for each constraint 𝜋(𝑢) = 𝑣, and 𝑏 ∈ B𝐷𝑣, add an equality constraint (𝑢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏 ◦ 𝜋) = (𝑣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏), and (3) collapse the equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, let us defne a notion that plays a major role in the characterisation of gadget reduction, pp-powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Though we will not use this fact directly, it is a special case of a Datalog interpretation defned below, and it relates the current work with other existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In brief, a pp-power is a transformation of the template that is adjoint to a gadget replacement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' a precise construction is described below after the defnition of pp-formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜏 be a signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A primitive positive 𝜏-formula (pp-formula) is a logical formula using only existential quantifcation, conjunctions, and equality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a formula that can be rewritten as 𝜙(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑛) ≡ ∃𝑥𝑛+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝑡1 ∧ 𝑡2 ∧ · · · ∧ 𝑡𝑚 where each 𝑡𝑖 is an atomic 𝜏-formula, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a formula of the form 𝑅(𝑥𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑖𝑙 ) or 𝑥𝑖1 = 𝑥𝑖2 where 𝑖𝑗 ∈ [𝑘].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The arity of 𝜙 is the word 𝑖1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑖𝑛 where 𝑖𝑗 is the type of 𝑥𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (An atomic formula 𝑅(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑙) is satisfed by 𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑙 in A if (𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑙) ∈ 𝑅A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=') 2In this and connected examples, we are deviating from the convention that requires that the template is fnite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 12 VICTOR DALMAU AND JAKUB OPRŠAL We note here that we are defning pp-formulae over typed (non-homogeneous) signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This means that each variable occurring in the formula must have one of the types of the signature and that, in addition, the atomic formulae respect the type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the arity (or type) of 𝑅 is the concatenation of the types of 𝑥𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑖𝑙 in each atomic formula 𝑅(𝑥𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑖𝑙 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Same applies to the other logical formalisms introduced in this paper such as Datalog programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We now defne logical interpretations in the special case of primitive positive logic, which are closely connected to gadget reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Fix two relational signatures 𝜎 and 𝜏, let 𝑛 be the number of domains in 𝜎, and let 𝑅1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑅𝑚 be the list of all relational symbols in 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A primitive positive interpretation (pp-interpretation) is an (𝑛 + 𝑚)-tuple 𝝓 = (𝜙1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝜙𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝜙𝑅1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝜙𝑅𝑚) of primitive positive 𝜏-formulae such that, for each relational symbol 𝑅 in 𝜎, the arity of 𝜙𝑅 is the concatenation of the arities of 𝜙𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝜙𝑖𝑘 where ar𝑅 = 𝑖1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑖𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The application of such a pp-interpretation 𝝓 to a 𝜏-structure is the 𝜎-structure 𝝓(A) = (𝜙A 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝜙A 𝑛 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝜙A 𝑅1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝜙A 𝑅𝑚) where 𝜙A 𝑖 denotes the set of all tuples in A that satisfy 𝜙𝑖, and 𝜙A 𝑅 is interpreted as a relation on 𝜙ar𝑅 (1) (A) × · · · × 𝜙ar𝑅 (𝑘) (A) in the natural way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', it consists of all tuples ((𝑎11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎1𝑖1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , (𝑎𝑘1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘𝑖𝑘)) ∈ 𝜙A ar𝑅 (1) × · · · × 𝜙A ar𝑅 (𝑘) such that A |= 𝜙𝑅(𝑎11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎1𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘𝑖𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We note that our defnition of an interpretation slightly difers from common defnitions, in particular, we do not allow factoring the newly defned domains by defnable equivalence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3 These pp-interpretations can describe gadget reductions in the following way: there is a correspondence between pp-interpretations and gadgets, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', if 𝝓 is a pp- interpretation corresponding to a gadget 𝜸, then 𝜸 (A) → B if and only if A → 𝝓(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This relation is called adjunction and it is enough to prove that such 𝜸 gives a reduction from CSP(𝝓(B)) to CSP(B) for each structure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For more details, see [KOWŽ22, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1] and [BKW17, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Logical reductions We frst described the class of reductions we consider in this paper in terms of logic, or more precisely Datalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We start with introducing Datalog and some of its known properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 3Our pp-interpretations are equivalent to pp-powers defned in [BOP18, Defnition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='6] in the following sense: every pp-interpretation of A is homomorphically equivalent to a pp-power of A, and conversely, every pp-power of A is homomorphically equivalent to a pp-interpretation of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, a structure is pp-constructible from A in the sense of [BOP18, Defnition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='4] if and only if it is homomorphically equivalent to a pp-interpretation of A (see [BOP18, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Datalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Datalog is a language of logic programs without functional symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜏 be a relational signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A Datalog program 𝜙 with input signature 𝜏 is constituted by a relational signature 𝛿 satisfying 𝜏 ⊆ 𝛿 (meaning that it has the same types as 𝜏 and each symbol of 𝜏 appears in 𝛿 with the same arity) along with a fnite collection of rules that are traditionally written in the form 𝑡0 ← 𝑡1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑡𝑟 where 𝑡0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝑡𝑟 are atomic 𝛿-formulae, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', formulae of the form 𝑅(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑛) where 𝑅 is a relational symbol in 𝛿 of arity 𝑛 and variables 𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑛, or 𝑥1 = 𝑥2 for variables 𝑥1 and 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In such a rule 𝑡0 is called the head and 𝑡1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑡𝑟 the body of the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Moreover, we require that neither the symbols in 𝜏 nor the equality appear in the head of any rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The symbols in 𝜏 are called input symbols, or EDBs (standing for extensional database predicates, while all the other symbols would be IDBs, intensional database predicates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Furthermore, one of the 𝛿-symbols is designed as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='4 A Datalog program receives as input a 𝜏-structure A and produces a relation with the same arity of the output predicate in the following way: Let B be the 𝛿-structure computed in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Start with setting 𝑅B = 𝑅A if 𝑅 ∈ 𝜏 and 𝑅B = ∅ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If there is a rule 𝑡0 ← 𝑡1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑡𝑟 and some assignment ℎ: 𝑋 → 𝐴, where 𝑋 are the variables occurring in the rule, such that all the atomic predicates hold in B then include (ℎ(𝑥1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,ℎ(𝑥𝑘)) in 𝑅B where 𝑡0 = 𝑅(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Repeat until we get to a fxed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Output the relation 𝑅B ⊆ 𝐴ar𝑅 (1) × · · · × 𝐴ar𝑅 (𝑘) where 𝑅 is the output predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will denote the output of such a Datalog program by 𝜙A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Commonly in the context of CSPs, Datalog is restricted so that the output is a nullary predicate, so that such a program outputs either true or false — we call such Datalog programs Datalog sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The arity of a Datalog program is the arity of the output predicate, and the width of a Datalog program is the maximal number of variables in a rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne Datalog interpretations in a similar way as pp-interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This hinges on interpretation of Datalog programs as formulae in a fragment of the infnitary logic L∞,𝜔 (for defnition see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [ABD09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' More precisely it is included in the existential positive fnite variable fragment of the said logic, and also in the fxed point logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A Datalog program 𝜙 whose output is a relation with arity 𝑖1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑖𝑘 is then seen as a formula of the same arity, so that A |= 𝜙(𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘) if and only if (𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘) ∈ 𝜙A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Fix two relational signatures 𝜎 and 𝜏, let 𝑛 be the number of types in 𝜎, and let 𝑅1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑅𝑚 be the list of all relational symbols in 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A Datalog interpretation is an (𝑛 + 𝑚)-tuple 𝝓 = (𝜙1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝜙𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝜙𝑅1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝜙𝑅𝑚) of Datalog programs with input signature 𝜏 such that, for each relational symbol 𝑅 in 𝜎, the arity of 𝜙𝑅 is the concatenation of the arities of 𝜙𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝜙𝑖𝑘 where ar𝑅 = 𝑖1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑖𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Such an interpretation is said to be of width 𝑘 if all 𝜙𝑖 and 𝜙𝑅𝑗 are of width 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The application of such a Datalog interpretation 𝝓 to a 𝜏-structure A is then defned in the same way as in Defnition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', it is the 𝜎-structure 𝝓(A) = (𝜙A 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝜙A 𝑛 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝜙A 𝑅, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝜙A 𝑅𝑚) 4In database theory, Datalog programs very often have several output predicates (and defne structures instead of relations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Such program can be viewed as a special case of a Datalog interpretation which we defne later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 14 VICTOR DALMAU AND JAKUB OPRŠAL where 𝜙A 𝑅 is interpreted as a relation on 𝜙ar𝑅 (1) (A) × · · · × 𝜙ar𝑅 (𝑘) (A) in the natural way (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', as in Defnition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us describe an easy Datalog interpretation (𝜙1,𝜙⊥) from graphs to structures of the same signature as the trivial template T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' That is the input signature consists of single type and a binary relation 𝐸 and the output signature consists also of one type and one nullary relation ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence, the input of both 𝜙1 and 𝜙⊥ is a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We let 𝜙1 be the program with a single rule 𝑉 (𝑥) ← 𝑥 = 𝑥 and output 𝑉 , and 𝜙⊥ be the program with rule ⊥ ← 𝐸(𝑥,𝑥) with output ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The interpretation 𝝓 then produces from a graph G = (𝐺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝐸G), the structure 𝝓(G) = (𝐺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ⊥𝝓(G)) where ⊥𝝓(G) is true if G contains a loop, and false otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence, 𝝓(G) → T if and only if G does not have a loop, and consequently, 𝝓 is a reduction from CSP(K∞) → CSP(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We again note a diference to Datalog interpretations defned in [ABD09, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3]: in the above defnition we do not allow parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We do this to ensure that Datalog interpretations are monotone in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝝓 be a Datalog interpretation, and A and B structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If A → B, then 𝝓(A) → 𝝓(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The claim follows from monotonicity of Datalog programs: assuming a homomorphism ℎ: A → B, the component-wise application of ℎ gives a well- defned mapping 𝜙A 𝑖 → 𝜙B 𝑖 for each 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is then straightforward to check that the collection of these mappings is a homomorphism 𝝓(A) → 𝝓(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Let us briefy mention that a CSP is said to be solved by Datalog if its complement is defnable by a Datalog sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Formalized in the following defnition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The apparent negation in the defnition will become clearer when we discuss using Datalog as reductions (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A PCSP(A, A′) is said to be solvable by Datalog if there is a Datalog sentence 𝜓 such that, for all X with X → A, 𝜙X is false, and for all X with X ̸→ A′, 𝜙X is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Datalog∪ reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Datalog interpretations are powerful reductions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' they can reduce a substantial class of CSPs, including 2SAT and Horn-3SAT, to the trivial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Also another particular example, the arc digraph construction, was used to improve the state-of-the-art hardness of approximate graph colouring [KOWŽ22, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Nevertheless, the monotone version of the interpretation that we defned above cannot fully emulate gadget reductions, in particular, they cannot express taking disjoint unions of domains and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We note that [ABD09] use parameters in their Datalog interpretations only to express these disjoint unions up to a fnite number of exceptions — we do not have that liberty here since in the multisorted setting, we would have to deal with an infnite number of exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Instead, we deal with this by extending Datalog interpretation with this operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We introduce a new operator called union gadget, that loosely speaking constructs a new structure by taking disjoint unions of domains of an input structure, and in a similar fashion defnes new relations as unions of relations of the original structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In the formal defnition, we need to take care to preserve types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume that 𝜏 and 𝜎 are two relational signatures, and let 𝑛 and 𝑚 be the number of types in 𝜏 and 𝜎 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A union gadget 𝝊 is defned by a pair of mappings (𝑑,𝑟), where 𝑑 maps 𝜏-types to 𝜎-types and 𝑟 maps 𝜏-symbols to LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 15 𝜎-symbols, such that for each 𝜏-symbol 𝑅 of arity 𝑖1 · · ·𝑖𝑘, the 𝜎-symbol 𝑟 (𝑅) is of arity 𝑑(𝑖1) · · ·𝑑(𝑖𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Such a union gadget can be then applied on a structure A to produce a 𝜎-structure 𝝊(A) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The 𝑖-th domain of 𝝊(A) is the disjoint union of 𝐴𝑗’s where 𝑑(𝑗) = 𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the set {(𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝑗) | 𝑑(𝑗) = 𝑖 and 𝑎 ∈ 𝐴𝑖}, and similarly, for a 𝜎-symbol 𝑆, 𝑆𝝊 (A) is the disjoint union of 𝑅A for 𝑅 ∈ 𝑟 −1(𝑆), more precisely, 𝑆𝝊 (A) = {((𝑎1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑(𝑖1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , (𝑎𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑(𝑖𝑘))) | 𝑟 (𝑅) = 𝑆,𝑎 ∈ 𝑅,𝑖1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑖𝑘 = ar𝑅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us describe a union gadget that produces a graph from a multisorted structure with two types, 0 and 1, and four binary relations 𝐸𝑖,𝑗, for 𝑖, 𝑗 ∈ {0, 1}, each 𝐸𝑖,𝑗 of arity 𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' There is only one choice for the maps 𝑑 and 𝑟 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Given a structure V in this signature, the union gadget 𝝊 produces a graph G with vertex set 𝑉0 ∪ 𝑉1 (assuming 𝑉0 and 𝑉1 are disjoint) and edges 𝐸 = 𝐸V 0,0 ∪ 𝐸V 0,1 ∪ 𝐸V 1,0 ∪ 𝐸V 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that this would not be possible by a Datalog interpretation: Consider the structure V where 𝑉0 = {0}, 𝑉1 = {1}, and 𝐸𝑖,𝑗 = {(𝑖, 𝑗)} if 𝑖 ≠ 𝑗 and 𝐸𝑖,𝑖 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The graph G = 𝝊(V) is then isomorphic to K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Nevertheless, it can be verifed that every Datalog defnable relation on V is either empty or a singleton, and hence a Datalog interpretation 𝝓 can only produce graphs with a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' None of such graphs is homomorphically equivalent to K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' As we will show below, it is enough to consider compositions of Datalog inter- pretations with union gadgets in this specifc order, so we defne our extension of Datalog as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A Datalog∪ reduction is a composition 𝝊 ◦ 𝝓 where 𝝊 is a union gadget and 𝝓 a Datalog interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We write PCSP(A, A′) ≤Datalog PCSP(B, B′) if there exists a Datalog∪ reduction that is a valid reduction between the two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Clearly, every Datalog interpretation is a Datalog∪ reduction since we take the trivial union gadget (both 𝑑 and 𝑟 being the identity map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will show that every gadget replacement can be also expressed as a Datalog∪ reduction up to homomorphic equivalence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', they produce homomorphically equivalent output on the same input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We also note that a union gadget can be expressed as a gadget though not necessarily strict gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We start by showing that Datalog∪ reductions compose, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume that 𝝍1 and 𝝍2 are two Datalog∪ reductions such that the output signature of 𝝍1 coincides with the input signature of 𝝍2, then there is a Datalog∪ reduction 𝝍 such that, for all structures A, 𝝍(A) and 𝝍2𝝍1(A) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The obvious consequence of the above theorem is that if PCSP(A, A′) ≤Datalog PCSP(B, B′) ≤Datalog PCSP(C, C′), then PCSP(A, A′) ≤Datalog PCSP(C, C′), justify- ing the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We prove the theorem by showing that both Datalog interpretations and union gadgets compose, and that a union gadget and a Datalog interpretation can be permuted in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) Let 𝝊 and 𝝊′ be union gadgets such that the output signature of 𝝊 and the input signature of 𝝊′ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' There is a union gadget 𝝂 such that, for all structures A, 𝝂(A) and 𝝊′𝝊(A) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) Let 𝝓 and 𝝓′ be Datalog interpretations such that the output signature of 𝝓 and the input signature of 𝝓′ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' There is a Datalog interpretation 𝝍 such that, for all structures A, 𝝓(A) and 𝝓′𝝓(A) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 16 VICTOR DALMAU AND JAKUB OPRŠAL (3) Let 𝝊 be a union gadget and 𝝓 be a Datalog interpretation such that the output signature of 𝝊 and the input signature of 𝝓 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' There exist a union gadget 𝝊′ and a Datalog interpretation 𝝓′ such that, for all structures A, 𝝊′𝝓′(A) and 𝝓𝝊(A) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) Let 𝝊 be defned by (𝑑,𝑟) and 𝝊′ by (𝑑′,𝑟 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is immediate that the union gadget defned by (𝑑′ ◦ 𝑑,𝑟 ′ ◦ 𝑟) gives isomorphic outputs to 𝝊′ ◦ 𝝊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) It is well-known that Datalog programs are closed under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We sketch how to extend it to Datalog interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We construct a new Datalog interpretation 𝝍 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For every Datalog program 𝜙 ′ 𝑗 in 𝝓′ we include a Datalog program 𝜓𝑗 in 𝝍 obtained from 𝜙 ′ 𝑗 and 𝝓 as follows: Start with the union of all programs in 𝝓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For each symbol 𝑅 of arity 𝑖1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑖𝑘 in 𝜙 ′ 𝑗, include in 𝜓𝑗 a new symbol 𝑅′ whose arity is the concatenation of arities of 𝜙𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝜙𝑖𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Then, for each rule 𝑟 of 𝜙 ′ 𝑗, include in 𝜓𝑗 a rule 𝑟 ′ obtained from 𝑟 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First, in every atomic formula in 𝑟 we replace its predicate 𝑅 by 𝑅′ and replace every variable 𝑥 occurring in it by a tuple (𝑥𝑗1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑗𝑘) of fresh variables where 𝑖 is the type of 𝑥, 𝑗1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝑗𝑘 = ar𝑆𝑖, and 𝑆𝑖 is the output symbol of 𝜙𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is immediate to verify that 𝝍 and 𝝓′ ◦ 𝝓 give isomorphic outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (3) Let us denote by 𝜏, 𝜎, and 𝜌 the input signature of 𝝊, the output signature of 𝝊 which coincides with the input signature of 𝝓, and the output signature of 𝝓, and let 𝝊 be defned by (𝑑,𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The proof is a simple typing exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Generally, we would want to adapt 𝝓 into 𝝓′ that runs directly on A by adding a rule 𝑟 (𝑅)(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑘) ← 𝑅(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑘) for each 𝜏-symbol 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' These rules obviously compute the union of 𝑅’s with 𝑟 (𝑅) = 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Nevertheless, there is a formal problem with them: the variables 𝑥𝑖 present in the rule are not properly typed since the union contains tuples of various arities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We circumvent this by adding copies of each relational symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜙 be a Datalog program with input signature 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne a new Datalog program, or more precisely, a collection of Datalog rules without an output predicate, which will be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us denote this ‘program’ by 𝜙 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Its input signature is 𝜏, and it has an IDB 𝑅𝑗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑗𝑘 of arity 𝑗1 · · · 𝑗𝑘 for each symbol 𝑅 (IDB or EDB) in 𝜙 of arity 𝑑(𝑗1) · · ·𝑑(𝑗𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We include, for each 𝜎-symbol 𝑅 of arity 𝑗1 · · · 𝑗𝑘 and 𝑆 = 𝑟 (𝑅), the following rule into 𝜙 ′: 𝑆 𝑗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑗𝑘 (𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑘) ← 𝑅(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑘) where each 𝑥𝑖 is of type 𝑗𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Further, for each function 𝑓 from 𝜎-types to 𝜏-types such that 𝑑𝑓 is the identity, and each rule 𝑡0 ← 𝑡1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑡𝑚 of 𝜙, we introduce to 𝜙 ′ a rule 𝑡 ′ 0 ← 𝑡 ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑡 ′ 𝑚 where 𝑡 ′ 𝑖 is obtained from 𝑡𝑖 by replacing the symbol 𝑆 with 𝑆 𝑓 (𝑖1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑓 (𝑖𝑘) where 𝑖1 · · ·𝑖𝑘 is the arity of 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that this rule is properly typed since if a variable 𝑥 in the original rule was of type 𝑖, it becomes a variable of type 𝑓 (𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Now, for every relation symbol 𝑆 𝑗 where 𝑆 is the output predicate of 𝜙, LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 17 we shall denote by 𝜙 𝑗 the Datalog program obtained from 𝜙 ′ by choosing 𝑆 𝑗 as the output predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We construct 𝝓′ by including all possible choices of outputs in the above construction, that is the output signature will have a type 𝑖 𝑗 for each 𝜌-type 𝑖, and 𝑗 = 𝑗1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝑗𝑘 where 𝜙𝑖 is of arity 𝑑(𝑗1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑(𝑗𝑘) defned by 𝜙 𝑗 𝑖 , and similarly, a symbol 𝑆 𝑗 for each 𝜌-symbol 𝑆 and 𝑗 = 𝑗1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝑗𝑚 where 𝜙𝑆 has arity 𝑑(𝑗1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑(𝑗𝑚) of corresponding arity defned by 𝜙 𝑗 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The union gadget 𝝊′ is defned by (𝑑′,𝑟 ′) where 𝑑′(𝑖 𝑗) = 𝑖 and 𝑟 ′(𝑆 𝑗) = 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is immediate to verify that 𝝊′𝝓′ and 𝝓𝝊 give isomorphic outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To bring a little light to the proof of case (3), let us describe the construction of 𝝓′ and 𝝊′ in the particular case of the Datalog interpretation 𝝓 from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10 and the union gadget 𝝊 from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us start with describing the component 𝜙 ′ ⊥ of 𝝓′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Recall that 𝜙⊥ is the Datalog sentence given by the rule ⊥ ← 𝐸(𝑥,𝑥), and that the input signature 𝜎 of 𝝊 has two types 0, 1 and four binary relations 𝐸𝑖,𝑗, for 𝑖, 𝑗 ∈ {0, 1}, of arity 𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The union gadget 𝝊 then produces digraphs in the only possible way by taking the disjoint union of all domains and of all relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne a new program 𝜙 ′ ⊥ with this input signature by including two rules: ⊥ ← 𝐸0,0(𝑥,𝑥) and ⊥ ← 𝐸1,1(𝑦,𝑦) and outputting ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Intuitively, instead of looking for a loop in 𝐸 which is a disjoint union of four relations, 𝜙 ′ ⊥ looks for a loop in each of the relations separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that 𝐸0,1 and 𝐸1,0 cannot contain a loop, and hence can be skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To fnish the defnition of 𝝓′, we need to defne two more Datalog programs 𝜙 ′ 0 and 𝜙 ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Recall that the component 𝜙1 of 𝝓 has a single rule 𝑉 (𝑥) ← 𝑥 = 𝑥 and outputs 𝑉 – each 𝜙 ′ 𝑖 therefore has a single rule 𝑉𝑖 (𝑥) ← 𝑥 = 𝑥 where 𝑥 is of type 𝑖 and output 𝑉𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we take the disjoint union gadget 𝝊′ defned by 𝑑(𝑖) = 1 and 𝑟 (⊥) = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is easy to check that, indeed, for each 𝜎-structure V, 𝝊′𝝓′(V) and 𝝓𝝊(V) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us now prove that Datalog∪ reductions compose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume that either of 𝝍1 and 𝝍2 is decomposed as 𝝍𝑖 = 𝝊𝑖◦𝝓𝑖, hence the composition is𝝊2◦𝝓2◦𝝊1◦𝝓1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='17(3) this produces isomorphic outputs as 𝝊2 ◦ 𝝊′ 1 ◦ 𝝓′ 2 ◦ 𝝓1 for some Datalog interpretation 𝝓′ 2 and a union gadget 𝝊′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='17(1–2), we can compose the two union gadgets and two Datalog interpretations while producing isomorphic outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ We continue with showing that a gadget can be expressed as a Datalog∪ reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The precise statement is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For every gadget 𝜸 there is a Datalog∪ reduction 𝝍 such that, for all A, 𝜸 (A) and 𝝍(A) are homomorphically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We prove this by frst producing a Datalog program that almost emulates a single relation in 𝜸 as per the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We slightly abuse the terminology and say that a tuple 𝑔 ∈ G1 × · · · × G𝑘 has arity 𝑖1 · · ·𝑖𝑘 where 𝑖𝑗 is the type of 𝑔𝑗 (recall that by convention 𝑔 = (𝑔1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑔𝑘)) even when the components of the tuple do not belong to the same structure, and we will say that such a tuple has the same arity as a symbol 𝑅 if ar𝑅 = 𝑖1 · · ·𝑖𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 18 VICTOR DALMAU AND JAKUB OPRŠAL Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜸 be a strict gadget with input signature 𝜏 and output signature 𝜎 composed of 𝜎-structures G𝑖, let 𝑅 be a 𝜎-symbol of arity 𝑘 and let 𝑔 ∈ G𝑖1 × · · · × G𝑖𝑘 be a tuple of the same arity as 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' There is a Datalog program 𝜙𝑅𝑔 of arity 𝑖1 · · ·𝑖𝑘 with the following property: For every 𝜏-structure A and every tuple 𝑎 ∈ 𝐴𝑖1 × · · · × 𝐴𝑖𝑘, 𝑎 ∈ (𝜙𝑅𝑔)A if and only if ([𝑎1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , [𝑎𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔𝑘]) ∈ 𝑅𝜸 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us defne program 𝜙𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In addition to all input predicates in 𝜎, 𝜙𝑔 has an IDB 𝐸ℎ1,ℎ2 of arity 𝑗1𝑗2 for each ℎ1 ∈ G𝑗1 and ℎ2 ∈ G𝑗2, and an IDB 𝑅𝑔 of arity 𝑖1 · · ·𝑖𝑘 which is designated as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Now, let us describe the rules of 𝜙𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) For every 𝜏 symbol 𝑆 of arity 𝑖𝑗, ℎ ∈ G𝑗, include the rule: 𝐸𝑝𝑆 (ℎ),ℎ(𝑥,𝑦) ← 𝑆(𝑥,𝑦) (2) Include the rules: 𝐸ℎ1,ℎ1 (𝑥,𝑥) ← 𝑥 = 𝑥 𝐸ℎ1,ℎ2(𝑥,𝑦) ← 𝐸ℎ2,ℎ1 (𝑦,𝑥) 𝐸ℎ1,ℎ3 (𝑥,𝑧) ← 𝐸ℎ1,ℎ2 (𝑥,𝑦), 𝐸ℎ2,ℎ3(𝑦,𝑧) for all ℎ1,ℎ2,ℎ3 in the disjoint union of all the domains of G𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (3) Finally, add the rule: 𝑅𝑔(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑘) ← 𝐸𝑔1,ℎ1(𝑥1,𝑦), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝐸𝑔𝑘,ℎ𝑘 (𝑥𝑘,𝑦) for each 𝜎-type 𝑗 all ℎ ∈ 𝑅G𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us show that 𝜙𝑔 has the required property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We start by observing that a relation 𝐸ℎ1,ℎ2(𝑥,𝑦) is derived in A if and only if [𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='ℎ1] = [𝑦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='ℎ2] — which follows since rules introduces in item (1) introduce into 𝐸 the equality constraints of the gadget, and the rules in item (2) then compute the transitive symmetric refexive closure of 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Observing this, it is straightforward to see that ([𝑎1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , [𝑎𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔𝑘]) ∈ 𝑅𝜸 (A) if and only if there exists 𝑎 ∈ 𝐴𝑗 and ℎ ∈ 𝑅G𝑗 such that [𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='ℎ𝑖] = [𝑎𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔𝑖] for all 𝑖, which equivalent to triggering the rule (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us explain in detail, how to obtain programs 𝜓𝐸𝑖,𝑗 (where 𝑖, 𝑗 ∈ {0, 1}) satisfying the claim of the above lemma for the relational symbol 𝐸 and the gadget replacement 𝜸 described in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We use the symbol 𝐼 instead of 𝐸 as in the above proof so that our notation does not clash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The programs 𝜓𝐸𝑖,𝑗 are composed of the same rules, with diferent outputs: 𝐼 0,1(𝑥,𝑦) ← 𝐸(𝑥,𝑦) 𝐼 1,0(𝑥,𝑦) ← 𝐸(𝑥,𝑦) 𝐼𝑖,𝑖 (𝑥,𝑥) ← 𝑥 = 𝑥 𝐼𝑖,𝑗 (𝑥,𝑦) ← 𝐼 𝑗,𝑖 (𝑦,𝑥) 𝐼𝑖,𝑘 (𝑥,𝑧) ← 𝐼𝑖,𝑗 (𝑥,𝑦), 𝐼 𝑗,𝑘 (𝑦,𝑧) 𝐸𝑖,𝑗 (𝑥,𝑦) ← 𝐼𝑖,0(𝑥,𝑧), 𝐼 𝑗,1(𝑦,𝑧) 𝐸𝑖,𝑗 (𝑥,𝑦) ← 𝐼𝑖,1(𝑥,𝑧), 𝐼 𝑗,0(𝑦,𝑧) where 𝑖, 𝑗, and 𝑘 above ranges over 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To recall the intuition, 𝐼𝑖,𝑗 (𝑥,𝑦) is derived if 𝑥𝑖 and 𝑦𝑗 were identifed in the third step of application of 𝜸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The output of each 𝜓𝐸𝑖,𝑗 is then 𝐸𝑖,𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Observe that 𝐼 0,1(𝑥,𝑦) and 𝐼 1,0(𝑥,𝑦) are derived in G if and only if 𝑥 and 𝑦 are connected by a path of odd length in G, and 𝐼 0,0(𝑥,𝑦) and 𝐼 1,1(𝑥,𝑦) are derived if and only if 𝑥 and 𝑦 are connected by a path of even length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, we get that, if 𝑖 ≠ 𝑗, 𝐸𝑖,𝑗 (𝑥,𝑦) is derived if and only if 𝑥 and 𝑦 are connected by a path of LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 19 even length, and 𝐸𝑖,𝑖 is derived if and only if 𝑥 and 𝑦 are connected by a path of odd length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This exactly corresponds to when 𝑥𝑖 and 𝑦𝑗 are identifed in the third step of application of 𝜸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We are ready to prove that Datalog∪ reductions can emulate gadget reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us assume that 𝜸 is strict which can be done without loss of generality due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='17 and the fact that reifcation is a Datalog interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We now construct a Datalog interpretation 𝝓 with output 𝜎′ (to be defned later) and a union gadget 𝝊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We assume that 𝜸 is composed of 𝜎-structures G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', G𝑚, and that these structures are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Signature 𝜎′ contains a type𝑔 for each𝑔 ∈ G1∪· · ·∪G𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝑗 be such that𝑔 ∈ G𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne 𝜙𝑔 to be the Datalog program composed of a single rule 𝐷𝑔(𝑥) ← 𝑥 = 𝑥, where 𝑥 is of type 𝑗, and the output symbol of 𝜙𝑔 is 𝐷𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The equality 𝑥 = 𝑥 in the body is just an artefact to satisfy the formal requirement that the body is non empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Further, 𝜎′ contains a relational symbol 𝑅𝑔 for each 𝜎-symbol 𝑅 of arity 𝑘 and each tuple 𝑔 = (𝑔1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑔𝑘) of the same arity as 𝑅 where each 𝑔𝑖 is an element of G1 ∪ · · · ∪ G𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The arity of 𝑅𝑔 is 𝑔1 · · ·𝑔𝑘 and it is defned by the program 𝜙𝑅𝑔 as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This concludes the defnition of the interpretation 𝝓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The consecutive union gadget 𝝊 is defned by maps (𝑑,𝑟) such that 𝑑(𝑔) is the 𝜎-type of 𝑔, and 𝑟 (𝑅𝑔) = 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us now prove that𝝊𝝓(A) and𝜸 (A) are homomorphically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Observe that we have the following chain of equivalences for 𝑘-tuples 𝑎 and 𝑔: ([𝑎1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , [𝑎𝑘,𝑔𝑘]) ∈ 𝑅𝜸 (A) ⇔ 𝑎 ∈ 𝜙A 𝑅𝑔 ⇔ 𝑎 ∈ (𝑅𝑔)𝝓(A) ⇔ ((𝑎1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , (𝑎𝑘,𝑔𝑘)) ∈ 𝑅𝝊𝝓(A) where the frst equivalence is by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='20 and the last equivalence is by the defnition of 𝝊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence, the mapping 𝑓 : 𝝊𝝓(A) → 𝝓(A) defned by 𝑓 (𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔) = [𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔] is a homomorphism, and any mapping 𝑔 such that 𝑔(𝑏) ∈ 𝑓 −1(𝑏) is a homomorphism in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us describe a Datalog∪ reduction obtained from the gadget replace- ment 𝜸 from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The reduction is composed of a Datalog interpretation 𝝍 and a union gadget 𝝂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The interpretation 𝝍 outputs structures with signature 𝜎 composed of two types 0 and 1 and four binary relations of the same arities as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' to unify the signatures, we identify the symbols 𝐸𝑖,𝑗 with 𝐸𝑖,𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The interpretation 𝝍 is (𝜓0,𝜓1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝜓𝐸0,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ) where𝜓𝐸𝑖,𝑗 are the programs from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='21 and 𝜓𝑖 for 𝑖 ∈ {0, 1} are programs with a single rule 𝐷𝑖 (𝑥) ← 𝑥 = 𝑥 and output 𝐷𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We claim that connecting this 𝝍 with the union gadget 𝝊 from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14 then produces homomorphically equivalent outputs to 𝜸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To show that, let us assume for simplicity that the input is a connected unoriented graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We distinguish two cases: (1) G is bipartite, and 𝜸 (G) ≃ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since G is connected it splits to two parts 𝐴 and 𝐵 uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Observe that 𝑥 and 𝑦 are in the same path if and only if 𝑥 and 𝑦 are connected by a path of even length, and they are in distinct parts if and only if 𝑥 and 𝑦 are connected by a path of odd length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence 𝝊𝝓(G) is the complete bipartite graph with parts (𝐴 × {0}) ∪ (𝐵 × {1}) and (𝐴 × {1}) ∪ (𝐵 × {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Clearly, it is homomorphically equivalent to K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) G contains an odd cycle, and 𝜸 (G) is a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In this case there is an odd path from 𝑥 to 𝑥 for each 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence all pairs of elements 𝑥 and 𝑦 are connected by 20 VICTOR DALMAU AND JAKUB OPRŠAL both odd and even path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This implies that 𝝊𝝓(G) is a clique with all loops, and hence clearly homomorphically equivalent to a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, we have that 𝝊 ◦ 𝝍 is a reduction from CSP(K2) to CSP(K∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us fnish this section with a fnal example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Recall Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='22 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10 which produce reductions CSP(K2) ≤Datalog CSP(K∞) ≤Datalog CSP(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='16 then asserts that CSP(K2) ≤Datalog CSP(K∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This reduction uses the composition 𝝍′ of interpretations 𝝓′ from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='18 and 𝝍 from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='22: the program 𝜓 ′ ⊥ is obtained from the Datalog rules described in the above example by adding two rules of 𝜙 ′ ⊥: ⊥ ← 𝐸0,0(𝑥,𝑥) and ⊥ ← 𝐸1,1(𝑥,𝑥) Note that 𝜓 ′ ⊥ in fact decides whether the input has an odd cycle or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The program that we implicitly constructed by joining our proofs is a more verbose version of much simpler Datalog program solving CSP(K2) described in [KV00, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To complete the Datalog interpretation 𝝍′, we extend this Datalog program 𝜓 ′ ⊥ with programs𝜓 ′ 0 and𝜓 ′ 1 that are identical to𝜓0 and𝜓1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we compose 𝝍′ with the disjoint union 𝝊′ defned by (𝑑,𝑟) with 𝑑(𝑖) = 1 for 𝑖 ∈ {0, 1} and 𝑟 (⊥) = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The composition 𝝊′ ◦ 𝝍′ is then a valid Datalog∪ reduction from CSP(K2) to CSP(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, let us remark that although we started with a strict gadget replacement 𝜸 and a very trivial Datalog interpretation 𝝓 of width 1 that does not uses recursion in any of its Datalog programs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝝓 is actually a pp-interpretation), the program 𝝍′ has width 3 and uses recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It can be shown, using [LLT07] and [KV00], that this cannot be avoided: there is no Datalog interpretation of width 2 or a pp-interpretation that could be used instead of 𝝓′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Combinatorial reductions In this subsection, we describe the combinatorial counterpart of Datalog∪ reduc- tions — called consistency reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This reduction is based on the local consistency algorithm for CSPs and the uniform gadget reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We also prove that, in the scope of promise CSPs, consistency reductions have the same power as Datalog∪ reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consistency reductions are defned by the two templates and a single parameter 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us assume that we are trying to reduce PCSP(A, ∗) to PCSP(B, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' the second part of the templates are irrelevant for the defnition of the reduction, but play an important role for the soundness which we do not characterise here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The 𝑘- consistency reduction is composed of two steps: (1) establishing 𝑘-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This step is possibly the most intuitive approach to solving CSPs, and it is used in many CSPs algorithms (including, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', Zhuk’s polynomial algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We describe it as a procedure that inputs a CSP instance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a pair of structures X and A, and outputs a label cover instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) the universal gadget replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The second step is the standard reduction from label cover to CSP(B) which we described in Defnition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We start with describing the frst step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let X and A be structures of the same signature and 𝐾 ⊆ 𝑋, a partial homomor- phism from 𝐾 to A is a mapping 𝑓 : 𝐾 → A that preserves types and relations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', it satisfes the defnition of a homomorphism when all variables are quantifed in 𝐾 instead of 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Intuitively, a partial homomorphism is simply a partial solution to the instance defned on the given set 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 21 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='24 (𝑘-consistency step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Fix an integer 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The input to the 𝑘- consistency procedure is a pair (X, D) where X is an instance of CSP(D) (it is useful to assume that𝑘 is bigger than the maximal arity of relations of X since the procedure below ignores all constraints of arity bigger than 𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Also we denote by � 𝑋 ≤𝑘 � the set of all at most 𝑘-element subsets of 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) For each 𝐾 ∈ � 𝑋 ≤𝑘 �, let F𝐾 be the set of all partial homomorphisms from 𝐾 to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) Ensure that for each 𝐿 ⊂ 𝐾 ∈ � 𝑋 ≤𝑘 �, the sets F𝐾 and F𝐿 are consistent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', remove from F𝐿 all ℎ: 𝐿 → 𝐷 that do not extend to a 𝑔: 𝐾 → 𝐷, 𝑔 ∈ F𝐾, remove from F𝐾 all 𝑔: 𝐾 → 𝐷 whose restriction 𝑔|𝐿 is not in F𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (3) Repeat step (2), while anything changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (4) Output the label cover instance with a variable 𝑣𝐾 with domain F𝐾 for each 𝐾 ∈ � 𝑋 ≤𝑘 �, and a constraint between 𝑣𝐾 and 𝑣𝐿 for each 𝐿 ⊂ 𝐾 of the form 𝜋(𝑣𝐾) = 𝑣𝐿 where 𝜋(𝑔) = 𝑔|𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We denote the output of this step by 𝜅𝑘 (X, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The 𝑘-consistency step is turned into a decision algorithm for CSPs by outputting no if one of the sets F𝐾 (and consequently each of them) is empty, and outputting yes if all F𝐾’s are non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The resulting algorithm coincides with [BK14, Algorithm 1 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='5] though we are using a single parameter 𝑘 instead of two parameters 𝑘 and 𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (Promise) CSPs solved by this algorithm are said to have bounded width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='5 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='25 (bounded width).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We say that a PCSP(A, A′) has width at most 𝑘 if the 𝑘-consistency algorithm solves this problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', if every X, such that F𝐾 ≠ ∅ for all 𝐾 ∈ � 𝑋 ≤𝑘 � where F𝐾 denotes the domains in 𝜅𝑘 (X, A), maps homomorphically to A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Such a problem is said to be of bounded width if there exists 𝑘 such that the problem has width at most 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us now describe how to turn the 𝑘-consistency step into a reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The resulting reduction depends on the two structures A and B, appearing in the tem- plates of the promise CSPs involved, and the parameter 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The higher 𝑘 we chose, the better reduction we obtain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', it will be a valid reduction for more choices of the second part of the templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The running time depends exponentially on 𝑘, hence 𝑘 has to be fxed in order to obtain a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='26 (𝑘-consistency reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Fix two relational structures A and B, and an integer 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The 𝑘-consistency reduction applied to an instance X of CSP(A) is the following construction: (K1) First run 𝑘-consistency step on input X, A as described above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', compute the label cover instance 𝜅𝑘 (X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (K2) Encode the label cover instance into an instance of CSP(B), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a structure with the same signature as B, using the universal gadget for B in the following way: Replace each variable 𝑣𝐾 of the label cover with domain F𝐾 by a copy of BF𝐾 whose elements are denoted by (𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏) where 𝑏 : F𝐾 → 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For each constraint 𝐿 ⊂ 𝐾 of the label cover, identify the elements (𝐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏) with (𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏′) where 𝑏′(𝑓 ) = 𝑏(𝑓 |𝐿), for all 𝑏 : F𝐿 → 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 5The name ‘bounded width’ is derived from the fact that the template of such CSP has bounded treewidth duality [FV98, Theorem 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 22 VICTOR DALMAU AND JAKUB OPRŠAL Similarly as for gadget replacements, we will denote by [𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏] the element obtained from (𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏) after identifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We denote the resulting structure by 𝜿A,B 𝑘 (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For 𝑘 > 1, we say that PCSP(A, A′) reduces PCSP(B, B′) by the 𝑘-consistency reduction, and write PCSP(A, A′) ≤𝑘-cons PCSP(B, B′) if 𝜿A,B 𝑘 is a reduction between these promise CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If such a 𝑘 exists, we say that that PCSP(A, A′) reduces to PCSP(B, B′) by a consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The 𝑘-consistency reduction produces from a structure of the same type as A, a structure 𝜿A,B 𝑘 (X) of the same signature as B through producing an intermediate instance of label cover 𝜅𝑘 (X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The main result of this section is that Datalog∪ reductions and consistency reductions have the same power in the scope of promise CSPs in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' PCSP(A, A′) ≤Datalog PCSP(B, B′) if and only if there exists 𝑘 > 1 such that PCSP(A, A′) ≤𝑘-cons PCSP(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We prove this theorem using the following special case which also relates the width of a Datalog interpretation with the parameter 𝑘 of the 𝑘-consistency reduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Fix 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' PCSP(A, A′) ≤𝑘-cons PCSP(B, B′) if and only if there exists a Datalog interpretation 𝝓 of width 𝑘 and a strict gadget 𝜸 such that 𝜸 ◦ 𝝓 is a valid reduction between these promise CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us briefy outline how Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='27 follows from the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='27 given Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28 Assume that PCSP(A, A′) reduces to PCSP(B, B′) by a Datalog∪ reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since a union gadget can be expressed as a gadget replacement and each gadget replacement decomposes as a reifcation and a strict gadget replacement, we can assume that this reduction is of the form 𝜸 ◦ 𝝓 for some Datalog interpretation 𝝓 and a strict gadget replacement 𝜸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Therefore, PCSP(A, A′) reduces to PCSP(B, B′) via the 𝑘-consistency reduction where 𝑘 is the width of 𝝓 by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Conversely, we get that if PCSP(A, A′) reduces to PCSP(B, B′) by a 𝑘-consistency reduction then it reduces by a Datalog∪ reduction by the other implication of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28 and Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='19 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ In order to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28, we will frst show that the𝑘-consistency reduction can be equivalently expressed as a composition of a Datalog interpretation of width 𝑘 and a gadget replacement — namely, it is essentially a composition of a canonical Datalog interpretation of width 𝑘 and the universal gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Formally, we will prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A and B be two structures, and 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' There is a Datalog interpre- tation 𝝓 of width 𝑘, and a strict gadget replacement 𝜸 such that, for all structures X of the same signature as A, 𝜸𝝓(X) and 𝜿A,B 𝑘 (X) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In the proof of this lemma, we will use some basic facts about the connection between Datalog and local consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This connection is well-studied (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [KV95, FV98]), and the above lemma is a consequence of known results through a new perspective which involves a rather technical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us frst outline a few notational simplifcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We adopt functional notation for tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝑤 ∈ 𝑋𝑛 be a tuple, we denote by im𝑤 the set of all its entries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', im𝑤 = {𝑤1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑤𝑛}, and we will view𝑤 as a function [𝑛] → im𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If 𝜋 : [𝑚] → [𝑛], LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 23 we denote by 𝑤 ◦ 𝜋 the tuple (𝑤𝜋 (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑤𝜋 (𝑚)), and if 𝑤 is an injective tuple, 𝑤−1 : im𝑤 → [𝑛] is the function returning the index of appearance of a component in 𝑤, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝑤−1(𝑤𝑖) = 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will allow to compose functions 𝑎: 𝐾 → 𝐿 and 𝑎′: 𝐾 ′ → 𝐿′ if 𝐿 ⊆ 𝐾 ′ resulting with a function 𝑎′ ◦ 𝑎: 𝐾 → 𝐿′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If 𝑅 is a 𝑛-ary relation, and 𝜋 : [𝑚] → [𝑛], we write 𝑅 ◦ 𝜋 for the 𝑚-ary relation {𝑎 ◦ 𝜋 | 𝑎 ∈ 𝑅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assuming F𝐾 for 𝐾 ∈ � 𝑋 ≤𝑘 � is a system output by the 𝑘-consistency procedure, and 𝑤 ∈ 𝑋𝑛 is such that im𝑤 has at most 𝑘 elements, we let F𝑤 = Fim 𝑤 ◦ 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' These sets satisfy F𝑤 = F𝐾 ◦ 𝑤 = {(𝑓 (𝑤1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑓 (𝑤𝑛)) | 𝑓 ∈ F𝐾} for all 𝑤 ∈ 𝑋𝑛 and 𝐾 such that im𝑤 ⊂ 𝐾, which is a direct consequence of the defnition and the consistency of F𝐾’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Similarly, we can show that F𝑤◦𝜋 = F𝑤 ◦ 𝜋 for all 𝑤 and 𝜋 which will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We write 𝑋 ≤𝑘 for � 𝑚≤𝑘 𝑋𝑚, and note that if 𝑤 ∈ 𝑋 ≤𝑘 then F𝑤 is defned (the converse is not true since there are tuples with higher arity and repeated entries with im𝑤 of size at most 𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In the rest of this section we will work with 𝑘-consistency using this notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we will work with functions 𝑑 : 𝑅 → 𝐵 where 𝑅 ⊆ 𝐴𝐾 for some 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will use the following notation, if 𝜋 : 𝐾 → 𝐿, 𝑆 ⊆ 𝐴𝐿, and 𝑆 ◦ 𝜋 ⊆ 𝑅, then 𝑑𝜋 : 𝑆 → 𝐵 is defned as 𝑑𝜋 (𝑎) = 𝑑(𝑎 ◦ 𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This is in agreement with an established notation for minors of a function of arity 𝐾 (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' One of the keys to showing that 𝑘-consistency procedure can replace a Datalog interpretation of width𝑘 is the following statement, which has been proved implicitly in [KV95] (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A and X be structures, 𝑘 > 1, and F𝐾 denote the sets output by 𝜅𝑘 (X, A) of the 𝑘-consistency procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For each Datalog program 𝜙 of width 𝑘, we have that if 𝑤 ∈ 𝜙X then F𝑤 ⊆ 𝜙A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that in the statement above, 𝑤 ∈ 𝜙X immediately gives that im𝑤 has at most 𝑘 elements even when the tuple might have larger arity since 𝜙 is of width 𝑘, and hence it can only introduce tuples with at most 𝑘 diferent entries into the relation 𝜙X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Next, we construct the canonical Datalog interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This interpretation is based on so-called canonical Datalog program 𝜙 of width 𝑘 for a 𝜏-structure A defned as follows: Its signature is the set of all relations that are defnable in A by a Datalog program of width 𝑘, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', all 𝑅 ⊆ 𝐴≤𝑘 for which there is a Datalog program 𝜓 of width 𝑘 with 𝑅 = 𝜓 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We treat each such relation as an abstract symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The rules of the canonical Datalog are all the rules that are satisfed in A when each rule is interpreted as an implication 𝑡0 ← 𝑡1 ∧ · · · ∧ 𝑡𝑟 where 𝑡0 is the head and 𝑡1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑡𝑟 is the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' With this program at hand, we can construct the canonical interpretation by choosing diferent output predicates, namely, we let 𝜙𝑖 to be the Datalog formula obtained from the canonical Datalog program by designating 𝑅𝑖 as an output, where 𝑅1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , 𝑅𝑚 is a list of all symbols of 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Further, we add a binary relation 𝑃𝑖,𝑗,𝜋 of arity 𝑖𝑗 for each 𝑖, 𝑗 and 𝜋 : [ar𝑅𝑗 ] → [ar𝑅𝑖] such that 𝑅𝑖 ◦ 𝜋 ⊆ 𝑅𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The relation 𝑃𝑖,𝑗,𝜋 contains all pairs of the form (𝑎,𝑎 ◦ 𝜋) where 𝑎 ∈ 𝑅𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Clearly, such a relation is defnable by the canonical Datalog program extended with the rule 𝑃𝑖,𝑗,𝜋 (𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑛,𝑥𝜋 (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝜋 (𝑚)) ← 𝑅𝑖 (𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑛) where𝑛 is the arity of 𝑅𝑖 and𝑚 is the arity of 𝑅𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that this Datalog interpretation satisfes 𝜙𝑖 (A) = 𝑅𝑖 for all 𝑖 where 𝑅𝑖 on the right-hand side is interpreted as the actual relation on 𝐴, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', each 𝑅𝑖 is actually defned by 𝜙𝑖 in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 24 VICTOR DALMAU AND JAKUB OPRŠAL The key property of the canonical Datalog program is that it is in a sense most general Datalog program of width 𝑘 with respect to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In particular, we will use the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝝓 be the canonical Datalog interpretation of width 𝑘 for A, X a structure, F𝐾 denote the sets output by 𝜅𝑘 (X, A), and let 𝜏 denote the output signature of 𝝓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If 𝑤 ∈ 𝑋 ≤𝑘, then there exists a 𝜏-type 𝑖 such that F𝑤 = 𝜙A 𝑖 and 𝑤 ∈ 𝜙X 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The lemma follows from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [KV95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We briefy sketch a direct argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We show that F𝑤 is Datalog defnable for each 𝑤 ∈ 𝑋 ≤𝑘, which can be argued by induction through the evaluation of the 𝑘-consistency algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In the frst step, F𝑤 is initiated as 𝐴𝑙 where 𝑙 is the length of 𝑤, which is clearly Datalog defnable in A, and is satisfed by 𝑤 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Then in each of the iterated steps, we alter F𝑤 by removing certain values, which is equivalently expressed by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a Datalog rule of the form F ′ 𝑤(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑙) ← F𝑤(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑙), F𝑤′(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑙′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' where F ′ 𝑤 denotes the new value for F𝑤, which is a rule in 𝑘 variables valid in A and hence a rule of the canonical Datalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Again, it is easy to observe that the corresponding relation is derived for 𝑤 on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ This observation, together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='30 describes the close relationship between the 𝑘-consistency procedure and the canonical Datalog interpretation: F𝑤 is the minimal 𝜙A 𝑖 (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' inclusion) such that 𝑤 ∈ 𝜙X 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The diference between the 𝑘-consistency step and the canonical Datalog interpretation (apart from the output of 𝑘-consistency step being a label cover instance and the output of the canonical Datalog a binary projective structure) is that in 𝝓(X), each 𝑤 might appear in several domains of 𝝓(X), hence it represents several elements of diferent types: one of each type 𝑖 where 𝑤 ∈ 𝜙X 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The domain of the corresponding variable 𝑤 of type 𝑖 is 𝜙A 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In the output of 𝑘-consistency, 𝑤 has only one copy with domain F𝑤 = 𝜙A 𝑖 for a suitable 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This diference is then smoothed by the use of the universal gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us for future reference note that the universal gadget for 𝝓(A) and B introduces an equality constraint between (𝑣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) and (𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒) where 𝑣 ∈ 𝜙X 𝑖 , 𝑤 ∈ 𝜙X 𝑗 , 𝑑 : 𝜙A 𝑖 → 𝐵, and 𝑒 : 𝜙A 𝑗 → 𝐵, if (𝑣,𝑤) ∈ 𝑃𝝓(X) 𝑖,𝑗,𝜋 (which is equivalent to 𝜙A 𝑖 ◦ 𝜋 ⊆ 𝜙A 𝑗 and 𝑣 ◦ 𝜋 = 𝑤) and 𝑑 = 𝑒𝜋 for some 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜸 be the universal gadget for 𝝓(A) and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We claim that 𝜸𝝓(X) is isomorphic to 𝜿A,B 𝑘 (X) for each X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We show this by constructing two mutually inverse homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In both cases, we defne the homomorphisms on the disjoint union of the gadgets before introducing and collapsing equality constraint, and subsequently argue that the value is not changed after collapsing said constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First, we construct ℎ: 𝜿A,B 𝑘 (X) → 𝜸𝝓(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne ℎ on the disjoint union of BF𝐾 ’s before collapsing the equality constraints, and then show that it preserves equality constraints introduced by the gadget replacement, hence inducing the required homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For each 𝐾, fx a bijection 𝑤𝐾 : [𝑙] → 𝐾 and let 𝑖𝐾 be such that F𝑤𝐾 = 𝜙A 𝑖𝐾 and 𝑤𝐾 ∈ 𝜙X 𝑖𝐾 , which exists by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Henceforth, we interpret the symbol 𝑤𝐾 as the element of type 𝑖𝑘 in 𝝓(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let ℎ(𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) = [𝑤𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑𝑤−1 𝐾 ] where 𝑑𝑤−1 𝐾 : 𝜙A 𝑖𝐾 → 𝐵 is defned to map 𝑎 to 𝑑(𝑎 ◦ 𝑤−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We claim that ℎ preserves the relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The proof of the claim is straight-forward: Each tuple in a relation LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 25 𝑅 of the disjoint union of BF𝐾 ’s is of the form (𝑑1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑑𝑙) ∈ 𝑅BF𝐾 for some 𝐾, and 𝑑 ↦→ 𝑑𝑤−1 𝐾 is a homomorphism BF𝐾 → B𝜙𝑖𝑘 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We check that ℎ preserves the equality constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', that ℎ(𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) = ℎ(𝐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒) whenever the universal gadget introduces an equality between (𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) and (𝐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', when 𝐿 ⊂ 𝐾 and 𝑑 = 𝑒1𝐿 where 1𝐿 : 𝐿 → 𝐾 is the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜋 = 𝑤−1 𝐾 ◦ 𝑤𝐿, which means that 𝑤𝐾 ◦ 𝜋 = 𝑤𝐿, and hence (𝑤𝐿,𝑤𝐾) ∈ 𝑃𝝓(X) 𝑖𝐿,𝑖𝐾,𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, the constraint (𝑤𝐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏) = (𝑤𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏𝜋) is introduced to 𝜸𝝓(X) for each 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' By substituting 𝑏 = 𝑒𝑤−1 𝐿 , we get that ℎ(𝐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒) = [𝑤𝐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒𝑤−1 𝐿 ] = [𝑤𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒𝜋◦𝑤−1 𝐿 ] = [𝑤𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒𝑤−1 𝐾 ◦1𝐿] = [𝑤𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑𝑤−1 𝐾 ] = ℎ(𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) as we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Second, we construct 𝑔: 𝜸𝝓(X) → 𝜿A,B 𝑘 (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Again, we defne 𝑔 on the disjoint union of B𝜙𝑖 (A)’s introduced to𝜸𝝓(X) by replacing 𝑤 ∈ 𝜙𝑖 (X) for all 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The mapping 𝑔 is defned as 𝑔(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) = [im𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑𝑤] where 𝑤 ∈ 𝜙X 𝑖 , 𝑑 : 𝜙A 𝑖 → 𝐵, and 𝑑𝑤 : F𝐾 → 𝐵 is defned by 𝑑𝑤(𝑓 ) = 𝑑(𝑓 ◦ 𝑤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since 𝑤 ∈ 𝜙X 𝑖 we have F𝑤 ⊆ 𝜙A 𝑖 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='30, and hence 𝑑𝑤 is a well-defned function with domain F𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Checking that 𝑔 is well-defned is straightforward and analogous to the argument that ℎ is well-defned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we argue that ℎ and 𝑔 are mutually inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The fact that 𝑔ℎ is the identity is immediate from the defnition since im𝑤𝐾 = 𝐾 and 𝑑𝑤𝐾 ◦𝑤−1 𝐾 = 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us show that ℎ𝑔 is the identity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We have ℎ𝑔(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) = ℎ(im𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑𝑤) = [𝑣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑𝑣−1◦𝑤] for 𝑤 ∈ 𝜙X 𝑖 where 𝑣 = 𝑤im 𝑤 is an element of type 𝑗 = 𝑖im 𝑤 and 𝜙A 𝑗 = F𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜋 = 𝑣−1 ◦ 𝑤, hence 𝑣 ◦ 𝜋 = 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Moreover, 𝜙A 𝑗 ◦ 𝜋 = F𝑣 ◦ 𝜋 = F𝑤 ⊆ 𝜙A 𝑖 where the last inclusion is given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='31, hence the required equality constraint (𝑣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑𝜋) = (𝑣 ◦𝜋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) is ensured by replacing 𝑃𝑗,𝑖,𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, ℎ𝑔(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑) = [𝑣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑𝜋] = [𝑣 ◦ 𝜋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑] = [𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑] as we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ An important consequence of the above lemma is that the𝑘-consistency reduction is expressible as a Datalog∪ reduction up to homomorphic equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This fact has a few consequences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the 𝑘-consistency reduction is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us now move to the proof of the other implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will use the following lemma to prove the completeness of the 𝑘-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A, B be relational structures, and 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If X → A for some structure X, then 𝜿A,B 𝑘 (X) → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝑔: X → A be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First, we observe that, for each 𝐾 ∈ � 𝑋 ≤𝑘 �, the restriction 𝑔|𝐾, satisfes 𝑔|𝐾 ∈ F𝐾: this is since the restrictions are locally consistent partial homomorphisms and will never be removed from F𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We use this observation to defne a homomorphism ℎ: 𝜿A,B 𝑘 (A) → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Again, we defne ℎ on the disjoint union of F𝐾’s for 𝐾 ∈ � 𝑋 ≤𝑘 � frst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let ℎ(𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏) = 𝑏(𝑔|𝐾), 𝑏 ∈ 𝐵F𝐾 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Clearly ℎ is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We need to show that for every 𝐾, 𝐿 ∈ � 𝑋 ≤𝑘 �, ℎ gives the same value on the elements glued in step (K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This is straightforward since if 𝐿 ⊂ 𝐾, and 𝑏(𝑓 ) = 𝑏′(𝑓 |𝐿), then ℎ(𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏) = 𝑏(𝑔|𝐾) = 𝑏′((𝑔|𝐾)𝐿) = 𝑏′(𝑔|𝐿) = ℎ(𝐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑏′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ The soundness of the 𝑘-consistency is achieved by the following lemma, which is a combination of two statements: frst, that 𝑘-consistency can be used instead 26 VICTOR DALMAU AND JAKUB OPRŠAL of any other Datalog interpretation of width 𝑘 in a similar way as the canonical Datalog interpretation, and that the universal gadgets can be used instead of any other gadget replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We present the proof as a common generalisation of these two statements instead of presenting the proofs of two cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A, B be a pair of structure 𝑘 > 1, 𝝓 be a Datalog interpretation of width 𝑘, 𝜸 be a strict gadget replacement, and assume 𝜸𝝓(A) → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Then for all X, 𝜸𝝓(X) → 𝜿A,B 𝑘 (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume that 𝜸 is composed of structures D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', D𝑛 and homomorphisms 𝑝𝑅, and let 𝑏 : 𝜸𝝓(A) → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne a homomorphism ℎ: 𝜸𝝓(X) → 𝜿A,B 𝑘 (X) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume that [𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔] ∈ 𝜸𝝓(X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝑤 ∈ 𝜙X 𝑖 for some type 𝑖 and 𝑔 ∈ D𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='30, we have that F𝑤 ⊆ 𝜙A 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We let ℎ(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔) = [im𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑] where 𝑑 : Fim 𝑤 → B is defned by 𝑑(𝑎) = 𝑏([𝑎 ◦ 𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' note that 𝑎 ◦ 𝑤 ∈ 𝜙A 𝑖 and hence (𝑎 ◦ 𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔) ∈ 𝜸𝝓(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We have to argue that ℎ is a homomorphism, and that it is well-defned, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', it preserves equality constraints introduced by the gadget replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First, we argue that the mapping ℎ is well-defned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let (𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔) and (𝑤 ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔′) be related by an equality constraint introduced to𝜸𝝓(X) by replacing a pair of elements of 𝝓(X) related by 𝑅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', (𝑤,𝑤 ′) ∈ 𝑅𝝓(X) and 𝑔′ = 𝑝𝑅(𝑔) for some relational symbol 𝑅 of arity 𝑖𝑖′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' More precisely, we have 𝑤 ∈ 𝜙X 𝑖 , 𝑤 ′ ∈ 𝜙X 𝑖′ and 𝑤𝑤 ′ ∈ 𝜙X 𝑅 where 𝑤𝑤 ′ denotes the concatenation of 𝑤 and 𝑤 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝐾 = im𝑤 ∪ im𝑤 ′ and observe that (𝑓 ◦𝑤, 𝑓 ◦𝑤 ′) ∈ 𝑅𝝓(A) for each 𝑓 ∈ F𝐾 since F𝑤𝑤′ ⊆ 𝜙A 𝑅 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We get that ℎ(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔) = [im𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑] = [𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒] where 𝑑(𝑎) = 𝑏([𝑎 ◦ 𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔]) and 𝑒(𝑓 ) = 𝑑(𝑓 |im 𝑤);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' the second equality is given by an equality constraint introduced to 𝜅A,B 𝑘 (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that 𝑒(𝑓 ) = 𝑏([𝑓 ◦ 𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔]) since the restriction of 𝑓 to im𝑤 is implicit in this expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Similarly, we have ℎ(𝑤 ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑝𝑅(𝑔)) = [𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒′] where 𝑒′(𝑓 ) = 𝑏([𝑎◦𝑤 ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑝𝑅(𝑔)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since (𝑓 ◦𝑤, 𝑓 ◦𝑤 ′) ∈ 𝑅𝝓(A), the equality constraint (𝑓 ◦𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔) = (𝑓 ◦𝑤 ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑝𝑅(𝑔)) is introduced to𝜸𝝓(A) by replacing this pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, 𝑏([𝑓 ◦𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔]) = 𝑏([𝑓 ◦𝑤 ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑝𝑅(𝑔)]) for all 𝑓 ∈ F𝐾, and hence 𝑒 = 𝑒′ concluding ℎ(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔) = [𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒] = [𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑒′] = ℎ(𝑤 ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑝𝑅(𝑔)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Second, we argue that ℎ is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Tuples in 𝑆𝜸𝝓(X) are of the form ([𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , [𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔𝑚]) where (𝑔1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑔𝑚) ∈ 𝑆D𝑗 and 𝑗 is the type of 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We get that ℎ(𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔𝑖) = [im𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑑𝑖] where 𝑑𝑖 (𝑎) = 𝑏([𝑎 ◦ 𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔𝑖]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Observe that ([𝑎 ◦ 𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' , [𝑎 ◦ 𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝑔𝑚]) ∈ 𝑆𝜸𝝓(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Therefore, we get the claim from the defnition of power and the fact that 𝑏 is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Finally, we can fnish the proof that 𝑘-consistency and Datalog∪ reductions have the same power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The implication (2)→(1) follows directly from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us prove (1)→(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We assume that 𝜸𝝓 is a reduction from PCSP(A, A′) to PCSP(B, B′), and claim that so is 𝜅A,B 𝑘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First, 𝜅A,B 𝑘 preserves positive instances by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We show soundness by the contrapositive, assuming 𝜅A,B 𝑘 (X) → B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that 𝜸𝝓(A) → B by completeness of 𝜸𝝓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, we can invoke Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='33 to get that 𝜸𝝓(X) → B′, and conclude that X → A′ by the sound- ness of 𝜸𝝓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 27 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28 has a few consequences, most prominently, since Datalog∪ re- ductions compose, we get that if one (promise) CSP reduced by a 𝑘-consistency reduction to a second (promise) CSP which in turn reduces to a third (promise) CSP by an 𝑙-consistency reduction, then the frst problem reduces to the third by a 𝑚-consistency reduction for some 𝑚 that depends on 𝑘, 𝑙, and the signatures involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ARC-CONSISTENCY REDUCTION In this section we characterise the applicability of a special case of a Datalog∪ reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Namely, a reduction that is obtained from the 𝑘-consistency reduction out- lined in the previous section by replacing the 𝑘-consistency step with arc-consistency step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We describe how is this procedure run on projective instances (instances of label cover) — in the full reduction it will be preceded by a reifcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1 (arc-consistency step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume I is a label cover instance with each variable 𝑣 having the domain 𝐷𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) For each variable 𝑣, set F𝑣 = 𝐷𝑣, (2) Ensure that for each constraint 𝜋(𝑣) = 𝑤, the sets F𝑣 and F𝑤 are consistent: remove from F𝑤 all 𝑑 that are not in the image of F𝑣 under 𝜋, remove from F𝑣 all 𝑑 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝜋(𝑑) ∉ F𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (3) Repeat step (2), while anything changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (4) Output the label cover instance with the same variables as X where each variable 𝑣 is assigned domain F𝑣 (instead of 𝐷𝑣), and with constraints of the form 𝜋|F𝑣 (𝑣) = 𝑤 for each original constraint 𝜋(𝑣) = 𝑤 (note that 𝜋 restricts to a function F𝑣 → F𝑤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We denote the resulting label cover instance by 𝜅arc(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Compare the arc-consistency step with the 𝑘-consistency step described in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3, the only signifcant diference is the initialisation: the local consistency can be also described as running the above arc-consistency on a label cover instance with a variable 𝑣𝐾 for each 𝐾 ∈ � 𝑋 ≤𝑘 � with the domain F𝐾 as defned by item (1) in the local consistency procedure, and a constraint 𝑅𝜋 for each 𝐿 ⊂ 𝐾 where 𝜋 : F𝐾 → F𝐿 is defned by 𝜋(ℎ) = ℎ|𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2 (arc-consistency reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The arc-consistency reduction applied an instance X of CSP(A) to obtain an instance of CSP(B) is then the following 3-step construction: (A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1) create a label cover instance I from X and A by reifcation and Defnition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2) apply the arc-consistency step as described above to get an instance 𝜅arc(I) of label cover;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' and (A2) turn the resulting instance of label cover into an instance of CSP(B) using the universal gadget, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', in the same way as in Step (K2) of 𝑘-consistency (see Defnition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We denote the resulting structure by 𝜅A,B arc (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We say that PCSP(A, A′) reduces to PCSP(B, B′) by the arc-consistency reduction, and write PCSP(A, A′) ≤arc-cons PCSP(B, B′), if 𝜅A,B arc is a proper reduction between these two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We characterise when arc-consistency reduction gives a proper reduction between two promise CSPs using the notions of polymorphisms, minion homomorphisms, 28 VICTOR DALMAU AND JAKUB OPRŠAL and a certain transformation on the polymorphism minion of the frst template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne the necessary notions in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Polymorphisms and minions We recall the known characterisation of gadget reductions, and the notions necessary for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' These notions will be also used in the characterisation of the arc-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A, A′ be a promise template, and 𝑋 a fnite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A polymorphism from A to A′ of arity 𝑋 is a homomorphism 𝑓 : A𝑋 → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The set of all 𝑋-ary polymorphisms from A to A′ is denoted by Pol(𝑋) (A, A′), and the collection of all these sets is denoted by Pol(A, A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If A = A′, we write Pol(A) instead of Pol(A, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To spell out what being a polymorphism means, let us repeat the preservation property without referring to the direct product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For each relational symbol 𝑅 of arity 𝑖1 · · ·𝑖𝑘, and a tuple (𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑘) ∈ 𝐴𝑋 𝑖1 × · · · × 𝐴𝑋 𝑖𝑘 of functions from 𝑋, write the elements of these tuples into a 𝑘 × |𝑋 | matrix by placing each 𝑎𝑖 onto its own row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The condition on 𝑓 being a polymorphism reads: if every column of this matrix is in the relation 𝑅A, then the 𝑘-tuple obtained by applying 𝑓 on each row, is in 𝑅A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Polymorphisms form an object that is called a minion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Minions, or more precisely functions minions are defned in [BBKO21, Defnition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Here we defne an abstraction of this notion which is better suited for the non-homogeneous setting and is necessary for a construction that we introduce below to characterise the arc-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' An abstract minion is a functor from the category of fnite sets to the category of sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a mapping M that assigns to each fnite set 𝑋 a set M (𝑋), and each function 𝜋 : 𝑋 → 𝑌 between two fnite sets 𝑋, 𝑌, a function M 𝜋 : M (𝑋) → M (𝑌), such that M 1𝑋 = 1M (𝑋 ) where 1𝑋 denotes the identity function on 𝑋, and M 𝜋 ◦ M 𝜎 = M 𝜋◦𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For 𝑓 ∈ M (𝑋) and 𝜋 : 𝑋 → 𝑌, we write 𝑓 𝜋 for M 𝜋 (𝑓 ) ∈ M (𝑌), and M (𝑛) for M ([𝑛]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This is done to unite the notation for an abstract minion and function minion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We further require that every minion M is non-empty and maps the empty set to itself, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', that M (𝑋) = ∅ if and only if 𝑋 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The polymorphisms of a template A, A′ form a minion, which we denote by the same symbol Pol(A, A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is the minion A where A (𝑋) = Pol(𝑋) (A, A′), and A 𝜋 for 𝜋 : 𝑋 → 𝑌 is defned to map 𝑓 ∈ Pol(𝑋) (A, B) to 𝑓 𝜋 ∈ Pol(𝑌) (A, B) where 𝑓 𝜋 𝑖 (𝑎) = 𝑓𝑖 (𝑎 ◦ 𝜋) for each type 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The polymorphism 𝑓 𝜋 is said to be a minor of 𝑓 defned by 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that since A → A′ for each promise template, we have that 𝑋 ≠ ∅ implies Pol(𝑋) (A, A′) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The converse follows from the assumption that every promise CSP in this paper has a negative instance: observe that X → A∅ for all structures X, and hence if X ̸→ A′ for some X, there is no homomorphism A∅ → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A minion homomorphism is a mapping between the two minions which preserves the minor taking operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' More precisely, a minion homomorphism from M to N is a natural transformation 𝜉 : M → N , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a collection of maps 𝜉𝑋 : M (𝑋) → N (𝑋), one for each fnite set 𝑋, such that for all 𝜋 : 𝑋 → 𝑌, we have N 𝜋 ◦ 𝜉𝑋 = 𝜉𝑌 ◦ M 𝜋, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝜉𝑋 (𝑓 )𝜋 = 𝜉𝑌 (𝑓 𝜋) for all 𝑓 ∈ M (𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 29 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Defne an abstract minion H to be the non-empty powerset functor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', H (𝑋) = {𝑈 ⊆ 𝑋 | 𝑈 ≠ ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The minor taking operation is then defned as 𝑈 𝜋 = 𝜋(𝑈 ) = {𝜋(𝑢) | 𝑢 ∈ 𝑈 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is straightforward to check that this defnition satisfes the properties in Defnition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Also it is not hard to observe that H is in fact isomorphic to the minion of polymorphisms of Horn-3SAT using a well-known classifcation of polymorphisms of Horn-3SAT (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [BKW17, Example 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we are ready to formulate the fundamental theorem of the algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We give a modern formulation that is essentially [BBKO21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1], though that theorem does not explicitly say that the log-space reduction that exists is a gadget reduction, and does not show the converse (the converse did not receive much attention until [KOWŽ22] though it was essentially folklore in the algebraic community).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The theorem builds on a long line of refnements of these reductions between CSPs and promise CSPs [JCG97, BJK05, BOP18, BG21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, none of the mentioned papers works with non-homogeneous structures, thought the proofs, in particular those of [BBKO21], apply essentially verbatim to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' See also [BJ03] for an algebraic treatment of non-homogeneous CSP including a description of encoding such a CSP in a single-sorted one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The following are equivalent for any two PCSPs with templates A, A′, and B, B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) There is a minion homomorphism Pol(B, B′) → Pol(A, A′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) PCSP(A, A′) is reducible to PCSP(B, B′) via a gadget reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Our characterisation of arc-consistency reductions relies heavily on the above the- orem and its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In fact, the gadget reduction provided in [BBKO21, Section 3] in presence of a minion homomorphism from Pol(B, B′) to Pol(A, A′) can equivalently be described as the arc-consistency replacement with omitted step (A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Naturally, this means that we can relate our constructions to [BBKO21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The above theorem is proved by a two step reduction through an intermediate problem denoted by PMC(M ) in [BBKO21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' the frst step corresponds to the step (A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1) above and formally produces a minor condition, and the second step which is equivalent to (A2) then transforms this minor condition to an instance of CSP(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Naturally, if we want to draw a parallel with this proof, we will need to interpret the step (A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2) as a transformation on those minor conditions which we will do in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us now explain what a minor condition is, and describe how the other two steps relate to the theory described in [BBKO21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Minor conditions form the third perspective on label cover instances, though written in a slightly diferent language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8 (Minor conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' An algebraic language is a collection of function symbols denoted by 𝑓 , 𝑔, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', together with their arities which are fnite sets 𝐷𝑓 , 𝐷𝑔, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We view a function symbol 𝑓 with arity 𝐷𝑓 as a placeholder for function of arity 𝐷𝑓 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', as 𝑓 : 𝑋 𝐷𝑓 → 𝑌 for some (at this point irrelevant) sets 𝑋 as 𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A minor condition is a fnite set of identities of the form 𝑓 = 𝑔𝜋 for some 𝜋 : 𝐷𝑔 → 𝐷𝑓 in some algebraic language, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', each identity can be written as a formal identity 𝑓 (𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑚) ≈ 𝑔(𝑥𝜋 (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝜋 (𝑛)) assuming 𝐷𝑓 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑚} and 𝐷𝑔 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑛};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' the symbol ≈ is used to distinguish formal identity from an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Such a condition is then said to be satisfed in a minion M if there is an assignment 𝜉 that assigns to each 𝑓 of arity𝑋 an element 𝜉(𝑓 ) ∈ M (𝑋) such that for each identity 30 VICTOR DALMAU AND JAKUB OPRŠAL 𝑓 = 𝑔𝜋, we have 𝜉(𝑓 ) = 𝜉(𝑔)𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' And it is called trivial if it is satisfed in the minion P of projections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the minion with P (𝑛) = [𝑛], and 𝑖𝜋 = 𝜋(𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We note that a condition is trivial if and only if it is satisfed in every minion since P maps to any other minion by a minion homomorphism, and minion ho- momorphisms preserve satisfaction of minor conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' for details see [BBKO21, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The translation between minor conditions and label cover instances uses the following dictionary: function symbol ∼ variable, arity ∼ domain, identity ∼ constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' More precisely, each identity 𝑓 = 𝑔𝜋 between 𝑓 of arity 𝐷𝑓 and 𝑔 of arity 𝐷𝑔 corresponds to a constraint 𝑓 = 𝜋(𝑔) where 𝑓 has domain 𝐷𝑓 and 𝑔 has domain 𝐷𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is also straightforward to reverse this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that the resulting label cover instance is solvable if and only if the condition is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' the key diference is that we can talk about ‘satisfying a minor condition in a minion’ while ‘solving label cover instance in a minion’ does not make much sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' With this distinction in mind, we will not distinguish between minor conditions and label cover instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' As we mentioned before, the frst step of the reduction in [BBKO21, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2] transforms an instance I of a CSP(A) into a minor condition denoted by Σ(A, I) — this condition then directly translates to the label cover instance obtained by reifcation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the label cover instance I obtained from I∗ and A∗ via Defnition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Below, we will adopt the notation Σ(A, I) and we interpret this label cover instance as a minor condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will use the following key property of this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9 ([BBKO21, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14(2)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If Pol(A, A′) satisfes Σ(A, X) then X → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The step (A2) corresponds to the indicator structure defned in [BBKO21, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This construction takes as input a minor condition (a label cover instance) Σ and a relational structure B, and produces a new relational structure that we will denote BΣ since it is obtained from Σ by the universal (power) gadget replacement for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='6 We will use the following properties of BΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10 ([BBKO21, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='16 & Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The following are equivalent for each minor condition Σ and every promise template B, B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' BΣ → B′, Σ is satisfed in Pol(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The fnal piece of the puzzle is a certain construction called the free structure of a minion M generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This is the last of four fundamental constructions that were used in [BBKO21] that we have not described yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will use it as a black box, only noting that the free structure of M generated by a fnite structure A is fnite as long as M (𝑛) is fnite for all 𝑛 [BBKO21, Defnition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1], and using the following fundamental lemma for the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This lemma can be taken as an alternative defnition of the free structure since it characterises such structure up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='11 ([BBKO21, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let M be a minion, let A, A′ be a promise template, and let F be the free structure of M generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Then there is a 1-to-1 cor- respondence between minion homomorphisms M → Pol(A, A′) and homomorphisms F → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will also use the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 6[BBKO21] used the notation IΣ (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 31 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='12 ([BBKO21, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let F be the free structure of a minion M generated by a structure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Then M satisfes Σ(A, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The characterisation of arc-consistency reduction The characterisation of the arc-consistency as a reduction uses the following construction on minions which from a minion produces another abstract minion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let M be an abstract minion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We say that an element 𝑓 ∈ M (𝑁 ) only depends on variables in 𝑋 ⊆ 𝑁 if there is 𝑔 ∈ M (𝑋) such that 𝑓 = 𝑔1𝑋 where 1𝑋 : 𝑋 → 𝑁 is the inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7 If this is the case we write 𝑓 ≺ 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8 Given a minion M , we defne a minion 𝜔(M ) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝜔(M )(𝑛) = {(𝑓 ,𝑋) | 𝑓 ∈ M (𝑛), 𝑓 ≺ 𝑋 } and for 𝜋 : [𝑛] → [𝑚], we let (𝑓 ,𝑋)𝜋 = (𝑓 𝜋, 𝜋(𝑋)) where 𝜋(𝑋) = {𝜋(𝑥) | 𝑥 ∈ 𝑋 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The construction 𝜔 is closely related to the minion H of polymorphisms of Horn-SAT (see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='6): 𝜔(M ) is a subminion of the product M × H of the two minions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that the frst projection (𝑓 ,𝑋) ↦→ 𝑓 is always a minion homomorphism from 𝜔(M ) to M and the second projection (𝑓 ,𝑋) ↦→ 𝑋 is a minion homomorphism from 𝜔(M ) to H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We also note that the 𝜔-image of the polymorphism minion of the trivial CSP, 𝜔 Pol(T), is isomorphic to H : Pol(T) has only one function of each arity which does not depend on any of its variables, let us call it 𝑡, then (𝑡,𝑋) ↦→ 𝑋 is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we are ready to state the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A, A′ and B, B′ be two promise templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) there is a minion homomorphism 𝜔 Pol(B, B′) → Pol(A, A′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) PCSP(A, A′) ≤arc-cons PCSP(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The theorem is proved by closely following the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7 as it is presented in [BBKO21, Sections 3] with inserting an additional reduction in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Namely, we replace the middle reduction which is trivial in [BBKO21] with the arc-consistency replacement, and we prove that the reduction is sound if (and only if) 𝜔 Pol(B, B′) → Pol(A, A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The novel ingredient is then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='15 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In order to use the strategy of [BBKO21] we describe the arc-consistency pro- cedure as a transformation of a minor condition: it iteratively decreases arity of function symbols in the given minor condition, until each identity contains the same variables on either of the sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Following our notation above, we will also denote by 𝜅arc(Σ) the minor condition obtained by applying arc-consistency to Σ, and for a symbol 𝑓 of Σ, we will denote by F𝑓 its updated arity in 𝜅arc(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The following lemma then explains the relation of the transformation 𝜅arc on minor conditions and the transformation 𝜔 on minions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will also use the symbol 1𝑋 for the inclusion map 1𝑋 : 𝑋 → 𝑌 where 𝑋 ⊆ 𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For all minor conditions Σ and minions M , 𝜅arc(Σ) is satisfed in M if and only if Σ is satisfed in 𝜔(M ) 7If M is a function minion on 𝐷, this is expressed as 𝑓 (𝑥) = 𝑔(𝑥 |𝑋 ) for all 𝑥 : 𝑁 → 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 8The set of essential variables of 𝑓 would be the smallest 𝑋 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' inclusion such that 𝑓 ≺ 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is not hard to prove that such a set must exist, but it is not necessary for our use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 32 VICTOR DALMAU AND JAKUB OPRŠAL Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume M satisfes 𝜅arc(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This implies that for each symbol 𝑓 in Σ, there is 𝑓 ′ ∈ M such that ar 𝑓 ′ = F𝑓 , and these 𝑓 ′ satisfy all identities of 𝜅arc(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Conse- quently, by putting 𝜇(𝑓 ) = (𝑓 ′)1F𝑓 we get that M satisfes the condition Σ as witnessed by 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Moreover, observe that 𝑓 ↦→ F𝑓 satisfes Σ in H , hence 𝑓 ↦→ (𝜇(𝑓 ), F𝑓 ) satisfes Σ in 𝜔(M ) — it is immediate from the defnition of 𝜇(𝑓 )’s that these elements indeed belong to 𝜔(M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For the other direction, assume Σ is satisfed in 𝜔(M ) by elements (𝜇(𝑓 ),𝑋𝑓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We note that for each minor identity 𝑓 = 𝑔𝜋 in Σ, since 𝜋(𝑋𝑓 ) = 𝑋𝑔, it follows that all variables in 𝑋𝑓 and 𝑋𝑔 are not removed in any of the iterations of the arc-consistency procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This implies that 𝑋𝑓 ⊆ F𝑓 , and that we can defne 𝑓 ′ of arity F𝑓 by 𝑓 ′(𝑥) = 𝑔1𝑋𝑓 where 𝑔 is a witness for 𝜇(𝑓 ) ≺ 𝑋𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is easy to check that the collection of 𝑓 ′’s satisfy 𝜅arc(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ We get back to the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Again, let us stress that with the exception of the above lemma, we are simply following the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7 as described in [BBKO21, Sections 3 & 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A = Pol(A, A′) and B = Pol(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First, we show that if there is a minion homomorphism 𝜉 : 𝜔(B) → A then PCSP(A, A′) reduces to PCSP(B, B′) via an arc-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Starting with an instance X of CSP(A), the arc-consistency reduction produces frst a minor condition Σ(A, X) in step (A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1), then the condition 𝜅arcΣ(A, X) in step (A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2), and fnally the instance B𝜅arcΣ(A,X) in step (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The completeness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', that the resulting instance maps to B if X → A is straightforward, let us focus on the soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume B𝜅arcΣ(A,X) → B′, then we get that: B satisfes 𝜅arc(Σ(A, X)) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10, 𝜔(B) satisfes Σ(A, X) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='15, A satisfes Σ(A, X) since a minion homomorphism preserves satis- faction of minor conditions, and fnally X → A′ by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This concludes the proof of the converse implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To show the other implication, assume that PCSP(A, A′) reduces to PCSP(B, B′) by the arc-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let F be the free structure of 𝜔(B) generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In order to obtain a minion homomorphism from 𝜔(B) to A , we use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='11 stating that such a minion homomorphism exists if and only if F → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We prove this by using the soundness of arc-consistency applied on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Observe that: 𝜔(B) satisfes Σ(A, F) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='12, B satisfes 𝜅arcΣ(A, F) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='15, and B𝜅arcΣ(A,F) → B′ by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since the last homomorphism witnesses that𝜅A,B arc (F) is not a negative instance of PCSP(B, B′), we get that F → A′ by the soundness of the arc-consistency reduction as we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ We briefy note that the above proof can be adapted for the cases when F is not fnite (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', when B, B′ and B(𝑛) are not fnite) using the standard compactness argument (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [BBKO21, Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14 together with [BK22, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1] provides an immediate corollary which gives a new sufcient condition for the existence of a local reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Namely, we can connect the notion of minion (𝑑,𝑟)-homomorphisms defned in [BK22, Def- nition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1] with the 𝜔 construction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' we refer to the cited paper for the defnition of this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 33 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume A, A′ and B, B′ are two promise templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If there exist 𝑑,𝑟 and a minion (𝑑,𝑟)-homomorphism from 𝜔 Pol(B, B′) to Pol(A, A′), then PCSP(A, A′) ≤Datalog PCSP(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' While (𝑑,𝑟)-homomorphisms between polymorphism minions are provably not necessary for existence of a Datalog∪ reduction — this would contradict Example ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='. It is not known if existence of a (𝑑,𝑟)-homomorphisms from the 𝜔-image is also a necessary for such a reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us mention an example of a CSP (and a minion) for which Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14 does not provide any new leverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The following minion Qconv, that was defned in [BBKO21, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2] to describe the power of basic linear programming relaxation (BLP) for promise CSPs, has a remarkable property that it is homomorphically equivalent to its 𝜔-image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This is underlined by the intuition that BLP is stronger than arc-consistency, and thus running arc-consistency before BLP does not increase the power of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The minion Qconv is defned as follows: Q(𝑋) conv consists of all rational probability distributions 𝜆 on 𝑋, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝜆: 𝑋 → Q with � 𝑥 ∈𝑋 𝜆(𝑥) = 1 and 𝜆(𝑥) ≥ 0 for all 𝑥 ∈ 𝑋, and for 𝜋 : 𝑋 → 𝑌, we let 𝜆𝜋 be the probability distribution of 𝜋(𝑥) when 𝑥 is sampled according to 𝜆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝜆𝜋 (𝑥) = � 𝑦∈𝜋−1(𝑥) 𝜆(𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We construct a minion homomorphism 𝜉 : Qconv → 𝜔(Qconv), a homomorphism in the other direction is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜉(𝜆) = (𝜆, supp(𝜆)) where supp(𝜆) is the set of all values 𝑥 ∈ 𝑋 that have a non-zero probability in 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is not hard to check that indeed 𝜆 ≺ supp(𝜆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To show that 𝜉 is indeed a minion homomorphism, we need to show that supp(𝜆𝜋) = 𝜋(supp(𝑓 )) for all 𝜋 : 𝑋 → 𝑌 which is rather easy: if 𝑥 has non-zero probability according to 𝜆 then 𝜋(𝑥) has a non-zero probability according to 𝜆𝜋, and conversely 𝑦 has a non-zero probability according to 𝜆𝜋 only if there is 𝑥 with 𝜋(𝑥) = 𝑦 and a non-zero probability in 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This concludes that 𝜉 is a minion homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' As a direct consequence, we get that if a promise CSP is reducible by arc- consistency reduction to another promise CSP that is solved by the basic linear programming relaxation (BLP), then it is itself solvable by BLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will also use this fact in the next section to derive some properties of Sherali-Adams hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Conic minions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Another group of examples of minions M that satisfy M → 𝜔(M ) are conic minions introduced in [CŽ23] to describe several algorithms includ- ing Sherali-Adams, Lasserre hierarchy of semi-defnite programming, and hierarchies of CLAP [CŽ22b] and a combination of linear programming and afne integer pro- gramming [BG20, BGWŽ20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Conic minions are a special case of so-called linear minions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For more clarity, we present this notion in a restricted setting of matrices over felds, see [CŽ23, Defnitions 16 & 20] for the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝑑 ≥ 1 be a possibly infnite cardinal (a dimension of a vector space) and fx a feld F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A linear minion of depth 𝑑 is a minion M , such that, for all 𝑛, M (𝑛) consists of some 𝑛 × 𝑑-matrices over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The minor taking operation are defned as 𝑀𝜋 = 𝑃𝜋𝑀 for 𝜋 : [𝑛] → [𝑚] where 𝑃𝜋 is the incidence matrix of the graph of 𝜋, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the (𝜋(𝑖),𝑖)-th entry of 𝑃𝜋 is 1 for all 𝑖 and all other entries are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10 9Most examples of linear minions fall under our defnition with the exception of H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 10This minor taking operations coincide with the identifcation minors of functions if we interpret an element 𝑀 ∈ M (𝑛) as linear map 𝑀𝑇 : F𝑛 → F𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 34 VICTOR DALMAU AND JAKUB OPRŠAL Such a linear minion M is said to be conic if 𝑀 ≠ 0 for all 𝑀 ∈ M and, for all 𝑀 ∈ M (𝑛) and 𝑋 ⊆ [𝑛], whenever � 𝑖 ∈𝑋 𝑀𝑇 e𝑖 = 0 we get 𝑀𝑇 e𝑖 = 0 for all 𝑖 ∈ 𝑋 where e𝑖 denotes the 𝑖-th vector of the canonical basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' See [CŽ23, Proposition 21] for examples of conic (and non-conic) minions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us now prove that every conic minion admits a homomorphism to its 𝜔-image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Naturally, this also raises the question whether a linear minion is conic if and only if it allows a homomorphism from its 𝜔-image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For every conic minion M , there is a minion homomorphism M → 𝜔(M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne a homomorphism 𝜉 : M → 𝜔(M ) by 𝜉(𝑀) = (𝑀;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' {𝑖 | 𝑀𝑇 e𝑖 ≠ 0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The mapping 𝜉 is well-defned since if 𝑀 does not depend on 𝑖 then 𝑀𝑇 e𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We claim that the fact that 𝜉 preserves minors follows from the conicity of 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Indeed if 𝑀 = 𝑁 𝜋, then 𝑀𝑇 e𝑗 = � 𝜋 (𝑖)=𝑗 𝑁𝑇 e𝑖, and hence 𝑀𝑇 e𝑗 = 0 if and only if 𝑁𝑇 e𝑖 = 0 for all 𝑖 such that 𝜋(𝑖) = 𝑗 since M is conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, we get that {𝜋(𝑖) | 𝑁𝑇 e𝑖 ≠ 0} = {𝑗 | (𝑁 𝜋)𝑇 e𝑗 ≠ 0} Since we only need to check preserving minors in the second coordinate (it is trivial in the frst), this equality gives the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ We believe that this general property of conic minions is the key to why hierar- chies of conic minions introduced in [CŽ23] have particularly nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Arc-consistency reductions are transitive: comonads and Kleisli arrows We argue that arc-consistency reductions compose by using the language of cate- gory theory that is incredibly elegant for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Essentially, this claim follows from an observation that 𝜔 is a comonad, and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14 which characterises the arc-consistency reduction in terms of co-Kleisli arrows of this comonad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us briefy outline the defnitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Although it is not the traditional way, we defne these notions together to highlight their connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume 𝜂 is an endofunctor of some category, and 𝐴, 𝐵 are two objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A co-Kleisli arrow is a morphism 𝑓 : 𝜂(𝐴) → 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='11 A functor 𝜂 is a comonad if its co-Kleisli arrows form a category with the same objects as the underlying category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', there is an associative binary operator ◦𝜂 that assigns to every pair of co-Kleisli arrows 𝑓 : 𝜂(𝐴) → 𝐵 and 𝑔: 𝜂(𝐵) → 𝐶 a co-Kleisli arrow 𝑔 ◦𝜂 𝑓 : 𝜂(𝐴) → 𝐶;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' for each object 𝐴, there is a co-Kleisli arrow 𝑒𝐴 : 𝜂(𝐴) → 𝐴 that acts as the identity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ◦𝜂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝜔 is a comonad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We start with defning the units 𝑒M where M is a minion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' this unit 𝑒M is simply the projection to the frst coordinate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝑒M (𝑓 ,𝑋) = 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' To defne the composition, it is easier to show how to get from a co-Kleisli arrow 𝜉 : 𝜔(M ) → N 11Usually, a morphism 𝜂(𝐴) → 𝐵 is called a co-Kleisli arrow only if 𝜂 is a comonad, and in that case the ‘co-’ prefx is often dropped since its implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 35 a minion homomorphism 𝜉♯ : 𝜔(M ) → 𝜔(N );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' this 𝜉♯ is defned by 𝜉♯(𝑓 ,𝑋) = (𝜉(𝑓 ,𝑋),𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The composition is then defned as 𝜉 ◦𝜔 𝜁 = 𝜉 ◦ 𝜁 ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is not hard to check that ◦𝜔 is associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that 𝑒♯ M it the identity on 𝜔(M ) which immediately gives that 𝜉 ◦𝜔 𝑒M = 𝜉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The other identity 𝜉 = 𝑒M ◦ 𝜉♯ is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Note that in the above proof, we showed that 𝜔(M ) → 𝜔(N ) if and only if 𝜔(M ) → N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, as an easy corollary of the above lemma and the fact that co-Kleisli arrows of a comonad compose, we get that arc-consistency reductions are transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If PCSP(A, A′) ≤arc-cons PCSP(B, B′) ≤arc-cons PCSP(C, C′), then PCSP(A, A′) ≤arc-cons PCSP(C, C′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' HIERARCHIES AND CSP ALGORITHMS Several hierarchies of algorithms are studied for the tractability of CSPs and promise CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The most prominent two are arguably the local consistency hierarchy and the Sherali-Adams hierarchy [SA90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Both of these hierarchies can be viewed as a reduction to a certain polynomial-time solvable CSP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Horn-SAT and linear programming, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will describe this in a bigger detail below, and explain that this reduction is actually a special case of a Datalog∪ reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In fact for these two hierarchies we show that a promise CSP is reducible by the 𝑘- consistency reduction to a CSP that defnes the hierarchy if and only if the promise CSP is solved by the 𝑘-th level of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A similar result can be obtained also for a few other hierarchies introduced in [CŽ23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' As a direct consequence we obtain that the class of promise CSPs solvable by some level of such a hierarchy is closed under consistency reductions, and in particular under minion homomorphisms between their templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We also suggest a new way to defne a hierarchy of any fxed promise CSP by simply defning it as all the problems reducible to the fxed problem via the 𝑘-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Such a hierarchy has a natural grading that corresponds to the parameter 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A slightly diferent general approach to hierarchies has been also suggested in [CŽ23], which defnes a hierarchy of ‘minion tests’ for a fxed minion M , generalising Sherali-Adams, Lasserre, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' While we defne the hierarchy in a diferent way, under a certain condition, namely that M → 𝜔(M ), where M is the polymorphism minion of the template of the base problem of the hierarchy, the two hierarchies coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is likely that if the above condition is not satisfed, the two hierarchies difer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In that case the algorithm provided through our hierarchy is always stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Often, it is useful that the problem we reduce to is an infnite template CSP — this is the case for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', linear programming, and for afne integer programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This creates a practical problem with the 𝑘-consistency reduction since it produces infnite instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Nevertheless, the templates we consider have a certain property that allows us to always produce an equivalent, but fnite, instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 36 VICTOR DALMAU AND JAKUB OPRŠAL 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Problems solvable by Datalog The bounded width hierarchy corresponds to problems solvable by Datalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The equivalence of the local consistency algorithm and Datalog is well-known [FV98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will show that the same coincides as a hierarchy of Horn-SAT, or the hierarchy of the trivial CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Recall that a (promise) CSP is said to have width 𝑘 if it is solvable by the 𝑘- consistency algorithm, and that a promise CSP is said to be solvable by Datalog if there is a Datalog sentence which is false on all positive instances, and true on all negative instances (see Defnitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='25 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that both the 𝑘- consistency reduction and the 𝑘-consistency decision algorithm use as a subroutine the𝑘-consistency step, and recall that the𝑘-consistency algorithm rejects an instance if and only if the 𝑘-consistency step derives that some variable has an empty domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' CSPs that have bounded width were characterised in [BK14] as exactly those CSPs which do not allow a gadget reduction from solving systems of linear equations over a fnite feld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The situation is more complicated for promise CSPs: 1-in-3- vs NAE- SAT does not have bounded width [AD22] and it does not allow a gadget reduction from solving systems linear equations (which can be shown using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us frst observe that promise CSPs are defnable by Datalog if and only if they reduce to the trivial CSP by a Datalog∪ reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This is achieved by a straightforward translation between the two Datalog programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A promise CSP is defnable by a Datalog program if and only if there is a Datalog∪ reduction that is a reduction from the promise CSP to the trivial CSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assuming 𝜙⊥ is a Datalog sentence, we defne a Datalog interpretation 𝝓 = (𝜙1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝜙𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='𝜙⊥) where 𝑛 is the number of types in the input signature of 𝜙⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Each 𝜙𝑖 is defned by the rule 𝜙𝑖 (𝑥) ← 𝑥 = 𝑥 where 𝑥 of type 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, in order to turn 𝝓 into a reduction to CSP(T), we compose it with the obvious disjoint union;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' defned by (𝑑,𝑟) where 𝑑(𝑖) = 1 for all 𝑖 ∈ [𝑛], and 𝑟 (⊥) = ⊥ for the single relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is straightforward to check that 𝜙⊥ solves PCSP(A, A′) if and only if this Datalog∪ reduction is a valid reduction from PCSP(A, A′) to CSP(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Observe that the width of 𝝓 in the above proof coincides with the width of 𝜙⊥, and that the disjoint union used above can be expressed as a strict gadget reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The following theorem shows that the hierarchies of Horn-SAT and the trivial CSP coincide, and they also coincides with CSPs of bounded width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The proof is an exercise of applying the results of previous sections using the fact that running arc-consistency on the output of 𝑘-consistency has no efect, and the fact that the polymorphisms of the trivial CSP and Horn-SAT are connected through the construction 𝜔 described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We note that if we would not insist on keeping the parameter 𝑘 constant, the theorem below would follow relatively easily from Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='27 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The following are equivalent for every promise template A, A′ and every 𝑘 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) PCSP(A, A′) is has width at most 𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) PCSP(A, A′) is solvable by Datalog with width 𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (3) PCSP(A, A′) reduces to the trivial CSP by the 𝑘-consistency reduction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (4) PCSP(A, A′) reduces to Horn-3SAT by the 𝑘-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 37 As we mentioned above, the equivalence of (1) and (2) is well-known for CSPs, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [FV98, BK14], the same argument works for promise CSPs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We also note that this could be argued in a similar way as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will frst show that (3) and (4) are equivalent, for which we use the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If 𝜔 Pol(B, B′) → Pol(A, A′) then every promise CSP that reduces to PCSP(A, A′) by the𝑘-consistency reduction reduces to PCSP(B, B′) by the𝑘-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume that PCSP(C, C′) reduces to PCSP(A, A′) by the 𝑘-consistency reduc- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', if A𝜅𝑘 (X,C) → A′ then X → C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will show that 𝑘-consistency is also a sound reduction to PCSP(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume that B𝜅𝑘 (X,C) → B′, which is equivalent to 𝜅𝑘 (X, C) being satisfed in Pol(B, B′) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='15, using the obser- vation that 𝜅arc𝜅𝑘 (X, C) = 𝜅𝑘 (X, C), we get that 𝜅𝑘 (X, C) is satisfed in 𝜔 Pol(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The minion homomorphism then implies that 𝜅𝑘 (X, C) is satisfed in Pol(A, A′), and the converse implication of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10 that A𝜅𝑘 (X,C) → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Therefore, we obtain X → C′ from the soundness of the reduction to PCSP(A, A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Note that a direct consequence of the above lemma and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14 is that if PCSP(C, C′) ≤𝑘-cons PCSP(A, A′) ≤arc-cons PCSP(B, B′) then PCSP(C, C′) ≤𝑘-cons PCSP(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We are now ready to return the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' As we mentioned above, the equivalence of (1) and (2) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The equivalence of (3) and (4) is a direct consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3, the fact that 𝜔(Pol(T)) is isomorphic to the polymorphism minion H of Horn-3SAT, and the trivial homomorphism 𝜔(H ) → Pol(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The implication (2)→(3) is a direct consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28 observing that the Datalog∪ reduction provided by the lemma is a composition of a Datalog interpretation of width 𝑘 and a particular disjoint union which is expressible strict gadget replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we argue that (3) implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The key to proving this implication is observing that a minor condition Σ is satisfed in Pol(T) if and only if it has no nullary symbols: Observe that T𝑋 is isomorphic to T if X ≠ ∅, and T∅ ̸→ T since the relation ⊥ of T∅ is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The claim follows immediately from this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume that PCSP(A, A′) reduces by the 𝑘-consistency reduction to CSP(T), and let X be an instance of CSP(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The 𝑘-consistency algorithm is defned to output yes on the input X of if and only if F𝐾 is non-empty in 𝜅𝑘 (X, A) for all 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This condition exactly correspond with 𝜅𝑘 (X, A) being satisfed in Pol(T), and hence with T𝜅𝑘 (X,A) → T (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence, if the 𝑘-consistency algorithm outputs yes, then X → A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Conversely, if the 𝑘-consistency algorithm outputs no, then X ̸→ A by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ A direct corollary of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2 is that PCSPs solvable by local consistency are closed under gadget reductions which was frst shown in [BBKO21, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='5], and in the non-promise setting in [LZ07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A, A′ and B, B′ be two promise templates such that Pol(A, A′) → Pol(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If PCSP(A, A′) is solvable by Datalog, then so is PCSP(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Sherali-Adams There are several slightly diferent ways to defne Sherali-Adams relaxation for CSPs and promise CSPs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [TŽ17, CŽ23, BD21, BB22] — the last of these references introduces Sherali-Adams in the most general setting, namely promise valued CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 38 VICTOR DALMAU AND JAKUB OPRŠAL Most of these defnitions difer in minor technical details, and all of them are based in some way on a paper of Sherali and Adams [SA90] which describes the hierarchy as a reduction from 0-1 integer programming to linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' All of the defnitions have a few things in common: They produce, from an instance of CSP, an instance of linear programming with 0-1 coefcients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This instance can be interpreted as an instance of CSP(Qconv) for some suitable template Qconv, which we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Our defnition below agrees with the (𝑘−1,𝑘)-SA system in [TŽ17] assuming that 𝑘 is at least the maximal arity of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We make a choice of not projecting larger arity constraints not to increase the width of the Datalog interpretation above 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Otherwise, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', when we would use the system in [TŽ17] directly, it would require a Datalog interpretation of width 𝑚, where 𝑚 is the maximal arity of the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This is related to the fact that interpreting the identity map as a Datalog interpretation still requires width 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Though it is not strictly necessary, we assume 𝑘 > 1, since smaller values of 𝑘 lead to rather trivial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In plain words, the goal of the 𝑘-th level of Sherali-Adams relaxation is to fnd a collection of probability distributions on partial solutions on each of the subsets of variables of size at most 𝑘 that have consistent marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='5 (𝑘-th level of Sherali-Adams hierarchy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Fix 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The 𝑘-th level of SA hierarchy is the following reduction from an instance X of CSP(A) to linear programming: (SA1) Create the following label cover instance: for each 𝐾 ∈ � 𝑄 ≤𝑘 �, introduce a variable 𝑣𝑘 with domain F𝐾 which is the set of all partial homomorphisms from 𝐾 to A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' for each 𝐿 ⊂ 𝐾 ∈ � 𝑄 ≤𝑘 �, introduce a constraint between 𝑣𝐾 and 𝑣𝐿 defned by the map 𝑔 = 𝑔|𝐿 from F𝐾 to F𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (SA2) Encode the label cover instance into linear program in a similar way as for basic linear programming relaxation with variables 𝜆𝐾,𝑓 ∈ [0, 1] for each 𝑓 ∈ F𝐾: ∑︁ 𝑓 ∈F𝐾 𝜆𝐾,𝑓 = 1 for all 𝐾 ∈ � 𝑄 ≤𝑘 �, (1) ∑︁ 𝑓 ∈F𝐾,𝑓 |𝐿=𝑔 𝜆𝐾,𝑓 = 𝜆𝐿,𝑔 for all 𝐿 ⊂ 𝐾 ∈ � 𝑄 ≤𝑘 � and 𝑔 ∈ F𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) Observe that the above system has a 0-1 solution if there is a homomorphism ℎ: X → A: simply assign 𝜆𝑓 ,𝐾 = 1 if 𝑓 = ℎ|𝐾 and 𝜆𝑓 ,𝐾 = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Linear programming can be viewed as a fxed template CSP though with infnite domain and infnitely many relations: Namely, the domain is Q, and the relations are all relations defned by afne inequalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', of the form 𝑎1𝑥1 + · · · + 𝑎𝑛𝑥𝑛 ≤ 𝑏 for some 𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑛,𝑏 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We have to address technicalities that arise due to the fact that the template of linear programming is infnite and has infnitely many relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In particular, in an instance of linear programming, the relations are usually given as the tuples of their coefcients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We note that all of these relations are in fact pp-defnable using the following three 𝑥 ≤ 𝑦, 𝑥1 + 𝑥2 = 𝑦, and 𝑦 = 1, and with some care, the length of these pp-defnitions is proportionate to the binary encoding of the coefcients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will henceforth ignore this issue with having infnitely many relations, and defne Qconv as the structure with the above relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 39 Another formal problem with the 𝑘-consistency reduction to linear programming when viewed as CSP(Qconv) is that the resulting instance is infnite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This issue is caused by the fact that the universal gadgets for Qconv are infnite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Nevertheless, linear programming and several other CSPs including afne integer programming, systems of linear equations over fnite felds, and semi-defnite programming, have a curious property that allows us to use fnite gadgets instead of the universal gadget with equivalent properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We informally call this property short code property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' These ‘smaller gadgets’ are exactly the gadgets used in the step (SA2) above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let Σ be a label cover instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We defne a system of linear equations and inequalities 𝜖LP(Σ) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We replace a variable 𝑣 with domain 𝐷 with variables 𝑥𝑣,𝑑 for 𝑑 ∈ 𝐷, an equation � 𝑑 ∈𝐷 𝑥𝑣,𝑑 = 1, and inequalities 𝑥𝑣,𝑑 ≥ 0 for all 𝑑 ∈ 𝐷, and we replace a label cover constraint 𝜋(𝑢) = 𝑣 defned by a map 𝜋 : 𝐷 → 𝐷′ is the collection of equations ∑︁ 𝑑 ∈𝜋−1(𝑑′) 𝑥𝑢,𝑑 = 𝑥𝑣,𝑑′, one for each 𝑑′ ∈ 𝐷′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The goal of the linear program is then to solve this system in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that 𝜖LP(Σ) can be expressed as a structure of the same type as Qconv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', a linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is nevertheless easier to work with it as a system of linear equations and inequalities with 0-1 coefcients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We note that Qconv, defned in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='17, is isomorphic to the polymorphism minion of Qconv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' a minion isomorphism 𝜉 : Qconv → Pol(Qconv) can be defned by 𝜉(𝜆)(𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑥𝑛) = 𝑛 ∑︁ 𝑖=1 𝜆(𝑖)𝑥𝑖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝜉(𝜆) is the convex combination with coefcients given by 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is straightfor- ward to check that 𝜉(𝜆) preserves the relations of Qconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' On the other hand, each polymorphism of Qconv is a convex combination, and hence its coefcients are a rational probability distribution which implies that 𝜉 is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It is clearly 1-to-1, and checking that 𝜉 preserves minors is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The purpose of the lemma below is to show that 𝜖LP is as good as the universal gadget for Qconv, and we will use it to freely exchange them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In particular, replacing Σ ↦→ QΣ conv with 𝜖LP in the 𝑘-consistency reduction to Qconv we obtain actually a polynomial time reduction to linear programming that is equivalent to the 𝑘- consistency reduction but does not produce infnite instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The following are equivalent for each minor condition Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) the system 𝜖LP(Σ) is solvable in Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) Qconv satisfes Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (3) QΣ conv → Qconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The equivalence of (2) and (3) follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10 and the fact that Qconv is isomorphic to Pol(Qconv) discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We prove that (1) and (2) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assume that 𝑥𝑓 ,𝑑 ∈ Q ∩ [0, 1] for 𝑓 and 𝑑 ∈ 𝐷𝑓 is a solution of 𝜖LP(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For each 𝑓 , we defne a probability distribution 𝜆𝑓 on 𝐷𝑓 by taking 𝑑 with probability 𝑥𝑓 ,𝑑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 𝜆𝑓 is a probability distribution since 𝑥𝑓 ,𝑑 ≥ 0 and � 𝑑 ∈𝐷𝑓 𝑥𝑓 ,𝑑 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The fact that each identity of Σ is satisfed is straightforward to check using that � 𝑑 ∈𝜋−1(𝑑′) 𝑥𝑓 ,𝑑 = 𝑥𝑓 ′,𝑑′ for 𝑓 𝜋 = 𝑓 ′ and 𝑑′ ∈ 𝐷𝑓 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This concludes that (1) implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The converse is given 40 VICTOR DALMAU AND JAKUB OPRŠAL by reversing this argument, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', checking that 𝑥𝑖,𝑓 = 𝜆𝑓 (𝑖) is a solution to 𝜖LP(Σ) if 𝜆𝑓 are probability distributions witnessing satisfaction of Σ in Qconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Once we have addressed the formal problems with the 𝑘-consistency reduction to linear programming, we can formulate the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A, A′ be a promise template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) PCSP(A, A′) is solvable by the 𝑘-th level of Sherali-Adams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) PCSP(A, A′) reduces to linear programming by the 𝑘-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The proof is a comparison of Sherali-Adams, as a reduction to linear programming, and the 𝑘-consistency reduction to linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' First, note that the 𝑘- consistency procedure can be decomposed using the frst step (SA1) of Sherali-Adams as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let 𝜃𝑘 (X, A) denote the label cover instance obtained by the step (SA1) on input X as an instance of CSP(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Then the output of 𝑘-consistency procedure 𝜅𝑘 (X, A) is the same as frst applying𝜃𝑘 and then running the arc consistency𝜅arc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', we have that 𝜅arc𝜃𝑘 (X, A) = 𝜅𝑘 (X, A) for all X and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Second, we compare the last step, which is 𝜖LP for Sherali-Adams and the universal gadget for the 𝑘-consistency reduction, using the short-code property (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Intuitively, the proof claims that the two gadgets, 𝜖LP and the universal gadget for Qconv, are equivalent, and that the arc-consistency step can be omitted since linear programming can emulate arc- consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The latter is a general property of promise CSPs whose polymorphism minion M admits a homomorphism M → 𝜔(M );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' note that Qconv → 𝜔(Qconv) is discussed in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Fix a promise template B, B′, and let B be the polymorphism minion of its template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If B → 𝜔(B), then the following are equivalent for any other promise template A, A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) X ↦→ B𝜃𝑘 (X,A) is a reduction from PCSP(A, A′) to PCSP(B, B′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) PCSP(A, A′) ≤𝑘-cons PCSP(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The implication (1)→(2) follows by a similar argument as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let us focus on the converse, and in particular on the soundness of the reductions since completeness is again straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' As we discussed above, the 𝑘-consistency reduction maps X to B𝜅arc𝜃𝑘 (X,A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' As- suming that the result maps to B′, we get by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10 that B satisfes𝜅arc𝜃𝑘 (X, A), and consequently, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='15, that 𝜔(B) satisfes 𝜃𝑘 (X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since 𝜔(B) → B and minion homomorphisms preserve satisfaction of minor conditions, also B satis- fes 𝜃𝑘 (X, A), and consequently by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10, B𝜃𝑘 (X,A) → B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we get that X → A′ by invoking the soundness of X ↦→ B𝜃𝑘 (X,A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ We are ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8 as a consequence of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For sanity, let us write Q instead of Qconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The theorem claims that the 𝑘-consistency reduction to linear programming, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', X ↦→ Q𝜅arc𝜃𝑘 (X,A), has the same power as the Sherali-Adams reduction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the mapping X ↦→ 𝜖LP𝜃𝑘 (X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9, for B = B′ = Q and B = Qconv, we get that PCSP(A, A′) reduces to CSP(Q) by the 𝑘-consistency reduction if and only if X ↦→ Q𝜃𝑘 (X,A) is a reduction between these problems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we have that Q𝜃𝑘 (X,A) → Q if and only if 𝜖LP𝜃𝑘 (X, A) is a system of equations solvable over Q ∩ [0, 1] by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, Sherali-Adams solves PCSP(A, A′) if and only if X → Q𝜃𝑘 (X,A) is a reduction from PCSP(A, A′) to linear programming, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 41 We also obtain an immediate corollary of Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='27, which we believe has not been shown before in the scope of promise CSPs, though the analogous statement in the non-promise setting follows from the characterisation of Sherali- Adams for CSPs [TŽ17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let A, A′ and B, B′ be two promise templates such that Pol(A, A′) → Pol(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' If PCSP(A, A′) is solvable by some level of Sherali-Adams hierarchy, then so is PCSP(B, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, we note that the key to the above arguments is the property that Qconv → 𝜔(Qconv), and the short code property of linear programming (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This in particular means that the arguments of this section generalise to many of the hierarchies of conic minions introduced in [CŽ23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In particular, we showed that a conic minion M allows a minion homomorphism M → 𝜔(M ) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' With some extra efort it can be shown that semi-defnite programming has also a short code property similar to the linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It follows that there is an analogue of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10 for the Lasserre hierarchy of semi-defnite programs for PCSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Naturally, the same would apply to hierarchies of CLAP and the combination of linear programming and afne integer programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hierarchies of groups and obstructions to consistency reductions Solving equations over a (fnite) Abelian group is a well-known CSP that is not solved by local consistency algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In this section, we argue that at least some of these problems may serve as obstruction for our consistency reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We will also describe some properties of hierarchies of these problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', the classes of problems that reduce by a consistency reduction to solving systems of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' One particularly interesting class is the class of all problems that reduce to solving systems of equations over integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let G be an Abelian group, by CSP(G) we mean the following problem: given a systems of equations of the form (3) 𝑎1𝑥1 + · · · + 𝑎𝑛𝑥𝑛 = 𝑏 where 𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ,𝑎𝑛 ∈ Z and 𝑏 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Decide whether the system is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Equivalently, it is formulated as the CSP with the template G, where the domain of G is 𝐺 and G has a relation 𝑅𝑎1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=',𝑎𝑛,𝑏 for each 𝑛, 𝑎𝑖’s, and 𝑏 as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Similarly as above, it is not hard to check that all these relations are generated by fnitely many relations, namely 𝑥1 + 𝑥2 = 𝑦 and 𝑦 = 𝑏 for each 𝑏 ∈ 𝐺, or equivalently, for each 𝑏 from some set of generators of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Note that CSP(G) for a non-Abelian group is NP-complete [GR02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We restrict to Abelian groups, and in particular, to cyclic groups: Every fnite Abelian group is a product of cyclic groups which allows us to ‘decompose’ CSP(G) for a fnite group G to several CSPs of Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Also note that the structure G assigned to an Abelian group G has an alternating polymorphism, and hence is solvable by afne integer programming relaxation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', reducible via a gadget reduction to CSP(Z) which is a CSP of a cyclic group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' for details see [BBKO21, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In the rest of the section, we assume that G is a cyclic group generated by an element denoted by 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', it is either isomorphic to the group Z𝑛 of addition modulo 𝑛 ∈ Z, or Z itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Similarly to linear programming, all the problems of the form CSP(G) have the short code property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The ‘short universal gadget’ reduction from a label cover instance to an instance of CSP(G) is similar to the systems of linear inequalities 42 VICTOR DALMAU AND JAKUB OPRŠAL we used for the linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' More precisely, let us denote by 𝜖(Σ) the system of equations obtained from 𝜖LP(Σ) defned in Defnition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='6 by dropping the inequalities 𝑥𝑣,𝑑 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The key diference here is that we are asking for a solution of this system in G instead of Q ∩ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The basic afne integer programming relaxation is an example of the reduction of the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' It can be described by a composition of the standard reduction to label cover, Σ(A, X), and 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The applicability of this basic level has been characterised in [BBKO21, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='19] using a minion Zaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Following our exposition of Sherali- Adams, we can see this minion in two equivalent ways, either as polymorphism minion of Z (the structure corresponding to Z), or as an abstract minion Zaf defned below in a slightly more general setting of cyclic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let G be a cyclic group generated by an element denoted by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The minion Gaf is defned as follows: G (𝑛) af = {𝑎 ∈ 𝐺𝑛 | �𝑛 𝑖=1 𝑎𝑖 = 1} with the minor-taking operations 𝑎 ↦→ 𝑎𝜋 defned as 𝑎𝜋 (𝑖) = � 𝑗 ∈𝜋−1(𝑖) 𝑎𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Similarly as for linear programming, it is straight-forward to check that Pol(G) is isomorphic to Gaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We also present the analogue of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Let G be a cyclic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The following are equivalent for each minor condition Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) 𝜖(Σ) is solvable in G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) Gaf satisfes Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (3) GΣ → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The proof is analogous to the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The equivalence of (2) and (3) is given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='10 and the fact that Gaf is isomorphic to Pol(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The equivalence of (1) and (2) is proved by following the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='7 only replac- ing ‘probability distribution 𝜆 on 𝐷’ with ‘a tuple 𝜆 ∈ 𝐺𝐷 such that � 𝑔∈𝐺 𝜆(𝑔) = 1’ which can intuitively be understood as “G-valued probability distribution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Using the same argumentation as in the previous subsection, we can again argue that a 𝑘-consistency reduction to CSP(G) is always equivalent to the following procedure, which we describe as a decision algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since solving equations over G is in P, and the reduction is polynomial-time computable, this algorithm runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (𝑘-consistency reduction to group equations) Fix 𝑘 > 1 and a cyclic group G generated by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Given an instance X of CSP(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (1) Run the 𝑘-consistency step as described in Defnition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='24 with the output being a label cover instance with variables 𝑣𝐾 for 𝐾 ∈ � 𝑄 ≤𝑘 � each with the domain F𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (2) Solve the following system of linear equations over G with variables 𝑥𝐾,𝑓 for each 𝑓 ∈ F𝐾: ∑︁ 𝑓 ∈F𝐾 𝑥𝐾,𝑓 = 1 for all 𝐾 ∈ � 𝑄 ≤𝑘 �, ∑︁ 𝑓 ∈F𝐾,𝑓 |𝐿=𝑔 𝑥𝐾,𝑓 = 𝑥𝐿,𝑔 for all 𝐿 ⊂ 𝐾 ∈ � 𝑄 ≤𝑘 � and 𝑔 ∈ F𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' (3) Output yes if the system is solvable, and no otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' LOCAL CONSISTENCY AS A REDUCTION BETWEEN CSPS 43 It is easy to check that if X → A then the above algorithm answers yes: if ℎ: X → A is a homomorphism, then the assignment 𝑥𝐾,ℎ|𝐾 = 1 extends to a solution of the system by assigning 0 to all other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence, in order to prove correctness of this algorithm, soundness needs to be analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The equivalence is formally given by the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' PCSP(A, A′) ≤𝑘-cons CSP(G) if and only if the 𝑘-consistency reduction to G-equations solves PCSP(A, A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ Using our framework, we get immediately that the above algorithm with 𝑘 > 2 and G = Z solves correctly all promise CSPs solvable by the 𝑘-consistency algorithm as well as systems of linear equations over a fnite Abelian group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', it solves the two prime examples of tractable CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We conjecture that it can be used as a replacement for Bulatov’s and Zhuk’s algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' For every fnite structure A, either 3-colouring reduces to CSP(A) by a gadget reduction, or there exists 𝑘 > 1 such that CSP(A) reduces to afne integer programming by the 𝑘-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' An intuition behind the above conjecture is an observation that solving systems of equations over a group is somewhat typical obstruction for an existence of a 𝑘-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' More precisely, we can prove the following from which we can derive further non-existence of a Datalog∪ reduction between certain CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' The core of the proof of the proposition below is an adaptation of the proof that CSP(Z𝑝) is not solved by any level of Sherali-Adams hierarchy with a twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' CSP(Z𝑝) does not reduce to CSP(Z𝑞) by a consistency reduction for any distinct primes 𝑝 and 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We start with a 𝑘-consistent but unsolvable instance of CSP(Z𝑝), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', with a system of equations modulo 𝑝 which is not solvable, but is accepted by the 𝑘- consistency algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Such a system exists due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [FV98, ABD09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Observe that, after each step in the 𝑘-consistency procedure, the sets F𝐾 are always afne subspaces of Z𝐾 𝑝 — since to begin with they are defned by some equations, and in each step, we intersect with a projection of another afne subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since the system is consistent, the subspaces are non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Further observe that the maps 𝜋 : F𝐾 → F𝐿 defned by restriction to 𝐿 are afne, and hence the size of the preimage of an element 𝑓 ∈ F𝐿 under 𝜋 does not depend on 𝑓 — it only depends on the dimension of the kernel of 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Hence the uniform probability on F𝐾, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', 𝜆: F𝐾 → Q defned as 𝜆(𝑓 ) = 1/𝑝𝑑 where 𝑑 is the dimension of F𝐾, solves the corresponding Sherali-Adams linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Since 𝑝 and 𝑞 are coprime, we can interpret the expression 1/𝑝𝑑 as an element of Z𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Consequently, this assignment solves the system in the second step of Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14 with G = Z𝑞, and hence the algorithm accepts a non-solvable instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' □ We note that the above proposition could be also proved by refning [ABD09, Theorem 3] (to adapt for composition with another Datalog reduction) and using the fact that CSP(Z𝑝) is not expressible in fxed-point logic with modulo 𝑞 rank operators (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', [Hol11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This alternative approach has been communicated to us by Anuj Dawar [Daw22] and precedes the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' As a consequence of the above proposition, we get that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', CSP(K3) does not reduce to CSP(Z2) by a consistency reduction, since CSP(Z3) reduces to CSP(K3) by a gadget reduction and consistency reductions are transitive by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' This concludes that the upper of the two inequalities in Figure 1b is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' More 44 VICTOR DALMAU AND JAKUB OPRŠAL generally, it can be shown that CSP(K3) does not reduce to CSP(Z) by a consistency reduction — this statement follows from the main result of [CŽ22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Further, it is known that CSP(Z𝑝)’s are a complete set of obstructions for a consistency reduction from a CSP to the trivial CSP, in the sense that CSP(A) reduces by 𝑘-consistency reduction to the trivial CSP if and only if CSP(Z𝑝) does not reduce to CSP(A) by a gadget reduction for any prime 𝑝 — this follows from the characterisation of CSPs solvable by local consistency [BK14] and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Could it be that equations over a group form a complete set of obstructions for consistency reductions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Assuming, it is the case and it is also true for the infnite template Z of solving equations over integers, we can explain the CSP dichotomy in the following way: Either CSP(A) allows a gadget reduction from CSP(G) for a non-Abelian group G, and hence it is NP-complete, or all groups G, such that CSP(A) allows a gadget reduction from CSP(G), are Abelian, and hence CSP(A) reduces to CSP(Z) by a consistency reduction and it is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, it is also possible that for each tractable CSP(A), there exist 𝑛 and 𝑘, that depend on A, such that CSP(A) reduces to CSP(Z𝑛) by the 𝑘-consistency reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Such a result would be a considerable step in providing a descriptive complexity of tractable CSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Finally, there are several algorithms related to Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='14 whose power is not fully described for fnite template CSPs, and are actually stronger than the one we suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A few of them are hierarchies of problems that can express CSP(Z): the question whether higher levels of LP+AIP solves all tractable CSPs was asked in [BGWŽ20], and higher levels of CLAP was suggested in [CŽ22b] (note that [CŽ22b] also asks whether CLAP itself could solve all fnite tractable CSPs which is not immediately implied by our conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Another algorithm, called cohomological 𝑘-consistency was suggested in [ÓCo22] — the cohomological 𝑘-consistency is es- sentially a stronger version of our algorithm: the diference is that it does not only ask whether the above system of linear equations has a solution over Z, but also attempts to fnd a solution, for each 𝑓 ∈ F𝐾, that satisfes 𝑥𝐾,𝑓 = 1, and iteratively removes values for which there is none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' In short, our conjecture implies that all CSPs tractable by [Bul17, Zhu20] can be solved by any of these algorithms, and the comparison of power of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=', cohomological 𝑘-consistency and higher levels of LP+AIP is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Acknowledgements Parts of this paper are based on part of an unpublished note by Marcin Wrochna which in particular contains the characterisation of the arc-consistency reduction, and several observations and statements that served as a ground for research pre- sented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' We are grateful to Marcin for allowing us to publish his results among ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' References [ABD09] Albert Atserias, Andrei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Bulatov, and Anuj Dawar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' Afne systems of equations and counting infnitary logic.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='9781611975994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' [Zhu20] Dmitriy Zhuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' A proof of the CSP dichotomy conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' ACM, 67(5):30:1–30:78, August 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} +page_content='1145/3402029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQfcgzB/content/2301.05084v1.pdf'} diff --git a/8dFLT4oBgHgl3EQftC_m/content/2301.12150v1.pdf b/8dFLT4oBgHgl3EQftC_m/content/2301.12150v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ca650c758173ec7affae0d0927bc201d32269b76 --- /dev/null +++ b/8dFLT4oBgHgl3EQftC_m/content/2301.12150v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8653509ad5702def198923fef1cf4594bb5bc98144de636b755efac6e654c9d5 +size 9654352 diff --git a/8dFLT4oBgHgl3EQftC_m/vector_store/index.faiss b/8dFLT4oBgHgl3EQftC_m/vector_store/index.faiss new file mode 100644 index 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b/9dAzT4oBgHgl3EQfFPqV/content/tmp_files/2301.01008v1.pdf.txt @@ -0,0 +1,1078 @@ +Topological Two-Dimensional Gravity +on Surfaces with Boundary +Jan Troostb +b Laboratoire de Physique de l’´Ecole Normale Sup´erieure +CNRS, ENS, Universit´e PSL, Sorbonne Universit´e, +Universit´e de Paris F-75005 Paris, France +E-mail: +jan.troost@ens.fr +Abstract +We solve two-dimensional gravity on surfaces with boundary in terms of contact +interactions and surface degenerations. The known solution of the bulk theory in terms +of a contact algebra is generalized to include boundaries and an enlarged set of boundary +operators. The latter allow for a linearization of the Virasoro constraints in terms of an +extended integrable KdV hierarchy. +arXiv:2301.01008v1 [hep-th] 3 Jan 2023 + +Contents +1 +Introduction +1 +2 +Open Topological Gravity +2 +3 +The Virasoro Algebra Representations +4 +3.1 +The Bulk Representation of the Virasoro Algebra +. . . . . . . . . . . . . . . . +4 +3.2 +The Extended Virasoro Representation . . . . . . . . . . . . . . . . . . . . . . +5 +3.3 +The Recursion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +3.4 +The Generalized Vertex Operators . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.5 +Amplitudes +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +4 +The Extended Partition Function +11 +4.1 +The Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +4.2 +A Few More Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +5 +Conclusions +13 +1 +Introduction +Two-dimensional gravity on closed Riemann surfaces was solved in terms of matrix models +[1–3], conformal field theory [4–6] and intersection theory [7,8]. While aspects of gravity on +Riemann surfaces with boundary were partially understood in terms of matrix models early +on [9, 10], a rigorous theory of topological gravity on Riemann surfaces with boundary was +only recently established [11]. Since then, various perspectives on these theories have been +developed [12–17]. The main approaches are through geometry and matrix models. The points +of view provided by these methods on the resulting integrable KdV hierarchy are qualitatively +distinct and usefully complementary. +Two-dimensional gravity on closed Riemann surfaces was also understood in a conformal +field theory approach closely related to string theory [18]. The theory was solved in terms of +Virasoro recursion relations. These relations were derived from a contact algebra for vertex +operators that carries all the topological information provided by the surface as well as the +bundles on the moduli space of surfaces [7]. +Our goal in this paper is to extend the contact algebra approach [18] to topological gravity +on Riemann surfaces with boundary. To that end, we study the contact algebra for operators +in the presence of boundaries as well as how the bulk algebra is represented on an extended +set of boundary vertex operators. Through representation theory and consistency conditions, +we fix all constants in the extended open Virasoro algebra, and manage to derive the Virasoro +recursion relation for the open and closed partition functions. Given a few initial correlators, +this allows to solve the theory. +The paper is structured as follows. In section 2 we review salient features of topological +gravity on Riemann surfaces with boundary [11]. The extended representation of the bulk +vertex operator contact algebra on the boundary vertex operators is constructed in section +3 using consistency arguments. In section 4 the constraints are translated into a differential +1 + +Virasoro algebra that acts on the generating function of topological correlators. At that point, +we make contact with the extended open string partition function [12] which is sufficient to +prove that the solution to the Virasoro constraints indeed coincides with the known solution +of open topological gravity. We conclude in section 5 with a summary and suggestions for +future research. +2 +Open Topological Gravity +In this section, we recall features of the solution of open and closed topological gravity, respec- +tively on Riemann surfaces with [11] or without boundary [7]. For open topological gravity, +we indicate a few features of the rigorous geometric solution [11]. For topological gravity on +Riemann surfaces (without boundary), we also briefly recall aspects of the solution in terms +of a conformal field theory [18] with contact interactions. We then start out on the path to +generalize that solution to Riemann surfaces with boundary.1 +Riemann Surfaces and Carriers of Curvature +Topological gravity on Riemann surfaces (without boundary) [7] satisfies the ghost number +conservation equation – or the dimension constraint on the integral over the moduli space of +surfaces –: +3g − 3 + nc = +nc +� +i=1 +nc +i . +(2.1) +The genus of the Riemann surface is g. The number of bulk vertex operator insertions is +nc and nc +i are the labels of the bulk vertex operators referring to the power of the tangent +line bundle at a point [7]. A central idea in [18] was to graft the curvature associated to the +Riemann surface itself onto the bulk vertex operators such that all topological properties of +the theory are captured by local operators – this in turn allows for the solution of the theory +in terms of contact interactions. When we associate a curvature 2(nc +i − 1)/3 to each bulk +vertex operator τnc +i of power nc +i, then ghost number conservation implies that: +The Integrated Curvature = 2g − 2 = +nc +� +i=1 +2 +3(nc +i − 1) , +(2.2) +namely that the curvature of the surface is faithfully represented. The puncture operator τ0 +has the smallest curvature contribution equal to −2/3, while the dilaton operator τ1 carries +no curvature at all. All other operators carry positive curvature (in this convention). +Riemann Surfaces with Boundary +The integration over the moduli space of Riemann surfaces with boundaries and with boundary +and bulk insertions leads to the dimensionality constraint valid for non-zero open correlation +functions [11]: +3g′ − 3 + no + 2nc = 2 +nc +� +i=1 +nc +i . +(2.3) +1See also [19] for an interesting alternative. +2 + +The doubled genus g′ is the genus of the Riemann surface that is obtained by gluing a given +Riemann surface with at least one boundary to its reflection. We therefore have the relation +g′ = 2g + b − 1 where b is the number of boundaries of the original surface and g its genus. +The number of boundary operator insertions σ is no [11]. In terms of the ordinary genus g +and number of boundaries b, we have: +6g − 6 + 3b + 2nc + no = 2 +nc +� +i=1 +nc +i , +(2.4) +in which we recognize the constraint (2.1) as the special case without boundaries. +Our first step in generalizing the solution of the closed theory in terms of contact interac- +tions [18] is to appropriately distribute curvature in the presence of boundaries and boundary +insertions. We continue to assign curvature to the bulk insertions as before [18] – see equation +(2.2). For simplicity, we momentarily imagine a single boundary, with a non-zero number no +of boundary insertions σ. The ghost number conservation equation (2.4) then suggests that +we should assign curvature −1/3 to each basic boundary insertion σ, in such a manner that +we find the equation: +Boundary Curvature = 1 = −no +3 , +(2.5) +in accord with our assignment for bulk curvature as well as the ghost number conservation +equation (2.4). The relative factor of a half compared to the basic bulk (puncture) operator +τ0 is due to the fact that the boundary operator increases the dimension of the moduli space +by real dimension one (compared to a bulk operator which increases the real dimension by +two). This reasoning can be generalized to the case of multiple boundaries with insertions. It +is sufficient to introduce an extra label corresponding to each boundary (with its associated +boundary insertions). We conclude that the boundary operator σ carries curvature −1/3. +Higher Powers +To prepare for reasonings to come, it may be useful to interject a thought experiment at this +point. Note that the closed string vertex operator τn can be thought of as a power of the +vertex operator τ1 in an approximate sense. The curvature it carries is then interpreted as +the curvature n × 2/3 from which we subtract 2/3. The curvature remains bounded from +below such that the vertex operators do not cut out such a large part of the surface for it +to disappear entirely.2 Similarly, if we were to attempt to define an arbitrary power of the +boundary operator σ to which we attached curvature −1/3, the operator would not have well- +defined correlation functions. A manner to remedy this obstruction is to add explicit powers +of the string coupling u to the operator: ρn = un−1σn. Now, the powers of the string coupling +are counted by the genus g and the number of boundaries b on the one hand, and the explicit +powers of u on the other hand. Suppose we study a correlation function of operators ρno +j and +τnc +i. It satisfies the equation: +2g − 2 + b + +no +� +j=1 +(no +j − 1) = +nc +� +i=1 +2(nc +i − 1) +3 ++ +no +� +j=1 +(2no +j +3 +− 1) . +(2.6) +2This is dictated by geometry or can be interpreted as a Seiberg bound [20]. +3 + +This is still the ghost conservation equation (2.4) but rewritten in such a way as to make the +explicit string coupling contributions visible on the left hand side. We made use of the fact +that the coupling u corresponds to the vacuum expectation value of the exponential of the +dilaton operator τ1 which couples to curvature. The operator ρn still caries ghost number n, +but it also carries curvature 2n/3 − 1, as we made manifest in our manner of writing equation +(2.6).3 While the boundary operators that we will soon encounter are more intricate still, +they share features with the operators ρn. +3 +The Virasoro Algebra Representations +In this subsection, we briefly remind the reader of an intuitive manner to solve topological +gravity on closed Riemann surfaces using contact terms [18]. We extend the approach to +include boundaries and boundary vertex operators which can be viewed as representing the +contact algebra. This section heavily relies on background provided in [18] to which we do +refer for more details. +3.1 +The Bulk Representation of the Virasoro Algebra +The method of [18] to solve topological gravity on closed Riemann surfaces is to represent all +the topological data in terms of local operators in a conformal field theory. As an example, we +already saw that the curvature (which codes the genus) was assigned to local bulk operator +insertions. Intersection numbers are then represented as integrals over the moduli space of +the Riemann surface of conformal field theory correlators.4 We denote the curvature carrying +bulk local operator insertions τn. Due to the topological nature of the theory, the contact +interactions between the local operators suffice to compute the intersection numbers on the +moduli space of Riemann surfaces. +The method of [18] to solve topological gravity uses the fact that the algebra of integrated +vertex operators is represented on localized bulk vertex operators (or states) in the form [18]: +� +Dϵ +τm|τn⟩ = An +m|τn+m−1⟩ , +(3.1) +where the localized vertex operator τn is assumed to lie in the disk Dϵ over which the vertex +operator τm is integrated. The representation arises from the contact term between the oper- +ators τm and τn. When we wish to compute the algebra of consecutive actions of the locally +integrated bulk vertex operators in the representation, we need to take into account that the +first integrated operator may enter into contact with the second integrated operator. To keep +track of this term, it is useful to define a measure of the non-commutativity of the operation +of localizing the vertex operator, and integrating over it [18]: +� +Dϵ +τm|τn⟩ − +� +Dϵ +τn|τm⟩ = Cnm|τn+m−1⟩ . +(3.2) +3For n ≥ 1 the boundary operator now has sufficient curvature to have well-defined correlation functions. +4This is heavily reminiscent of string theory (see e.g. [21]) and we allow string theory nomenclature to creep +into our language. +4 + +Then, when we consider the action of two integrated vertex operators on a localized operator, +we find a consistency condition between the representation coefficients A and the measure of +non-commutativity C [18]: +Am+k−1 +n +Ak +m − An+k−1 +m +Ak +n + CnmAk +m+n−1 += +0 . +(3.3) +The coefficient An +m is calculated in [18] and it equals the curvature of the insertion plus one: +An +m = 2 +3(n − 1) + 1 , +(3.4) +and we retain that we have the contact contribution +� +Dϵ +τm|τn⟩ = 2n + 1 +3 +|τn+m−1⟩ . +(3.5) +In turn this implies that the measure of non-commutativity C is proportional to the difference +in the curvature of the insertions: +Cmn = 2 +3(m − n) . +(3.6) +Note that when we identify the coefficients An of the representation on the bulk vertex operator +space with an operator Ln−1, then the commutation relation (3.3) shows that we have a +representation of the Virasoro algebra: +[Ln, Lm] = 2 +3(m − n)Lm+n . +(3.7) +Thus, the contact algebra is a Virasoro algebra, represented on the space of bulk operator +insertions. This is an essential tool in the solution to the bulk topological gravity theory [18], +and we wish to extend it to Riemann surfaces with boundary. +3.2 +The Extended Virasoro Representation +In the presence of a boundary, we first address the question what happens when a bulk vertex +operator is integrated over a small ring Rϵ near an empty boundary. We propose that the +integrated vertex operator in that case generates an operator on the boundary: +� +Rϵ +τn| ⟩b = u c(n)|σb +n−1⟩ . +(3.8) +We have introduced operators σb +n that live on a boundary of the Riemann surface. We have +stripped off one factor of the string coupling constant u on the right hand side – we think of the +bulk vertex operators as carrying one power of the coupling constant more than the boundary +operators.5 We have allowed for a representation coefficient c(n) that is undetermined for +now. The curvature of the operator σb +n equals the curvature of the bulk vertex operator minus +one, to compensate for the string coupling constant prefactor. Therefore, the curvature of the +5This is standard in string theory. Alternatively, it can be viewed as a consequence of the relative contri- +bution of bulk and boundary vertex operators to the dimension of moduli space. +5 + +operator σb +n−1 equals 2(n − 1)/3 − 1. We allow for operators with n ≥ 2 and set other terms +to zero. +Thus, we have introduced a new space parameterized by the operators σb +n. Our next step +is to assume that the integrated bulk vertex operators also act on this space and provide a +new representation of the Virasoro algebra. We need to make sure that the resulting operator +carries the sum of the curvatures of the operators on the left hand side, and we propose that +the contact algebra coefficient is again fixed to equal the curvature of the operator plus one – +see equation (3.4). We thus find: +� +Rϵ +τm|σb +n⟩ = 2n +3 |σb +m+n−1⟩ . +(3.9) +This natural proposal partially fixes the normalization of the boundary vertex operators. We +still need to check whether the integrated vertex operators satisfy the Virasoro algebra. The +action (3.9) is indeed a representation of the Virasoro algebra, as before. For the action (3.8) +to also enter into a representation of the Virasoro algebra, the coefficient c(n) needs to be a +linear function of n. Finally, we use a choice of overall normalization of the boundary vertex +operators to set c(n) = n+a +3 +where a is a constant to be determined. We will later argue that +consistency requires a = 0 and we therefore find the action on an empty boundary: +� +Rϵ +τn| ⟩b = u n +3|σb +n−1⟩ . +(3.10) +In summary, we have extended the space of boundary operators considerably, and we have +represented the Virasoro contact algebra on that space. +3.3 +The Recursion Relation +For topological gravity on closed Riemann surfaces, the representation of the contact algebra +was leveraged into a recursion relation for the topological correlators [18]. The integral over +bulk vertex operators was split into an integral over small disks where other operators reside, +neighbourhoods of nodes and uneventful regions. The fact that integrals of bulk operators +over the whole Riemann surface should commute, combined with the contact algebra, gave +rise to consistency conditions on the contributions of nodes which in turn provided a recursion +relation for correlators. Our claim is that the same reasoning applies to the integrated bulk +vertex operators on Riemann surfaces with boundary. We again need to take into account the +possible development of nodes on the Riemann surface, as well as possible generalized contact +terms with the boundary, which we described previously. +To ease into the generalized recursion relation, let us recall the closed recursion relation +first [18]6: +⟨τn+1 +� +i∈C +τni⟩c += +� +j +2nj + 1 +3 +⟨τn+nj +� +i̸=j +τni⟩c +(3.11) ++u2 +18( +n−1 +� +k=0 +(⟨τkτn−k−1 +� +i∈C +τni⟩c + +� +C=C1∪C2 +⟨τk +� +i∈C1 +τni⟩c⟨τk−i−1 +� +j∈C2 +τnj⟩c) . +6We normalize the bulk correlators as ⟨τ0τ0τ0⟩c = 1 and ⟨τ1⟩c = 1/24. We often set the string coupling u +to one. +6 + +Figure 1: Two degenerations of Riemann surfaces are depicted. The left figure represents a +surface splitting into two surfaces. The sum of genera is conserved. The right figure shows a +genus two Riemann surface that turns into a genus one Riemann surface, lowering the genus +by one. +The set C is a set of bulk operator insertions. The first term on the right hand side arises +from the bulk contact algebra representation (3.5) while the second line has its origins in the +fact that a Riemann surface can develop nodes which give rise to a Riemann surface of one +genus less, or which splits the Riemann surface into two closed Riemann surfaces. See Figure +1 and reference [18]. +The generalization to the case of extended open correlators is: +⟨τn+1 +� +i∈C +τni +� +l∈O +σb +nl⟩o,ext += +� +j +2nj + 1 +3 +⟨τn+nj +� +i̸=j +τni +� +l +σnl⟩o,ext + +� +j +2nj +3 ⟨ +� +i +τniσn+nj +� +l̸=j +σnl⟩o,ext ++un + 1 +3 +⟨σb +n +� +i∈C +τni +� +l∈O +σb +nl⟩o,ext +(3.12) ++u2 +18( +n−1 +� +k=0 +(⟨τkτn−k−1 +� +i,j∈CO +τnk⟩o,ext ++ +� +(e,f) +� +CO=CO1∪CO2 +⟨τk +� +i,j∈CO1 +τniσb +nj⟩e⟨τk−i−1 +� +l,m∈CO2 +τnlσb +nm⟩f) . +The first line corresponds to the fact that we are considering an integrated bulk operator τn+1. +It gives rise to the contact terms in the second line from the bulk contact term (3.5) and the +boundary contact term (3.9). The third line arises from the naked boundary term (3.10). The +fourth line arises from pinching off a handle. The fifth line requires explanation. We sum +over the sectors (e, f) which can be either (open,closed), (closed,open) or (open,open).7 The +first two arise when we split the surface into a closed Riemann surface and a Riemann surface +with boundary.8 In that case, the open string sector will contain all the boundary insertions, +necessarily. The third value, (open,open) arises when a node splits the Riemann surface into +two Riemann surfaces with boundary. The set CO indicates the set of all bulk and boundary +insertions, and we sum over their possible distributions CO1 and CO2 on the two disjoint +7We exclude the case with no boundaries from our definition of extended open correlators. See equation +(3.11) for the purely closed correlators. +8We effectively obtain a factor of two from these first two sectors. +7 + +surfaces.9 +Note that the second line in the right hand side contains a correlator that is of one order +less in the string coupling constant, and the third line a correlator that is down by two orders +in the string coupling constant u. +3.4 +The Generalized Vertex Operators +To make further progress, we must discuss the nature of the extended set of boundary vertex +operators σb +n in more detail. We recall that in the geometric open topological theory [11], we +found a single boundary vertex operator σ of curvature −1/3 in section 2. This matches the +curvature of σb +1 and we will indeed identify the two operators: σ = σb +1.10 The curvature of +the general operator σb +n is 2n/3 − 1. To make such operators on the boundary, we can use a +power of the operator σ as well as the string coupling constant (effectively of curvature one). +A natural guess is that there is a component ρn = u−1(uσ)n to the boundary vertex operator +σb +n (as previewed in section 2). However, we also need to allow for more drastic processes. +Up to now, a number of complications were implicit in our extended boundary vertex +operators. To start with, we concentrate on the simplest extended operator, namely σb +2. It +naively corresponds to an insertion of uσσ. However, to understand further possibilities, we +need to study the boundary analogue of nodes. +A strip (or open string propagator) can +be squeezed near the boundary of the moduli space of open Riemann surfaces, in various +manners. Either the number of boundaries can decrease as in an annulus to disk transition, +or the number of boundaries can increase as in a disk to two disks transition.11 See figure +2. When the integrated bulk vertex operator is close to these singular configurations, it can +either give rise to boundary vertex operators that sit on a single boundary or it can give rise +to boundary vertex operators that sit on two different boundaries of disconnected surfaces. +The boundary vertex operator σb +2 must capture both these possibilities. Thus, we propose the +equation: +⟨. . . σb +2 . . . ⟩o,ext = b1u⟨. . . σσ⟩o,ext + b2u⟨. . . σ⟩⟨σ . . . ⟩o,ext . +(3.13) +This equation shows that the generalized vertex operator σb +2 exhibits a non-local characteristic. +We recall that in the case of a node degeneration (see Figure 1), there was a universality +between losing a handle and splitting a surface – both terms have equal coefficient in the +second line of equation (3.11). We propose a similar universality here for the two terms in +which the boundary operators remain on the same boundary, or split – compare Figures 1 and +2 – and set the two constants in the above equation equal, namely b1 = b = b2. To determine +the overall constant b, we calculate an amplitude. +9If one labels boundaries, and their associated boundary operators, a finer combinatorics and summation +is necessary. +10There is a possible normalization factor between these two operators. +Our previous choice of overall +normalization of the boundary operators makes sure that this identification is spot on in standard conventions. +11There is a third degeneration process in which the genus drops by one. +When one labels boundary +components, it will play a role. See e.g. [22] for a discussion in open/closed string field theory. +8 + +Figure 2: Two degenerations of Riemann surfaces with boundary are drawn. The left figure +represents a disk splitting into two disks. The right figure shows an annulus that turns into a +disk. +3.5 +Amplitudes +To understand the content of the recursion relation further, we need initial conditions, which +we take from the most basic geometric calculations [11]. We have that the boundary three- +point function is the only non-zero disk amplitude with only boundary σ insertions, and +normalize it to one:12 +⟨σσσ⟩o,ext = 1 . +(3.14) +The other initial condition is that the bulk-boundary one-point function on the disk equals: +⟨τ0σ⟩o,ext = 1 . +(3.15) +To save on indices, we will drop the upper index on the correlator from now on – it should be +clear from the context which correlator we have in mind. +To understand the structure of the vertex operator σb +m≥2, we can use the puncture equation, +namely, the recursion relation (3.12) for n = −1: +⟨τ0 +� +i∈C +τni +� +l∈O +σb +nl⟩ = +� +j +2nj + 1 +3 +⟨τnj−1 +� +i̸=j +τni +� +l +σnl⟩ + +� +j +2nj +3 ⟨ +� +i +τniσnj−1 +� +l̸=j +σnl⟩ . +(3.16) +Let us also be explicit about the dilaton equation: +⟨τ1 +� +i∈C +τni +� +l∈O +σb +nl⟩ = +� +j +2nj + 1 +3 +⟨ +� +i +τni +� +l +σnl⟩ + +� +j +2nj +3 ⟨ +� +i +τni +� +l +σnl⟩ . +(3.17) +We are ready to calculate a first amplitude in two manners, using either the puncture equation, +or the factorization equation (3.13): +⟨τ0σ2σσ⟩ += +4 +3⟨σσσ⟩ += +2b⟨τ0σ⟩⟨σσσ⟩ . +(3.18) +In the first line we used the puncture equation (3.16). In the second line, we used the ansatz +(3.13) and allowed for the two possible ways in which the vertex operators can split over two +12This is a disk amplitude. We have set u = 1 once more. +9 + +correlators to give a non-vanishing result.13 Note that in the second line a factor of the string +coupling constant implicitly cancelled between the two disk amplitudes and the expression for +the operator σb +2. Using the normalization of the initial conditions, we find: +b = 2 +3 . +(3.19) +This fixes our reading of the extended vertex operator σb +2 once and for all. +For the next +extended operator σb +3 we propose a similar universal ansatz consistent with curvature conser- +vation and splitting off a single vertex operator σ: +⟨. . . σb +3 . . . ⟩ = c(u⟨. . . σσ2⟩ + u⟨. . . σ2⟩⟨σ . . . ⟩) . +(3.20) +We can again determine the constant c using either the puncture or the factorization equation +to determine one and the same amplitude consistently: +⟨τ0σ3σ4⟩ += +2⟨σ2σ4⟩ = 8⟨σ3⟩⟨σ3⟩ += +c⟨τ0σ⟩⟨σ2σ4⟩ + 6c⟨τ0σ2σ2⟩⟨σ3⟩ += +4c⟨τ0σ⟩⟨σ3⟩⟨σ3⟩ + 8c⟨σ3⟩⟨σ3⟩ , +(3.21) +and find that again c = 2/3 – the constant is fixed once more in terms of the bulk-boundary +one-point function ⟨τ0σ⟩. Continuing recursively in this manner, e.g. exploiting the correlation +functions ⟨τ0σb +nσ2(n−1)⟩, we find: +⟨. . . σb +n⟩ = u 2 +3(⟨. . . σσb +n−1⟩ + ⟨. . . σb +n−1⟩⟨σ . . . ⟩) . +(3.22) +Thus, we have determined the intricate nature of the extended boundary vertex operators σb +n +and how they recursively code the splitting of boundaries of open Riemann surfaces. +Tying up a loose end: fixing the constant a +We tie up a loose end at the hand of another amplitude. The amplitude illustrates a splitting +of open Riemann surfaces involving two disk one-point functions. We calculate the amplitude +⟨τ3τ0σσ⟩ in two manners. We can apply recursion to the operator τ3, or to the operator τ0 +first. In this calculation, we restore the possible constant a that we introduced in subsection +3.2 and use an appropriately modified recursion relation. We demonstrate that the constant +can be determined by consistency. Using the a-modified recursion relation, we find: +⟨τ3τ0σ2⟩ += +7 +3⟨τ2σ2⟩ += +1 +3⟨τ2σ2⟩ + 2 +9⟨τ0σ⟩⟨τ1τ0σ⟩ + 4 +3⟨τ0σ3σ⟩ + 3 + a +3 +⟨σ2τ0σ2⟩ . +(3.23) +This implies: +⟨τ2σ2⟩ += +1 +9⟨τ0σ⟩⟨τ0σ⟩ + 4 +3⟨σ2σ⟩ + 3 + a +3 +2 +3⟨σ3⟩ . +(3.24) +13Ghost number conservation applies to each factor separately. +10 + +We can compute the latter correlator in another manner, using the puncture equation and +the modified recursion relation: +⟨τ2σ2⟩ += +1 +9⟨τ0σ⟩⟨τ0σ⟩ + 4 +3⟨σ2σ⟩ + 2 + a +3 +⟨σ3⟩ . +(3.25) +Using our previous results, we find full consistency if and only if a = 0. Thus, we tied up the +loose end in subsection 3.2. +4 +The Extended Partition Function +In this section we introduce the generating function of extended open string correlators and +prove that the recursion relations for the correlators imply Virasoro constraints on the gen- +erating function. +This allows us to make our results more rigorous by connecting to the +mathematics literature on the integrable structure of the intersection theory on moduli spaces +of Riemann surfaces with boundary [12]. We conclude the section with a few example ampli- +tudes. +4.1 +The Generating Function +We recall the generating functions of closed as well as open topological gravity correlation +functions [11]: +F c += +� +g≥0,n≥1,2g−2+n>0 +u2g−2 +n! +� +ki≥0 +⟨τk1 . . . τkn⟩c +gtk1 . . . tkn +F o,geom += +� +g′,k,l≥0,2g′−2+k+2l>0 +� +ai≥0 +ug′−1 +k!l! ⟨τa1 . . . τalσk⟩o +g′sk +l� +i=1 +tai . +(4.1) +In view of our enlarged space of boundary vertex operators, we also introduce a generating +function for extended open topological gravity correlation functions: +F o,ext += +� +g′,k,l≥0,2g′−2+k+2l>0 +� +ai,bi≥0 +ug′−1 +k!l! ⟨τa1 . . . τalσb +b1 . . . σb +bk⟩o,ext +g +� +i +tai +� +j +sbj . +(4.2) +The Extended Virasoro Constraints +We define Virasoro generators +Ln += +� +i≥0 +2i + 1 +2 +ti∂ti+n − 3 +2∂tn+1 + u2 +12 +n−1 +� +i=0 +∂ti∂tn−i−1 + 3 +4 +t2 +0 +u2δn,−1 + 1 +16δn,0 +(4.3) +Lext +n += +Ln + +� +i≥0 +(i + 1)si+1∂sn+i+1 + un + 1 +2 +∂sn + 3 +2 +s1 +u δn,−1 + 3 +4δn,0 +(4.4) +11 + +for n ≥ −1. These are defined such that the recursion relation (3.11) on the closed as well as +the recursion relation (3.12) on the extended open correlators leads to the constraints: +Ln exp F c += +0 +Lext +n exp(F c + F o,ext) += +0 . +(4.5) +The extra constants terms in the closed Virasoro algebra (4.3) are due to the initialization +cases ⟨τ 3 +0 ⟩c = 1 = 24 ⟨τ1⟩ at genus zero and one respectively, while the initial conditions +⟨σ3⟩ = 1 = ⟨τ0σ⟩ on the disk lead to the extra constants in the extended Virasoro algebra +(4.4), which satisfies14 +[Lm, Ln] = (m − n)Lm+n . +(4.6) +At this stage, we are able to make contact with rigorous results – these constraints on an +extended partition function of open topological correlators defined through an extended (or +unconstrained) integrable KdV hierarchy were found to hold in [12].15 The relation between +the operators σb +n and σ as well as the string coupling constant is neatly captured by a relation +between derivatives of the extended partition function: +∂sn = (2u +3 )n−1∂n +s1 . +(4.7) +This equation was proven from the KdV integrable hierarchy perspective in [12]. Using this +equation, and setting extended open times sn≥2 to zero, this relation between derivatives imply +the higher order Virasoro constraints on the geometric open topological partition function, +where the open Virasoro generators are [11]: +Lo +n = Ln + (2u +3 )n∂n+1 +s1 ++ n + 1 +2 +u(2u +3 )n−1∂n +s1 + δn,−1 +3 +2 +s1 +u + δn,0 +3 +4 . +(4.8) +The Virasoro constraints and the initialization condition are sufficient to determine the full +partition function [11,12]. Through the generating function of extended correlators, we have +connected our arguments with rigorous results on intersection theory on moduli spaces of +Riemann surfaces with boundary [11,12]. +4.2 +A Few More Amplitudes +For illustrative purposes, we calculate a few more amplitudes. They render the integrable +hierarchy structure, the Virasoro constraints and how to solve them more concrete. +4.2.1 +Amplitudes on The Disk +We have already indicated that on the disk only the third power of the elementary boundary +vertex operator σ has a non-zero correlation function and equals one, ⟨σ3⟩ = 1. The disk +bulk-boundary one-point function ⟨τ0σ⟩ is also one by a choice of normalization. Amplitudes +14These generators are rescaled by a factor of 2/3 compared to section 3 in order to reach a standard +normalization for the Virasoro algebra. +15 The translation of variables and normalizations is: Lthere,ext +n += (3/2)nLext +n , tthere +n += 3−n(2n + 1)!!tn and +sthere +n−1 = (2/3)n−1n!sn. +12 + +involving extended boundary vertex operators are computed through the reduction formula +(3.22). A non-trivial example is: +⟨τ2σ5⟩ += +10 +3 ⟨σb +2σ4⟩ = 40 +3 , +(4.9) +where we used the recursion relations (3.12) and (3.22) as well as the 6 choices of factoriza- +tion. After taking into account the different normalization in footnote 15, this agrees with a +more generic formula in [11]. Another interesting correlation function is ⟨τ2τ0σσ⟩. It can be +computed through the puncture equation (in the first line below) and/or the L1 constraint +(in the second line below): +⟨τ2τ0σσσ⟩ += +5 +3⟨τ1σσσ⟩ = 10 +3 ⟨σσσ⟩ += +1 +3⟨τ1σσσ⟩ + 2⟨τ0σ2σσ⟩ += +2 +3⟨σσσ⟩ + 2 × 8 +3⟨σσσ⟩ = 10 +3 ⟨σσσ⟩ . +(4.10) +The two ways of computing are in agreement. +4.2.2 +Higher Order Amplitudes +Amplitudes that are higher order in the string coupling exhibit qualitatively new phenomena. +We illustrate a few. We first compute amplitudes corresponding to cylinder diagrams, with two +boundaries and genus zero. An interesting amplitude that involves a closed-open factorization +due to a node can once again be computed in two manners: +⟨τ2τ0τ0σ⟩ += +5 +3⟨τ1τ0σ⟩ = 5 +3⟨τ0σ⟩ += +2 +3⟨τ1τ0σ⟩ + 1 +9⟨τ 3 +0 ⟩c⟨τ0σ⟩ + 2 +3⟨τ0τ0σ2⟩ += +2 +3⟨τ0σ⟩ + 1 +9⟨τ 3 +0 ⟩c⟨τ0σ⟩ + 8 +9⟨τ0σ⟩⟨τ0σ⟩ . +(4.11) +Both ways of computing the correlator lead to the same result, given the normalization of +the closed three-point function ⟨τ 3 +0 ⟩c as well as the bulk-boundary one-point function ⟨τ0σ⟩. +Finally, we compute an order O(u1) amplitude. +It involves the one-loop closed one-point +function ⟨τ1⟩c: +⟨τ3σ⟩ = 2 +3⟨σ3⟩ + ⟨σ2σ⟩ + 1 +9(1 + ⟨τ1⟩)⟨τ0σ⟩ +=((2 +3)3 + 2 +3)⟨σ3⟩ + 1 +9(1 + ⟨τ1⟩)⟨τ0σ⟩ . +(4.12) +Needless to say, many more results can be generated, e.g. by computer. We provided a few +telling illustrations that provide insight into the foundation of the integrable hierarchy. +5 +Conclusions +In the spirit of the solution of the bulk theory [18] and building on earlier mathematical +work [11, 12], we have solved two-dimensional topological gravity on Riemann surfaces with +13 + +boundary. By making use of an extended set of boundary vertex operators, we rendered the +representation of the contact algebra on the boundary linear. Only in a second step the more +complicated degeneration of surfaces with boundary is taken into account and the non-linear +realization of the (half) Virasoro algebra is found [12]. The picture in which the solution of +the theory is provided through contact interactions is a welcome intuitive complement to the +geometric and matrix model approaches. +While we have provided a compelling global picture, there are many details that remain +to be worked out. It would be good to find the geometric counterpart to the extended set of +boundary operators. The link between (the expectation values of) the conformal field theory +fields implicit in our analysis [18] and the sections of vector bundles of open topological +gravity can be clarified (e.g. by exploiting references [15, 20]). The analysis of the contact +terms in terms of an integration over a degeneration region of the moduli space of open +Riemann surfaces would be interesting. It will also be instructive to compare our analysis +to the geometric derivation of the topological recursion relation through closed and open +factorization [11], intuitively reviewed in [15]. +Another research direction is to exploit the insights developed here and apply them to +more general theories. The generalization to the extended closed theory [23]) comes to mind, +but mostly to open spin r curves. 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Sakai, “JT gravity, KdV equations and macroscopic loop operators,” +JHEP 01 (2020), 156 doi:10.1007/JHEP01(2020)156 [arXiv:1911.01659 [hep-th]]. +16 + diff --git a/9dAzT4oBgHgl3EQfFPqV/content/tmp_files/load_file.txt b/9dAzT4oBgHgl3EQfFPqV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4473b873f54077f0e60f056ab96297835ad91ad6 --- /dev/null +++ b/9dAzT4oBgHgl3EQfFPqV/content/tmp_files/load_file.txt @@ -0,0 +1,798 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf,len=797 +page_content='Topological Two-Dimensional Gravity on Surfaces with Boundary Jan Troostb b Laboratoire de Physique de l’´Ecole Normale Sup´erieure CNRS, ENS, Universit´e PSL, Sorbonne Universit´e, Universit´e de Paris F-75005 Paris, France E-mail: jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='troost@ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='fr Abstract We solve two-dimensional gravity on surfaces with boundary in terms of contact interactions and surface degenerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The known solution of the bulk theory in terms of a contact algebra is generalized to include boundaries and an enlarged set of boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The latter allow for a linearization of the Virasoro constraints in terms of an extended integrable KdV hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 9 4 The Extended Partition Function 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='1 The Generating Function .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2 A Few More Amplitudes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 12 5 Conclusions 13 1 Introduction Two-dimensional gravity on closed Riemann surfaces was solved in terms of matrix models [1–3], conformal field theory [4–6] and intersection theory [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' While aspects of gravity on Riemann surfaces with boundary were partially understood in terms of matrix models early on [9, 10], a rigorous theory of topological gravity on Riemann surfaces with boundary was only recently established [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Since then, various perspectives on these theories have been developed [12–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The main approaches are through geometry and matrix models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The points of view provided by these methods on the resulting integrable KdV hierarchy are qualitatively distinct and usefully complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Two-dimensional gravity on closed Riemann surfaces was also understood in a conformal field theory approach closely related to string theory [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The theory was solved in terms of Virasoro recursion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' These relations were derived from a contact algebra for vertex operators that carries all the topological information provided by the surface as well as the bundles on the moduli space of surfaces [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Our goal in this paper is to extend the contact algebra approach [18] to topological gravity on Riemann surfaces with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' To that end, we study the contact algebra for operators in the presence of boundaries as well as how the bulk algebra is represented on an extended set of boundary vertex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Through representation theory and consistency conditions, we fix all constants in the extended open Virasoro algebra, and manage to derive the Virasoro recursion relation for the open and closed partition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Given a few initial correlators, this allows to solve the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' In section 2 we review salient features of topological gravity on Riemann surfaces with boundary [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The extended representation of the bulk vertex operator contact algebra on the boundary vertex operators is constructed in section 3 using consistency arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' In section 4 the constraints are translated into a differential 1 Virasoro algebra that acts on the generating function of topological correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' At that point, we make contact with the extended open string partition function [12] which is sufficient to prove that the solution to the Virasoro constraints indeed coincides with the known solution of open topological gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We conclude in section 5 with a summary and suggestions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 2 Open Topological Gravity In this section, we recall features of the solution of open and closed topological gravity, respec- tively on Riemann surfaces with [11] or without boundary [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' For open topological gravity, we indicate a few features of the rigorous geometric solution [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' For topological gravity on Riemann surfaces (without boundary), we also briefly recall aspects of the solution in terms of a conformal field theory [18] with contact interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We then start out on the path to generalize that solution to Riemann surfaces with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='1 Riemann Surfaces and Carriers of Curvature Topological gravity on Riemann surfaces (without boundary) [7] satisfies the ghost number conservation equation – or the dimension constraint on the integral over the moduli space of surfaces –: 3g − 3 + nc = nc � i=1 nc i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='1) The genus of the Riemann surface is g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The number of bulk vertex operator insertions is nc and nc i are the labels of the bulk vertex operators referring to the power of the tangent line bundle at a point [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' A central idea in [18] was to graft the curvature associated to the Riemann surface itself onto the bulk vertex operators such that all topological properties of the theory are captured by local operators – this in turn allows for the solution of the theory in terms of contact interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' When we associate a curvature 2(nc i − 1)/3 to each bulk vertex operator τnc i of power nc i, then ghost number conservation implies that: The Integrated Curvature = 2g − 2 = nc � i=1 2 3(nc i − 1) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2) namely that the curvature of the surface is faithfully represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The puncture operator τ0 has the smallest curvature contribution equal to −2/3, while the dilaton operator τ1 carries no curvature at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' All other operators carry positive curvature (in this convention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Riemann Surfaces with Boundary The integration over the moduli space of Riemann surfaces with boundaries and with boundary and bulk insertions leads to the dimensionality constraint valid for non-zero open correlation functions [11]: 3g′ − 3 + no + 2nc = 2 nc � i=1 nc i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='3) 1See also [19] for an interesting alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 2 The doubled genus g′ is the genus of the Riemann surface that is obtained by gluing a given Riemann surface with at least one boundary to its reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We therefore have the relation g′ = 2g + b − 1 where b is the number of boundaries of the original surface and g its genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The number of boundary operator insertions σ is no [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' In terms of the ordinary genus g and number of boundaries b, we have: 6g − 6 + 3b + 2nc + no = 2 nc � i=1 nc i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4) in which we recognize the constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='1) as the special case without boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Our first step in generalizing the solution of the closed theory in terms of contact interac- tions [18] is to appropriately distribute curvature in the presence of boundaries and boundary insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We continue to assign curvature to the bulk insertions as before [18] – see equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' For simplicity, we momentarily imagine a single boundary, with a non-zero number no of boundary insertions σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The ghost number conservation equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4) then suggests that we should assign curvature −1/3 to each basic boundary insertion σ, in such a manner that we find the equation: Boundary Curvature = 1 = −no 3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='5) in accord with our assignment for bulk curvature as well as the ghost number conservation equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The relative factor of a half compared to the basic bulk (puncture) operator τ0 is due to the fact that the boundary operator increases the dimension of the moduli space by real dimension one (compared to a bulk operator which increases the real dimension by two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' This reasoning can be generalized to the case of multiple boundaries with insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' It is sufficient to introduce an extra label corresponding to each boundary (with its associated boundary insertions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We conclude that the boundary operator σ carries curvature −1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Higher Powers To prepare for reasonings to come, it may be useful to interject a thought experiment at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Note that the closed string vertex operator τn can be thought of as a power of the vertex operator τ1 in an approximate sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The curvature it carries is then interpreted as the curvature n × 2/3 from which we subtract 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The curvature remains bounded from below such that the vertex operators do not cut out such a large part of the surface for it to disappear entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2 Similarly, if we were to attempt to define an arbitrary power of the boundary operator σ to which we attached curvature −1/3, the operator would not have well- defined correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' A manner to remedy this obstruction is to add explicit powers of the string coupling u to the operator: ρn = un−1σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Now, the powers of the string coupling are counted by the genus g and the number of boundaries b on the one hand, and the explicit powers of u on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Suppose we study a correlation function of operators ρno j and τnc i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' It satisfies the equation: 2g − 2 + b + no � j=1 (no j − 1) = nc � i=1 2(nc i − 1) 3 + no � j=1 (2no j 3 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='6) 2This is dictated by geometry or can be interpreted as a Seiberg bound [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 3 This is still the ghost conservation equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4) but rewritten in such a way as to make the explicit string coupling contributions visible on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We made use of the fact that the coupling u corresponds to the vacuum expectation value of the exponential of the dilaton operator τ1 which couples to curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The operator ρn still caries ghost number n, but it also carries curvature 2n/3 − 1, as we made manifest in our manner of writing equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='3 While the boundary operators that we will soon encounter are more intricate still, they share features with the operators ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 3 The Virasoro Algebra Representations In this subsection, we briefly remind the reader of an intuitive manner to solve topological gravity on closed Riemann surfaces using contact terms [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We extend the approach to include boundaries and boundary vertex operators which can be viewed as representing the contact algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' This section heavily relies on background provided in [18] to which we do refer for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='1 The Bulk Representation of the Virasoro Algebra The method of [18] to solve topological gravity on closed Riemann surfaces is to represent all the topological data in terms of local operators in a conformal field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' As an example, we already saw that the curvature (which codes the genus) was assigned to local bulk operator insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Intersection numbers are then represented as integrals over the moduli space of the Riemann surface of conformal field theory correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4 We denote the curvature carrying bulk local operator insertions τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Due to the topological nature of the theory, the contact interactions between the local operators suffice to compute the intersection numbers on the moduli space of Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The method of [18] to solve topological gravity uses the fact that the algebra of integrated vertex operators is represented on localized bulk vertex operators (or states) in the form [18]: � Dϵ τm|τn⟩ = An m|τn+m−1⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='1) where the localized vertex operator τn is assumed to lie in the disk Dϵ over which the vertex operator τm is integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The representation arises from the contact term between the oper- ators τm and τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' When we wish to compute the algebra of consecutive actions of the locally integrated bulk vertex operators in the representation, we need to take into account that the first integrated operator may enter into contact with the second integrated operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' To keep track of this term, it is useful to define a measure of the non-commutativity of the operation of localizing the vertex operator, and integrating over it [18]: � Dϵ τm|τn⟩ − � Dϵ τn|τm⟩ = Cnm|τn+m−1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2) 3For n ≥ 1 the boundary operator now has sufficient curvature to have well-defined correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 4This is heavily reminiscent of string theory (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' [21]) and we allow string theory nomenclature to creep into our language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 4 Then, when we consider the action of two integrated vertex operators on a localized operator, we find a consistency condition between the representation coefficients A and the measure of non-commutativity C [18]: Am+k−1 n Ak m − An+k−1 m Ak n + CnmAk m+n−1 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='3) The coefficient An m is calculated in [18] and it equals the curvature of the insertion plus one: An m = 2 3(n − 1) + 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4) and we retain that we have the contact contribution � Dϵ τm|τn⟩ = 2n + 1 3 |τn+m−1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='5) In turn this implies that the measure of non-commutativity C is proportional to the difference in the curvature of the insertions: Cmn = 2 3(m − n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='6) Note that when we identify the coefficients An of the representation on the bulk vertex operator space with an operator Ln−1, then the commutation relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='3) shows that we have a representation of the Virasoro algebra: [Ln, Lm] = 2 3(m − n)Lm+n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='7) Thus, the contact algebra is a Virasoro algebra, represented on the space of bulk operator insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' This is an essential tool in the solution to the bulk topological gravity theory [18], and we wish to extend it to Riemann surfaces with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2 The Extended Virasoro Representation In the presence of a boundary, we first address the question what happens when a bulk vertex operator is integrated over a small ring Rϵ near an empty boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We propose that the integrated vertex operator in that case generates an operator on the boundary: � Rϵ τn| ⟩b = u c(n)|σb n−1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='8) We have introduced operators σb n that live on a boundary of the Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We have stripped off one factor of the string coupling constant u on the right hand side – we think of the bulk vertex operators as carrying one power of the coupling constant more than the boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='5 We have allowed for a representation coefficient c(n) that is undetermined for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The curvature of the operator σb n equals the curvature of the bulk vertex operator minus one, to compensate for the string coupling constant prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Therefore, the curvature of the 5This is standard in string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Alternatively, it can be viewed as a consequence of the relative contri- bution of bulk and boundary vertex operators to the dimension of moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 5 operator σb n−1 equals 2(n − 1)/3 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We allow for operators with n ≥ 2 and set other terms to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Thus, we have introduced a new space parameterized by the operators σb n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Our next step is to assume that the integrated bulk vertex operators also act on this space and provide a new representation of the Virasoro algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We need to make sure that the resulting operator carries the sum of the curvatures of the operators on the left hand side, and we propose that the contact algebra coefficient is again fixed to equal the curvature of the operator plus one – see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We thus find: � Rϵ τm|σb n⟩ = 2n 3 |σb m+n−1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='9) This natural proposal partially fixes the normalization of the boundary vertex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We still need to check whether the integrated vertex operators satisfy the Virasoro algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='9) is indeed a representation of the Virasoro algebra, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' For the action (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='8) to also enter into a representation of the Virasoro algebra, the coefficient c(n) needs to be a linear function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Finally, we use a choice of overall normalization of the boundary vertex operators to set c(n) = n+a 3 where a is a constant to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We will later argue that consistency requires a = 0 and we therefore find the action on an empty boundary: � Rϵ τn| ⟩b = u n 3|σb n−1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='10) In summary, we have extended the space of boundary operators considerably, and we have represented the Virasoro contact algebra on that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='3 The Recursion Relation For topological gravity on closed Riemann surfaces, the representation of the contact algebra was leveraged into a recursion relation for the topological correlators [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The integral over bulk vertex operators was split into an integral over small disks where other operators reside, neighbourhoods of nodes and uneventful regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The fact that integrals of bulk operators over the whole Riemann surface should commute, combined with the contact algebra, gave rise to consistency conditions on the contributions of nodes which in turn provided a recursion relation for correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Our claim is that the same reasoning applies to the integrated bulk vertex operators on Riemann surfaces with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We again need to take into account the possible development of nodes on the Riemann surface, as well as possible generalized contact terms with the boundary, which we described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' To ease into the generalized recursion relation, let us recall the closed recursion relation first [18]6: ⟨τn+1 � i∈C τni⟩c = � j 2nj + 1 3 ⟨τn+nj � i̸=j τni⟩c (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='11) +u2 18( n−1 � k=0 (⟨τkτn−k−1 � i∈C τni⟩c + � C=C1∪C2 ⟨τk � i∈C1 τni⟩c⟨τk−i−1 � j∈C2 τnj⟩c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 6We normalize the bulk correlators as ⟨τ0τ0τ0⟩c = 1 and ⟨τ1⟩c = 1/24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We often set the string coupling u to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 6 Figure 1: Two degenerations of Riemann surfaces are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The left figure represents a surface splitting into two surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The sum of genera is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The right figure shows a genus two Riemann surface that turns into a genus one Riemann surface, lowering the genus by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The set C is a set of bulk operator insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The first term on the right hand side arises from the bulk contact algebra representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='5) while the second line has its origins in the fact that a Riemann surface can develop nodes which give rise to a Riemann surface of one genus less, or which splits the Riemann surface into two closed Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' See Figure 1 and reference [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The generalization to the case of extended open correlators is: ⟨τn+1 � i∈C τni � l∈O σb nl⟩o,ext = � j 2nj + 1 3 ⟨τn+nj � i̸=j τni � l σnl⟩o,ext + � j 2nj 3 ⟨ � i τniσn+nj � l̸=j σnl⟩o,ext +un + 1 3 ⟨σb n � i∈C τni � l∈O σb nl⟩o,ext (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='12) +u2 18( n−1 � k=0 (⟨τkτn−k−1 � i,j∈CO τnk⟩o,ext + � (e,f) � CO=CO1∪CO2 ⟨τk � i,j∈CO1 τniσb nj⟩e⟨τk−i−1 � l,m∈CO2 τnlσb nm⟩f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The first line corresponds to the fact that we are considering an integrated bulk operator τn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' It gives rise to the contact terms in the second line from the bulk contact term (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='5) and the boundary contact term (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The third line arises from the naked boundary term (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The fourth line arises from pinching off a handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The fifth line requires explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We sum over the sectors (e, f) which can be either (open,closed), (closed,open) or (open,open).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='7 The first two arise when we split the surface into a closed Riemann surface and a Riemann surface with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='8 In that case, the open string sector will contain all the boundary insertions, necessarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The third value, (open,open) arises when a node splits the Riemann surface into two Riemann surfaces with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The set CO indicates the set of all bulk and boundary insertions, and we sum over their possible distributions CO1 and CO2 on the two disjoint 7We exclude the case with no boundaries from our definition of extended open correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' See equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='11) for the purely closed correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 8We effectively obtain a factor of two from these first two sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 7 surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='9 Note that the second line in the right hand side contains a correlator that is of one order less in the string coupling constant, and the third line a correlator that is down by two orders in the string coupling constant u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4 The Generalized Vertex Operators To make further progress, we must discuss the nature of the extended set of boundary vertex operators σb n in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We recall that in the geometric open topological theory [11], we found a single boundary vertex operator σ of curvature −1/3 in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' This matches the curvature of σb 1 and we will indeed identify the two operators: σ = σb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='10 The curvature of the general operator σb n is 2n/3 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' To make such operators on the boundary, we can use a power of the operator σ as well as the string coupling constant (effectively of curvature one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' A natural guess is that there is a component ρn = u−1(uσ)n to the boundary vertex operator σb n (as previewed in section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' However, we also need to allow for more drastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Up to now, a number of complications were implicit in our extended boundary vertex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' To start with, we concentrate on the simplest extended operator, namely σb 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' It naively corresponds to an insertion of uσσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' However, to understand further possibilities, we need to study the boundary analogue of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' A strip (or open string propagator) can be squeezed near the boundary of the moduli space of open Riemann surfaces, in various manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Either the number of boundaries can decrease as in an annulus to disk transition, or the number of boundaries can increase as in a disk to two disks transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='11 See figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' When the integrated bulk vertex operator is close to these singular configurations, it can either give rise to boundary vertex operators that sit on a single boundary or it can give rise to boundary vertex operators that sit on two different boundaries of disconnected surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The boundary vertex operator σb 2 must capture both these possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Thus, we propose the equation: ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σb 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' ⟩o,ext = b1u⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σσ⟩o,ext + b2u⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σ⟩⟨σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' ⟩o,ext .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='13) This equation shows that the generalized vertex operator σb 2 exhibits a non-local characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We recall that in the case of a node degeneration (see Figure 1), there was a universality between losing a handle and splitting a surface – both terms have equal coefficient in the second line of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We propose a similar universality here for the two terms in which the boundary operators remain on the same boundary, or split – compare Figures 1 and 2 – and set the two constants in the above equation equal, namely b1 = b = b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' To determine the overall constant b, we calculate an amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 9If one labels boundaries, and their associated boundary operators, a finer combinatorics and summation is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 10There is a possible normalization factor between these two operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Our previous choice of overall normalization of the boundary operators makes sure that this identification is spot on in standard conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 11There is a third degeneration process in which the genus drops by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' When one labels boundary components, it will play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' [22] for a discussion in open/closed string field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 8 Figure 2: Two degenerations of Riemann surfaces with boundary are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The left figure represents a disk splitting into two disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The right figure shows an annulus that turns into a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='5 Amplitudes To understand the content of the recursion relation further, we need initial conditions, which we take from the most basic geometric calculations [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We have that the boundary three- point function is the only non-zero disk amplitude with only boundary σ insertions, and normalize it to one:12 ⟨σσσ⟩o,ext = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='14) The other initial condition is that the bulk-boundary one-point function on the disk equals: ⟨τ0σ⟩o,ext = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='15) To save on indices, we will drop the upper index on the correlator from now on – it should be clear from the context which correlator we have in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' To understand the structure of the vertex operator σb m≥2, we can use the puncture equation, namely, the recursion relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='12) for n = −1: ⟨τ0 � i∈C τni � l∈O σb nl⟩ = � j 2nj + 1 3 ⟨τnj−1 � i̸=j τni � l σnl⟩ + � j 2nj 3 ⟨ � i τniσnj−1 � l̸=j σnl⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='16) Let us also be explicit about the dilaton equation: ⟨τ1 � i∈C τni � l∈O σb nl⟩ = � j 2nj + 1 3 ⟨ � i τni � l σnl⟩ + � j 2nj 3 ⟨ � i τni � l σnl⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='17) We are ready to calculate a first amplitude in two manners, using either the puncture equation, or the factorization equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='13): ⟨τ0σ2σσ⟩ = 4 3⟨σσσ⟩ = 2b⟨τ0σ⟩⟨σσσ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='18) In the first line we used the puncture equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' In the second line, we used the ansatz (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='13) and allowed for the two possible ways in which the vertex operators can split over two 12This is a disk amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We have set u = 1 once more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 9 correlators to give a non-vanishing result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='13 Note that in the second line a factor of the string coupling constant implicitly cancelled between the two disk amplitudes and the expression for the operator σb 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Using the normalization of the initial conditions, we find: b = 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='19) This fixes our reading of the extended vertex operator σb 2 once and for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' For the next extended operator σb 3 we propose a similar universal ansatz consistent with curvature conser- vation and splitting off a single vertex operator σ: ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σb 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' ⟩ = c(u⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σσ2⟩ + u⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σ2⟩⟨σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' ⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='20) We can again determine the constant c using either the puncture or the factorization equation to determine one and the same amplitude consistently: ⟨τ0σ3σ4⟩ = 2⟨σ2σ4⟩ = 8⟨σ3⟩⟨σ3⟩ = c⟨τ0σ⟩⟨σ2σ4⟩ + 6c⟨τ0σ2σ2⟩⟨σ3⟩ = 4c⟨τ0σ⟩⟨σ3⟩⟨σ3⟩ + 8c⟨σ3⟩⟨σ3⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='21) and find that again c = 2/3 – the constant is fixed once more in terms of the bulk-boundary one-point function ⟨τ0σ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Continuing recursively in this manner, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' exploiting the correlation functions ⟨τ0σb nσ2(n−1)⟩, we find: ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σb n⟩ = u 2 3(⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σσb n−1⟩ + ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σb n−1⟩⟨σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' ⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='22) Thus, we have determined the intricate nature of the extended boundary vertex operators σb n and how they recursively code the splitting of boundaries of open Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Tying up a loose end: fixing the constant a We tie up a loose end at the hand of another amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The amplitude illustrates a splitting of open Riemann surfaces involving two disk one-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We calculate the amplitude ⟨τ3τ0σσ⟩ in two manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We can apply recursion to the operator τ3, or to the operator τ0 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' In this calculation, we restore the possible constant a that we introduced in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2 and use an appropriately modified recursion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We demonstrate that the constant can be determined by consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Using the a-modified recursion relation, we find: ⟨τ3τ0σ2⟩ = 7 3⟨τ2σ2⟩ = 1 3⟨τ2σ2⟩ + 2 9⟨τ0σ⟩⟨τ1τ0σ⟩ + 4 3⟨τ0σ3σ⟩ + 3 + a 3 ⟨σ2τ0σ2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='23) This implies: ⟨τ2σ2⟩ = 1 9⟨τ0σ⟩⟨τ0σ⟩ + 4 3⟨σ2σ⟩ + 3 + a 3 2 3⟨σ3⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='24) 13Ghost number conservation applies to each factor separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 10 We can compute the latter correlator in another manner, using the puncture equation and the modified recursion relation: ⟨τ2σ2⟩ = 1 9⟨τ0σ⟩⟨τ0σ⟩ + 4 3⟨σ2σ⟩ + 2 + a 3 ⟨σ3⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='25) Using our previous results, we find full consistency if and only if a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Thus, we tied up the loose end in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 4 The Extended Partition Function In this section we introduce the generating function of extended open string correlators and prove that the recursion relations for the correlators imply Virasoro constraints on the gen- erating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' This allows us to make our results more rigorous by connecting to the mathematics literature on the integrable structure of the intersection theory on moduli spaces of Riemann surfaces with boundary [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We conclude the section with a few example ampli- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='1 The Generating Function We recall the generating functions of closed as well as open topological gravity correlation functions [11]: F c = � g≥0,n≥1,2g−2+n>0 u2g−2 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' � ki≥0 ⟨τk1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' τkn⟩c gtk1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' tkn F o,geom = � g′,k,l≥0,2g′−2+k+2l>0 � ai≥0 ug′−1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' ⟨τa1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' τalσk⟩o g′sk l� i=1 tai .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='1) In view of our enlarged space of boundary vertex operators, we also introduce a generating function for extended open topological gravity correlation functions: F o,ext = � g′,k,l≥0,2g′−2+k+2l>0 � ai,bi≥0 ug′−1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' ⟨τa1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' τalσb b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' σb bk⟩o,ext g � i tai � j sbj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2) The Extended Virasoro Constraints We define Virasoro generators Ln = � i≥0 2i + 1 2 ti∂ti+n − 3 2∂tn+1 + u2 12 n−1 � i=0 ∂ti∂tn−i−1 + 3 4 t2 0 u2δn,−1 + 1 16δn,0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='3) Lext n = Ln + � i≥0 (i + 1)si+1∂sn+i+1 + un + 1 2 ∂sn + 3 2 s1 u δn,−1 + 3 4δn,0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4) 11 for n ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' These are defined such that the recursion relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='11) on the closed as well as the recursion relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='12) on the extended open correlators leads to the constraints: Ln exp F c = 0 Lext n exp(F c + F o,ext) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='5) The extra constants terms in the closed Virasoro algebra (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='3) are due to the initialization cases ⟨τ 3 0 ⟩c = 1 = 24 ⟨τ1⟩ at genus zero and one respectively, while the initial conditions ⟨σ3⟩ = 1 = ⟨τ0σ⟩ on the disk lead to the extra constants in the extended Virasoro algebra (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='4), which satisfies14 [Lm, Ln] = (m − n)Lm+n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='6) At this stage, we are able to make contact with rigorous results – these constraints on an extended partition function of open topological correlators defined through an extended (or unconstrained) integrable KdV hierarchy were found to hold in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='15 The relation between the operators σb n and σ as well as the string coupling constant is neatly captured by a relation between derivatives of the extended partition function: ∂sn = (2u 3 )n−1∂n s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='7) This equation was proven from the KdV integrable hierarchy perspective in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Using this equation, and setting extended open times sn≥2 to zero, this relation between derivatives imply the higher order Virasoro constraints on the geometric open topological partition function, where the open Virasoro generators are [11]: Lo n = Ln + (2u 3 )n∂n+1 s1 + n + 1 2 u(2u 3 )n−1∂n s1 + δn,−1 3 2 s1 u + δn,0 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='8) The Virasoro constraints and the initialization condition are sufficient to determine the full partition function [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Through the generating function of extended correlators, we have connected our arguments with rigorous results on intersection theory on moduli spaces of Riemann surfaces with boundary [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2 A Few More Amplitudes For illustrative purposes, we calculate a few more amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' They render the integrable hierarchy structure, the Virasoro constraints and how to solve them more concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='1 Amplitudes on The Disk We have already indicated that on the disk only the third power of the elementary boundary vertex operator σ has a non-zero correlation function and equals one, ⟨σ3⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The disk bulk-boundary one-point function ⟨τ0σ⟩ is also one by a choice of normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Amplitudes 14These generators are rescaled by a factor of 2/3 compared to section 3 in order to reach a standard normalization for the Virasoro algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 15 The translation of variables and normalizations is: Lthere,ext n = (3/2)nLext n , tthere n = 3−n(2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='tn and sthere n−1 = (2/3)n−1n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 12 involving extended boundary vertex operators are computed through the reduction formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' A non-trivial example is: ⟨τ2σ5⟩ = 10 3 ⟨σb 2σ4⟩ = 40 3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='9) where we used the recursion relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='22) as well as the 6 choices of factoriza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' After taking into account the different normalization in footnote 15, this agrees with a more generic formula in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Another interesting correlation function is ⟨τ2τ0σσ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' It can be computed through the puncture equation (in the first line below) and/or the L1 constraint (in the second line below): ⟨τ2τ0σσσ⟩ = 5 3⟨τ1σσσ⟩ = 10 3 ⟨σσσ⟩ = 1 3⟨τ1σσσ⟩ + 2⟨τ0σ2σσ⟩ = 2 3⟨σσσ⟩ + 2 × 8 3⟨σσσ⟩ = 10 3 ⟨σσσ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='10) The two ways of computing are in agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='2 Higher Order Amplitudes Amplitudes that are higher order in the string coupling exhibit qualitatively new phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We illustrate a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We first compute amplitudes corresponding to cylinder diagrams, with two boundaries and genus zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' An interesting amplitude that involves a closed-open factorization due to a node can once again be computed in two manners: ⟨τ2τ0τ0σ⟩ = 5 3⟨τ1τ0σ⟩ = 5 3⟨τ0σ⟩ = 2 3⟨τ1τ0σ⟩ + 1 9⟨τ 3 0 ⟩c⟨τ0σ⟩ + 2 3⟨τ0τ0σ2⟩ = 2 3⟨τ0σ⟩ + 1 9⟨τ 3 0 ⟩c⟨τ0σ⟩ + 8 9⟨τ0σ⟩⟨τ0σ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='11) Both ways of computing the correlator lead to the same result, given the normalization of the closed three-point function ⟨τ 3 0 ⟩c as well as the bulk-boundary one-point function ⟨τ0σ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Finally, we compute an order O(u1) amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' It involves the one-loop closed one-point function ⟨τ1⟩c: ⟨τ3σ⟩ = 2 3⟨σ3⟩ + ⟨σ2σ⟩ + 1 9(1 + ⟨τ1⟩)⟨τ0σ⟩ =((2 3)3 + 2 3)⟨σ3⟩ + 1 9(1 + ⟨τ1⟩)⟨τ0σ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='12) Needless to say, many more results can be generated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' by computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' We provided a few telling illustrations that provide insight into the foundation of the integrable hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' 5 Conclusions In the spirit of the solution of the bulk theory [18] and building on earlier mathematical work [11, 12], we have solved two-dimensional topological gravity on Riemann surfaces with 13 boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' By making use of an extended set of boundary vertex operators, we rendered the representation of the contact algebra on the boundary linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Only in a second step the more complicated degeneration of surfaces with boundary is taken into account and the non-linear realization of the (half) Virasoro algebra is found [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The picture in which the solution of the theory is provided through contact interactions is a welcome intuitive complement to the geometric and matrix model approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' While we have provided a compelling global picture, there are many details that remain to be worked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' It would be good to find the geometric counterpart to the extended set of boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The link between (the expectation values of) the conformal field theory fields implicit in our analysis [18] and the sections of vector bundles of open topological gravity can be clarified (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' by exploiting references [15, 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The analysis of the contact terms in terms of an integration over a degeneration region of the moduli space of open Riemann surfaces would be interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' It will also be instructive to compare our analysis to the geometric derivation of the topological recursion relation through closed and open factorization [11], intuitively reviewed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Another research direction is to exploit the insights developed here and apply them to more general theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The generalization to the extended closed theory [23]) comes to mind, but mostly to open spin r curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' Geometric [24], integrable [25, 26], matrix model [27, 28] and conformal field theory insights [29] could be complemented by the perspective developed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' The study of these topological theories of gravity is worthwhile in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQfFPqV/content/2301.01008v1.pdf'} +page_content=' It occasion- ally fruitfully interfaces with recent developments.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..d22de1699e19ed8916bbe20c1e92341acd1c17e7 --- /dev/null +++ b/A9E0T4oBgHgl3EQfxwJo/content/tmp_files/2301.02650v1.pdf.txt @@ -0,0 +1,1694 @@ +Model-Agnostic Hierarchical Attention for 3D Object Detection +Manli Shu1* +Le Xue2 +Ning Yu2 +Roberto Martín-Martín2,3 +Juan Carlos Niebles2 +Caiming Xiong2 +Ran Xu2 +1 University of Maryland, 2 Salesforce Research, 3 UT Austin +Abstract +Transformers as versatile network architectures have re- +cently seen great success in 3D point cloud object detec- +tion. However, the lack of hierarchy in a plain transformer +makes it difficult to learn features at different scales and +restrains its ability to extract localized features. Such limita- +tion makes them have imbalanced performance on objects +of different sizes, with inferior performance on smaller ones. +In this work, we propose two novel attention mechanisms +as modularized hierarchical designs for transformer-based +3D detectors. To enable feature learning at different scales, +we propose Simple Multi-Scale Attention that builds multi- +scale tokens from a single-scale input feature. For localized +feature aggregation, we propose Size-Adaptive Local At- +tention with adaptive attention ranges for every bounding +box proposal. Both of our attention modules are model- +agnostic network layers that can be plugged into existing +point cloud transformers for end-to-end training. We evalu- +ate our method on two widely used indoor 3D point cloud +object detection benchmarks. By plugging our proposed mod- +ules into the state-of-the-art transformer-based 3D detector, +we improve the previous best results on both benchmarks, +with the largest improvement margin on small objects.1 +1. Introduction +3D point cloud data provides accurate geometric and spa- +tial information, which are important to computer vision ap- +plications such as autonomous driving and augmented reality. +Different from image data, which has a grid-like structure, +point clouds consist of unordered irregular points. Due to +such unique properties of point clouds, previous works have +proposed various deep network architectures for point cloud +understanding [7,8,22–25,34,37,48,50]. With the success +of transformers in natural language processing [4, 26, 40] +and 2D vision [5,14,39], attention-based architectures for +point clouds [20,38,46,49,52,53,55] are explored in recent +*Work done during an internship at Salesforce. manlis@umd.edu. +1The code and models will be available at https://github.com/ +salesforce/Hierarchical_Point_Attention. +Groundtruths +Predictions +Attention weights +(high to low) +Plain Attention +Our Attention +Groundtruth object +Figure 1. Visualization of the attention weights. With our hierar- +chical attentions, the object center has higher attention weights with +points that belong to the object, and the predicted bounding box +is better aligned with the groundtruth. Our multi-scale attention +extracts feature at different scales, which helps distinguish object +boundaries. Our size-adaptive local attention aggregates features at +the object level and helps refine the bounding box proposals. +works and have seen great success in 3D point cloud object +detection [16, 19, 32, 45]. Several properties of transform- +ers make them ideal for learning on raw point clouds. For +example, their permutation-invariant property is necessary +for modeling unordered sets like point clouds, and their at- +tention mechanism can model long-range relationships that +help capture the global context for point cloud learning. +Despite the advantages of transformers for point clouds, +we find the state-of-the-art transformer detector to have im- +balanced performance across different object sizes, with the +lowest average precision on small objects (see Section 4.3). +We speculate the inferior performance on small objects can +be due to two factors. Firstly, to make the computation feasi- +ble, transformer detectors use point cloud features consisting +of a small set of points compared to the original point cloud. +1 +arXiv:2301.02650v1 [cs.CV] 6 Jan 2023 + +The extensively downsampled point cloud loses geometric +details, which has a larger impact on small objects. Secondly, +plain transformers (e.g., Transformer [40], ViT [5]) extract +features at the global scale throughout the network, which +does not support explicit localized feature learning. +Motivated by the above observations, we expect existing +point cloud transformers to benefit from a hierarchical fea- +ture learning strategy, which allows multi-scale feature learn- +ing and supports localized feature aggregation. Nonetheless, +considering the computation intensity of point cloud trans- +formers, it is inefficient to use higher-resolution (i.e., higher +point density) point cloud features throughout the network. +Furthermore, due to the irregularity of point clouds, it is +non-trivial to integrate hierarchical design and multi-scale +features into transformers for point cloud object detection. +Our approach. +In this work, we aim to improve +transformer-based 3D object detectors with modularized +hierarchical designs. We propose two attention modules for +multi-scale feature learning and size-adaptive local feature +aggregation. Our attention modules are model-agnostic and +can be plugged into existing point cloud transformers for +end-to-end training. +We first propose Simple Multi-Scale Attention (MS-A). It +builds higher resolution point features from the single-scale +input feature with a learnable upsampling strategy and use +both features in the attention function. To reduce computa- +tion and parameter overhead, we transform the multi-scale +features into multi-scale tokens and perform multi-scale to- +ken aggregation [30] within a multi-head attention module. +The second module is Size-Adaptive Local Attention (Local- +A), which learns localized object-level features for each +object candidate. It assigns larger attention regions to ob- +ject candidates with larger bounding box proposals. The +local attention regions are defined by their corresponding +intermediate bounding box proposals. +We evaluate our method on two widely used indoor 3D +object detection benchmarks: ScanNetV2 [3] and SUN RGB- +D [36]. We plug our attention modules into the state-of- +the-art transformer-based 3D detector and perform end-to- +end training. Our method improves the previous best result +by over 1% in mAP@0.25 and over 2% in mAP@0.50 on +ScanNetV2. Furthermore, our size-aware evaluation shows +we have the most performance gain among small objects +with a 2.5% increase in mAPS. We summarize our main +contributions as follows: +• We propose Simple Multi-Scale Attention (MS-A) to en- +able multi-scale feature learning on single-scale features. +• We present Size-Adaptive Local Attention (Local-A) for +local feature aggregation within bounding box proposals. +• We conduct experiments on two widely used indoor 3D +detection benchmarks and surpass the previous best results +on both benchmarks. +2. Related Work +Network architectures for point cloud learning. +Exist- +ing network architectures for point cloud learning can be +roughly divided into two categories based on their point +cloud representation: grid-based and point-based, yet in +between, there also exist hybrid architectures that operate on +both representations [9,33,51,53,57]. Grid-based methods +project the irregular point clouds into grid-like structures, +such as 3D voxels. With the grid-like structure, existing +works have proposed a variety of 3D-convolution-based ar- +chitectures [7,18,31,42]. Point-based methods, on the other +hand, directly learn features from the raw point cloud. Within +this category, graph-based method [8, 35, 44, 47, 56] use a +graph to model the relationships among the points. Another +line of work models a point cloud as a set of points, and +extracts features through set abstraction [17,23,25,41]. Re- +cent works explore transformer architecture for point-based +learning [16,19,20,53,55], where each point is fed into the +transformer as a token and the attention mechanism learns +point features at a global scale. While previous methods +improve point cloud learning by developing new backbones +and modifying the overall network architecture, our work +focuses on the attention mechanism of the point cloud trans- +former. Instead of proposing new architectures for point +cloud learning, we aim to provide a model-agnostic solution. +Point cloud object detection. +One major challenge in +point cloud object detection is extracting object features. +In 2D object detection, a common practice for extracting ob- +ject features is to use a region proposal network (RPN) [29] +to generate dense bounding box proposals (i.e., object can- +didate) in a top-down manner and then extract features for +each object candidate. However, in 3D vision, generating +dense 3D bounding box proposals for point cloud data is +inefficient due to the irregularity and sparsity of point clouds. +Previous work [2,57] addresses this issue by projecting point +clouds into 2D bird’s-eye views or voxels and then applying +RPN. However, such projection operations can result in the +loss of geometric information or introduce quantization er- +rors. Another line of work seeks to generate 3D proposals +in a bottom-up manner (i.e., point-based) [16, 19, 21, 34]. +VoteNet [21] samples a set of points from a point cloud +as the initial object candidates and then assigns points to +each object candidate through voting. Object features of +each candidate are learned by aggregating features within +its corresponding vote cluster (i.e., group). Instead of voting +and grouping, follow-up works [16, 19] propose to use a +transformer to automatically model the relationship between +the object candidates and the point cloud. Although point- +based methods do not have quantization errors caused by +voxelization, to make the computation feasible, a point cloud +needs to be extensively downsampled at the beginning of the +2 + +model. Such downsampling also causes a loss of geomet- +ric information, while it is important for object detection to +have fine-grained features to make accurate predictions. Our +work is based on point-based transformer detectors. We ad- +dress the downsampling issue by building higher-resolution +features without increasing the computation budget. +Hierarchical designs for 2D and 3D vision transformers. +Extensive work has been done to adapt transformers for +vision recognition. One direction is to borrow the hierar- +chical design and inductive biases from convolutional neu- +ral networks (ConvNet) [10]. In the 2D vision, one line +of ConvNet-based hierarchical design [6,11,43] produces +multi-scale feature maps for 2D images by progressively +decreasing the resolution and expanding feature channels. +Swin Transformer [15] adopts the idea of weight-sharing of +ConvNet and proposes efficient self-attention with shifted +windows. Shunted self-attention [30] attends to features at +different scales through multi-scale token aggregation. In the +3D vision, hierarchical designs for point cloud transformers +are explored in previous works, where self-attentions are ap- +plied to local regions (specified by k nearest neighbors [55] +or a given radius [20]), and downsampling operations are +performed after every encoding stage following the hierar- +chical design of PointNet++ [25]. Patchformer [53] proposes +a multi-scale attention block that performs extracts features +at multiple granularities, but it requires voxelization on the +point cloud. Different from previous works, we pack our +hierarchical design into model-agnostic attention modules +that can be plugged into any existing architecture and enable +both multi-scale and localized feature learning. +3. Method +In this section, we first discuss the background, including +a brief introduction to the task of point cloud object detection, +an overview of point-based 3D detection methods, and the +attention mechanism. Next, we dive into the detailed designs +of our proposed attention modules. +3.1. Background +Point cloud object detection. +Given a point cloud Praw +with a set of P points P = {pi}P +i=1, each point pi ∈ R3 +is represented by its 3-dimensional coordinate. 3D object +detection on point cloud aims to predict a set of bounding +boxes for the objects in the scene, including their locations +(as the center of the bounding box), size and orientation +of the bounding box, and the semantic class of the corre- +sponding object. Note that due to the computation limit, the +point cloud is downsampled at the early stage of a model +to a subset of Praw, which contains N (N << P) points. +P = SA(Praw) = {pi}N +i=1 contains the aggregated groups +of points around N group centers, where SA (set abstraction) +is the aggregation function, and the group centers are sam- +pled from the raw point cloud using Furthest Point Sample +(FPS) [23], a random sampling algorithm that provides good +coverage of the entire point cloud. +Point-based 3D object detectors. +Our method is built on +point-based 3D object detectors [16,21,34], which detect 3D +objects in point clouds in a bottom-up manner. Compared to +other 3D detectors that generate box proposals in a top-down +manner on the bird’s-eye view or voxelized point clouds [2], +point-based methods work directly on the irregular point +cloud and do not cause loss of information or quantization +errors. In addition, point-based methods are suitable for +more efficient single-stage object detection [1,13,28]. +The feature representation of the input point cloud +{zi}N +i=1, zi ∈ Rd is first obtained using a backbone model +(e.g., PointNet++ [25]), where d is the feature dimen- +sion. Point-based detectors generate bounding box predic- +tions starting with M (M < N) initial object candidates +{qi}M +i=1, qi ∈ RC, sampled from the point cloud as object +centers. A common approach for sampling the candidates is +Furthest Point Sample (FPS). Once get the initial candidates, +the detector then extracts features for every object candidate. +Attention-based methods [16] learn features by doing self- +attention among the object candidates, and cross-attention be- +tween the candidates (i.e., query) and point features {zi}N +i=1. +The learned features of the object candidates will then be +passed to prediction heads, which predict the attributes of +the bounding box for each object candidate. The attributes of +a 3D bounding box include its location (box center) ˆc ∈ R3, +size (H/W/D dimensions) ˆd ∈ R3, orientation (heading an- +gles) ˆa ∈ R, and the semantic label of the object ˆs. With +these parameterizations, we can represent a bounding box +proposal as ˆb = {ˆc, ˆd, ˆa,ˆs}. The detailed parameterizations +of a bounding box are included in Appendix A.2. +Attention mechanism +is the basic building block of trans- +formers. The attention function takes in query (Q), key (K), +and value (V ) as the input. The output of the attention func- +tion is a weighted sum of the value with the attention weight +being the scaled dot-product between the key and query: +Attn(Q, K, V ) = softmax(QKT +√dh +)V, +(1) +where dh is the hidden dimension of the attention layer. +For self-attention, Q ∈ Rdh, K ∈ Rdh and V ∈ Rdv are +transformed from the input X ∈ Rd via linear projection +with parameter matrix W Q +i +∈ Rd×dh, W K +i +∈ Rd×dh, and +W V +i +∈ Rd×dv respectively. For cross-attention, Q, K, and +V can have different sources. +In practice, transformers adopt the multi-head attention +design, where multiple attention functions are applied in +3 + +… +Object Features +Point Features +Upsampled Point Features +… +… +Q +K1×, V1× +K2×, V2× +Learnable +Upsample +Concat & Linear +Attention head=0 +Attention head=h/2 +Object Features +Prediction +Head +Bounding Box +Proposals +Point Features +Sampled Point Features +{Q} (batch) +{K, V} (batch) +(a) +Simple Multi-Scale Attention +(b) +Size-Adaptive Local Attention +Multi-Head +Attention +Pad & +Truncate +Figure 2. An illustration of our hierarchical attention modules. (a). Simple Multi-Scale Attention (MS-A) learns features at different +scales within the multi-head cross-attention module. It constructs high resolution (i.e., point density) point features from the single-scale +input point features and uses keys and values of both scales. (b). Size-Adaptive Local Attention (Local-A) extracts localized features for +each object candidate by restricting the attention range to be inside its bounding box proposal. The attention range (the token lengths of key +and value) is adaptive for each object candidate (query) and we perform padding or truncating to allow batch processing +. +parallel across different attention heads. The input of each +attention head is a segment of the layer’s input. Specifically, +the query, key, and value are split along the hidden dimension +into (Qi, Ki, Vi)h +i=1, with Qi ∈ Rdh/h, Ki ∈ Rdh/h, Vi ∈ +Rdv/h, where h is the number of attention heads. The final +output of the multi-head attention layer is the projection of +the concatenated outputs of all attention heads: +MultiHead(Q,K, V ) = Concat({Attn(Q0, K0, V0); +...; Attn(Qh−1, Kh−1, Vh−1)})W O, +(2) +where the first term denotes the concatenation of the output +and W O is the output projection matrix. +3.2. Simple Multi-Scale Attention +When applying transformers to point-based 3D object +detection, the cross-attention models the relationship be- +tween object candidates and all other points within the point +cloud. The intuition is that, for each object candidate, every +point within the point cloud (i.e., scene) either belongs to +the object or can provide context information for the object. +Therefore, it makes sense to gather all point features for +every object candidate, and the importance of a point to the +object candidate can be determined by the attention weight. +However, due to the computation overhead of the atten- +tion function, the actual number of points (i.e., tokens) that a +model is learned on is set as 1024 [16,19], whereas the raw +point cloud usually contains tens of thousands points [3,36]. +Such extensive downsampling on the point cloud causes a +loss of detailed geometric information and fine-grained fea- +tures, which are important for dense prediction tasks like +object detection. +To this end, we propose Simple Multi-Scale Attention +(MS-A), which builds higher-resolution (i.e., higher point +density) feature maps from the single-scale feature input. It +then uses features of both scales as the key and value in the +cross-attention between object candidates and other points. +the multi-scale feature aggregation is realized through multi- +scale token aggregation, where we use the key and value of +different scales in different subsets of attention heads. Our +goal is to create a higher-resolution feature map that provides +fine-grained geometric details of the point cloud. +The first step of our multi-scale attention is to obtain a +higher-resolution feature map from the single-scale input. +We propose a learnable upsampling operation. Given the +layer’s input point cloud feature {zi}N +i=1, zi ∈ Rd, we want +to create a feature map with 2N points. To get the locations +(i.e., coordinates) of the 2N points, we use FPS to sample +4 + +2N points from the raw point cloud {pi}2N +i=1, pi ∈ R3. Next, +for each sampled point pi, we search for the top three of +its nearest neighbors (in the euclidean distance) in the input +feature map {zi}N +i=1, denoted as {z0 +i , z1 +i , z2 +i }. Then we cal- +culate a weighted interpolation of the three-point features, +weighted by the inverse of their distance to the sample point. +The interpolated feature is then projected into the feature rep- +resentation of sampled point. The upsampled point feature +map can be written as: +{˜zi}2N +i=1, ˜zi = Φθ(interpolate({z0 +i , z1 +i , z2 +i })) +(3) +Here, Φθ is learnable projection function parameterized by +θ. We choose MLP as our projection function. +After the upsampling, we have two sets of point features +of different scale {zi}N +i=1, {˜zi}2N +i=1. To avoid computation +increase, we perform multi-head cross-attention on both sets +of point features in a single pass by using features of different +scales on different attention heads. We divide attention heads +evenly into two groups, and use zi}N +i=1 to obtain K and V +in the first group while using the other for the second group. +Both groups share the same set of queries transformed from +{qi}M +i=1. Since the input and output of this module are the +same as a plain attention module, we can plug MS-A into any +attention-based model to enable feature learning at different +scales. In practice, we apply MS-A only at the first layer +of a transformer which makes minimal modifications to the +network and introduces little computation overhead. +3.3. Size-Adaptive Local Attention +Although the attention mechanism can model the relation- +ship between every point pair, it is not guaranteed the learned +model will pay more attention to points that are important +to an object (e.g., those belonging to the object) than the +ones that are not. The lack of hierarchy in transformers, on +the other hand, does not support explicit localized feature +extraction. Different from existing local attentions that are +performed within a fixed region, we propose Size-Adaptive +Local Attention (Local-A) that defines local regions based +on the size of bounding box proposals. +We first generate intermediate bounding box proposals +{ˆbi}M +i=1 with the features of object candidates ({qi}M +i=1). +We then perform cross-attention between every candidate qi +and the points sampled from within its corresponding box +proposal ˆbi. Therefore, we have customized size-adaptive +local regions for every query point. For every input object +candidate qil ∈ Rd, it is updated Local-A as: +qi +l+1 = Attn(Ql +i, Ki, Vi), where +(4) +Ql +i = qi +lW Q, Ki = ZiW K, Vi = ZiW V with +(5) +Zi = {zk +i | pos(zk +i) in ˆbi}, ˆbi = Predl +box(qi +l). +(6) +In the Eq.( 6), we use pos(·) to denote the coordinate of a +point in the 3D space, and Zi is a set of points inside box +ˆbi. Note that the point features {zi}N +i=1 are extracted by the +backbone network and are not updated during the feature +learning of object candidates. Predl +box is the prediction head +at layer l that generate intermediate box predictions. +Since object candidates (i.e., query) will have different +sets of keys and values depending on the size of their bound- +ing box proposals, the number of K and V tokens also +differs for each object candidate. To allow batch computa- +tion, we set a maximum number of points (Nlocal) for the +sampling process and use Nlocal as a fixed token length for +every query point. For bounding boxes that contain less than +Nlocal points, we pad the point sequence with an unused +token to Nlocal and mask the unused tokens out in the cross- +attention function; for those containing more than Nlocal +points, we randomly discard them and truncate the sequence +to have Nlocal points as keys and values. Lastly, in the case +where the bounding box is empty, we perform ball query [23] +around the object candidate to sample Nlocal points. +Same as MS-A, Local-A does not pose additional require- +ments on modules input, therefore we can apply it at any +layer of a transformer. Specifically, we apply Local-A at the +end of a transformer where bounding box proposals are in +general more accurate. +4. Experiments +In this section, we first evaluate our method on two widely +used indoor point cloud detection datasets, ScanNetV2 and +SUN RGB-D. Next, we provide qualitative and quantita- +tive analyses of our method, including visualizations of the +bounding box predictions and attention weights, and eval- +uations using our proposed size-aware metrics. Lastly, we +include ablation studies on the design choices of our atten- +tion modules. We include more experiments and ablation +studies in Appendix A.1, including analyses on the infer- +ence speed and the number of parameters of each individual +attention module. +4.1. Main Results +Datasets. +ScanNetV2 [3] consists of 1513 reconstructed +meshes of hundreds of indoor scenes. It contains rich anno- +tations for various 3D scene understanding tasks, including +object classification, semantic segmentation, and object de- +tection. For point cloud object detection, it provides axis- +aligned bounding boxes with 18 object categories. We follow +the official dataset split by using 1201 samples for training +and 312 samples for testing. SUN RGB-D [36] is a single- +view RGB-D dataset with 10335 samples. For 3D object +detection, it provides oriented bounding box annotations +with 37 object categories, while we follow the standard eval- +uation protocol [21] and only use the 10 common categories. +The training split contains 5285 samples and the testing set +contains 5050 samples. +5 + +Methods +#Params +Backbone +ScanNet V2 +mAP@0.25 +mAP@0.50 +VoteNet [21] +- +PointNet++ +62.9 +39.9 +H3DNet [54] +- +PointNet++ +64.4 +43.4 +H3DNet [54] +- +4×PointNet++ +67.2 +48.1 +3DETR [19] +- +transformer +65.0 +47.0 +Pointformer [20] +- +transformer +64.1 +42.6 +Group-Free6,256 [16] +13.0M +PointNet++ +67.3 (66.3) +48.9 (48.5) +w/ MS + Local (Ours) +15.0M +PointNet++ +67.9 (67.1) (↑ 0.6) +51.4 (49.8) (↑ 2.5) +RepSurf-U6,256 [27] +13.1M +PointNet++ +68.8 ( - ) +50.5 ( - ) +RepSurf-U6,256 (reproduce) +13.1M +PointNet++ +68.0 (67.4) +50.2 (48.7) +w/ MS + Local (Ours) +15.1M +PointNet++ +69.5 (68.8) (↑ 1.5) +52.5 (51.1) (↑ 2.3) +Group-Free12,512 [16] +26.9M +PointNet++w2x +69.1 (68.6) +52.8 (51.8) +w/ MS + Local (Ours) +28.9M +PointNet++w2x +70.3 (69.2) (↑ 1.2) +54.6 (53.2) (↑ 1.8) +RepSurf-U12,512 [27] +27.1M +PointNet++w2x +71.2 ( - ) +54.8 ( - ) +RepSurf-U12,512 (reproduce) +27.1M +PointNet++w2x +70.8 (70.2) +54.4 (53.6) +w/ MS + Local (Ours) +29.1M +PointNet++w2x +71.7 (71.0) (↑ 0.9) +56.5 (54.8) (↑ 2.1) +Table 1. Performance of object detection on ScanNetV2. We follow the standard protocol [21] by reporting the best results over 5 × 5 +trials (5 trainings, each with 5 testings) and including the averaged results in the bracket. Group-FreeL,O denotes the variant with L decoder +layers and O object candidates. The same notation applies to RepSurf-U. The detection code of RepSurf is not published, so we implement +our version of RepSurf-U and apply our method to it. We include the results of our implementation of RepSurf-U. +Methods +mAP@0.25 +mAP@0.50 +VoteNet [21] +59.1 +35.8 +H3DNet [54] +- +- +H3DNet [54] +60.1 +39.0 +3DETR [19] +59.1 +32.7 +Pointformer [20] +61.1 +36.6 +Group-Free6,256 [16] +63.0 (62.6) +45.2 (44.4) +w/ MS + Local (Ours) +63.8 (63.2) (↑ 0.8) +46.6 (45.7) (↑ 1.4) +RepSurf-U6,256 [27] +64.3 ( - ) +45.9 ( - ) +RepSurf-U6,256 (repd.) +64.0 (63.3) +45.7 (45.2) +w/ MS + Local (Ours) +64.5 (63.8) (↑ 0.5) +47.5 (46.1) (↑ 1.8) +Table 2. +Performance of object detection on SUN RGB-D. +“repd." stands for the reproduced results of our implementation. +“-" means the official result is not available. +Evaluation metrics. +For both datasets, we follow the stan- +dard evaluation protocol [21] and use the mean Average Pre- +cision (mAP) as the evaluation metric. We report mAP scores +under two different Intersection over Union (IoU) thresholds: +mAP@0.25 and mAP@0.5. In addition, in Section 4.3, to +evaluate model performance across different object sizes, we +follow the practice in 2D vision [12] and implement our own +size-aware metrics that measure the mAP on small, medium, +and large objects respectively. On account of the randomness +of point cloud training and inference, we train a model 5 +times and test each model 5 times. We report both the best +and the average results among the 25 trials. +Baselines. +We validate our method by applying it to ex- +isting transformer point cloud detectors. Group-Free [16] +extracts features for object candidates using a transformer +decoder with plain attention. We include two configura- +tions of Group-Free in our comparison: Group-Free6,256 +samples a total of 256 object candidates for feature learning +and bounding box prediction, using a transformer decoder +with 6 layers; Group-Free12,512 is the largest configuration, +which has 12 transformer layers and 512 object candidates. +RepSurf-U [27] proposes a novel multi-surface (umbrella +curvature) representation of point clouds that can explicitly +describe the local geometry. For object detection, RepSurf-U +adopts the transformer decoder of Group-Free and replaces +its backbone with one that extracts features on both point +clouds and the surface representations. The official imple- +mentation and the averaged results of RepSurf-U for object +detection are not publicly available, so we include the results +of our own implementation of RepSurf-U. +We also include the performance of previous point-based +3D detectors for comparison. VoteNet [21] aggregates fea- +tures for object candidates through end-to-end optimizable +Hough Voting. H3DNet [54] proposes a hybrid set of ge- +ometric primitives for object detection and trains multiple +individual backbones for each primitive. 3DETR [19] solves +point cloud object detection as a set-to-set problem using +a transformer encoder-decoder network. Pointformer [20] +proposes a hierarchical transformer-based point cloud back- +bone and adopts the voting algorithm of VoteNet for object +detection. +Implementation details. +For a baseline model with L +transformer layers, we enable multi-scale feature learning +by replacing the cross-attention of the 1-st layer with MS-A. +After the L-th layer, we append an additional transformer +6 + +layer to perform local feature aggregation, which consists +of Local-A and a feedforward layer. We follow the original +training settings of the baseline models [16,27]. The detailed +hyperparameter settings can be found in Appendix A.2. +Results. +From Table 1, on ScanNetV2, we observe consis- +tent improvements in point cloud transformer detectors when +equipped with our attention modules. By applying MS-A +and Local-A to Group-Free, we achieve on-par performance +with the state-of-the-art RepSurf-U detector. In addition, we +can further improve RepSurf-U by over 1% in mAP@0.25 +and over 2% in mAP@0.50 on varying model configurations. +Table 2 shows a similar trend on SUN RGB-D, where our +attention modules boost the mAP@0.50 of group-Free to sur- +pass RepSurf-U, and can further improve the state-of-the-art +method by 0.5% in mAP@0.25 and 1.8% in mAP@0.50. +4.2. Qualitative Results +In Figure 3, we provide qualitative results on both datasets. +The visualized results are of our methods applied to the +Group-Free detectors. The qualitative results suggest that +our model is able to detect and classify objects of different +scales even in complex scenarios containing more than 10 +objects (e.g., the example in the bottom row). By looking +into cross-attention weights in the transformer detector, we +find that object candidates tend to have higher correlations +with points that belong to their corresponding objects. +4.3. Performance on objects of different sizes. +In addition to the standard evaluation metrics, we are in- +terested in examining models’ performance across different +object sizes. Inspired by the size-aware metrics in 2D de- +tection [12], we implement our own version of size-aware +metrics for 3D detection. We conduct this analysis on Scan- +NetV2, on which we calculate the volume for all the objects +in all samples. We set the threshold for mAPS as the 30th +percentile of the volume of all objects, and use the 70th +percentile as the threshold for mAPL. More details about +these metrics are included in Appendix A.2. +MS-A +Local-A +mAPS +mAPM +mAPL +- +- +63.1 +76.6 +83.2 + +- +65.0 +77.5 +83.9 +- + +65.2 +78.6 +83.9 + + +65.6 (↑ 2.5) +79.0 (↑ 2.4) +84.3 (↑ 1.1) +Table 3. +Performance on different size categories on Scan- +NetV2. We define the S/M/L thresholds based on the statistics +(volume distribution) of ScanNetV2 objects. The configuration in +the first row denotes the Group-Free12,512 baseline. +In Table 3, we evaluate our methods using size-aware +metrics. We report the average result over 25 trials. The first +row denotes the Group-Free12,512 baseline. Firstly, by com- +paring the mAPS to mAPL, we notice that it has imbalanced +performance across different object sizes. Looking at the +improvement margins, we find our method to have the most +performance gain on small and medium-sized objects. The +result suggests that hierarchical designs can aid fine-grained +and localized feature learning for point cloud transformer +detectors and helps models detect smaller objects. +4.4. Ablation Study +In this subsection, we first conduct an ablation study on +the stand-alone effects of our multi-scale attention and size- +adaptive local attention. Next, we include empirical analyses +of the design choices of our attention modules. If not other- +wise specified, experiments in this subsection are conducted +on ScanNetV2 with the Group-Free12,512 baseline. With- +out loss of generality, the results in this subsection are the +averaged numbers over 25 trials. +The stand-alone effects of MS-A and Local-A. +Table 4 +shows the stand-alone performance of our proposed attention +modules. Compared to the plain attention baseline, both of +our attentions are proved to be effective. When combined +together, we find the two modules to be complementary to +each other and bring more significant performance gain. +MS-A +Local-A +mAP@0.25 +mAP@0.50 +- +- +68.6 +51.8 + +- +68.9 +52.5 +- + +68.9 +52.9 + + +69.2 +53.2 +Table 4. The stand-alone effect of our attention modules. The +configuration in the first row denotes the Group-Free12,512 baseline. +The results are averaged over 25 trials. +The maximum number of points (Nlocal) in Local-A. +In Local-A, for each object candidate (i.e., query), we sam- +ple a set of points within its corresponding bounding box +proposal and use the point features as the key and value +for this object candidate in the cross-attention function. As +introduced in Section 3.3, we cap the number of sampled +points with Nlocal to allow batch computation. +We provide an empirical analysis of the effects of Nlocal +on Local-A. From Table 5, we find that too little number +of points (e.g., Nlocal = 8) for Local-A results in a per- +formance drop. On the other hand, as Nlocal continues to +increase, we do not observe a significant performance gain +compared to Nlocal = 16. Intuitively, a small Nlocal means +the points within each bounding box are sampled sparsely, +which can be too sparse to provide enough information about +7 + +Scene +Groundtruth +Prediction +Attention +Figure 3. Qualitative results on SUN RGB-D (top) and ScanNetV2 (bottom). The color of a bounding box in the middle two columns +stands for the semantic label of the object. In the last column, we draw both the groundtruth (in green) and the prediction (in blue) of the +object. We highlight the points that belong to an object for better visualization. In the last column, we visualize the attention weight of the +last transformer layer (before applying Local-A). We visualize the cross-attention weight between an object candidate and the point cloud. +Nlocal +mAP@0.25 +mAP@0.50 +mAPS +mAPM +mAPL +8 +67.8 +51.1 +64.6 +78.0 +82.8 +16 +68.9 +52.9 +65.2 +78.6 +83.9 +24 +68.9 +53.0 +65.4 +78.5 +84.0 +32 +68.3 +52.1 +64.7 +77.8 +84.3 +Table 5. The effect of Nlocal in Local-A. When there are enough +points, a larger Nlocal means the points are sampled more densely +within each bounding box proposal. +any object. This explains why Nlocal = 8 does not work +well. However, on the other hand, a large Nlocal may only +benefit large objects and has little effect on smaller objects, +because the latter are padded with unused tokens. +MS-A with different feature resolutions. +In Section 3, +we propose learnable upsampling for MS-A to build higher- +resolution point features from the single-scale input. In the +same spirit, a parameterized downsampling procedure can +be realized through conventional set abstraction [23], which +aggregated point features within local groups and produce +a feature map with fewer points (i.e., lower resolution). In- +tuitively, a higher point density of the feature map provides +more fine-grained features. To study the effects of feature +maps of different granularity, we conduct an empirical analy- +sis on MS-A using different sets of multi-scale feature maps +representing point clouds of varying granularity. +In Table 6, we examined the performance of two multi- +scale choices in comparison with the single-scale baseline. +The result suggests that coarse features (s = 0.5×) do not +benefit transformer detectors. This is expected because trans- +formers do not have limited receptive fields and thus do not +Feature Scales s +mAP@0.25 +mAP@0.50 +[1×] +68.6 +51.8 +[1×, 2×] +68.9 +52.5 +[0.5×, 1×, 2×] +67.9 +51.7 +Table 6. Simple Multi-Scale Attention with different feature +scales. Feature scale = s× means the feature map contains s × N +points, with N being the original number of points. A larger s +denotes a feature map with higher point density (i.e., resolution) +rely on a coarse-grained feature map to learn global context. +5. Conclusion +In this work, we present Simple Multi-Scale Attention and +Size-Adaptive Local Attention, two model-agnostic modules +that bring in hierarchical designs to existing transformer- +based 3D detectors. We enable multi-scale feature learning +and explicit localized feature aggregation through improved +attention functions, which are generic modules that can be +applied to any existing attention-based network for end-to- +end training. We improve the state-of-the-art transformer +detector on two challenging indoor 3D detection benchmarks, +with the largest improvement margin on small objects. +As our attention modules promote fine-grained feature +learning, which is important to various dense prediction +vision tasks, one direction for future work is to adapt our +attention modules for other point cloud learning problems +such as segmentation. Another direction is to introduce more +efficient attention mechanisms to the multi-scale attention to +further bring down the computation overhead. +8 + +References +[1] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas +Usunier, Alexander Kirillov, and Sergey Zagoruyko. End- +to-end object detection with transformers. In ECCV, 2020. +3 +[2] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia. +Multi-view 3d object detection network for autonomous driv- +ing. In CVPR, 2017. 2, 3 +[3] Angela Dai, Angel X. Chang, Manolis Savva, Maciej Hal- +ber, Thomas A. 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In ICCV, 2021. 2 +[57] Yin Zhou and Oncel Tuzel. Voxelnet: End-to-end learning +for point cloud based 3d object detection. In CVPR, 2018. 2 +A. Appendix +A.1. More Experiments +The placement of Simple Multi-scale Attention. +We de- +sign simple multi-scale attention as a compact network layer +to enable hierarchical feature learning. As it can be inserted +at any place within a network, we are interested in finding out +how the placement of the multi-scale attention layer affects +a model’s performance. +Layers +mAP@0.25 +mAP@0.25 +[0] +68.9 +52.5 +[0, 4, 8] +68.8 +52.4 +[0, 3, 6, 9] +68.9 +52.3 +[0, 2, 4, 6, 8, 10] +68.7 +52.6 +Table 7. Different placements of the simple multi-scale atten- +tion layer. Layers = [i] means we replace the ith layer of the +transformer decoder with our MS-A layer. The best results are in +bold, and the second-best results are underlined. +We consider different strategies to place MS-A within +the transformer decoder of Group-Free. We divide the 12- +layer decoder into several stages and place MS-A at the +first layer of each stage. Specifically, our default setting +uses a single MS-A at the first decoder layer, and we try +placing MS-A by dividing the decoder evenly into 3, 4, and +6 stages. From the results in Table 7, we do not observe a +significant benefit of using more than one multi-scale layer. +We conjecture this is because the up-scaled feature map in +our multi-scale attention is obtained through interpolation +10 + +and simple linear projection. The up-scale point feature +obtained in this way may mainly provide more accurate +geometric information with a higher point density, while may +not have much semantic difference than the original input +feature. We expect such fine-grained geometric information +to be particularly helpful at the beginning of the decoding +stage (i.e., Layer= 0), yet may be less useful as the object +features go deeper in the decoder and become more abstract. +Per-category mAP on ScanNetV2 and SUN RGB-D. +We include the detailed per-category mAP on both datasets +in Table 9, Table 10, Table 11, and Table 12. For the results +in this paragraph, we follow the baselines [16,19,20,27] and +report the result of the best trial. +Inference Speed. +We analyze the parameter and computa- +tion overhead of each of our attention modules. We measure +the inference speeds for all model configurations on the same +machine with a single A100 GPU. In Table 8, we can see that +replacing plain attention with MS-A results in little parame- +ter increase. While applying Local-A leads to a larger param- +eter increase, the Local-A module itself contains the same +number of parameters as a plain cross-attention. The param- +eter increase is mainly due to the additional feed-forward +layer and learnable positional embeddings, etc. In terms of +inference speed, we find MS-A to cause more substantial +latency in inference. Such latency is caused by applying +the attention function on the key/value with 2 times more +tokens (from 1024 to 2048). A future direction is to incorpo- +rate more efficient attention mechanisms into the multi-scale +attention function. +MS-A +Local-A +#Params +Inference Speed +(M) +(ms/frame) +- +- +26.9 +186 + +- +27.0 (+0.1) +225 +- + +28.8 (+1.9) +191 + + +28.9 (+2.0) +232 +Table 8. Ablating the parameter and computation overhead of +individual attention modules. +A.2. Implementation Details. +We include implementation details covering several as- +pects in this paragraph. In addition, we include our source +code in the supplementary material, containing the full im- +plementation of our attention modules. +Training Details. +Group-Free baseline. When applying our method to this +baseline, we follow the original training settings. Specifi- +cally, on ScanNetV2, the models are trained for 400 epochs +on 4 GPUs with a batchsize of 32 (8 on each GPU). We use +the same optimizer with the same learning rates and weight +decays as the baseline training. On SUN RGB-D, models +are trained for 600 epochs on 4 GPUs with the same learning +rate and weight decay as the baseline training on this dataset. +RepSurf-U baseline. The official implementation and +training details of this baseline are not published. We im- +plement our own version of RepSurf-U detector, for which +we mostly follow the training setup of Group-Free and have +done a grid search for the hyperparameters. Different from +Group-Free, we train RepSurf-U models on ScanNetV2 and +SUN RGB-D using a weight decay of 0.01 for all model +parameters, because we find it to achieve better performance +on our reproduced RepSurf-U. The learning rate and other +hyperparameters remain the same as Group-Free on both +datasets. When applying our method to the reproduced +model, we do not change the hyperparameter configurations. +Bounding box parameterization. +In this paragraph, we +include a brief introduction to the bounding box parameteri- +zation used in our baselines. First, the predicted box center +ˆc for each object candidate q is obtained by adding an offset +to the coordinate of q. In this way, by predicting the cen- +ter, the actual prediction made by a detector is this offset +value. The size ˆd of a box is the height, width, and depth +dimension. One way for predicting ˆd is to directly predict +the values of H, W, and D. Another way is to divide a range +of sizes into several bins and make a classification prediction +that determines which “bin" the object belongs to. The final +size prediction is obtained by adding the quantized size (i.e., +the bin) with a “residual" term which is also predicted by +the model with another prediction head. The bounding box +orientation ˆa is also parameterized as the combination of a +quantized value and a residual term. Lastly, the prediction of +the semantic label is a common classification problem that +parameterizes a semantic label as a one-hot vector. +Size-Aware Evaluation Metrics. +For a quantitative anal- +ysis of the model’s performance on objects of different +sizes. We implement our own size-aware evaluation met- +rics, namely mAPS, mAPM and mAPL. For each metric, +we only calculate the mAP score among objects that fall +into the corresponding size category (i.e., small, medium, or +large). We conduct the size-aware evaluation on ScanNetV2, +where we determine the threshold for dividing object size +categories based on the statistics of this dataset. Specifically, +we take the 1201 training samples and record the volume +(v = H × W × D) of every groundtruth bounding box +of every sample (see Figure 4). Among a total of 15733 +goundtruth bounding boxes, we take the 30th (v = 0.155) +and 70th (v = 0.526) percentile as the thresholds for divid- +ing small and large objects. +11 + +00.155 0.526 +1 +2 +3 +4 +5 +Volume of the object bounding box +0 +200 +400 +600 +800 +1000 +Number of objects +Figure 4. Volume distribution of the object groundtruth bounding boxes in ScanNetV2. We highlight the threshold of small objects +(v <= 0.155, the 30th percentile) and large objects (v > 0.526, the 70th percentile) +methods +backbone +cab +bed chair sofa tabl door wind bkshf +pic +cntr desk curt fridg showr toil +sink bath ofurn mAP +VoteNet [21] +PointNet++ +47.7 88.7 89.5 89.3 62.1 54.1 40.8 +54.3 +12.0 63.9 69.4 52.0 52.5 +73.3 +95.9 52.0 92.5 42.4 +62.9 +H3DNet [54] +4×PointNet++ +49.4 88.6 91.8 90.2 64.9 61.0 51.9 +54.9 +18.6 62.0 75.9 57.3 57.2 +75.3 +97.9 67.4 92.5 53.6 +67.2 +3DETR [19] +transformer +49.4 83.6 90.9 89.8 67.6 52.4 39.6 +56.4 +15.2 55.9 79.2 58.3 57.6 +67.6 +97.2 70.6 92.2 53.0 +65.0 +Pointformer [20] +Pointformer +46.7 88.4 90.5 88.7 65.7 55.0 47.7 +55.8 +18.0 63.8 69.1 55.4 48.5 +66.2 +98.9 61.5 86.7 47.4 +64.1 +GroupFree6,256 +PointNet++ +54.1 86.2 92.0 84.8 67.8 55.8 46.9 +48.5 +15.0 59.4 80.4 64.2 57.2 +76.3 +97.6 76.8 92.5 55.0 +67.3 +w/ MS + Local +PointNet++ +55.9 88.6 93.6 90.8 68.2 59.0 44.2 +50.3 +14.6 63.0 85.0 62.8 58.5 +68.6 +97.6 73.2 92.4 56.4 +67.9 +RepSurf-U6,256 +PointNet++ +55.5 87.7 93.4 85.9 69.1 57.3 48.8 +50.0 +16.5 61.0 81.6 66.2 59.0 +77.5 +99.2 78.2 94.0 56.8 +68.8 +RepSurf-U6,256 (repd.) +PointNet++ +57.4 89.6 93.2 87.4 70.2 58.8 46.6 +47.4 +18.1 63.4 78.2 70.4 46.5 +81.0 +99.8 69.4 90.8 55.5 +68.0 +w/ MS + Local +PointNet++ +51.2 89.5 93.4 87.5 71.8 60.5 49.0 +57.7 +21.9 65.2 82.1 70.3 53.3 +80.2 +98.2 68.8 91.9 58.2 +69.5 +GroupFree12,512 +PointNet++w2x 52.1 91.9 93.6 88.0 70.7 60.7 53.7 +62.4 +16.1 58.5 80.9 67.9 47.0 +76.3 +99.6 72.0 95.3 56.4 +69.1 +w/ MS + Local +PointNet++ +53.7 91.9 93.4 88.8 72.1 61.3 52.8 +58.6 +17.4 70.8 83.3 69.9 56.5 +75.6 +98.5 70.3 94.4 56.9 +70.3 +RepSurf-U12,512 +PointNet++w2x 54.6 94.0 96.2 90.5 73.2 62.7 55.7 +64.5 +18.6 60.9 83.1 69.9 49.4 +78.4 +99.4 74.5 97.6 58.3 +71.2 +RepSurf-U12,512 (repd.) PointNet++w2x 54.5 90.7 93.4 87.6 76.3 64.4 54.4 +61.4 +19.0 62.2 84.0 69.2 48.8 +79.2 +99.8 75.9 92.2 62.0 +70.8 +w/ MS + Local +PointNet++w2x 58.0 89.3 94.1 86.5 74.3 62.4 60.2 +57.9 +21.7 67.9 85.3 74.4 53.5 +75.9 +99.6 74.6 91.6 63.7 +71.7 +Table 9. Performance of mAP@0.25 in each category on ScanNetV2. +methods +backbone +cab +bed chair sofa tabl door wind bkshf +pic +cntr desk curt fridg showr toil +sink bath ofurn mAP +VoteNet [21] +PointNet++ +14.6 77.8 73.1 80.5 46.5 25.1 16.0 +41.8 +2.5 +22.3 33.3 25.0 31.0 +17.6 +87.8 23.0 81.6 18.7 +39.9 +H3DNet [54] +4×PointNet++ +20.5 79.7 80.1 79.6 56.2 29.0 21.3 +45.5 +4.2 +33.5 50.6 37.3 41.4 +37.0 +89.1 35.1 90.2 35.4 +48.1 +GroupFree6,256 +PointNet++ +23.0 78.4 78.9 68.7 55.1 35.3 23.6 +39.4 +7.5 +27.2 66.4 43.3 43.0 +41.2 +89.7 38.0 83.4 37.3 +48.9 +w/ MS + Local +PointNet++ +27.3 80.8 83.3 85.3 60.2 39.7 21.7 +40.4 +7.6 +41.7 61.5 42.9 42.3 +26.2 +96.1 38.5 89.5 39.7 +51.4 +RepSurf-U6,256 +PointNet++ +24.9 79.6 80.1 70.4 56.4 36.7 25.5 +41.4 +8.8 +28.7 68.0 45.2 45.0 +42.7 +91.3 40.1 85.1 39.2 +50.5 +RepSurf-U6,256 (repd.) +PointNet++ 1. +24.3 82.6 82.6 71.3 55.9 38.3 18.6 +40.3 +11.2 44.0 60.7 45.1 35.7 +36.6 +97.1 34.6 84.6 39.8 +50.2 +w/ MS + Local +PointNet++ +27.1 80.9 83.0 77.1 58.0 45.8 24.8 +50.8 +10.5 31.9 67.7 44.6 40.6 +34.9 +97.7 38.3 87.3 44.6 +52.5 +GroupFree12,512 +PointNet++w2x 26.0 81.3 82.9 70.7 62.2 41.7 26.5 +55.8 +7.8 +34.7 67.2 43.9 44.3 +44.1 +92.8 37.4 89.7 40.6 +52.8 +w/ MS + Local +PointNet++ +31.0 81.0 85.0 79.4 61.1 44.5 27.9 +50.6 +10.1 45.0 61.2 54.1 39.5 +43.5 +91.7 45.9 89.3 42.4 +54.6 +RepSurf-U12,512 +PointNet++w2x 28.5 83.5 84.8 72.6 64.0 43.6 28.3 +57.8 +9.6 +37.0 69.7 45.9 46.4 +46.1 +94.9 39.1 92.1 42.6 +54.8 +RepSurf-U12,512 (repd.) PointNet++w2x 27.6 82.7 85.3 68.8 60.6 44.0 27.3 +56.7 +9.6 +39.6 63.7 53.8 43.0 +42.4 +99.8 38.8 88.7 47.3 +54.4 +w/ MS + Local +PointNet++w2x 29.3 83.6 85.7 78.7 66.2 45.6 30.4 +59.8 +10.4 34.2 60.0 60.8 48.1 +45.3 +99.9 44.5 87.1 48.4 +56.5 +Table 10. Performance of mAP@0.50 in each category on ScanNetV2. +12 + +methods +backbone +bathtub bed bkshf chair desk drser nigtstd sofa table toilet mAP +VoteNet [21] +PointNet++ +75.5 +85.6 +31.9 +77.4 24.8 27.9 +58.6 +67.4 51.1 +90.5 +59.1 +H3DNet [54] +4×PointNet++ +73.8 +85.6 +31.0 +76.7 29.6 33.4 +65.5 +66.5 50.8 +88.2 +60.1 +3DETR [19] +transformer +69.8 +84.6 +28.5 +72.4 34.3 29.6 +61.4 +65.3 52.6 +91.0 +61.1 +Pointformer [20] +Pointformer +80.1 +84.3 +32.0 +76.2 27.0 37.4 +64.0 +64.9 51.5 +92.2 +61.1 +GroupFree6,256 +PointNet++ +80.0 +87.8 +32.5 +79.4 32.6 36.0 +66.7 +70.0 53.8 +91.1 +63.0 +w/ MS + Local +PointNet++ +83.2 +86.7 +34.5 +79.0 31.9 39.3 +66.0 +70.6 55.6 +90.8 +63.8 +RepSurf-U6,256 +PointNet++ +81.1 +89.3 +34.4 +80.4 33.5 37.3 +68.1 +71.4 54.8 +92.3 +64.3 +RepSurf-U6,256 (repd.) +PointNet++ +79.5 +87.5 +33.8 +79.4 32.7 40.2 +69.0 +70.3 55.4 +92.1 +64.0 +w/ MS + Local +PointNet++ +79.9 +87.0 +36.8 +79.5 33.8 41.4 +67.4 +71.2 55.3 +92.4 +64.5 +Table 11. Performance of mAP@0.25 in each category on SUN RGB-D. +methods +backbone +bathtub bed bkshf chair desk drser nigtstd sofa table toilet mAP +VoteNet [21] +PointNet++ +45.4 +53.4 +6.8 +56.5 +5.9 +12.0 +38.6 +49.1 21.3 +68.5 +35.8 +H3DNet [54] +4×PointNet++ +47.6 +52.9 +8.6 +60.1 +8.4 +20.6 +45.6 +50.4 27.1 +69.1 +39.0 +GroupFree6,256 +PointNet++ +64.0 +67.1 +12.4 +62.6 14.5 21.9 +49.8 +58.2 29.2 +72.2 +45.2 +w/ MS + Local +PointNet++ +66.2 +67.4 +10.8 +63.6 15.0 24.7 +56.7 +56.1 30.8 +74.3 +46.6 +RepSurf-U6,256 +PointNet++ +65.2 +67.5 +13.2 +63.4 15.0 22.4 +50.9 +58.8 30.0 +72.7 +45.9 +RepSurf-U6,256 (repd.) +PointNet++ +61.4 +66.8 +11.3 +64.0 14.8 24.2 +51.8 +59.0 31.6 +71.7 +45.7 +w/ MS + Local +PointNet++ +62.2 +67.6 +16.6 +65.0 15.0 24.2 +57.0 +59.0 30.9 +77.7 +47.5 +Table 12. Performance of mAP@0.50 in each category on SUN RGB-D. +13 + diff --git a/A9E0T4oBgHgl3EQfxwJo/content/tmp_files/load_file.txt b/A9E0T4oBgHgl3EQfxwJo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..228dd342162dce672f8ea3ea44c95292dbe11e3e --- /dev/null +++ b/A9E0T4oBgHgl3EQfxwJo/content/tmp_files/load_file.txt @@ -0,0 +1,1522 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf,len=1521 +page_content='Model-Agnostic Hierarchical Attention for 3D Object Detection Manli Shu1* Le Xue2 Ning Yu2 Roberto Martín-Martín2,3 Juan Carlos Niebles2 Caiming Xiong2 Ran Xu2 1 University of Maryland, 2 Salesforce Research, 3 UT Austin Abstract Transformers as versatile network architectures have re- cently seen great success in 3D point cloud object detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' However, the lack of hierarchy in a plain transformer makes it difficult to learn features at different scales and restrains its ability to extract localized features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Such limita- tion makes them have imbalanced performance on objects of different sizes, with inferior performance on smaller ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In this work, we propose two novel attention mechanisms as modularized hierarchical designs for transformer-based 3D detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' To enable feature learning at different scales, we propose Simple Multi-Scale Attention that builds multi- scale tokens from a single-scale input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For localized feature aggregation, we propose Size-Adaptive Local At- tention with adaptive attention ranges for every bounding box proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Both of our attention modules are model- agnostic network layers that can be plugged into existing point cloud transformers for end-to-end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We evalu- ate our method on two widely used indoor 3D point cloud object detection benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' By plugging our proposed mod- ules into the state-of-the-art transformer-based 3D detector, we improve the previous best results on both benchmarks, with the largest improvement margin on small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Introduction 3D point cloud data provides accurate geometric and spa- tial information, which are important to computer vision ap- plications such as autonomous driving and augmented reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Different from image data, which has a grid-like structure, point clouds consist of unordered irregular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Due to such unique properties of point clouds, previous works have proposed various deep network architectures for point cloud understanding [7,8,22–25,34,37,48,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' With the success of transformers in natural language processing [4, 26, 40] and 2D vision [5,14,39], attention-based architectures for point clouds [20,38,46,49,52,53,55] are explored in recent Work done during an internship at Salesforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' manlis@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 1The code and models will be available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='com/ salesforce/Hierarchical_Point_Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Groundtruths Predictions Attention weights (high to low) Plain Attention Our Attention Groundtruth object Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Visualization of the attention weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' With our hierar- chical attentions, the object center has higher attention weights with points that belong to the object, and the predicted bounding box is better aligned with the groundtruth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Our multi-scale attention extracts feature at different scales, which helps distinguish object boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Our size-adaptive local attention aggregates features at the object level and helps refine the bounding box proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' works and have seen great success in 3D point cloud object detection [16, 19, 32, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Several properties of transform- ers make them ideal for learning on raw point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For example, their permutation-invariant property is necessary for modeling unordered sets like point clouds, and their at- tention mechanism can model long-range relationships that help capture the global context for point cloud learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Despite the advantages of transformers for point clouds, we find the state-of-the-art transformer detector to have im- balanced performance across different object sizes, with the lowest average precision on small objects (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We speculate the inferior performance on small objects can be due to two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Firstly, to make the computation feasi- ble, transformer detectors use point cloud features consisting of a small set of points compared to the original point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='02650v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='CV] 6 Jan 2023 The extensively downsampled point cloud loses geometric details, which has a larger impact on small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Secondly, plain transformers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', Transformer [40], ViT [5]) extract features at the global scale throughout the network, which does not support explicit localized feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Motivated by the above observations, we expect existing point cloud transformers to benefit from a hierarchical fea- ture learning strategy, which allows multi-scale feature learn- ing and supports localized feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Nonetheless, considering the computation intensity of point cloud trans- formers, it is inefficient to use higher-resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', higher point density) point cloud features throughout the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Furthermore, due to the irregularity of point clouds, it is non-trivial to integrate hierarchical design and multi-scale features into transformers for point cloud object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In this work, we aim to improve transformer-based 3D object detectors with modularized hierarchical designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We propose two attention modules for multi-scale feature learning and size-adaptive local feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Our attention modules are model-agnostic and can be plugged into existing point cloud transformers for end-to-end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We first propose Simple Multi-Scale Attention (MS-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' It builds higher resolution point features from the single-scale input feature with a learnable upsampling strategy and use both features in the attention function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' To reduce computa- tion and parameter overhead, we transform the multi-scale features into multi-scale tokens and perform multi-scale to- ken aggregation [30] within a multi-head attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The second module is Size-Adaptive Local Attention (Local- A), which learns localized object-level features for each object candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' It assigns larger attention regions to ob- ject candidates with larger bounding box proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The local attention regions are defined by their corresponding intermediate bounding box proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We evaluate our method on two widely used indoor 3D object detection benchmarks: ScanNetV2 [3] and SUN RGB- D [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We plug our attention modules into the state-of- the-art transformer-based 3D detector and perform end-to- end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Our method improves the previous best result by over 1% in mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 and over 2% in mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 on ScanNetV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Furthermore, our size-aware evaluation shows we have the most performance gain among small objects with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5% increase in mAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We summarize our main contributions as follows: We propose Simple Multi-Scale Attention (MS-A) to en- able multi-scale feature learning on single-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We present Size-Adaptive Local Attention (Local-A) for local feature aggregation within bounding box proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We conduct experiments on two widely used indoor 3D detection benchmarks and surpass the previous best results on both benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Related Work Network architectures for point cloud learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Exist- ing network architectures for point cloud learning can be roughly divided into two categories based on their point cloud representation: grid-based and point-based, yet in between, there also exist hybrid architectures that operate on both representations [9,33,51,53,57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Grid-based methods project the irregular point clouds into grid-like structures, such as 3D voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' With the grid-like structure, existing works have proposed a variety of 3D-convolution-based ar- chitectures [7,18,31,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Point-based methods, on the other hand, directly learn features from the raw point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Within this category, graph-based method [8, 35, 44, 47, 56] use a graph to model the relationships among the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Another line of work models a point cloud as a set of points, and extracts features through set abstraction [17,23,25,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Re- cent works explore transformer architecture for point-based learning [16,19,20,53,55], where each point is fed into the transformer as a token and the attention mechanism learns point features at a global scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' While previous methods improve point cloud learning by developing new backbones and modifying the overall network architecture, our work focuses on the attention mechanism of the point cloud trans- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Instead of proposing new architectures for point cloud learning, we aim to provide a model-agnostic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Point cloud object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' One major challenge in point cloud object detection is extracting object features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In 2D object detection, a common practice for extracting ob- ject features is to use a region proposal network (RPN) [29] to generate dense bounding box proposals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', object can- didate) in a top-down manner and then extract features for each object candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' However, in 3D vision, generating dense 3D bounding box proposals for point cloud data is inefficient due to the irregularity and sparsity of point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Previous work [2,57] addresses this issue by projecting point clouds into 2D bird’s-eye views or voxels and then applying RPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' However, such projection operations can result in the loss of geometric information or introduce quantization er- rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Another line of work seeks to generate 3D proposals in a bottom-up manner (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', point-based) [16, 19, 21, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' VoteNet [21] samples a set of points from a point cloud as the initial object candidates and then assigns points to each object candidate through voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Object features of each candidate are learned by aggregating features within its corresponding vote cluster (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Instead of voting and grouping, follow-up works [16, 19] propose to use a transformer to automatically model the relationship between the object candidates and the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Although point- based methods do not have quantization errors caused by voxelization, to make the computation feasible, a point cloud needs to be extensively downsampled at the beginning of the 2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Such downsampling also causes a loss of geomet- ric information, while it is important for object detection to have fine-grained features to make accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Our work is based on point-based transformer detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We ad- dress the downsampling issue by building higher-resolution features without increasing the computation budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Hierarchical designs for 2D and 3D vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Extensive work has been done to adapt transformers for vision recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' One direction is to borrow the hierar- chical design and inductive biases from convolutional neu- ral networks (ConvNet) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In the 2D vision, one line of ConvNet-based hierarchical design [6,11,43] produces multi-scale feature maps for 2D images by progressively decreasing the resolution and expanding feature channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Swin Transformer [15] adopts the idea of weight-sharing of ConvNet and proposes efficient self-attention with shifted windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Shunted self-attention [30] attends to features at different scales through multi-scale token aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In the 3D vision, hierarchical designs for point cloud transformers are explored in previous works, where self-attentions are ap- plied to local regions (specified by k nearest neighbors [55] or a given radius [20]), and downsampling operations are performed after every encoding stage following the hierar- chical design of PointNet++ [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Patchformer [53] proposes a multi-scale attention block that performs extracts features at multiple granularities, but it requires voxelization on the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Different from previous works, we pack our hierarchical design into model-agnostic attention modules that can be plugged into any existing architecture and enable both multi-scale and localized feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Method In this section, we first discuss the background, including a brief introduction to the task of point cloud object detection, an overview of point-based 3D detection methods, and the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Next, we dive into the detailed designs of our proposed attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Background Point cloud object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Given a point cloud Praw with a set of P points P = {pi}P i=1, each point pi ∈ R3 is represented by its 3-dimensional coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 3D object detection on point cloud aims to predict a set of bounding boxes for the objects in the scene, including their locations (as the center of the bounding box), size and orientation of the bounding box, and the semantic class of the corre- sponding object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Note that due to the computation limit, the point cloud is downsampled at the early stage of a model to a subset of Praw, which contains N (N << P) points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' P = SA(Praw) = {pi}N i=1 contains the aggregated groups of points around N group centers, where SA (set abstraction) is the aggregation function, and the group centers are sam- pled from the raw point cloud using Furthest Point Sample (FPS) [23], a random sampling algorithm that provides good coverage of the entire point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Point-based 3D object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Our method is built on point-based 3D object detectors [16,21,34], which detect 3D objects in point clouds in a bottom-up manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Compared to other 3D detectors that generate box proposals in a top-down manner on the bird’s-eye view or voxelized point clouds [2], point-based methods work directly on the irregular point cloud and do not cause loss of information or quantization errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In addition, point-based methods are suitable for more efficient single-stage object detection [1,13,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The feature representation of the input point cloud {zi}N i=1, zi ∈ Rd is first obtained using a backbone model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', PointNet++ [25]), where d is the feature dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Point-based detectors generate bounding box predic- tions starting with M (M < N) initial object candidates {qi}M i=1, qi ∈ RC, sampled from the point cloud as object centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' A common approach for sampling the candidates is Furthest Point Sample (FPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Once get the initial candidates, the detector then extracts features for every object candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Attention-based methods [16] learn features by doing self- attention among the object candidates, and cross-attention be- tween the candidates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', query) and point features {zi}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The learned features of the object candidates will then be passed to prediction heads, which predict the attributes of the bounding box for each object candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The attributes of a 3D bounding box include its location (box center) ˆc ∈ R3, size (H/W/D dimensions) ˆd ∈ R3, orientation (heading an- gles) ˆa ∈ R, and the semantic label of the object ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' With these parameterizations, we can represent a bounding box proposal as ˆb = {ˆc, ˆd, ˆa,ˆs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The detailed parameterizations of a bounding box are included in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Attention mechanism is the basic building block of trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The attention function takes in query (Q), key (K), and value (V ) as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The output of the attention func- tion is a weighted sum of the value with the attention weight being the scaled dot-product between the key and query: Attn(Q, K, V ) = softmax(QKT √dh )V, (1) where dh is the hidden dimension of the attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For self-attention, Q ∈ Rdh, K ∈ Rdh and V ∈ Rdv are transformed from the input X ∈ Rd via linear projection with parameter matrix W Q i ∈ Rd×dh, W K i ∈ Rd×dh, and W V i ∈ Rd×dv respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For cross-attention, Q, K, and V can have different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In practice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' transformers adopt the multi-head attention design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' where multiple attention functions are applied in 3 … Object Features Point Features Upsampled Point Features … … Q K1×,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' V1× K2×,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' V2× Learnable Upsample Concat & Linear Attention head=0 Attention head=h/2 Object Features Prediction Head Bounding Box Proposals Point Features Sampled Point Features {Q} (batch) {K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' V} (batch) (a) Simple Multi-Scale Attention (b) Size-Adaptive Local Attention Multi-Head Attention Pad & Truncate Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' An illustration of our hierarchical attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Simple Multi-Scale Attention (MS-A) learns features at different scales within the multi-head cross-attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' It constructs high resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', point density) point features from the single-scale input point features and uses keys and values of both scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Size-Adaptive Local Attention (Local-A) extracts localized features for each object candidate by restricting the attention range to be inside its bounding box proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The attention range (the token lengths of key and value) is adaptive for each object candidate (query) and we perform padding or truncating to allow batch processing .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' parallel across different attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The input of each attention head is a segment of the layer’s input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Specifically, the query, key, and value are split along the hidden dimension into (Qi, Ki, Vi)h i=1, with Qi ∈ Rdh/h, Ki ∈ Rdh/h, Vi ∈ Rdv/h, where h is the number of attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The final output of the multi-head attention layer is the projection of the concatenated outputs of all attention heads: MultiHead(Q,K, V ) = Concat({Attn(Q0, K0, V0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Attn(Qh−1, Kh−1, Vh−1)})W O, (2) where the first term denotes the concatenation of the output and W O is the output projection matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Simple Multi-Scale Attention When applying transformers to point-based 3D object detection, the cross-attention models the relationship be- tween object candidates and all other points within the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The intuition is that, for each object candidate, every point within the point cloud (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', scene) either belongs to the object or can provide context information for the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Therefore, it makes sense to gather all point features for every object candidate, and the importance of a point to the object candidate can be determined by the attention weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' However, due to the computation overhead of the atten- tion function, the actual number of points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', tokens) that a model is learned on is set as 1024 [16,19], whereas the raw point cloud usually contains tens of thousands points [3,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Such extensive downsampling on the point cloud causes a loss of detailed geometric information and fine-grained fea- tures, which are important for dense prediction tasks like object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' To this end, we propose Simple Multi-Scale Attention (MS-A), which builds higher-resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', higher point density) feature maps from the single-scale feature input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' It then uses features of both scales as the key and value in the cross-attention between object candidates and other points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' the multi-scale feature aggregation is realized through multi- scale token aggregation, where we use the key and value of different scales in different subsets of attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Our goal is to create a higher-resolution feature map that provides fine-grained geometric details of the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The first step of our multi-scale attention is to obtain a higher-resolution feature map from the single-scale input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We propose a learnable upsampling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Given the layer’s input point cloud feature {zi}N i=1, zi ∈ Rd, we want to create a feature map with 2N points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' To get the locations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', coordinates) of the 2N points, we use FPS to sample 4 2N points from the raw point cloud {pi}2N i=1, pi ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Next, for each sampled point pi, we search for the top three of its nearest neighbors (in the euclidean distance) in the input feature map {zi}N i=1, denoted as {z0 i , z1 i , z2 i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Then we cal- culate a weighted interpolation of the three-point features, weighted by the inverse of their distance to the sample point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The interpolated feature is then projected into the feature rep- resentation of sampled point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The upsampled point feature map can be written as: {˜zi}2N i=1, ˜zi = Φθ(interpolate({z0 i , z1 i , z2 i })) (3) Here, Φθ is learnable projection function parameterized by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We choose MLP as our projection function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' After the upsampling, we have two sets of point features of different scale {zi}N i=1, {˜zi}2N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' To avoid computation increase, we perform multi-head cross-attention on both sets of point features in a single pass by using features of different scales on different attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We divide attention heads evenly into two groups, and use zi}N i=1 to obtain K and V in the first group while using the other for the second group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Both groups share the same set of queries transformed from {qi}M i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Since the input and output of this module are the same as a plain attention module, we can plug MS-A into any attention-based model to enable feature learning at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In practice, we apply MS-A only at the first layer of a transformer which makes minimal modifications to the network and introduces little computation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Size-Adaptive Local Attention Although the attention mechanism can model the relation- ship between every point pair, it is not guaranteed the learned model will pay more attention to points that are important to an object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', those belonging to the object) than the ones that are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The lack of hierarchy in transformers, on the other hand, does not support explicit localized feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Different from existing local attentions that are performed within a fixed region, we propose Size-Adaptive Local Attention (Local-A) that defines local regions based on the size of bounding box proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We first generate intermediate bounding box proposals {ˆbi}M i=1 with the features of object candidates ({qi}M i=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We then perform cross-attention between every candidate qi and the points sampled from within its corresponding box proposal ˆbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Therefore, we have customized size-adaptive local regions for every query point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For every input object candidate qil ∈ Rd, it is updated Local-A as: qi l+1 = Attn(Ql i, Ki, Vi), where (4) Ql i = qi lW Q, Ki = ZiW K, Vi = ZiW V with (5) Zi = {zk i | pos(zk i) in ˆbi}, ˆbi = Predl box(qi l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' (6) In the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' ( 6), we use pos(·) to denote the coordinate of a point in the 3D space, and Zi is a set of points inside box ˆbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Note that the point features {zi}N i=1 are extracted by the backbone network and are not updated during the feature learning of object candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Predl box is the prediction head at layer l that generate intermediate box predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Since object candidates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', query) will have different sets of keys and values depending on the size of their bound- ing box proposals, the number of K and V tokens also differs for each object candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' To allow batch computa- tion, we set a maximum number of points (Nlocal) for the sampling process and use Nlocal as a fixed token length for every query point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For bounding boxes that contain less than Nlocal points, we pad the point sequence with an unused token to Nlocal and mask the unused tokens out in the cross- attention function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' for those containing more than Nlocal points, we randomly discard them and truncate the sequence to have Nlocal points as keys and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Lastly, in the case where the bounding box is empty, we perform ball query [23] around the object candidate to sample Nlocal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Same as MS-A, Local-A does not pose additional require- ments on modules input, therefore we can apply it at any layer of a transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Specifically, we apply Local-A at the end of a transformer where bounding box proposals are in general more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Experiments In this section, we first evaluate our method on two widely used indoor point cloud detection datasets, ScanNetV2 and SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Next, we provide qualitative and quantita- tive analyses of our method, including visualizations of the bounding box predictions and attention weights, and eval- uations using our proposed size-aware metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Lastly, we include ablation studies on the design choices of our atten- tion modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We include more experiments and ablation studies in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1, including analyses on the infer- ence speed and the number of parameters of each individual attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Main Results Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' ScanNetV2 [3] consists of 1513 reconstructed meshes of hundreds of indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' It contains rich anno- tations for various 3D scene understanding tasks, including object classification, semantic segmentation, and object de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For point cloud object detection, it provides axis- aligned bounding boxes with 18 object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We follow the official dataset split by using 1201 samples for training and 312 samples for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' SUN RGB-D [36] is a single- view RGB-D dataset with 10335 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For 3D object detection, it provides oriented bounding box annotations with 37 object categories, while we follow the standard eval- uation protocol [21] and only use the 10 common categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The training split contains 5285 samples and the testing set contains 5050 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 5 Methods #Params Backbone ScanNet V2 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 VoteNet [21] PointNet++ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 H3DNet [54] PointNet++ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 H3DNet [54] 4×PointNet++ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 3DETR [19] transformer 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 Pointformer [20] transformer 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 Group-Free6,256 [16] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0M PointNet++ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5) w/ MS + Local (Ours) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0M PointNet++ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1) (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 (49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8) (↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5) RepSurf-U6,256 [27] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1M PointNet++ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 ( - ) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 ( - ) RepSurf-U6,256 (reproduce) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1M PointNet++ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7) w/ MS + Local (Ours) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1M PointNet++ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8) (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1) (↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3) Group-Free12,512 [16] 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9M PointNet++w2x 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8) w/ MS + Local (Ours) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9M PointNet++w2x 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 (69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2) (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2) (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8) RepSurf-U12,512 [27] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1M PointNet++w2x 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 ( - ) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 ( - ) RepSurf-U12,512 (reproduce) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1M PointNet++w2x 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6) w/ MS + Local (Ours) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1M PointNet++w2x 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0) (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8) (↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Performance of object detection on ScanNetV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We follow the standard protocol [21] by reporting the best results over 5 × 5 trials (5 trainings, each with 5 testings) and including the averaged results in the bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Group-FreeL,O denotes the variant with L decoder layers and O object candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The same notation applies to RepSurf-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The detection code of RepSurf is not published, so we implement our version of RepSurf-U and apply our method to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We include the results of our implementation of RepSurf-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Methods mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 VoteNet [21] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 H3DNet [54] H3DNet [54] 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 3DETR [19] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 Pointformer [20] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 Group-Free6,256 [16] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 (44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4) w/ MS + Local (Ours) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2) (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7) (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4) RepSurf-U6,256 [27] 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 ( - ) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 ( - ) RepSurf-U6,256 (repd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=') 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2) w/ MS + Local (Ours) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8) (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1) (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8) Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Performance of object detection on SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' “repd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='" stands for the reproduced results of our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' “-" means the official result is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For both datasets, we follow the stan- dard evaluation protocol [21] and use the mean Average Pre- cision (mAP) as the evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We report mAP scores under two different Intersection over Union (IoU) thresholds: mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 and mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In addition, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3, to evaluate model performance across different object sizes, we follow the practice in 2D vision [12] and implement our own size-aware metrics that measure the mAP on small, medium, and large objects respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' On account of the randomness of point cloud training and inference, we train a model 5 times and test each model 5 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We report both the best and the average results among the 25 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We validate our method by applying it to ex- isting transformer point cloud detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Group-Free [16] extracts features for object candidates using a transformer decoder with plain attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We include two configura- tions of Group-Free in our comparison: Group-Free6,256 samples a total of 256 object candidates for feature learning and bounding box prediction, using a transformer decoder with 6 layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Group-Free12,512 is the largest configuration, which has 12 transformer layers and 512 object candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' RepSurf-U [27] proposes a novel multi-surface (umbrella curvature) representation of point clouds that can explicitly describe the local geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For object detection, RepSurf-U adopts the transformer decoder of Group-Free and replaces its backbone with one that extracts features on both point clouds and the surface representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The official imple- mentation and the averaged results of RepSurf-U for object detection are not publicly available, so we include the results of our own implementation of RepSurf-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We also include the performance of previous point-based 3D detectors for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' VoteNet [21] aggregates fea- tures for object candidates through end-to-end optimizable Hough Voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' H3DNet [54] proposes a hybrid set of ge- ometric primitives for object detection and trains multiple individual backbones for each primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 3DETR [19] solves point cloud object detection as a set-to-set problem using a transformer encoder-decoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Pointformer [20] proposes a hierarchical transformer-based point cloud back- bone and adopts the voting algorithm of VoteNet for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For a baseline model with L transformer layers, we enable multi-scale feature learning by replacing the cross-attention of the 1-st layer with MS-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' After the L-th layer, we append an additional transformer 6 layer to perform local feature aggregation, which consists of Local-A and a feedforward layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We follow the original training settings of the baseline models [16,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The detailed hyperparameter settings can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' From Table 1, on ScanNetV2, we observe consis- tent improvements in point cloud transformer detectors when equipped with our attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' By applying MS-A and Local-A to Group-Free, we achieve on-par performance with the state-of-the-art RepSurf-U detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In addition, we can further improve RepSurf-U by over 1% in mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 and over 2% in mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 on varying model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Table 2 shows a similar trend on SUN RGB-D, where our attention modules boost the mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 of group-Free to sur- pass RepSurf-U, and can further improve the state-of-the-art method by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5% in mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8% in mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Qualitative Results In Figure 3, we provide qualitative results on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The visualized results are of our methods applied to the Group-Free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The qualitative results suggest that our model is able to detect and classify objects of different scales even in complex scenarios containing more than 10 objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', the example in the bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' By looking into cross-attention weights in the transformer detector, we find that object candidates tend to have higher correlations with points that belong to their corresponding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Performance on objects of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In addition to the standard evaluation metrics, we are in- terested in examining models’ performance across different object sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Inspired by the size-aware metrics in 2D de- tection [12], we implement our own version of size-aware metrics for 3D detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We conduct this analysis on Scan- NetV2, on which we calculate the volume for all the objects in all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We set the threshold for mAPS as the 30th percentile of the volume of all objects, and use the 70th percentile as the threshold for mAPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' More details about these metrics are included in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' MS-A Local-A mAPS mAPM mAPL 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 \x14 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 \x14 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 \x14 \x14 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 (↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 (↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Performance on different size categories on Scan- NetV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We define the S/M/L thresholds based on the statistics (volume distribution) of ScanNetV2 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The configuration in the first row denotes the Group-Free12,512 baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In Table 3, we evaluate our methods using size-aware metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We report the average result over 25 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The first row denotes the Group-Free12,512 baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Firstly, by com- paring the mAPS to mAPL, we notice that it has imbalanced performance across different object sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Looking at the improvement margins, we find our method to have the most performance gain on small and medium-sized objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The result suggests that hierarchical designs can aid fine-grained and localized feature learning for point cloud transformer detectors and helps models detect smaller objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Ablation Study In this subsection, we first conduct an ablation study on the stand-alone effects of our multi-scale attention and size- adaptive local attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Next, we include empirical analyses of the design choices of our attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' If not other- wise specified, experiments in this subsection are conducted on ScanNetV2 with the Group-Free12,512 baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' With- out loss of generality, the results in this subsection are the averaged numbers over 25 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The stand-alone effects of MS-A and Local-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Table 4 shows the stand-alone performance of our proposed attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Compared to the plain attention baseline, both of our attentions are proved to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' When combined together, we find the two modules to be complementary to each other and bring more significant performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' MS-A Local-A mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 \x14 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 \x14 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 \x14 \x14 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The stand-alone effect of our attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The configuration in the first row denotes the Group-Free12,512 baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The results are averaged over 25 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The maximum number of points (Nlocal) in Local-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In Local-A, for each object candidate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', query), we sam- ple a set of points within its corresponding bounding box proposal and use the point features as the key and value for this object candidate in the cross-attention function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' As introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3, we cap the number of sampled points with Nlocal to allow batch computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We provide an empirical analysis of the effects of Nlocal on Local-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' From Table 5, we find that too little number of points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', Nlocal = 8) for Local-A results in a per- formance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' On the other hand, as Nlocal continues to increase, we do not observe a significant performance gain compared to Nlocal = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Intuitively, a small Nlocal means the points within each bounding box are sampled sparsely, which can be too sparse to provide enough information about 7 Scene Groundtruth Prediction Attention Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Qualitative results on SUN RGB-D (top) and ScanNetV2 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The color of a bounding box in the middle two columns stands for the semantic label of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In the last column, we draw both the groundtruth (in green) and the prediction (in blue) of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We highlight the points that belong to an object for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In the last column, we visualize the attention weight of the last transformer layer (before applying Local-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We visualize the cross-attention weight between an object candidate and the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Nlocal mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 mAPS mAPM mAPL 8 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 16 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 24 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 32 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The effect of Nlocal in Local-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' When there are enough points, a larger Nlocal means the points are sampled more densely within each bounding box proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' any object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' This explains why Nlocal = 8 does not work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' However, on the other hand, a large Nlocal may only benefit large objects and has little effect on smaller objects, because the latter are padded with unused tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' MS-A with different feature resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In Section 3, we propose learnable upsampling for MS-A to build higher- resolution point features from the single-scale input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In the same spirit, a parameterized downsampling procedure can be realized through conventional set abstraction [23], which aggregated point features within local groups and produce a feature map with fewer points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', lower resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In- tuitively, a higher point density of the feature map provides more fine-grained features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' To study the effects of feature maps of different granularity, we conduct an empirical analy- sis on MS-A using different sets of multi-scale feature maps representing point clouds of varying granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In Table 6, we examined the performance of two multi- scale choices in comparison with the single-scale baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The result suggests that coarse features (s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5×) do not benefit transformer detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' This is expected because trans- formers do not have limited receptive fields and thus do not Feature Scales s mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 [1×] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 [1×, 2×] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5×, 1×, 2×] 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Simple Multi-Scale Attention with different feature scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Feature scale = s× means the feature map contains s × N points, with N being the original number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' A larger s denotes a feature map with higher point density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', resolution) rely on a coarse-grained feature map to learn global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Conclusion In this work, we present Simple Multi-Scale Attention and Size-Adaptive Local Attention, two model-agnostic modules that bring in hierarchical designs to existing transformer- based 3D detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We enable multi-scale feature learning and explicit localized feature aggregation through improved attention functions, which are generic modules that can be applied to any existing attention-based network for end-to- end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We improve the state-of-the-art transformer detector on two challenging indoor 3D detection benchmarks, with the largest improvement margin on small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' As our attention modules promote fine-grained feature learning, which is important to various dense prediction vision tasks, one direction for future work is to adapt our attention modules for other point cloud learning problems such as segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Another direction is to introduce more efficient attention mechanisms to the multi-scale attention to further bring down the computation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 8 References [1] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' End- to-end object detection with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 3 [2] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Multi-view 3d object detection network for autonomous driv- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In CVPR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 2, 3 [3] Angela Dai, Angel X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Chang, Manolis Savva, Maciej Hal- ber, Thomas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Funkhouser, and Matthias Nießner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Scannet: Richly-annotated 3d reconstructions of indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In CVPR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 2, 4, 5 [4] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 1, 2, 3 [54] Zaiwei Zhang, Bo Sun, Haitao Yang, and Qixing Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' H3dnet: 3d object detection using hybrid geometric primi- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 6, 12, 13 [55] Hengshuang Zhao, Li Jiang, Jiaya Jia, Philip H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Torr, and Vladlen Koltun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Point transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 1, 2, 3 [56] Haoran Zhou, Yidan Feng, Mingsheng Fang, Mingqiang Wei, Jing Qin, and Tong Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Adaptive graph convolution for point cloud analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 2 [57] Yin Zhou and Oncel Tuzel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Voxelnet: End-to-end learning for point cloud based 3d object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In CVPR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' More Experiments The placement of Simple Multi-scale Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We de- sign simple multi-scale attention as a compact network layer to enable hierarchical feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' As it can be inserted at any place within a network, we are interested in finding out how the placement of the multi-scale attention layer affects a model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Layers mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 [0] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 [0, 4, 8] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 [0, 3, 6, 9] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 [0, 2, 4, 6, 8, 10] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Different placements of the simple multi-scale atten- tion layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Layers = [i] means we replace the ith layer of the transformer decoder with our MS-A layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The best results are in bold, and the second-best results are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We consider different strategies to place MS-A within the transformer decoder of Group-Free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We divide the 12- layer decoder into several stages and place MS-A at the first layer of each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Specifically, our default setting uses a single MS-A at the first decoder layer, and we try placing MS-A by dividing the decoder evenly into 3, 4, and 6 stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' From the results in Table 7, we do not observe a significant benefit of using more than one multi-scale layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We conjecture this is because the up-scaled feature map in our multi-scale attention is obtained through interpolation 10 and simple linear projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The up-scale point feature obtained in this way may mainly provide more accurate geometric information with a higher point density, while may not have much semantic difference than the original input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We expect such fine-grained geometric information to be particularly helpful at the beginning of the decoding stage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', Layer= 0), yet may be less useful as the object features go deeper in the decoder and become more abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Per-category mAP on ScanNetV2 and SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We include the detailed per-category mAP on both datasets in Table 9, Table 10, Table 11, and Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For the results in this paragraph, we follow the baselines [16,19,20,27] and report the result of the best trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Inference Speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We analyze the parameter and computa- tion overhead of each of our attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We measure the inference speeds for all model configurations on the same machine with a single A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In Table 8, we can see that replacing plain attention with MS-A results in little parame- ter increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' While applying Local-A leads to a larger param- eter increase, the Local-A module itself contains the same number of parameters as a plain cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The param- eter increase is mainly due to the additional feed-forward layer and learnable positional embeddings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In terms of inference speed, we find MS-A to cause more substantial latency in inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Such latency is caused by applying the attention function on the key/value with 2 times more tokens (from 1024 to 2048).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' A future direction is to incorpo- rate more efficient attention mechanisms into the multi-scale attention function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' MS-A Local-A #Params Inference Speed (M) (ms/frame) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 186 \x14 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1) 225 \x14 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9) 191 \x14 \x14 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0) 232 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Ablating the parameter and computation overhead of individual attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We include implementation details covering several as- pects in this paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In addition, we include our source code in the supplementary material, containing the full im- plementation of our attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Training Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Group-Free baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' When applying our method to this baseline, we follow the original training settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Specifi- cally, on ScanNetV2, the models are trained for 400 epochs on 4 GPUs with a batchsize of 32 (8 on each GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We use the same optimizer with the same learning rates and weight decays as the baseline training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' On SUN RGB-D, models are trained for 600 epochs on 4 GPUs with the same learning rate and weight decay as the baseline training on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' RepSurf-U baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The official implementation and training details of this baseline are not published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We im- plement our own version of RepSurf-U detector, for which we mostly follow the training setup of Group-Free and have done a grid search for the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Different from Group-Free, we train RepSurf-U models on ScanNetV2 and SUN RGB-D using a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='01 for all model parameters, because we find it to achieve better performance on our reproduced RepSurf-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The learning rate and other hyperparameters remain the same as Group-Free on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' When applying our method to the reproduced model, we do not change the hyperparameter configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Bounding box parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In this paragraph, we include a brief introduction to the bounding box parameteri- zation used in our baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' First, the predicted box center ˆc for each object candidate q is obtained by adding an offset to the coordinate of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' In this way, by predicting the cen- ter, the actual prediction made by a detector is this offset value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The size ˆd of a box is the height, width, and depth dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' One way for predicting ˆd is to directly predict the values of H, W, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Another way is to divide a range of sizes into several bins and make a classification prediction that determines which “bin" the object belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The final size prediction is obtained by adding the quantized size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', the bin) with a “residual" term which is also predicted by the model with another prediction head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' The bounding box orientation ˆa is also parameterized as the combination of a quantized value and a residual term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Lastly, the prediction of the semantic label is a common classification problem that parameterizes a semantic label as a one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Size-Aware Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For a quantitative anal- ysis of the model’s performance on objects of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We implement our own size-aware evaluation met- rics, namely mAPS, mAPM and mAPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' For each metric, we only calculate the mAP score among objects that fall into the corresponding size category (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=', small, medium, or large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We conduct the size-aware evaluation on ScanNetV2, where we determine the threshold for dividing object size categories based on the statistics of this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Specifically, we take the 1201 training samples and record the volume (v = H × W × D) of every groundtruth bounding box of every sample (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Among a total of 15733 goundtruth bounding boxes, we take the 30th (v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='155) and 70th (v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='526) percentile as the thresholds for divid- ing small and large objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 11 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='526 1 2 3 4 5 Volume of the object bounding box 0 200 400 600 800 1000 Number of objects Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Volume distribution of the object groundtruth bounding boxes in ScanNetV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' We highlight the threshold of small objects (v <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='155, the 30th percentile) and large objects (v > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='526, the 70th percentile) methods backbone cab bed chair sofa tabl door wind bkshf pic cntr desk curt fridg showr toil sink bath ofurn mAP VoteNet [21] PointNet++ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 RepSurf-U12,512 (repd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=') PointNet++w2x 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 48.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Performance of mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 in each category on ScanNetV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' methods backbone cab bed chair sofa tabl door wind bkshf pic cntr desk curt fridg showr toil sink bath ofurn mAP VoteNet [21] PointNet++ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 H3DNet [54] 4×PointNet++ 20.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 GroupFree6,256 PointNet++ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 78.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Performance of mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 in each category on ScanNetV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 12 methods backbone bathtub bed bkshf chair desk drser nigtstd sofa table toilet mAP VoteNet [21] PointNet++ 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 H3DNet [54] 4×PointNet++ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 29.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 Pointformer [20] Pointformer 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 GroupFree6,256 PointNet++ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 66.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 RepSurf-U6,256 (repd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=') PointNet++ 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 w/ MS + Local PointNet++ 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Performance of mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='25 in each category on SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' methods backbone bathtub bed bkshf chair desk drser nigtstd sofa table toilet mAP VoteNet [21] PointNet++ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 H3DNet [54] 4×PointNet++ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 GroupFree6,256 PointNet++ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 w/ MS + Local PointNet++ 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='6 RepSurf-U6,256 PointNet++ 65.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 RepSurf-U6,256 (repd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=') PointNet++ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='8 11.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='5 Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' Performance of mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content='50 in each category on SUN RGB-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E0T4oBgHgl3EQfxwJo/content/2301.02650v1.pdf'} diff --git a/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf b/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fc07df2cce18c7a25cacf0cc9cac91cad81cde76 --- /dev/null +++ b/ANFKT4oBgHgl3EQfVi5k/content/2301.11788v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f7ebe37fb75b7f85c5a052e292b93d2cb72a53bf6ad734ea69ee4350640b1375 +size 785323 diff --git a/ANFKT4oBgHgl3EQfVi5k/vector_store/index.faiss b/ANFKT4oBgHgl3EQfVi5k/vector_store/index.faiss new file mode 100644 index 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Z. Pretel1, ∗ +1Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud, +150 URCA, Rio de Janeiro CEP 22290-180, RJ, Brazil +(Dated: January 10, 2023) +Within the framework of f(R, T) theories of gravity, we investigate the hydrostatic equilibrium of +anisotropic neutron stars with a physically relevant equation of state (EoS) for the radial pressure. +In particular, we focus on the f(R, T) = R + 2βT model, where β is a minimal coupling constant. +In the slowly rotating approximation, we derive the modified TOV equations and the expression for +the relativistic moment of inertia. The main properties of neutron stars, such as radius, mass and +moment of inertia, are studied in detail. Our results revel that the main consequence of the 2βT term +is a substantial increase in the surface radius for low enough central densities. Nevertheless, such +a term slightly modifies the total gravitational mass and moment of inertia of the slowly rotating +stars. Furthermore, the changes are noticeable when anisotropy is incorporated into the stellar fluid, +and it is possible to obtain higher masses that are consistent with the current observational data. +I. +INTRODUCTION +Despite the great success of General Relativity (GR) in +predicting various gravitational phenomena tested in the +solar system [1] and in strong-field situations (such as the +final stage of compact-object binaries [2, 3]), it could not +help to identify the nature of dark energy and other puz- +zles. In other words, there are still many open problems +in modern cosmology and it is well known that GR is not +the only theory of gravity [4]. Indeed, it has been shown +that GR is not renormalizable as a quantum field theory +unless higher-order curvature invariants are included in +its action [5, 6]. Furthermore, GR requires modifications +at small time and length scales or at energies comparable +with the Planck energy scales. In that regard, it has been +argued that the early-time inflation and the late-time ac- +celerated expansion of the Universe can be an effect of +the modification of the geometric theory formulated by +Einstein [7–10]. +One of the simplest ways to modify GR is by re- +placing the Ricci scalar R in the standard Einstein- +Hilbert action by an arbitrary function of R, this is, the +so-called f(R) theories of gravity [11, 12]. +Extensive +and detailed reviews on the cosmological implications +of such theories can be found in Refs. [13–16]. On the +other hand, at astrophysical level, these theories basically +change the Tolman-Oppenheimer-Volkoff (TOV) equa- +tions and hence the astrophysical properties of compact +stars, such as mass-radius relations, maximum masses, or +moment of inertia are somehow altered. See Ref. [17] for +a broad overview about relativistic and non-relativistic +stars within the context of modified theories of gravity +formulated in both metric and metric-affine approaches. +In most of the works reported in the literature about +internal structure of compact stars in GR and modified +theories of gravity it is very common to assume that such +stars are made up of an isotropic perfect fluid. Never- +∗ juanzarate@cbpf.br +theless, there are strong arguments indicating that the +impact of anisotropy (this is, unequal radial and tangen- +tial pressures) cannot be neglected when we deal with +nuclear matter at very high densities and pressures, for +instance, see Refs. [18–24] and references therein. In that +regard, it has been shown that the presence of anisotropy +can lead to significant changes in the main characteris- +tics of compact stars [21–23, 25–31]. Within the frame- +work of extended theories of gravity, it is also important +to mention that non-rotating anisotropic compact stars +have been recently studied by some authors in Refs. [32– +50]. In addition, in the context of scalar-tensor theory +of gravity, slowly rotating anisotropic neutron stars have +been investigated in Ref. [51]. +Harko and collaborators [52] have proposed a gener- +alization of f(R) modified theories of gravity in order +to introduce a coupling between geometry and matter, +namely f(R, T) gravity, where T denotes the trace of the +energy-momentum tensor. Indeed, the simplest and most +studied model involving a minimal matter-gravity cou- +pling is given by f(R, T) = R+2βT gravity. The cosmo- +logical aspects of this model have been recently explored +in Refs. [53–57], while other authors have investigated +the astrophysical consequences of the 2βT term on the +equilibrium structure of isotropic [58–65] and anisotropic +[37–42] compact stars. A characteristic of this model is +that R = 0 outside a compact star, and hence the ex- +terior spacetime is still described by the Schwarzschild +exterior solution. +As a result, it has been shown that +for high enough central densities the contributions of the +2βT term are irrelevant, whereas below a certain cen- +tral density value the radius of an isotropic compact star +undergoes substantial deviations from GR [62, 63]. +To determine the equilibrium configurations and mo- +ment of inertia of slowly rotating anisotropic stars up to +first order in the angular velocity, we will employ a phys- +ically motivated functional relation σ (defined as the dif- +ference between radial and tangential pressure) for the +anisotropy profile known in the literature as quasi-local +ansatz [25]. Moreover, we will follow a procedure anal- +ogous to that carried out by Hartle in GR [66] in order +arXiv:2301.02881v1 [gr-qc] 7 Jan 2023 + +2 +to obtain the modified version of the differential equation +which governs the difference between the angular velocity +of the star and the angular velocity of the local inertial +frames. +To achieve our results, the present work is organized +as follows: In Sec. II we briefly review f(R, T) gravity +and we present the corresponding relativistic equations +for the f(R, T) = R + 2βT model. In Sec. III we de- +rive the modified TOV equations for anisotropic stellar +configurations by adopting a non-rotating and slowly ro- +tating metric. Section IV presents a well-known EoS to +describe neutron stars as well as the anisotropy ansatz. +In Sec. V we discuss our numerical results, and finally, +our conclusions are presented in Sec. VI. In this paper +we will use a geometric unit system and the sign conven- +tion (−, +, +, +). However, our results will be given in +physical units. +II. +BASIC FORMALISM OF f(R, T) GRAVITY +A more general formulation of f(R) modified theories +of gravity consists in the inclusion of an explicit gravity- +matter coupling by means of an arbitrary function of the +Ricci scalar R and the trace of the energy-momentum +tensor T. Thus, the modified Einstein-Hilbert action in +f(R, T) gravity is given by [52] +S = +1 +16π +� +f(R, T)√−gd4x + +� +Lm +√−gd4x, +(1) +where g is the determinant of the spacetime metric gµν +and Lm denotes the Lagrangian density for matter fields. +The corresponding field equations in f(R, T) gravity can +be obtained from the variation of the action (1) with +respect to the metric: +fR(R, T)Rµν − 1 +2f(R, T)gµν + [gµν□ − ∇µ∇ν]fR(R, T) += 8πTµν − (Tµν + Θµν)fT (R, T), +(2) +where Rµν is the Ricci tensor, Tµν the energy-momentum +tensor, fR ≡ ∂f/∂R, fT ≡ ∂f/∂T, □ ≡ ∇µ∇µ is the +d’Alembertian operator with ∇µ standing for the covari- +ant derivative, and the tensor Θµν is defined in terms of +the variation of Tµν with respect to the metric, namely +Θµν ≡ gαβ δTαβ +δgµν += −2Tµν + gµνLm − 2gαβ +∂2Lm +∂gµν∂gαβ . +(3) +Just as in f(R) gravity [11, 12], in f(R, T) theories the +Ricci scalar is also a dynamical entity which is described +by a differential equation obtained by taking the trace of +the field equations (2), this is +3□fR(R, T) + RfR(R, T) − 2f(R, T) += 8πT − (T + Θ)fT (R, T), +(4) +where we have denoted Θ = Θ µ +µ . In addition, the four- +divergence of Eq. (2) yields [67] +∇µTµν = +fT (R, T) +8π − fT (R, T) +� +(Tµν + Θµν)∇µ ln fT (R, T) ++ ∇µΘµν − 1 +2gµν∇µT +� +. +(5) +In order to obtain numerical solutions that describe +compact stars, one has to specify the particular model of +f(R, T) gravity. In that regard, we consider the simplest +model involving a minimal matter-gravity coupling pro- +posed by Harko et al. [52], i.e. f(R, T) = R + 2βT grav- +ity, which has been the most studied model of f(R, T) +gravity at both astrophysical and cosmological scale. As +a consequence, Eqs. (2), (4) and (5) can be written as +follows +Gµν = 8πTµν + βTgµν − 2β(Tµν + Θµν), +(6) +R = −8πT − 2β(T − Θ), +(7) +∇µTµν = +2β +8π − 2β +� +∇µΘµν − 1 +2gµν∇µT +� +, +(8) +where Gµν is the Einstein tensor. +III. +MODIFIED TOV EQUATIONS +A. +Non-rotating stars +We shall assume that the matter source is described +by an anisotropic perfect fluid with energy density ρ, ra- +dial pressure pr and tangential pressure pt. Under theses +assumptions, the energy-momentum tensor is given by +Tµν = (ρ + pt)uµuν + ptgµν − σkµkν, +(9) +with uµ being the four-velocity of the fluid and which +satisfies the normalization property uµuµ = −1, kµ is a +unit radial four-vector so that kµkµ = 1, and σ ≡ pt − pr +is the anisotropy factor. +In addition, we consider that the interior spacetime +of the spherically symmetric stellar configuration is de- +scribed by the standard line element +ds2 = −e2ψdt2 + e2λdr2 + r2(dθ2 + sin2 θdφ2), +(10) +where xµ = (t, r, θ, φ) are the Schwarzschild-like coordi- +nates, and the metric potentials ψ and λ are functions +only of the radial coordinate in a hydrostatic equilib- +rium situation. Consequently, we can write uµ = e−ψδµ +0 , +kµ = e−λδµ +1 and the trace of the energy-momentum ten- +sor (9) takes the form T = −ρ + 3pr + 2σ. +Within the context of anisotropic fluids in f(R, T) +gravity, the most adopted choice in the literature for +the matter Lagrangian density is given by Lm = P, +where P ≡ (pr + 2pt)/3. +For more details about this + +3 +choice, see Refs. [37–40, 42]. Under this consideration, +Θµν = −2Tµν + Pgµν and Eqs. (6), (7) and (8) become +Gµν = 8πTµν + βTgµν + 2β(Tµν − Pgµν), +(11) +R = −8πT − 2β(3T − 4P), +(12) +∇µTµν = +2β +8π + 2β ∂ν +� +P − 1 +2T +� +. +(13) +For the metric (10) and energy-momentum tensor (9), +the non-zero components of the field equations (11) are +explicitly given by +1 +r2 +d +dr(re−2λ) − 1 +r2 = −8πρ + β +� +−3ρ + pr + 2 +3σ +� +, +(14) +e−2λ +�2 +r ψ′ + 1 +r2 +� +− 1 +r2 = 8πpr + β +� +−ρ + 3pr + 2 +3σ +� +, +(15) +e−2λ +� +ψ′′ + ψ′2 − ψ′λ′ + 1 +r (ψ′ − λ′) +� += 8π(pr + σ) + β +� +−ρ + 3pr + 8 +3σ +� +, +(16) +where the prime represents differentiation with respect +to the radial coordinate. Moreover, Eq. (13) implies that +dpr +dr = − (ρ + pr)ψ′ + 2 +r σ ++ +β +8π + 2β +d +dr +� +ρ − pr − 2 +3σ +� +. +(17) +Eq. (14) leads to +re−2λ = r − +� +r2 +� +8πρ + β +� +3ρ − pr − 2 +3σ +�� +dr, +(18) +or alternatively, +e−2λ = 1 − 2m +r , +(19) +where m(r) represents the gravitational mass within a +sphere of radius r, given by +m(r) = 4π +� r +0 +¯r2ρ(¯r)d¯r ++ β +2 +� r +0 +¯r2 +� +3ρ(¯r) − pr(¯r) − 2 +3σ(¯r) +� +d¯r. +(20) +At the surface, where the radial pressure vanishes, +M ≡ m(rsur) is the total mass of the anisotropic compact +star. From our anisotropic version (20), here we can see +that by making σ = 0 one recovers the mass function for +the isotropic case given in Ref. [63]. In view of Eq. (19), +from Eq. (15) we obtain +ψ′ = +�m +r2 + 4πrpr + βr +2 +� +−ρ + 3pr + 2 +3σ +�� +× +� +1 − 2m +r +�−1 +, +(21) +and hence the relativistic structure of an anisotropic com- +pact star within the context of f(R, T) = R+2βT gravity +is described by the modified TOV equations: +dm +dr = 4πr2ρ + βr2 +2 +� +3ρ − pr − 2 +3σ +� +, +(22) +dpr +dr = − ρ + pr +1 + a +�m +r2 + 4πrpr + βr +2 +� +3pr − ρ + 2 +3σ +�� +× +� +1 − 2m +r +�−1 ++ +a +1 + a +dρ +dr ++ +2 +1 + a +�σ +r − a +3 +dσ +dr +� +, +(23) +dψ +dr = +1 +ρ + pr +� +−(1 + a)dpr +dr + adρ +dr + 2 +�σ +r − a +3 +dσ +dr +�� +, +(24) +where we have defined a ≡ β/(8π + 2β). As expected, +the modified TOV equations in the isotropic scenario are +retrieved when pr = pt [63]. Furthermore, when the min- +imal coupling constant vanishes (this is, β = 0), we can +recover the standard TOV equations for anisotropic stars +in GR [23]. +Given an EoS for the radial pressure pr = pr(ρ) and +an anisotropy relation for σ, Eqs. (22) and (23) can be +integrated by guaranteeing regularity at the center of the +star and for a given value of central energy density. In +addition, according to Eq. (12), we notice that R = 0 in +the outer region of the star. This means that we can still +use the Schwarzschild vacuum solution to describe the +exterior spacetime so that the interior solution is matched +at the boundary r = rsur to the exterior Schwarzschild +solution. Thus, the system of equations (22)-(24) can be +solved by imposing the following boundary conditions +m(0) = 0, +ρ(0) = ρc, +ψ(rsur) = 1 +2 ln +� +1 − 2M +rsur +� +. +(25) +B. +Slowly rotating stars +In the slowly rotating approximation [66], i.e., when +rotational corrections appear at first order in the angu- +lar velocity of the stars Ω, the spacetime metric (10) is +replaced by its slowly rotating counterpart [66, 68] +ds2 = − e2ψ(r)dt2 + e2λ(r)dr2 + r2(dθ2 + sin2 θdφ2) +− 2ω(r, θ)r2 sin2 θdtdφ, +(26) +where ω(r, θ) stands for the angular velocity of the lo- +cal inertial frames dragged by the stellar rotation. +In +other words, if a particle is dropped from rest at a great +distance from the rotating star, the particle would expe- +rience an ever increasing drag in the direction of rotation +of the star as it approaches. In fact, here it is convenient +to define the difference ϖ ≡ Ω − ω as the coordinate an- +gular velocity of the fluid element at (r, θ) seen by the +freely falling observer [66]. + +4 +Since Ω is the angular velocity of the fluid as seen by an +observer at rest at some spacetime point (t, r, θ, φ), one +finds that the four-velocity up to linear terms in Ω is given +by uµ = (e−ψ, 0, 0, Ωe−ψ). To this order, the spherical +symmetry is still preserved and it is possible to extend +the validity of the TOV equations (22)-(24). Neverthe- +less, the 03-component of the field equations contributes +an additional differential equation for angular velocity +ω(r, θ). By retaining only first-order terms in the angu- +lar velocity, we have T03 = −[ϖ(ρ + pt) + ωpt]r2 sin2 θ +and hence Eq. (11) gives the following expression +G03 = − +� +2(4π + β)(ρ + pt)ϖ + 8πωpt ++β +� +−ρ + 1 +3pr + 8 +3pt +� +ω +� +r2 sin2 θ, +(27) +or alternatively, +eψ−λ +r4 +∂ +∂r +� +e−(ψ+λ)r4 ∂ϖ +∂r +� ++ +1 +r2 sin3 θ +∂ +∂θ +� +sin3 θ∂ϖ +∂θ +� += 4(4π + β)(ρ + pt)ϖ. +(28) +Following the procedure carried out by Hartle in GR +[66] and Staykov et al. in R2-gravity [68], we expand ϖ +in the form +ϖ(r, θ) = +∞ +� +l=1 +ϖl(r) +� −1 +sin θ +dPl +dθ +� +, +(29) +where Pl are Legendre polynomials. In view of Eq. (29), +we can write +∂ +∂θ +� +sin3 θ∂ϖ +∂θ +� += +� +l +ϖl(r) +� +(cos2 θ − sin2 θ)dPl +dθ +− sin θ cos θd2Pl +dθ2 − sin2 θd3Pl +dθ3 +� += +� +l +ϖl(r) [l(l + 1) − 2] sin2 θdPl +dθ , +(30) +where we have used the Legendre differential equation +d2Pl +dθ2 + cos θ +sin θ +dPl +dθ + l(l + 1)Pl = 0. +(31) +Thus, after substituting Eqs. (29) and (30) into (28), +we get +eψ−λ +r4 +d +dr +� +e−(ψ+λ)r4 dϖl +dr +� +− l(l + 1) − 2 +r2 +ϖl += 4(4π + β)(ρ + pt)ϖl. +(32) +At great distances from the stellar surface, where +spacetime must be asymptotically flat, the solution of +Eq. (32) assumes the form ϖl(r) → c1r−l−2 + c2rl−1. +Furthermore, the dragging angular velocity is expected +to be ω → 2J/r3 (or alternatively, ϖ → Ω − 2J/r3) for +r → ∞, where J is the angular momentum carried out +by the star (see Ref. [69] for more details). Therefore, by +comparison we can see that all coefficients in the Legen- +dre expansion vanish except for l = 1. This means that +ϖ is a function of r only, and Eq. (32) reduces to +eψ−λ +r4 +d +dr +� +e−(ψ+λ)r4 dϖ +dr +� += 4(4π + β)(ρ + pt)ϖ, +(33) +and taking into account that e−(ψ+λ) = 1 at the edge of +the star and beyond, the last equation can be integrated +to give +� +r4 dϖ +dr +� +rsur += 4(4π + β) +� rsur +0 +(ρ + pt)r4eλ−ψϖdr. (34) +From Eq. (34) we can obtain the relativistic moment +of inertia of a slowly rotating anisotropic compact star in +f(R, T) = R + 2βT gravity by means of expression +I = 2 +3(4π + β) +� rsur +0 +(ρ + pr + σ)eλ−ψr4 �ϖ +Ω +� +dr, +(35) +and hence the angular momentum J = IΩ can be written +as +J = 2 +3(4π + β) +� rsur +0 +ρ + pr + σ +� +1 − 2m/r +(Ω − ω)e−ψr4dr. (36) +It can be seen that the above result then reduces to the +pure general relativistic expression when β = 0. Further- +more, when both parameters β and σ vanish, Eq. (36) +reduces to the expression given in Ref. [69] for isotropic +compact stars in Einstein gravity. Analogously as in GR, +the differential equation (33) will be integrated from the +origin at r = 0 with an arbitrary choice of the central +value ϖ(0) and with vanishing slope, i.e., dϖ/dr = 0. +Once the solution for ϖ(r) is found, we can then com- +pute the moment of inertia via the integral (35). +IV. +EQUATION OF STATE AND ANISOTROPY +ANSATZ +Just as the construction of anisotropic compact stars +in GR, to close the system of Eqs. (22)-(24), one needs to +specify a barotropic EoS (which relates the radial pres- +sure to the mass density by means of equation pr = pr(ρ)) +and also assign an anisotropy function σ since there is +now an extra degree of freedom pt. Alternatively, it is +possible to assign an EoS for radial pressure and another +for tangential pressure. +For instance, an approach for +the study of anisotropic fluids has been recently carried +out within the context of Newtonian gravity in Ref. [70] +and in conventional GR [71], where both the radial and +tangential pressures satisfy a polytropic EoS. +In this work, we will follow the first procedure de- +scribed in the previous paragraph in order to deal + +5 +with anisotropic neutron stars within the framework of +f(R, T) gravity. Indeed, for radial pressure we use a well- +known and physically relevant EoS which is compatible +with the constraints of the GW170817 event (the first de- +tection of gravitational waves from a binary neutron star +inspiral [72]), namely, the soft SLy EoS [73]. This EoS +is based on the SLy effective nucleon-nucleon interaction, +which is suitable for the description of strong interactions +in the nucleon component of dense neutron-star matter. +Such unified EoS describes both the neutron-star crust +and the liquid core (which is assumed to be a “minimal” +npeµ composition), and it can be represented by the fol- +lowing analytical expression +ζ(ξ) = a1 + a2ξ + a3ξ3 +1 + a4ξ +f(a5(ξ − a6)) ++ (a7 + a8ξ)f(a9(a10 − ξ)) ++ (a11 + a12ξ)f(a13(a14 − ξ)) ++ (a15 + a16ξ)f(a17(a18 − ξ)), +(37) +where ζ ≡ log(pr/dyn cm−2), ξ ≡ log(ρ/g cm−3), and +f(x) ≡ 1/(ex + 1). The values ai are fitting parameters +and can be found in Ref. [74]. +In addition, we adopt the anisotropy ansatz proposed +by Horvat et al. [25] to model anisotropic matter inside +compact stars, namely +σ = αprµ = αpr(1 − e−2λ), +(38) +with µ(r) ≡ 2m/r being the compactness of the star. The +advantage of this ansatz is that the stellar fluid becomes +isotropic at the origin since µ ∼ r2 when r → 0. It is also +commonly known as quasi-local ansatz in the literature +[25], where α controls the amount of anisotropy inside +the star and in principle can assume positive or negative +values [23, 25, 33, 50, 51, 75, 76]. Note that in the Newto- +nian limit, when the pressure contribution to the energy +density is negligible, the effect of anisotropy vanishes in +the hydrostatic equilibrium equation. Regardless of the +particular functional form of the anisotropy model, here +we must emphasize that physically relevant solutions cor- +respond to pr, pt ≥ 0 for r ≤ rsur. +V. +NUMERICAL RESULTS AND DISCUSSION +Given an EoS for the radial pressure, we numerically +integrate the modified TOV equations (22)-(24) with +boundary conditions (25) from the stellar center to the +surface r = rsur where the radial pressure vanishes. In +addition, we have to specify a particular value for the cou- +pling constant β and for anisotropy parameter α which +appears in Eq. (38). For instance, for a central mass den- +sity ρc = 2.0×1018 kg/m3 with SLy EoS (37), Fig. 1 illus- +trates the mass function and anisotropy factor as func- +tions of the radial coordinate for β = −0.01 and several +values of α. The left plot reveals an increase in gravita- +tional mass and a decrease in radius as α increases. More- +over, from the right plot we can see that the anisotropy +vanishes at the center (which is a required condition in +order to guarantee regularity), is more pronounced in the +intermediate regions, and it vanishes again at the stellar +surface. +For the anisotropy function (38), the left panel of Fig. 2 +displays the mass-radius relations for anisotropic neutron +stars with SLy EoS in f(R, T) = R+2βT gravity for three +particular values of the coupling constant β and different +values of α. Here the total gravitational mass of each +configuration is given by M = m(rsur), and the isotropic +case in Einstein gravity has been included for compari- +son purposes by a black solid line. The mass-radius re- +lation exhibits substantial deviations from GR mainly +in the low-mass region. On the other hand, anisotropy +introduces considerable changes only in the high-mass re- +gion. We remark that the 2βT term together with the +presence of anisotropies (with positive values of α) al- +low us to obtain maximum masses bigger than 2.0 M⊙. +As a consequence, the introduction of anisotropies in +f(R, T) = R + 2βT gravity gives rise to massive neutron +stars that are in good agreement with the millisecond +pulsar observations [77, 78]. From NICER and XMM- +Newton data [79], the radius measurement for a 1.4 M⊙ +neutron star is 12.45 ± 0.65 km and, according to the +mass-radius diagram, our results consistently describe +this star when β = −0.01 (see blue curves). +Further- +more, it should be noted that the parameter β = −0.01 +is the one that best fits the mass-radius constraint from +the GW170817 event (see the filled cyan region). Nev- +ertheless, the massive pulsar J0740+6620 (whose radius +is 12.35 ± 0.75 km [79]) could be described only when +β = −0.03 and α = 0.4. +It is worth commenting that the value of the parame- +ter α could be constrained, but that will depend on the +particular compact star observed in the Universe. For in- +stance, the range α ∈ [−0.4, −0.2] consistently describes +the millisecond pulsar J1614-2230 regardless of the value +of β. However, for highly massive neutron stars whose +masses are greater than 2.0 M⊙, positive values of α will +be required. For PSR J0740+6620, whose gravitational +mass is 2.08 M⊙, the best value for α is 0.2. In fact, this +constraint will depend not only on the modified theory +of gravity but also on the equation of state adopted for +the radial pressure. +According to the right panel of Fig. 2, the parameter +β slightly modifies the total gravitational mass, however, +the effect of anisotropy introduces more relevant changes. +To better analyze the effects that arise as a result of the +modification of Einstein’s theory as well as the incorpo- +ration of anisotropies, in Fig. 3 we show the behavior +of the surface radius as a function of the central den- +sity. From the left plot we can conclude that the radius +is significantly altered due to the 2βT term in the low- +central-density region, while anisotropy slightly modifies +the radius of the stars. The right plot corresponds to the +pure general relativistic case and it can be observed that +the radius undergoes more significant modifications with +respect to its isotropic counterpart if the values for |α| + +6 +0 +2 +4 +6 +8 +10 +0.0 +0.5 +1.0 +1.5 +2.0 +r [km] +m [M⊙] +-0.6 -0.3 +0 +0.3 +0.6 +10.85 +10.92 +10.99 +α +rsur [km] +α +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0 +2 +4 +6 +8 +10 +-4 +-2 +0 +2 +4 +6 +r [km] +σ [1033 Pa] +α +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +FIG. 1. +Radial behaviour of the mass function (left panel) and the anisotropy factor (right panel) in the framework of +f(R, T) = R + 2βT gravity for β = −0.01 and different values of α. SLy EoS (37) is valid from 1011 kg/m3 up to the maximum +density reachable within neutron stars [73], and in these plots we have considered ρc = 2.0 × 1018 kg/m3. The isotropic case is +recovered when the anisotropy parameter vanishes (this is, α = 0). We can observe that the gravitational mass increases and +the radius decreases as α increases. In addition, the anisotropy is more pronounced in the intermediate regions and vanishes +at the stellar center as expected. +are larger than those considered in the left plot. +Eq. (33) is first solved in the interior region from the +center to the surface of the star by considering an arbi- +trary value for ϖ and with vanishing slope at r = 0. Then +the same equation is solved in exterior spacetime from the +surface to a sufficiently far distance from the star where +ϖ(r) → Ω. In Fig. 4 we display the radial profile of these +solutions for the central mass density considered above. +We observe that ϖ(r) is an increasing function of the ra- +dial coordinate, whereas ω(r) is a decreasing function and +hence the largest rate of dragging of local inertial frames +always occurs at the stellar center. Furthermore, appre- +ciable effects (mainly in the interior region of the stellar +configuration) can be noted on frame-dragging angular +velocity due to the inclusion of anisotropies. +Once ϖ(r) is known for each stellar configuration, we +can then determine the moment of inertia by means of +Eq. (35). Figure 5 presents the moment of inertia as a +function of the total gravitational mass in GR and within +the context of f(R, T) = R + 2βT gravity for β = −0.01. +It can be observed that the moment of inertia undergoes +irrelevant changes from GR, however, it can change sig- +nificantly due to anisotropies in the high-mass region. +VI. +CONCLUSIONS +In this work we have investigated slowly rotating +anisotropic neutron stars in f(R, T) = R + 2βT grav- +ity, where the degree of modification with respect to GR +is measured by the coupling constant β. The modified +TOV equations and moment of inertia have been derived +within the context of anisotropic fluids by retaining only +first-order terms in the angular velocity as measured by +a distant observer (Ω). Notice that, within this linear +approximation, the moment of inertia can be calculated +from the structure of a non-rotating configuration since +the TOV equations describing the static background are +still valid. In addition, we have adopted the anisotropy +ansatz proposed by Horvat and collaborators [25], where +appears a dimensionless parameter α which measures the +degree of anisotropy within the neutron star. +We have analyzed the consequences of the extra term +2βT together with anisotropies on the properties of neu- +tron stars such as radius, mass, frame-dragging angular +velocity and moment of inertia. Indeed, our results re- +veal that the radius deviates considerably from GR in +the low-central-density region, however, the total gravi- +tational mass and the moment of inertia undergo slight +modifications due to the influence of the effects generated +by the minimal matter-gravity coupling. Furthermore, +the presence of anisotropy generates substantial changes +both in the mass and in the moment of inertia with re- +spect to the isotropic case. The appreciable effects due +to the inclusion of anisotropy occur mainly in the higher- +central-density region, this is, for large masses (near the +maximum-mass configuration). +ACKNOWLEDGMENTS +JMZP acknowledges financial support from the PCI +program of the Brazilian agency “Conselho Nacional de +Desenvolvimento Cient´ıfico e Tecnol´ogico”–CNPq. + +7 +PSR J1614-2230 +PSR J0740+6620 +GW170817 +β = 0 +β = -0.01 +β = -0.02 +β = -0.03 +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +10 +12 +14 +16 +0.5 +1.0 +1.5 +2.0 +rsur [km] +M [M⊙] +17.8 +18.0 +18.2 +18.4 +18.6 +18.8 +0.5 +1.0 +1.5 +2.0 +Log ρc [kg/m3] +M [M⊙] +1.32 +1.36 +1.4 +17.94 +17.98 +18.02 +FIG. 2. +Mass-radius diagrams (left panel) and mass-central density relations (right panel) for anisotropic neutron stars with +SLy EoS (37) in f(R, T) = R + 2βT gravity for β = −0.01 (blue curves), β = −0.02 (orange curves) and β = −0.03 (in green). +The solid lines correspond to α = 0 (that is, isotropic solutions), and the pure GR case (β = 0) is shown in both plots as a +benchmark by a black line. The magenta horizontal band stands for the observational measurement for the millisecond pulsar +J1614-2230 reported in Ref. [77]. The filled cyan region is the mass-radius constraint from the GW170817 event. The Radius +of PSR J0740+6620 from NICER and XMM-Newton Data [79] is indicated by the top brown dot with their respective error +bars. Moreover, the bottom brown dot represents the radius estimate for a 1.4 M⊙ neutron star [79]. +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +17.6 +17.8 +18.0 +18.2 +18.4 +18.6 +18.8 +10 +15 +20 +25 +30 +Log ρc [kg/m3] +rsur [km] +17.76 +17.82 +17.88 +13 +15 +17 +α = -1.0 +α = -0.5 +α = 0 +α = 0.5 +α = 1.0 +17.6 +17.8 +18.0 +18.2 +18.4 +18.6 +9 +10 +11 +12 +13 +14 +Log ρc [kg/m3] +rsur [km] +FIG. 3. +Surface radius as a function of the central mass density. On the left panel, different styles and colors of the curves +correspond to different values of the parameters β and α as in Fig. 2. The most substantial deviations from GR take place at +low central densities, whereas for large central densities the changes are very slight due to the 2βT term. On the right panel +we display the modifications of the radius due to the inclusion of anisotropies when β = 0, where we have considered larger +values for |α| in order to appreciate the changes in radius as a consequence of anisotropy. We can mainly observe three regions +where the radius can decrease or increase depending on the value of α. + +8 +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +0 +10 +20 +30 +40 +0.4 +0.6 +0.8 +1.0 +r [km] +ϖ/Ω +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +0 +10 +20 +30 +40 +0.0 +0.2 +0.4 +0.6 +r [km] +ω/Ω +FIG. 4. +Left panel: Numerical solution of the differential equation (33) for a given central mass density ρc = 2.0 × 1018 kg/m3 +in f(R, T) = R + 2βT gravity with β = −0.01 and different values of the free parameter α. The dotted lines represent the +solutions of the exterior region, and as expected ϖ → Ω at great distances from the stellar surface. Right panel: Ratio of +frame-dragging angular velocity to the angular velocity of the stars, namely ω(r)/Ω = 1 − ϖ(r)/Ω. Notice that the solution of +the exterior problem provides an asymptotic behavior of ω(r). +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +0.5 +1.0 +1.5 +2.0 +0.5 +1.0 +1.5 +2.0 +M [M⊙] +I [1038 kg.m2] +α = -0.4 +α = -0.2 +α = 0 +α = 0.2 +α = 0.4 +1.6 +1.7 +1.8 +1.9 +2.0 +1.7 +1.8 +1.9 +2.0 +2.1 +M [M⊙] +I [1038 kg.m2] +FIG. 5. +Left panel: Moment of inertia of slowly rotating anisotropic neutron stars as a function of the total mass within the +context of f(R, T) = R + 2βT gravity for β = −0.01 in blue. Different styles of the curves correspond to different values of the +anisotropy parameter α. Results based on Einstein’s theory have been included for comparison purposes and are represented +by the black curves. 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Lett. 918, L28 (2021). + diff --git a/CNE1T4oBgHgl3EQfDwNk/content/tmp_files/load_file.txt b/CNE1T4oBgHgl3EQfDwNk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c9e1b82631603102281b292aef3be97a9a3b665 --- /dev/null +++ b/CNE1T4oBgHgl3EQfDwNk/content/tmp_files/load_file.txt @@ -0,0 +1,878 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf,len=877 +page_content='Moment of inertia of slowly rotating anisotropic neutron stars in f(R, T) gravity Juan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Pretel1, ∗ 1Centro Brasileiro de Pesquisas F´ısicas, Rua Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Xavier Sigaud, 150 URCA, Rio de Janeiro CEP 22290-180, RJ, Brazil (Dated: January 10, 2023) Within the framework of f(R, T) theories of gravity, we investigate the hydrostatic equilibrium of anisotropic neutron stars with a physically relevant equation of state (EoS) for the radial pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In particular, we focus on the f(R, T) = R + 2βT model, where β is a minimal coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In the slowly rotating approximation, we derive the modified TOV equations and the expression for the relativistic moment of inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The main properties of neutron stars, such as radius, mass and moment of inertia, are studied in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Our results revel that the main consequence of the 2βT term is a substantial increase in the surface radius for low enough central densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Nevertheless, such a term slightly modifies the total gravitational mass and moment of inertia of the slowly rotating stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Furthermore, the changes are noticeable when anisotropy is incorporated into the stellar fluid, and it is possible to obtain higher masses that are consistent with the current observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' INTRODUCTION Despite the great success of General Relativity (GR) in predicting various gravitational phenomena tested in the solar system [1] and in strong-field situations (such as the final stage of compact-object binaries [2, 3]), it could not help to identify the nature of dark energy and other puz- zles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In other words, there are still many open problems in modern cosmology and it is well known that GR is not the only theory of gravity [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Indeed, it has been shown that GR is not renormalizable as a quantum field theory unless higher-order curvature invariants are included in its action [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Furthermore, GR requires modifications at small time and length scales or at energies comparable with the Planck energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In that regard, it has been argued that the early-time inflation and the late-time ac- celerated expansion of the Universe can be an effect of the modification of the geometric theory formulated by Einstein [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' One of the simplest ways to modify GR is by re- placing the Ricci scalar R in the standard Einstein- Hilbert action by an arbitrary function of R, this is, the so-called f(R) theories of gravity [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Extensive and detailed reviews on the cosmological implications of such theories can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [13–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' On the other hand, at astrophysical level, these theories basically change the Tolman-Oppenheimer-Volkoff (TOV) equa- tions and hence the astrophysical properties of compact stars, such as mass-radius relations, maximum masses, or moment of inertia are somehow altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [17] for a broad overview about relativistic and non-relativistic stars within the context of modified theories of gravity formulated in both metric and metric-affine approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In most of the works reported in the literature about internal structure of compact stars in GR and modified theories of gravity it is very common to assume that such stars are made up of an isotropic perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Never- ∗ juanzarate@cbpf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='br theless, there are strong arguments indicating that the impact of anisotropy (this is, unequal radial and tangen- tial pressures) cannot be neglected when we deal with nuclear matter at very high densities and pressures, for instance, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [18–24] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In that regard, it has been shown that the presence of anisotropy can lead to significant changes in the main characteris- tics of compact stars [21–23, 25–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Within the frame- work of extended theories of gravity, it is also important to mention that non-rotating anisotropic compact stars have been recently studied by some authors in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [32– 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In addition, in the context of scalar-tensor theory of gravity, slowly rotating anisotropic neutron stars have been investigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Harko and collaborators [52] have proposed a gener- alization of f(R) modified theories of gravity in order to introduce a coupling between geometry and matter, namely f(R, T) gravity, where T denotes the trace of the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Indeed, the simplest and most studied model involving a minimal matter-gravity cou- pling is given by f(R, T) = R+2βT gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The cosmo- logical aspects of this model have been recently explored in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [53–57], while other authors have investigated the astrophysical consequences of the 2βT term on the equilibrium structure of isotropic [58–65] and anisotropic [37–42] compact stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' A characteristic of this model is that R = 0 outside a compact star, and hence the ex- terior spacetime is still described by the Schwarzschild exterior solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' As a result, it has been shown that for high enough central densities the contributions of the 2βT term are irrelevant, whereas below a certain cen- tral density value the radius of an isotropic compact star undergoes substantial deviations from GR [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' To determine the equilibrium configurations and mo- ment of inertia of slowly rotating anisotropic stars up to first order in the angular velocity, we will employ a phys- ically motivated functional relation σ (defined as the dif- ference between radial and tangential pressure) for the anisotropy profile known in the literature as quasi-local ansatz [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Moreover, we will follow a procedure anal- ogous to that carried out by Hartle in GR [66] in order arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='02881v1 [gr-qc] 7 Jan 2023 2 to obtain the modified version of the differential equation which governs the difference between the angular velocity of the star and the angular velocity of the local inertial frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' To achieve our results, the present work is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' II we briefly review f(R, T) gravity and we present the corresponding relativistic equations for the f(R, T) = R + 2βT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' III we de- rive the modified TOV equations for anisotropic stellar configurations by adopting a non-rotating and slowly ro- tating metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Section IV presents a well-known EoS to describe neutron stars as well as the anisotropy ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' V we discuss our numerical results, and finally, our conclusions are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In this paper we will use a geometric unit system and the sign conven- tion (−, +, +, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' However, our results will be given in physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' BASIC FORMALISM OF f(R, T) GRAVITY A more general formulation of f(R) modified theories of gravity consists in the inclusion of an explicit gravity- matter coupling by means of an arbitrary function of the Ricci scalar R and the trace of the energy-momentum tensor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Thus, the modified Einstein-Hilbert action in f(R, T) gravity is given by [52] S = 1 16π � f(R, T)√−gd4x + � Lm √−gd4x, (1) where g is the determinant of the spacetime metric gµν and Lm denotes the Lagrangian density for matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The corresponding field equations in f(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' T) gravity can be obtained from the variation of the action (1) with respect to the metric: fR(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' T)Rµν − 1 2f(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' T)gµν + [gµν□ − ∇µ∇ν]fR(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' T) = 8πTµν − (Tµν + Θµν)fT (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (2) where Rµν is the Ricci tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Tµν the energy-momentum tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' fR ≡ ∂f/∂R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' fT ≡ ∂f/∂T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' □ ≡ ∇µ∇µ is the d’Alembertian operator with ∇µ standing for the covari- ant derivative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' and the tensor Θµν is defined in terms of the variation of Tµν with respect to the metric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' namely Θµν ≡ gαβ δTαβ δgµν = −2Tµν + gµνLm − 2gαβ ∂2Lm ∂gµν∂gαβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (3) Just as in f(R) gravity [11, 12], in f(R, T) theories the Ricci scalar is also a dynamical entity which is described by a differential equation obtained by taking the trace of the field equations (2), this is 3□fR(R, T) + RfR(R, T) − 2f(R, T) = 8πT − (T + Θ)fT (R, T), (4) where we have denoted Θ = Θ µ µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In addition, the four- divergence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (2) yields [67] ∇µTµν = fT (R, T) 8π − fT (R, T) � (Tµν + Θµν)∇µ ln fT (R, T) + ∇µΘµν − 1 2gµν∇µT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (5) In order to obtain numerical solutions that describe compact stars, one has to specify the particular model of f(R, T) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In that regard, we consider the simplest model involving a minimal matter-gravity coupling pro- posed by Harko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [52], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' f(R, T) = R + 2βT grav- ity, which has been the most studied model of f(R, T) gravity at both astrophysical and cosmological scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' As a consequence, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (2), (4) and (5) can be written as follows Gµν = 8πTµν + βTgµν − 2β(Tµν + Θµν), (6) R = −8πT − 2β(T − Θ), (7) ∇µTµν = 2β 8π − 2β � ∇µΘµν − 1 2gµν∇µT � , (8) where Gµν is the Einstein tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' MODIFIED TOV EQUATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Non-rotating stars We shall assume that the matter source is described by an anisotropic perfect fluid with energy density ρ, ra- dial pressure pr and tangential pressure pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Under theses assumptions, the energy-momentum tensor is given by Tµν = (ρ + pt)uµuν + ptgµν − σkµkν, (9) with uµ being the four-velocity of the fluid and which satisfies the normalization property uµuµ = −1, kµ is a unit radial four-vector so that kµkµ = 1, and σ ≡ pt − pr is the anisotropy factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In addition, we consider that the interior spacetime of the spherically symmetric stellar configuration is de- scribed by the standard line element ds2 = −e2ψdt2 + e2λdr2 + r2(dθ2 + sin2 θdφ2), (10) where xµ = (t, r, θ, φ) are the Schwarzschild-like coordi- nates, and the metric potentials ψ and λ are functions only of the radial coordinate in a hydrostatic equilib- rium situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Consequently, we can write uµ = e−ψδµ 0 , kµ = e−λδµ 1 and the trace of the energy-momentum ten- sor (9) takes the form T = −ρ + 3pr + 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Within the context of anisotropic fluids in f(R, T) gravity, the most adopted choice in the literature for the matter Lagrangian density is given by Lm = P, where P ≡ (pr + 2pt)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' For more details about this 3 choice, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [37–40, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Under this consideration, Θµν = −2Tµν + Pgµν and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (6), (7) and (8) become Gµν = 8πTµν + βTgµν + 2β(Tµν − Pgµν), (11) R = −8πT − 2β(3T − 4P), (12) ∇µTµν = 2β 8π + 2β ∂ν � P − 1 2T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (13) For the metric (10) and energy-momentum tensor (9), the non-zero components of the field equations (11) are explicitly given by 1 r2 d dr(re−2λ) − 1 r2 = −8πρ + β � −3ρ + pr + 2 3σ � , (14) e−2λ �2 r ψ′ + 1 r2 � − 1 r2 = 8πpr + β � −ρ + 3pr + 2 3σ � , (15) e−2λ � ψ′′ + ψ′2 − ψ′λ′ + 1 r (ψ′ − λ′) � = 8π(pr + σ) + β � −ρ + 3pr + 8 3σ � , (16) where the prime represents differentiation with respect to the radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Moreover, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (13) implies that dpr dr = − (ρ + pr)ψ′ + 2 r σ + β 8π + 2β d dr � ρ − pr − 2 3σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (17) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (14) leads to re−2λ = r − � r2 � 8πρ + β � 3ρ − pr − 2 3σ �� dr, (18) or alternatively, e−2λ = 1 − 2m r , (19) where m(r) represents the gravitational mass within a sphere of radius r, given by m(r) = 4π � r 0 ¯r2ρ(¯r)d¯r + β 2 � r 0 ¯r2 � 3ρ(¯r) − pr(¯r) − 2 3σ(¯r) � d¯r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (20) At the surface, where the radial pressure vanishes, M ≡ m(rsur) is the total mass of the anisotropic compact star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' From our anisotropic version (20), here we can see that by making σ = 0 one recovers the mass function for the isotropic case given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In view of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (19), from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (15) we obtain ψ′ = �m r2 + 4πrpr + βr 2 � −ρ + 3pr + 2 3σ �� × � 1 − 2m r �−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (21) and hence the relativistic structure of an anisotropic com- pact star within the context of f(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' T) = R+2βT gravity is described by the modified TOV equations: dm dr = 4πr2ρ + βr2 2 � 3ρ − pr − 2 3σ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (22) dpr dr = − ρ + pr 1 + a �m r2 + 4πrpr + βr 2 � 3pr − ρ + 2 3σ �� × � 1 − 2m r �−1 + a 1 + a dρ dr + 2 1 + a �σ r − a 3 dσ dr � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (23) dψ dr = 1 ρ + pr � −(1 + a)dpr dr + adρ dr + 2 �σ r − a 3 dσ dr �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (24) where we have defined a ≡ β/(8π + 2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' As expected, the modified TOV equations in the isotropic scenario are retrieved when pr = pt [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Furthermore, when the min- imal coupling constant vanishes (this is, β = 0), we can recover the standard TOV equations for anisotropic stars in GR [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Given an EoS for the radial pressure pr = pr(ρ) and an anisotropy relation for σ, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (22) and (23) can be integrated by guaranteeing regularity at the center of the star and for a given value of central energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In addition, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (12), we notice that R = 0 in the outer region of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' This means that we can still use the Schwarzschild vacuum solution to describe the exterior spacetime so that the interior solution is matched at the boundary r = rsur to the exterior Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Thus, the system of equations (22)-(24) can be solved by imposing the following boundary conditions m(0) = 0, ρ(0) = ρc, ψ(rsur) = 1 2 ln � 1 − 2M rsur � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (25) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Slowly rotating stars In the slowly rotating approximation [66], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=', when rotational corrections appear at first order in the angu- lar velocity of the stars Ω, the spacetime metric (10) is replaced by its slowly rotating counterpart [66, 68] ds2 = − e2ψ(r)dt2 + e2λ(r)dr2 + r2(dθ2 + sin2 θdφ2) − 2ω(r, θ)r2 sin2 θdtdφ, (26) where ω(r, θ) stands for the angular velocity of the lo- cal inertial frames dragged by the stellar rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In other words, if a particle is dropped from rest at a great distance from the rotating star, the particle would expe- rience an ever increasing drag in the direction of rotation of the star as it approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In fact, here it is convenient to define the difference ϖ ≡ Ω − ω as the coordinate an- gular velocity of the fluid element at (r, θ) seen by the freely falling observer [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 4 Since Ω is the angular velocity of the fluid as seen by an observer at rest at some spacetime point (t, r, θ, φ), one finds that the four-velocity up to linear terms in Ω is given by uµ = (e−ψ, 0, 0, Ωe−ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' To this order, the spherical symmetry is still preserved and it is possible to extend the validity of the TOV equations (22)-(24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Neverthe- less, the 03-component of the field equations contributes an additional differential equation for angular velocity ω(r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' By retaining only first-order terms in the angu- lar velocity, we have T03 = −[ϖ(ρ + pt) + ωpt]r2 sin2 θ and hence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (11) gives the following expression G03 = − � 2(4π + β)(ρ + pt)ϖ + 8πωpt +β � −ρ + 1 3pr + 8 3pt � ω � r2 sin2 θ, (27) or alternatively, eψ−λ r4 ∂ ∂r � e−(ψ+λ)r4 ∂ϖ ∂r � + 1 r2 sin3 θ ∂ ∂θ � sin3 θ∂ϖ ∂θ � = 4(4π + β)(ρ + pt)ϖ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (28) Following the procedure carried out by Hartle in GR [66] and Staykov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' in R2-gravity [68], we expand ϖ in the form ϖ(r, θ) = ∞ � l=1 ϖl(r) � −1 sin θ dPl dθ � , (29) where Pl are Legendre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In view of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (29), we can write ∂ ∂θ � sin3 θ∂ϖ ∂θ � = � l ϖl(r) � (cos2 θ − sin2 θ)dPl dθ − sin θ cos θd2Pl dθ2 − sin2 θd3Pl dθ3 � = � l ϖl(r) [l(l + 1) − 2] sin2 θdPl dθ , (30) where we have used the Legendre differential equation d2Pl dθ2 + cos θ sin θ dPl dθ + l(l + 1)Pl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (31) Thus, after substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (29) and (30) into (28), we get eψ−λ r4 d dr � e−(ψ+λ)r4 dϖl dr � − l(l + 1) − 2 r2 ϖl = 4(4π + β)(ρ + pt)ϖl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (32) At great distances from the stellar surface, where spacetime must be asymptotically flat, the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (32) assumes the form ϖl(r) → c1r−l−2 + c2rl−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Furthermore, the dragging angular velocity is expected to be ω → 2J/r3 (or alternatively, ϖ → Ω − 2J/r3) for r → ∞, where J is the angular momentum carried out by the star (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [69] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Therefore, by comparison we can see that all coefficients in the Legen- dre expansion vanish except for l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' This means that ϖ is a function of r only, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (32) reduces to eψ−λ r4 d dr � e−(ψ+λ)r4 dϖ dr � = 4(4π + β)(ρ + pt)ϖ, (33) and taking into account that e−(ψ+λ) = 1 at the edge of the star and beyond, the last equation can be integrated to give � r4 dϖ dr � rsur = 4(4π + β) � rsur 0 (ρ + pt)r4eλ−ψϖdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (34) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (34) we can obtain the relativistic moment of inertia of a slowly rotating anisotropic compact star in f(R, T) = R + 2βT gravity by means of expression I = 2 3(4π + β) � rsur 0 (ρ + pr + σ)eλ−ψr4 �ϖ Ω � dr, (35) and hence the angular momentum J = IΩ can be written as J = 2 3(4π + β) � rsur 0 ρ + pr + σ � 1 − 2m/r (Ω − ω)e−ψr4dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (36) It can be seen that the above result then reduces to the pure general relativistic expression when β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Further- more, when both parameters β and σ vanish, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (36) reduces to the expression given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [69] for isotropic compact stars in Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Analogously as in GR, the differential equation (33) will be integrated from the origin at r = 0 with an arbitrary choice of the central value ϖ(0) and with vanishing slope, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=', dϖ/dr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Once the solution for ϖ(r) is found, we can then com- pute the moment of inertia via the integral (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' EQUATION OF STATE AND ANISOTROPY ANSATZ Just as the construction of anisotropic compact stars in GR, to close the system of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (22)-(24), one needs to specify a barotropic EoS (which relates the radial pres- sure to the mass density by means of equation pr = pr(ρ)) and also assign an anisotropy function σ since there is now an extra degree of freedom pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Alternatively, it is possible to assign an EoS for radial pressure and another for tangential pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' For instance, an approach for the study of anisotropic fluids has been recently carried out within the context of Newtonian gravity in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [70] and in conventional GR [71], where both the radial and tangential pressures satisfy a polytropic EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In this work, we will follow the first procedure de- scribed in the previous paragraph in order to deal 5 with anisotropic neutron stars within the framework of f(R, T) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Indeed, for radial pressure we use a well- known and physically relevant EoS which is compatible with the constraints of the GW170817 event (the first de- tection of gravitational waves from a binary neutron star inspiral [72]), namely, the soft SLy EoS [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' This EoS is based on the SLy effective nucleon-nucleon interaction, which is suitable for the description of strong interactions in the nucleon component of dense neutron-star matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Such unified EoS describes both the neutron-star crust and the liquid core (which is assumed to be a “minimal” npeµ composition), and it can be represented by the fol- lowing analytical expression ζ(ξ) = a1 + a2ξ + a3ξ3 1 + a4ξ f(a5(ξ − a6)) + (a7 + a8ξ)f(a9(a10 − ξ)) + (a11 + a12ξ)f(a13(a14 − ξ)) + (a15 + a16ξ)f(a17(a18 − ξ)), (37) where ζ ≡ log(pr/dyn cm−2), ξ ≡ log(ρ/g cm−3), and f(x) ≡ 1/(ex + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The values ai are fitting parameters and can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In addition, we adopt the anisotropy ansatz proposed by Horvat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [25] to model anisotropic matter inside compact stars, namely σ = αprµ = αpr(1 − e−2λ), (38) with µ(r) ≡ 2m/r being the compactness of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The advantage of this ansatz is that the stellar fluid becomes isotropic at the origin since µ ∼ r2 when r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' It is also commonly known as quasi-local ansatz in the literature [25], where α controls the amount of anisotropy inside the star and in principle can assume positive or negative values [23, 25, 33, 50, 51, 75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Note that in the Newto- nian limit, when the pressure contribution to the energy density is negligible, the effect of anisotropy vanishes in the hydrostatic equilibrium equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Regardless of the particular functional form of the anisotropy model, here we must emphasize that physically relevant solutions cor- respond to pr, pt ≥ 0 for r ≤ rsur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' NUMERICAL RESULTS AND DISCUSSION Given an EoS for the radial pressure, we numerically integrate the modified TOV equations (22)-(24) with boundary conditions (25) from the stellar center to the surface r = rsur where the radial pressure vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In addition, we have to specify a particular value for the cou- pling constant β and for anisotropy parameter α which appears in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' For instance, for a central mass den- sity ρc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0×1018 kg/m3 with SLy EoS (37), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 1 illus- trates the mass function and anisotropy factor as func- tions of the radial coordinate for β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='01 and several values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The left plot reveals an increase in gravita- tional mass and a decrease in radius as α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' More- over, from the right plot we can see that the anisotropy vanishes at the center (which is a required condition in order to guarantee regularity), is more pronounced in the intermediate regions, and it vanishes again at the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' For the anisotropy function (38), the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 2 displays the mass-radius relations for anisotropic neutron stars with SLy EoS in f(R, T) = R+2βT gravity for three particular values of the coupling constant β and different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Here the total gravitational mass of each configuration is given by M = m(rsur), and the isotropic case in Einstein gravity has been included for compari- son purposes by a black solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The mass-radius re- lation exhibits substantial deviations from GR mainly in the low-mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' On the other hand, anisotropy introduces considerable changes only in the high-mass re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' We remark that the 2βT term together with the presence of anisotropies (with positive values of α) al- low us to obtain maximum masses bigger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' As a consequence, the introduction of anisotropies in f(R, T) = R + 2βT gravity gives rise to massive neutron stars that are in good agreement with the millisecond pulsar observations [77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' From NICER and XMM- Newton data [79], the radius measurement for a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 M⊙ neutron star is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='65 km and, according to the mass-radius diagram, our results consistently describe this star when β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='01 (see blue curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Further- more, it should be noted that the parameter β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='01 is the one that best fits the mass-radius constraint from the GW170817 event (see the filled cyan region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Nev- ertheless, the massive pulsar J0740+6620 (whose radius is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='75 km [79]) could be described only when β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='03 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' It is worth commenting that the value of the parame- ter α could be constrained, but that will depend on the particular compact star observed in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' For in- stance, the range α ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2] consistently describes the millisecond pulsar J1614-2230 regardless of the value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' However, for highly massive neutron stars whose masses are greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 M⊙, positive values of α will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' For PSR J0740+6620, whose gravitational mass is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='08 M⊙, the best value for α is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In fact, this constraint will depend not only on the modified theory of gravity but also on the equation of state adopted for the radial pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' According to the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 2, the parameter β slightly modifies the total gravitational mass, however, the effect of anisotropy introduces more relevant changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' To better analyze the effects that arise as a result of the modification of Einstein’s theory as well as the incorpo- ration of anisotropies, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 3 we show the behavior of the surface radius as a function of the central den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' From the left plot we can conclude that the radius is significantly altered due to the 2βT term in the low- central-density region, while anisotropy slightly modifies the radius of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The right plot corresponds to the pure general relativistic case and it can be observed that the radius undergoes more significant modifications with respect to its isotropic counterpart if the values for |α| 6 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 r [km] m [M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='85 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='92 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='99 α rsur [km] α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 0 2 4 6 8 10 4 2 0 2 4 6 r [km] σ [1033 Pa] α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Radial behaviour of the mass function (left panel) and the anisotropy factor (right panel) in the framework of f(R, T) = R + 2βT gravity for β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='01 and different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' SLy EoS (37) is valid from 1011 kg/m3 up to the maximum density reachable within neutron stars [73], and in these plots we have considered ρc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 × 1018 kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The isotropic case is recovered when the anisotropy parameter vanishes (this is, α = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' We can observe that the gravitational mass increases and the radius decreases as α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In addition, the anisotropy is more pronounced in the intermediate regions and vanishes at the stellar center as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' are larger than those considered in the left plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (33) is first solved in the interior region from the center to the surface of the star by considering an arbi- trary value for ϖ and with vanishing slope at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Then the same equation is solved in exterior spacetime from the surface to a sufficiently far distance from the star where ϖ(r) → Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 4 we display the radial profile of these solutions for the central mass density considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' We observe that ϖ(r) is an increasing function of the ra- dial coordinate, whereas ω(r) is a decreasing function and hence the largest rate of dragging of local inertial frames always occurs at the stellar center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Furthermore, appre- ciable effects (mainly in the interior region of the stellar configuration) can be noted on frame-dragging angular velocity due to the inclusion of anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Once ϖ(r) is known for each stellar configuration, we can then determine the moment of inertia by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Figure 5 presents the moment of inertia as a function of the total gravitational mass in GR and within the context of f(R, T) = R + 2βT gravity for β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' It can be observed that the moment of inertia undergoes irrelevant changes from GR, however, it can change sig- nificantly due to anisotropies in the high-mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' CONCLUSIONS In this work we have investigated slowly rotating anisotropic neutron stars in f(R, T) = R + 2βT grav- ity, where the degree of modification with respect to GR is measured by the coupling constant β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The modified TOV equations and moment of inertia have been derived within the context of anisotropic fluids by retaining only first-order terms in the angular velocity as measured by a distant observer (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Notice that, within this linear approximation, the moment of inertia can be calculated from the structure of a non-rotating configuration since the TOV equations describing the static background are still valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' In addition, we have adopted the anisotropy ansatz proposed by Horvat and collaborators [25], where appears a dimensionless parameter α which measures the degree of anisotropy within the neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' We have analyzed the consequences of the extra term 2βT together with anisotropies on the properties of neu- tron stars such as radius, mass, frame-dragging angular velocity and moment of inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Indeed, our results re- veal that the radius deviates considerably from GR in the low-central-density region, however, the total gravi- tational mass and the moment of inertia undergo slight modifications due to the influence of the effects generated by the minimal matter-gravity coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Furthermore, the presence of anisotropy generates substantial changes both in the mass and in the moment of inertia with re- spect to the isotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The appreciable effects due to the inclusion of anisotropy occur mainly in the higher- central-density region, this is, for large masses (near the maximum-mass configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' ACKNOWLEDGMENTS JMZP acknowledges financial support from the PCI program of the Brazilian agency “Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico”–CNPq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 7 PSR J1614-2230 PSR J0740+6620 GW170817 β = 0 β = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='01 β = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='02 β = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='03 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 10 12 14 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 rsur [km] M [M⊙] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 Log ρc [kg/m3] M [M⊙] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='94 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='98 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='02 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Mass-radius diagrams (left panel) and mass-central density relations (right panel) for anisotropic neutron stars with SLy EoS (37) in f(R, T) = R + 2βT gravity for β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='01 (blue curves), β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='02 (orange curves) and β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='03 (in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The solid lines correspond to α = 0 (that is, isotropic solutions), and the pure GR case (β = 0) is shown in both plots as a benchmark by a black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The magenta horizontal band stands for the observational measurement for the millisecond pulsar J1614-2230 reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The filled cyan region is the mass-radius constraint from the GW170817 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The Radius of PSR J0740+6620 from NICER and XMM-Newton Data [79] is indicated by the top brown dot with their respective error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Moreover, the bottom brown dot represents the radius estimate for a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 M⊙ neutron star [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='8 10 15 20 25 30 Log ρc [kg/m3] rsur [km] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='76 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='82 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='88 13 15 17 α = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 9 10 11 12 13 14 Log ρc [kg/m3] rsur [km] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Surface radius as a function of the central mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' On the left panel, different styles and colors of the curves correspond to different values of the parameters β and α as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The most substantial deviations from GR take place at low central densities, whereas for large central densities the changes are very slight due to the 2βT term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' On the right panel we display the modifications of the radius due to the inclusion of anisotropies when β = 0, where we have considered larger values for |α| in order to appreciate the changes in radius as a consequence of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' We can mainly observe three regions where the radius can decrease or increase depending on the value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 8 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 r [km] ϖ/Ω α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 r [km] ω/Ω FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Left panel: Numerical solution of the differential equation (33) for a given central mass density ρc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 × 1018 kg/m3 in f(R, T) = R + 2βT gravity with β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='01 and different values of the free parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The dotted lines represent the solutions of the exterior region, and as expected ϖ → Ω at great distances from the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Right panel: Ratio of frame-dragging angular velocity to the angular velocity of the stars, namely ω(r)/Ω = 1 − ϖ(r)/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Notice that the solution of the exterior problem provides an asymptotic behavior of ω(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 M [M⊙] I [1038 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='m2] α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 α = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='2 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='1 M [M⊙] I [1038 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='m2] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Left panel: Moment of inertia of slowly rotating anisotropic neutron stars as a function of the total mass within the context of f(R, T) = R + 2βT gravity for β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='01 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Different styles of the curves correspond to different values of the anisotropy parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Results based on Einstein’s theory have been included for comparison purposes and are represented by the black curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' We can appreciate that the moment of inertia is modified very slightly by the 2βT term, however, the anisotropies introduce relevant changes in the large-mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' The right plot is a magnification of the left one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Will, Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 17, 4 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' (LIGO Scientific and Virgo Collabo- rations), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' 116, 221101 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Saridakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=', arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content='12582 [gr-qc] (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [5] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Stelle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' D 16, 953 (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Vilkovisky, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} +page_content=' Quantum Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfDwNk/content/2301.02881v1.pdf'} 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mode 100644 index 0000000000000000000000000000000000000000..b9e85a94da66f9353dc4ac1e19ba0b4d9984c4f8 --- /dev/null +++ b/DNE2T4oBgHgl3EQfoQhP/content/tmp_files/2301.04016v1.pdf.txt @@ -0,0 +1,3538 @@ +1 +Causal Inference for Recommendation: Foundations, +Methods and Applications +Shuyuan Xu, Jianchao Ji, Yunqi Li, Yingqiang Ge, Juntao Tan, Yongfeng Zhang +Abstract—Recommender systems are important and powerful tools for various personalized services. Traditionally, these systems use +data mining and machine learning techniques to make recommendations based on correlations found in the data. However, relying +solely on correlation without considering the underlying causal mechanism may lead to various practical issues such as fairness, +explainability, robustness, bias, echo chamber and controllability problems. Therefore, researchers in related area have begun +incorporating causality into recommendation systems to address these issues. In this survey, we review the existing literature on causal +inference in recommender systems. We discuss the fundamental concepts of both recommender systems and causal inference as well +as their relationship, and review the existing work on causal methods for different problems in recommender systems. Finally, we +discuss open problems and future directions in the field of causal inference for recommendations. +Index Terms—Recommender Systems, Causal Inference +! +1 +INTRODUCTION +R +ECOMMENDER systems have been recognized as one +of the most effective tools to alleviate the information +overloading, and have been widely deployed in many real- +world systems, such as e-commerce platforms (e.g., Ama- +zon, eBay), social networks (e.g., Facebook, Twitter), video- +sharing platforms (e.g., Youtube, TikTok) and streaming +services (e.g., Netflix, Hulu). In general, these systems use +advanced techniques to learn users’ preferences from his- +torical data, along with collected user, item, and content +information. And the development of these techniques has +advanced rapidly in recent years. +Generally speaking, recommendation algorithms can +be categorized into three major types: collaborative filter- +ing, content-based recommendation and hybrid methods +[1, 2, 3]. Collaborative filtering (CF) models are based on +a key idea that similar users may share similar interest and +similar items may be liked by similar users. Early memory- +based CF models, such as user-based CF [4, 5] and item- +based CF [6, 7], take the row or column vectors of the +user-item rating matrix as the user and item vector rep- +resentations, and calculate the similarity between users or +items for recommendation based on pre-defined similarity +functions such as cosine similarity and Pearson correlation +coefficient. To extract latent semantic meanings from the +matrix, researchers later explored learned user and item vec- +tor representations. This started with Latent Factor Models +(LFM) such as matrix factorization [8], probabilistic matrix +factorization [9] and factorization machines [10], which are +widely adopted models in practice. In these models, each +user and item is learned as a latent representation to calcu- +late the matching score of each user-item pair, usually based +on inner-product. The development of deep learning and +neural networks has further extended CF models. For exam- +ple, [11, 12, 13, 14] adopts simple user and item representa- +tions (e.g., one-hot vectors) and learns a complex matching +The authors are with the Department of Computer Science, Rutgers Univer- +sity, USA. +Emails: +{shuyuan.xu, +jianchao.ji, +yunqi.li, +yingqiang.ge, +juntao.tan, +yongfeng.zhang}@rutgers.edu +function. [15, 16, 17, 18, 19] learn complex user and item +representations and adopt a simple matching function (e.g., +inner product). User representations can also be directly +calculated from historical interactions, such as in sequential +recommendation [20, 21]. Content-based recommendation +will utilize rich information about users and items, or +even context information, to enhance recommendation. In +order to learn the similarities among items based on the +side information, the representation approaches applied by +the content-based recommendations have been developed +from simple models such as TF-IDF [22] to deep learning +based models such as DNN [23], CNN [24], etc. Hybrid ap- +proaches combine collaborative filtering and content-based +methods, which exploit the benefit of both methods and +avoid their certain limitations [1, 2, 25]. +The foundation of traditional recommendation algo- +rithms is mining or learning the correlative pattern from +data. For example, many collaborative filtering models aim +to learn the user-item correlative pattern, some content- +based recommendation models aim to learn the feature- +feature correlative pattern. However, the real-world appli- +cations are driven by underlying causal mechanisms, pure +correlative learning without considering the causation will +lead to some practical issues. We take the classic “beer and +diapers” problems as an example. Pure correlative learning +will learn the strong correlation pattern between beer and +diapers, thus recommend beer for customers bought diapers +or vice versa. However, the underlying mechanism is that +young fathers usually buy beer and diapers together, and +recommending beer or diapers without considering the +underlying mechanism will cause confusion and further +hurt user’s satisfaction. Therefore, it is important to advance +from correlative learning to causal learning. +Formally speaking, causal inference studies the causal +relation between cause and effect, where cause takes respon- +sible for the effect. Two famous and popular frameworks +are the potential outcome framework (also known as the +Neyman–Rubin Potential Outcomes or the Rubin Causal +Model) [26] and the structural causal model (SCM) [27, 28]. +arXiv:2301.04016v1 [cs.IR] 8 Jan 2023 + +2 +Both causal frameworks contribute to the development +of causal recommendations. By leveraging the underlying +causal mechanisms in recommender systems, causal rec- +ommendation is able to handle different practical issues, +including explainability, fairness, robustness, uplift, and +unbiasedness. +Contribution of this survey. In this survey, we aim to +provide a comprehensive review of causal inference for rec- +ommendation. We first introduce the fundamental knowl- +edge of recommender systems and then discuss existing +work of causal inference for recommendation. Specifically, +we explore the causal inference in recommender systems in +two dimensions. The first dimension follows the pipeline +of causal inference, including concepts, notations, and tech- +niques in causal inference, and the connection between +causal inference and recommender systems. The second +dimension follows the practical problems in recommenda- +tion, including problem introduction, causal methods, and +open problems. More specifically, we include explainability, +fairness, robustness, uplift-based, unbiasedness in recom- +mendation. Finally, we highlight several open problems +in causal inference for recommendation that remain to be +addressed. +Difference with Existing Surveys. Several surveys in +recommender systems or causal inference have been pub- +lished in recent years. For example, Zhang et al. [29] and +Chen et al. [30] review explainable recommendation, Li et al. +[31] and Wang et al. [32] review fairness in recommendation, +Ge et al. [33], Wang et al. [34] and Fan et al. [35] summarize +trustworthy recommender systems, Chen et al. [36] review +bias in recommendation, Zhang et al. [2] review the deep +learning based recommendation algorithms, Ko et al. [37] +provide a comprehensive review of recommender systems, +Yao et al. [38] provide a comprehensive review of causal +inference methods, Guo et al. [39] and Vowels et al. [40] +summarize existing methods on causal structural learning +and causal discovery. Gao et al. [41] summarize existing +work on causal inference in recommender systems. Unlike +Gao et al. [41] mainly introduce existing work in perspective +of recommender systems, our survey provide systematic +review in perspective of both causal inference and recom- +mender systems. +Organization. This survey is organized as follows: Sec- +tion 2 introduces the preliminaries of recommender systems. +From Section 3 to 7, we introduce fundamental knowledge +of causal inference and the connection with recommender +systems. Section 8 to 12 introduce existing causal meth- +ods on explainable recommendation, fairness in recommen- +dation, uplift-based recommendation, robust recommenda- +tion, unbiased recommendation, respective. In Section 13, +we discuss some open problems and future directions in +causal inference for recommendation. Section 14 concludes +this survey. +2 +PRELIMINARIES FOR RECOMMENDER SYSTEMS +In general, recommender systems aim to model user prefer- +ences based on collected information, including user profile, +item profile, and user-item interactions, and further predict +users’ future interactions. User profile represents the reg- +istered information of the user, which may include user +id, user age, user gender, user income, etc. Recommender +systems may only use partial information for recommen- +dation (e.g., using user id only). The term “items” rep- +resents different objects in differnt recommender systems +(e.g., product in e-commerce, other users in social networks, +videos in online video platform, etc.). According to different +definition of “item”, item profile may include different item +features. For example, products in e-commerce may take +brand, category, price, image , etc. in item profile; videos in +online video platform item profile may take video length, +content description, etc. in video recommendation; other +users in social networks may take corresponding user pro- +files as item profile. Similarly, recommender systems may +only partial information of item profile for recommendation. +Interactions refer to possible user behaviors towards items +according to defined task (e.g., click, purchase, rate, add-to- +cart, review for e-commerce recommendation, like, dislike, +share for video recommendation, etc.). In general recom- +mender systems, interactions are typically represented in +two ways, one is explicit feedback, the other is implicit +feedback. Explicit feedback, such as ratings and reviews, is +the explicit representation of users’ preference (e.g., rating +score as 5 means that user like this item), while implicit +feedback, such as click, is collected during user-system in- +teraction process and implicitly represent users’ preference +(e.g., user’s click behavior means that it is likely that user +likes the corresponding item). +Traditional recommendation algorithms can be roughly +categorized into collaborative filtering, content-based rec- +ommendation and hybrid models. The basic idea of col- +laborative filtering (CF) is that similar users may share +similar interests and similar items may be likede by similar +users. CF methods can be further divided into memory- +based CF and model-based CF. memory-based CF makes +predictions by a simple similarity measurement over his- +torical data. For example, user-based CF [4, 5] or item- +based CF [6, 7] takes the row or column vector of the user- +item rating matrix as the representation of each user or +item and calculate the similarity by a simple measurement +such as cosine similarity. Model-based CF leverage a model +to learn the representation of users and items to make +predictions. It starts from Latent Factor Models, such as +matrix factorization [8], probabilistic matrix factorization +[9], tensor factorization [42], etc. Deep learning and neural +networks have further extend CF models. Deep CF methods +can be further divided into similarity learning approach and +representation learning approach. The similarity learning +approaches [15, 16, 17, 18, 19] leverage simple representa- +tion of users and items (e.g., one-hot vectors) and learns +a complex matching function to make prediction on each +user-item pair. The representation learning approaches learn +complex representation of users and items, and then apply +a simple matching function (e.g., inner product) to calcu- +late the prediction scores. Content-based recommendation +[23, 24, 43, 44, 45, 46, 47], on the other hand, replies on +rich user and item profile to recommend items similar to +the ones the user prefered in the past. For example, in a +movie recommender system, the model tries to understand +the features (e.g., actors, directors, genres, tags, etc.) of +movies that a user has rate highly in the past. Then, only +the movies that match the preferred features of the user + +3 +would be recommended. Hybrid models combine collabora- +tive filtering and content-based methods, which exploit the +benefit of both methods and avoid their certain limitations +[1, 2, 25, 48, 49]. Moreover, several works, such as [50, 51], +have empirically demonstrated that the hybrid approaches +are able to achieve more accurate recommendation than +pure collaborative and content-based methods. +Besides above traditional recommendation algorithms, +there are some other recommendation algorithms. Sequen- +tial recommendation [52] (also related to session-based or +session-aware recommendation), which leverage the times- +tamp information of interactions to suggest items, have +become increasingly popular in academic research and in- +dustrial application. Traditional sequential recommendation +models employ simple machine learning approaches to +model sequential data, such as Markov chain [53], session- +based KNN [54]. With the development of deep learning +techniques, many deep models obtain tremendous achieve- +ments in sequential recommendation, including RNN [55], +LSTM [56], CNN [57, 58], attention models [59] and memory +networks[60]. Moreover, with increasing success achieved +by foundation models (e.g., Large Language Models) on +natural language tasks (e.g., T5 [61], GPT-3 [62], OPT [63], +PaLM [64]), recommender system community, leverage the +unique characteristic of recommender systems, has devel- +oped the research on personalized foundation models. For +example, P5 [65], as a pretrain, personalized prompt,and +predict paradigm for recommendation, formulates recom- +mendation as a language understanding and generation +task to serve as a foundation model for many recommen- +dation tasks. +The recommendation models learn users’ preference +based on collected information, and make recommendation +based on learned preference. Specifically, a recommender +system will provide a personalized recommendation list +along with possible explanations to a specific user. Rec- +ommender systems will first predict user’s preference to- +wards a set of candidate items. Then the system will rank +candidate items to provide personalized recommendation +list. It is worth mentioning that the ranking process is not +necessarily solely based on the predicted scores provided by +the recommendation algorithm. It is possible to re-rank the +list based on different demands, such as diversity, fairness, +some business purpose, etc. After generating personalized +recommendation list, some recommendation systems may +provide explanations along with recommendations. The ex- +planations can be either generated simultaneously with the +recommendation or after the recommendation, depending +on the recommendation model is explainable or black-box. +To evaluate the performance of recommender systems, +it is important to define the characteristics of a good rec- +ommender system and quantify the characteristics. For a +recommendation model with ability of predicting rating +scores, a excellent model should be able to predict accu- +rate ratings. Therefore, RMSE or MSE is used to evaluate +the recommendation performance. By considering the accu- +racy of ranking list and whether the user’s prefered items +recommended by the list, some commonly used metrics +include Precision, Recall, F-Measure, NDCG, ROC Curve, +AUC, MRR, etc. Besides above metrics used to evaluate +recommendation performance, some metrics are used to +evaluate the recommendation model in perspective of other +purpose. For example, Absolute Difference (AD) [66] is used +to evaluate the fairness of recommender systems. +3 +CAUSAL NOTATIONS IN RECOMMENDATION +Causal inference is a critical research topic stemmed from +statistics [28, 67, 68], and has been widely used in many do- +mains for decades, such as computer science, public policy, +economic, etc. In this section, we introduce causal notations +and demonstrate how to apply them in recommendation. +3.1 +What is Causation +Causation (also refer to as causality) is a terminology that +is usually compared to and discussed with correlation. +Although both correlation and causation explore the rela- +tionship between variables, it is well known that “corre- +lation does not imply causation” [68]. Causation takes a +step further than correlation. Intuitively, causation explicitly +applies to the case that event A causes event B. On the other +hand, correlation is a much simple relation that event A is +related to event B, but one event does not necessarily cause +another event to happen. For example, a study has shown +that the data of monthly ice cream sales is highly related +to the number of monthly shark attacks across the United +States. Although the two variables are highly correlated, it +is impossible to conclude that consuming ice cream causes +shark attacks (or vice versa). It is more likely that both ice +cream sales and shark attacks increase in the summer due +to other factors such as warm weather, which leads to both +variables being correlated. Similar examples can be found +in recommendations. The beer-and-diapers story is a good +example to illustrate the difference between causation and +correlation in recommendation. There is an observation that +beer and diapers sell well together. Based on pure correla- +tive learning, beer should be recommended for customers +who bought diapers or vice versa because of the strong +correlation pattern between beers and diapers. However, +the underlying causal mechanism is that young fathers may +pick up some diapers while buying beer. Therefore, directly +recommending items without considering the underlying +causation may lead to confusion and scarified recommen- +dation performance. In general, understanding causation +helps us to better understand how the world works and +can improve the performance of recommendation systems. +To theoretically study the causation, it is required to +understand the mathematical representation of causation. +In general, there are two commonly used frameworks for +causal inference, one is the potential outcomes framework +(also known as the Neyman–Rubin Potential Outcomes or +the Rubin Causal Model) [26] and the other is the structural +causal model framework [27, 28] proposed by Pearl. Existing +works usually introduce two framework separately, how- +ever, we think both frameworks are logically equivalent [28] +and follow the similar intuition. In the following sections, +we will introduce those two frameworks following the +intuitive idea of causation, including the connections and +differences of two frameworks. +3.2 +Key mathematical notations of Causation. +Causal inference refers to a process of drawing a conclusion +that a specific treatment was the “cause” of the outcome that + +4 +was observed [69]. In this case, the atomic goal is to estimate +the outcome if any specific treatment has been applied. Both +frameworks use mathematical notations to represent the +desired value. For Rubin Causal Model, the basic element +is called potential outcome. +Definition 1. (Potential Outcome) A potential outcome is the +outcome for an individual under a possible treatment +Let X (X ∈ {x1, x2, · · · , xn}) denotes the treatment, +where n is the total number of possible treatments. Most of +the literature considers the binary treatment, for example, +taking medicine is denoted as X = 1 and not taking +medicine is denoted as X = 0. Under the binary treatment, +the group of individuals with treatment X = 1 is named +as the treated group, and the group of individuals with +treatment X = 0 is called as the control group. Generally, the +potential outcome of treatment with value xi is denoted as +Y (X = xi), which can be simplified as Y (xi). The average +potential outcome of treatment with value xi can be denoted +as E[Y (xi)]. For any individual, only one treatment can be +applied while keeping other variables unchanged, thus only +one potential outcome can be observed. Therefore, potential +outcomes can be further divided into two categories, the +observed one is named as observed outcome while the +remaining unobserved potential outcomes are named as +counterfactual outcomes. +In recommendation, the outcome is usually defined as +user behavior (e.g., click, purchase) or user preference (e.g., +rating). Unbiased recommendation models define the treat- +ment as exposure, in which the observed feedback Y (i.e., +observed outcome) can be modeled as the product of two +unobserved variables exposure O and relevance R (i.e., +Y = O · R) [70, 71, 72, 73, 74]. More specifically, in recom- +mender systems, Y = 0 can be either negative samples (i.e., +R = 0) or potential positive samples (i.e., R = 1, O = 0), +which lead to data bias in recommendation. To achieve +personalized recommendation, the models usually estimate +the potential outcome Yu,v for a certain user-item pair (u, v) +(i.e., Yu,v = Ou,v · Ru,v). By correctly estimating potential +outcome Yu,v(Ou,v = 1) (i.e., Ru,v = Yu,v(Ou,v = 1)), +the designed model is able to achieve unbiased recom- +mendation. Uplift-based recommendation models define the +treatment as recommendation (i.e., 1 for recommended, 0 for +not recommended) [75]. For each observed user-item pair, +only one treatment can be observed (i.e., recommended or +not recommended). Therefore, it is a challenge of estimating +the counterfactual outcome to calculate the uplift value for +recommendation. To achieve fairness, the treatment can also +be defined as the sensitive attribute [76](e.g., 1 for privileged +group and 0 for disadvantaged group). +Besides the treatment variable and the outcome variable, +some other variables can be observed, and they can be +further categorized as pre-treatment variables and the post- +treatment variables [38]. +Definition 2. (Pre-treatment Variables) Pre-treatment variables +are the variables that are not affected by the treatment, which are +also named as background variables. +Definition 3. (Post-treatment Variables) Post-treatment vari- +ables are the variables that are affected by the treatment. +Different recommendation scenario may include differ- +ent information and causal mechanisms, thus the specific +definition of pre-treatment variables and post-treatment +variables may vary. +In addition to the potential outcome, Pearl uses another +popular representation which distinguishes correlation and +causation using do-operation [27, 68] from the perspective +of probability. Supposed that X denotes the treatment and +Y denotes the outcome, correlation and causation pursue +different probabilities. Specifically, correlation estimates the +conditional probability P(Y |X) from observational data to +determine the correlative relation between X and Y . By +contrast, causal inference estimates P(Y |do(X = xi)) rep- +resenting the outcome under a possible treatment xi, where +do-operation intuitively denotes applying the treatment in- +stead of observing the treatment. The average outcome of +applying treatment xi can be represented as E[Y |do(X = +xi)]. A specific probability P(Y = y|do(X = x)) can be +simplified as P(y|do(x)). As we mentioned before, existing +causal frameworks follow the same intuition, therefore, +the mathematical notations of do-operations and potential +outcomes can be converted to each other in most cases. +For example, in unbiased recommendation model, the treat- +ment is usually defined as exposure. The results for a +user-item pair (u, v) under exposure can be expressed as +P(Y |u, v, do(X = 1)) where Y is the outcome and X is the +exposure variable. If we define variable V as exposed items, +then it can also be represented by P(Y |u, do(V = v)). Sim- +ilarly, the causal notations in uplift-based recommendation +and fairness for recommendation can also be expressed by +do-operations. +By defining do-operations, intervention as a basic con- +cept in causal inference can be formally defined. As we +mentioned above, the do-operation denotes applying the +treatment, which can be also defined as the intervention +on the treatment variable. We will introduce more details +in section 6. Counterfactual is an important concepts in +both the potential outcome framework and structural causal +model, which represents the difference with factual. More +specifically, counterfactual represents the scenario that the +treatment variable had a different value compared with +the observed value in the factual world. For example, con- +sidering the treatment as taking drugs and the outcome +as recovery, a patient who took drugs and recovered may +wonder if he would have been recovered if he hadn’t taken +the drugs. In this case, in the factual world, the patient took +drugs and recovered, and in the counterfactual world, the +patient did not take drugs and we wonder if he would +recover. Similar example can be observed in recommender +systems, for uplift-based recommendation, the treatment is +defined as recommendation, the outcome is defined as user +behaviors, and the system aims to maximize the increment +of user behavior caused by recommendation. However, the +item cannot be both recommended and not recommended +in the factual world, therefore, it is necessary to apply coun- +terfactual into recommendation. Counterfactual has been +widely applied into recommender systems to address prac- +tical issues and made great success. We will demonstrate +details in this survey. + +5 +4 +CAUSAL ASSUMPTIONS IN RECOMMENDATION +In this section, we will introduce commonly used assump- +tions in causal inference [38]. +Definition 4. (Stable Unit Treatment Value Assumption +(SUTVA)) The potential outcomes for any individual do not +vary with the treatment assigned to other individual, and, for +each individual, there are no difference forms or versions of each +treatment level, which lead to different potential outcomes. +This assumption emphasizes the independence of each +individual, which means that there are no interconnections +between individuals. In recommendation, the individual +usually represents the user. The traditional recommendation +implicitly assumes the independence between users, which +satisfies the SUTVA assumption. However, this assumption +does not always hold in practical recommender systems. +For example, in the recommendation for social networks, +the users may connect with each other through the network +structure. Some recommendation models do not have ex- +plicit users, for example, in session-based recommendation. +In this case, the individual may be considered as the ses- +sions, which temporally connect with each other. +Definition 5. (Ignorability) Given the variables W, which +are not affected by the treatment, treatment assignment X is +independent to the potential outcomes, i.e., Y (1), Y (0) ⊥⊥ X|W. +The ignorability assumption is also named as the un- +confoundedness assumption. This assumption defines the +treatment assignment under certain condition. Specifically, +for individuals with the same variables W, the treatment +assignment is random. This assumption is accepted by +many recommendation algorithms, however, in real-world +recommender systems, there may exists unobserved vari- +ables affect both the treatment and outcome, which has been +studied by existing works [77, 78]. +Definition 6. (Positivity) For any value of variables W, which +are not affected by the treatment, the treatment assignment is not +deterministic: +P(X = x|W = w) > 0, +∀x and w. +(1) +This assumption guarantees the feasibility and signifi- +cance of estimating the treatment effect. If for some values +of W, the treatment assignment is deterministic, then the +outcomes of at least one treatment can not be observed for- +ever for these values. In this case, estimating the treatment +effect is impractical and meaningless. This assumptions hold +in recommendation algorithm design. For each user, every +items have the chance to be exposed to the users. Items +that cannot be exposed are not within the research scope +of recommender systems. +With above three assumptions, the connection between +the observed outcomes and the potential outcomes can be +established. +E[Y (x)|W = w] =E[Y (x)|W = w, X = x] +=E[Y |W = w, X = x] +(2) +Apart from above three commonly used assumptions, +there is another way to represent the assumed mechanism. +(e) +(d) +(a) Chain +(b) Fork +(c) Collider +Fig. 1. X, Y , Z represent three variables. (a)-(c) show three funda- +mental causal graphs. (d) show and example of causal graph, and (e) +represents the manipulated graph of (d) when intervene on variable X. +Definition 7. (Structural Causal Model (SCM)) A SCM +consists of a set of endogenous (V ) and a set of exogenous (U) +variables connected by a set of functions (F) that determine the +values of the variables in V based on the values of the variables in +U. +SCM is the key concept in the Pearl’s causal framework, +which provides stronger assumptions (than potential out- +comes framework) about the mechanisms behind the sce- +narios, which indicates the relationships between variables +other than the treatment and the outcome. Each SCM is as- +sociated with a graphical model G, represented as a Directed +Acyclic Graph (DAG), where each node is a variable in U +or V and each edge is a function f. Each edge corresponds +to a causal assumption: If the variable Y is the child of a +variable X, then it is assumed that X is the direct cause of +Y ; If the variable Y is the descendant of a variable X, then +it is assumed that X is the potential cause of Y . The causal +graph is the key difference between potential outcomes +framework and structural causal model framework, where +potential outcomes framework does not consider the causal +graph to depict causal relationships. However, we think that +both frameworks are built on some assumptions, and the +causal graph is just a stronger assumption, which cannot +completely separate two frameworks. We introduce three +fundamental causal graph in Figure 1. +Causal graph is a straightforward way to represents +the underlying mechanism of recommender systems, and +three typical causal graphs in Figure 1 often appear in +the mechanisms of recommender systems. For example, the +chain structure in Figure 1(a) appears in [77], where item +decides intrinsic item features and intrinsic item features +further decides user preference; the fork structure in Figure +1(b) appears in [79], where item popularity is considered +as a common cause of both item exposure and interaction +probability; the collider structure in Figure 1(c) appears +in [80], where user click is the common outcome of both +user interest and conformity. For SCM, an existing work +[81] has shown that the traditional recommendation and +causal recommendation can be unified through a causal +view, where the recommendation models aim to estimate + +6 +(a) +(b) +(c) +(d) +Fig. 2. Many traditional recommendation and causal recommendation +can be unified under different causal graphs. In the graphs, U is user, +V is item, X is user interaction history, Y is preference score. (a) +Causal graph for non-personalized models. (b) Causal graph for simi- +larity matching-based CF models. (c) Causal graph that considers the +causality from user to item [82]. (d) Causal graph used in [81]. +P(Y |U, do(X)) (i.e., Y represents the user preference, U +denotes users, V denotes items) but with different causal +graphs. More details can be found in Figure 2. +As we mentioned before, the intervention on the treat- +ment variable can be interpreted as applying do-operation +on the treatment variable. Intuitively, the do-operation +means directly intervention, which cut off the influence +from other variables to the treatment. Therefore, considering +two variable X and Y , the desired interventional proba- +bility P(y|do(x)) can be intuitively calculated as Pm(y|x), +which is the observed probability on the manipulated graph. +Specifically, the manipulated graph removes all the income +edges to the treatment variable. For example, considering +a simple causal graph as Figure 1 (d), where X is the treat- +ment, Y is the outcome, and Z is the confounder, P(y|do(x)) +on the original causal graph G is the same as Pm(y|x) on the +manipulated graph Gm shown as Figure 1 (e). An example +in recommendation is taking intervention on item exposure, +which generate the data of randomized experiments (i.e., +data generation process follows the manipulated causal +graph). We will introduce more details about randomized +experiments in Section 6. Similar to the intervention on +causal graphs, the intervention on structural equations take +the intervened value as the input to calculate the output of +the structural equations. +The introduced assumptions bridge the gap between +the observed correlation and the estimated causation. We +will introduce some commonly used methods based on +introduced assumptions. +5 +CAUSAL EFFECTS IN RECOMMENDATION +After introducing the basic representation of the causal +representation, many different kinds of causal effects can +be defined using basic representations. +A basic causal effect is called as the treatment effect (i.e., +the outcome change if another treatment has been applied). +More specifically, the treatment effect can be measured at +the population, treated group, subgroup and individual +levels. Here we define the treatment effect under binary +treatment to make it clear, and it can be extended to multiple +treatments by comparing their potential outcomes [38]. We +takes the potential outcome as an example, and the do- +operation can be applied in similar ways. +The treatment effect at the population level is named as +the Average Treatment Effect (ATE) (some reference also +name it as the Average Causal Effect [68] or the Total Effect +[83]), which is defined as: +ATE = E[Y (1)] − E[Y (0)] +(3) +The treatment effect at the treated group level is called +Average Treatment effect on the Treated Group (ATT) +(some reference also name it as Effect of Treatment on the +Treated (ETT)[27, 68]), which is defined as: +ATT = E[Y (1)|X = 1] − E[Y (0)|X = 1] +(4) +where Y (1)|X = 1 and Y (0)|X = 1 represent the potential +outcomes under both treatments of the treated group. +For the subgroup level, the treatment effect is named +as Conditional Average Treatment Effect (CATE), which is +defined as: +CATE = E[Y (1)|W = w] − E[Y (0)|W = w] +(5) +where W denotes the variables (i.e., grouping by multiple +variables) defining the subgroup which are not affected by +the treatment, and Y (1)|W = w and Y (0)|W = w are +the potential outcomes under both treatments within the +subgroup with W = w. +At the individual level, the treatment effect is called +as Individual Treatment Effect (ITE), which can be rep- +resented as: +ITE = Yi(1) − Yi(0) +(6) +where Yi(1) and Yi(0) are the potential outcomes for treat- +ment X = 1 and X = 0 of individual i respectively. The +ITE is considered equivalent as the CATE [84, 85] if each +subgroup represents an individual. +The treatment effect on different level has been used as +quantitative evaluation in recommender systems to handle +many issues. For example, ITE is used to estimate the uplift +value of recommendation [75, 86, 87, 88]; ATE can be used to +evaluate explanations [89] and estimate unbiased preference +[90]; ATT is used to evaluate counterfactual fairness [91]; etc. +In addition to the treatment effect at different levels +we introduced above, there are some causal effects for +mediation analysis. A mediation model seeks to explain +the process that underlines a causal relationship between +the treatment and the outcome via the inclusion of a third +variable, known as a mediator variable. Let X, Y , and +M denote treatment, outcome, and mediator respectively. +We will introduce three types of effects under the binary +treatment for mediation analysis. +First, Controlled Direct Effect (CDE) measure the ex- +pected increase in Y as the treatment changes, while the +mediator is set to a specific value m for the entire popula- +tion, which can be defined as: +CDE(m) = E[Y |do(X = 1, M = m)]−E[Y |do(X = 0, M = m)] +(7) +Second, Natural Direct Effect (NDE) measures the ex- +pected increase in the outcome as the treatment changes, +while the mediator is set to whatever value it would have + +7 +attained prior to the change, i.e., X = 0, which can be +defined as: +NDE = E[Y |do(X = 1, M = M0)] − E[Y |do(X = 0, M = M0)] +(8) +where M0 represents the value of mediator under treatment +as 0. +Finally, Natural Indirect Effect (NIE) measures the ex- +pected increase in outcome when the treatment is held +constant at X = 0, while M changes to whatever value +it would have attained under X = 1, which can be defined +as: +NIE = E[Y |do(X = 0, M = M1)] − E[Y |do(X = 0, M = M0)] +(9) +where M1 represents the value of mediator under treatment +as 1. NIE captures the portion of the effect which can be +explained by mediation alone. +The above direct and indirect effects play an important +role in recommendation as well. The direct and indirect +effects help the models quantitatively evaluate path-specific +effects to detect and remove undesired effects. For example, +they can be used to identify direct and indirect discrimina- +tion to achieve or explain fairness [92, 93], they can be used +to identify and remove some bias [94, 95], etc. +6 +CAUSAL ESTIMATION METHODS IN RECOMMEN- +DATION +Having defined the causal effects, the next logical step is +to ask how can we estimate those effects. One way is to +perform a randomized experiment. +6.1 +Randomized Experiments. +To measure the average treatment effect, an ideal way is to +apply different treatment to the same group of individuals. +However, the ideal solution is impractical in real-world +situation. It can only be approximate by a randomized ex- +periment. Specifically, a randomized experiment randomly +assigns individuals into the treated group or the control +group. The estimated ATE can be obtained by the difference +of the average outcomes of two groups. To understand +why a randomized experiment is the golden standard for +estimating the average treatment effect, it is necessary to +understand how correlation is different from causation. +E[Y |X = 1] − E[Y |X = 0] +1=E[Y (1)|X = 1] − E[Y (0)|X = 0] +2= E[Y (1)|X = 1] − E[Y (0)|X = 1] +� +�� +� +ATT ++ E[Y (0)|X = 1] − E[Y (0)|X = 0] +� +�� +� +bias +(10) +Here, step 1 follows the fact that Y (1) is the observed +outcome when conditioning on X = 1 and Y (0) is the +observed outcome when conditioning on X = 0; step 2 +adds and subtracts E[Y (0)|X = 1] to construct the ATT term +and the bias term. The bias term in Eq.(10) creates the gap +between the correlation and causation. The randomized ex- +periment eliminate the bias term by randomly assigning in- +dividuals into the treated group or the control group. More +specifically, the random assignment makes the potential out- +comes are independent from the treatment Y (1), Y (0) ⊥⊥ X +(it does not imply that outcomes are independent from the +treatment), thus E[Y (0)|X = 1] = E[Y (0)|X = 0]. Given +Y (1), Y (0) ⊥⊥ X, Eq.(10) can be rewrite as: +E[Y |X = 1] − E[Y |X = 0] = E[Y (1)] − E[Y (0)] +(11) +Therefore, a randomized experiment can simply estimate +ATE as the difference of the average outcomes of the treated +group and the control group. In recommendation, the ran- +domized experiments are usually used to handle the bias +[82, 96, 97, 98, 99, 100, 101, 102]. Specifically, by taking item +exposure as the treatment, the randomized experiments +follow the random recommendation policy instead of the +deployed policy, in which return the unbiased data (i.e., also +called as uniform data) for recommendation. +A randomized experiment is not a one-size-fits-all solu- +tion for causal inference. In reality, randomized experiments +are always time-consuming and expensive, thus the study +usually involve small number of individuals, which may +not be representative of the population. Meanwhile, ethi- +cal concerns largely limit the applications of the random- +ized experiments such as environmental health studies. In +addition, the randomized experiments cannot explain the +causation on the individual level. Therefore, given the wide +availability of observational data, the observational study is +a shortcut for causal inference. +6.2 +Observational Data +Although the observational study could be a shortcut for +causal inference, there are some issues of the observational +data should be carefully considered during designing the +causal models. The existence of confounders is a critical +problem in the observational data. +Definition 8. (Confounders) Confounders are variables that +affect both the treatment assignment and the outcome. +Due to the existence of confounders, some spurious +effect may be observed (taking relationship between ice +cream consumption and shark attacks as an example). Con- +founders widely exists in recommender systems. The exis- +tence of confounders often results in different bias based on +the definition of confounders. For example, taking item pop- +ularity as a confounder, it will lead to popularity bias [79]. +In addition to some observed and measurable confounders, +such as item popularity, some unobserved or immeasurable +confounders (i.e., which violate the ignorability assumption +in Section 4) exist in real-world recommendation and have +been widely studied by the community [74, 78, 81]. +Simpson’s paradox is another phenomenon that could +be observed in the observational data. From Table 1, it can +be observed that in both male and female groups, taking the +drug has a better recovery rate; but in the total population, +not taking drug has a better recovery rate. This phenomenon +is usually caused by confounders. The Simpson’s para- +dox can be also observed in recommender systems [103]. + +8 +TABLE 1 +Results of a study into a new drug, with gender being taken into +account [68]. a/b represents a out of b recovered. +Drug +No Drug +Male +81/87 (93%) +234/270 (87%) +Female +192/263 (73%) +55/80 (69%) +Total +273/350 (78%) +289/350 (83%) +Macdonald [103] observes the Simpson’s paradox in offline +evaluation for recommendation, and propose a method to +mitigate the paradox in offline evaluation. +Compared to the experimental data, observational data +only provides the information about what has occurred, but +the why a specific treatment is token is unknown. Given +that the treatment assignment mechanism is unknown, the +bias term in Eq.(10) cannot be eliminated or quantitatively +measured. Therefore, the bias caused by unknown treatment +assignment is also a critical issue that should be carefully +handled in model design. +6.3 +Methods Relying on Assumptions +In some complex scenarios, it is risky to assume the causal +mechanism based on prior knowledge. In this case, SUTVA, +ignorability and positivity assumptions support some meth- +ods to estimate the potential outcomes. +One commonly used method is based on the idea of +reweighting. As we mentioned before, due to the unknown +treatment assignment mechanism, there may exists the bias +problem. By assigning appropriate weight to each sample in +the observational data, a pseudo-population can be created +on which the distributions of the treated group and the +control group are similar. There are two commonly used +reweighting methods: inverse propensity scoring and con- +founder balancing. +Definition 9. (Propensity Score) The propensity score is defined +as the conditional probability of treatment given background +variables: +e(w) = P(X = 1|W = w) +(12) +Given the propensity scores defined above, inverse +propensity scoring methods [104, 105] assign a weight based +on propensity score to each observed samples. Thus the es- +timated ATE based on the observed samples can be rewrite +as: +ATEIP S = 1 +n1 +� +i,xi=1 +yi +e(wi) − 1 +n0 +� +j,xj=0 +yj +1 − e(wj) +(13) +Inverse propensity scoring (IPS) is often used to de- +sign unbiased estimator for recommender systems [90, 106], +where the propensity score can be pre-defined or learned +from the data. +Although the use of propensity score is effective to +reduce the bias, there are some issues during applying IPS +in practice. First, the correctness of the IPS estimator highly +relies on the correctness of the propensity score estimator. +To handle this dilemma, some augmented IPS methods are +proposed, such as doubly robust estimator [107]. Another +drawback is that the IPS estimator has variance problem, +that the estimator is unstable if the estimated propensity +scores are small. To overcome this drawback, some methods +propose to clip the propensity score [70] or trim samples +with small propensity scores [108]. +Another reweighting method is confounder balancing +[109, 110, 111]. The motivation is that the confounders can +be balanced by the moments, which uniquely determine the +distribution of variables. Thus the sample weights can be +learned to estimate the causal effect through reweighting. +The confounder balancing based methods are used for stable +learning [112] and robust recommendation [113]. +In addition to reweighting methods, stratification is an- +other representative method. The idea of stratification is to +split the entire population into homogeneous subgroups, +which makes the treated group and the control group are +similar in each subgroup. Ideally, in this case, the samples in +the same subgroup can be viewed as sampled from the data +under randomized experiments. Macdonald [103] adopt this +idea to mitigate Simpson’s paradox in offline evaluation for +recommendation. +In some applications, the causal mechanism is safely +assumed based on prior knowledge or expert knowledge. In +this case, the causal mechanism can be represented as a SCM +as we introduced before. Although structural causal model +framework requires stronger assumptions than potential +outcomes framework, it also enable reasoning through the +graph. Using a SCM, the key difference between causation +and correlation is do-operations, which is the basic element +to estimate the causal effect. As we mentioned, the do- +operation can estimated by manipulated graph. However, +the data from the manipulated graph is generated from +randomized experiments. The approaches based on the data +generated by the original causal graph are useful in practice. +Applying backdoor adjustment is a popular approach. +Definition 10. (Back-door Criterion) A set of variables Z sat- +isfies the backdoor criterion related to an ordered pair of variables +(X, Y ) in a causal graph G if Z satisfies both (1) No node in Z +is a descendant of X and (2) Z blocks every path between X and +Y that contains an arrow into X. +Through identifying a set of variables satisfying the +back-door criterion, the causal effect can be estimated using +back-door adjustment formula. +Definition 11. (Back-door Adjustment) If a set of variables +Z satisfy the back-door criterion related to an ordered pair of +variables (X, Y ), and if P(x, z) > 0, then the causal effect of +X on Y is identifiable and is given by +P(y|do(x)) = +� +z +P(y|x, z)P(z) +(14) +Given the population of the observed data, if we divide +the subgroup based on value of Z, Eq.(14) can be considered +as calculating the causal effect by the weighted sum of +each subgroup, which is very similar to the stratification +methods. Additionally, Eq.(14) can be rewrite as: +P(y|do(x)) = +� +z +P(y, x, z) +P(x|z) +(15) +where P(x|z) is known as the “propensity score”, therefore, +the back-door adjustment is also an alternative representa- +tion of IPS methods. The back-door adjustment is widely + +9 +(b) +(a) +Fig. 3. (a) An example of applying back-door adjustment on the causal +graph. (b) An example of causal graph with an unobserved confounder, +in which the causal values can be estimated by the front-door adjust- +ment. +used to address issues in recommendation, such as bias +issues [79, 114], echo chambers [115], etc. +When we consider the do-operations, the interventions +are not limited to actions that force a variable or a group of +variables to take on specific value. In general, interventions +may involve dynamic policies in which the treatment vari- +able X is made to respond in a specified way to some set +Z of other variables, which is denoted as x = g(z). In this +case, the estimated causal effect P(Y = y|do(X = g(Z)) can +be calculated as: +P(Y = y|do(X = g(Z))) += +� +z +P(Y = y|do(X = g(Z)), Z = z)P(Z = z|do(X = g(Z))) += +� +z +P(Y = y|do(X = g(Z)), Z = z)P(Z = z) += +� +z +P(Y = y|do(X = x), Z = z)|x=g(z)P(Z = z) +(16) +In recommendation, the feedback data is collected from +a deployed recommendation algorithm, thus the recom- +mendation policy exists in the data generation process. +Considering the dynamic policy as the recommendation +policy, conditional intervention can also be applied to design +causal recommendation models [81]. In recommendation +scenario, observing an interaction in the feedback data does +not imply that the interaction is destined to happen, thus +the causal adjustment methods is sometimes applied with +counterfactual reasoning [77, 81, 115]. +Apart from above adjustment formulas, there are some +rules are valid for interventional probabilities, which are +called as the rules of do-calculus. Before introducing the +specific rules, we first introduce some notations. Let X, Y , +Z, and W be arbitrary disjoint sets of nodes in a causal +DAG G. GX denotes the graph obtained by deleting from G +all arrows pointing to nodes in X. Likewise, GX denotes as +the graph obtained by deleting from G all arrows emerging +from nodes in X. The rules of do-calculus can be represented +using above notations. +Definition 12. (The rules of do-calculus) The following three +rules are valid for every interventional distribution compatible +with a causal graph G +Rule 1 (Insertion/deletion of observations): +P(y|do(x), z, w) = P(y|do(x), w) +if +(Y ⊥⊥ Z|X, W)GX +(17) +Rule 2 (Action/observation exchange): +P(y|do(x), do(z), w) = P(y|do(x), z, w) +if +(Y ⊥⊥ Z|X, W)GXZ +(18) +Rule 3 (Insertion/deletion of actions): +P(y|do(x), do(z), w) = P(y|do(x), w) +if +(Y ⊥⊥ Z|X, W)GXZ(W ) +(19) +where Z(W) is the set of Z-nodes that are not ancestors of +any W-nodes in GX. +With the help of the rules of do-calculus and introduced +adjustment formulas, the interventional probabilities can be +estimated by the observational data. +6.4 +Methods with Relaxed Assumptions +Although above methods relying on introduced assump- +tions basically satisfy the requirement of estimating causal +effect from the observational data, in practice, for some +specific applications, the introduced assumptions may not +always hold. There are some methods trying to estimate the +causal effect with relaxed assumptions. +SUTVA assumes that individuals are independent and +identical distributed. However, in some real-world applica- +tions, such as social networks, SUTVA cannot hold anymore +since individuals are inherently interconnected with each +other through the network structure. To handle this issue +in real applications, a commonly used approach is applying +a model, which capture the interconnection, into a causal +inference model. For examples, applying graph convolu- +tional networks into a causal inference model to handle the +network data [116]. +The ignorability assumption assumes that the treament +assignment is independent to the potential outcomes given +the background variables. However, it is impossible to +identify and collect all the background variables in real +world, thus the ignorability assumption is hard to satisfy. +In other words, there may exist unobserved confounders +as we mentioned before. Only using observational data to +estimate the causal effect is difficult, an alternative way is +to combine the limited experimental data and observational +data together [117]. In recommendation, the unbiased data is +collected from randomized experiments, using a small part +of unbaised data and a large part of observed feedback is a +popular way to design unbiased recommendation models. +Another solution is based on the assumed SCM, which +models the unobserved confounders into the causal graph +(an example is shown in Figure 3(c)). Similar to applying +the back-door adjustment, we first identify a set of variables +satisfying the front-door criterion. +Definition 13. (Front-door Criterion) Given an ordered pair of +variables (X, Y ) in a causal graph G, a set of variables Z satisfies +the front-door criterion with respect to (X, Y ) if Z satisfies the +following conditions: +– Z intercepts all directed paths from X to Y . +– There is no unblocked back-door path from X to Z. +– X blocks all back-door paths from Z to Y . +Given a set of variables that satisfies the front-door +criterion, we can identify the causal effect with unobserved +confounders [68]. + +10 +Definition 14. (Front-door Adjustment) If a set of variables +Z satisfy the front-door criterion related to an ordered pair of +variables (X, Y ), and if P(x, z) > 0, then the causal effect of X +on Y is identifiable and is given by +P(y|do(x)) = +� +z +P(z|x) +� +x′ +P(y|x′, z)P(x′) +(20) +The existence of unobserved confounders is widely rec- +ognized by the community [74, 77, 78, 118], there are some +works [77, 118] that attempt to apply front-door adjustment +in recommendation. +Using instrumental variables is a possible way to get +around the ignorability assumption and conduct causal +inference. Instrumental variables are defined as variables +that only affect the outcome via the treatment variables. +Typical instrumental variables methods [119, 120] adopt +two-stage models: the first stage reconstructs the treatment +variable based on the instrumental variable and the second +stage reconstructs the outcome based on the treatment from +the first stage. In recommender systems, Si et al. [121] +adopt the instrumental variable to design a model-agnostic +recommendation framework using search data. +7 +CAUSAL DISCOVERY IN RECOMMENDATION +The above methods aim to learn the causal effect, there is +another branch of causal models targeting at learning causal +relations, which is also known as causal discovery. Except +for few works only aim to identify treatment and outcome +[122], most of the works aim to discover causal graphs. +Following [39, 123], traditional methods can be divided into +three categories: constraint-based, socre-based and those +based on functional causal models. +Constraint-based Algorithms learn a set of causal graphs +that satisfy the conditional independence embedded in the +data and statistical tests are utilized to verify if a candidate +graph satisfies the independence. Score-based algorithms +learn causal graphs by maximizing the scoring function +S(X, G), which returns the score of the causal graph G given +data X. Algorithms based on Functional causal models +(FCMs) usually define a variable as a function of its directed +causes and some noise term (e.g., linearly weighted by the +adjacency matrix of the causal graph [124]) and optimize the +designed objective to learn the parameters of the functions. +We only briefly introduce the causal discovery methods, +interested readers may refer to [39, 123] for more details. +Most existing works in causal recommendation are based +on pre-defined causal graph representing the underlying +causal mechanisms. The pre-defined causal graphs are usu- +ally defined based on expert knowledge, which may be +inaccurate and quite simple (i.e., only involve few vari- +ables). Leveraging causal discovery in recommendation will +handle these issues. There exist few works [125, 126] design +recommender systems with causal discovery techniques +based on continuous optimization [127]. The learned causal +mechanism will increase the explainability of recommender +systems and guide the model design for other aspects, such +as fairness, unbiasedness. +8 +CAUSAL EXPLAINABILITY IN RECOMMENDATION +With the development of machine learning, accuracy is +no longer the only only pursuit. Moreover, transparency +and trustworthiness start to obtain increasing attention. +For example, heathcare AI is required to provide not only +accurate diagnoses, but also supporting explanations to +convince patients. Recommender systems, with humans in +the loop, also require transparency. Explainable recommen- +dation, which emerged and developed with the pursuit of +transparency and trustworthiness of recommender systems, +has been increasing popular in both academia and indus- +try. It aims to provide explanations for the recommended +items, which will benefit the community in many ways. +For consumers, the explainable recommendation is able to +help them make better decisions. For the platform, it may +improve the transparency, persuasiveness, trustworthiness +and user’s satisfaction of the system. For model developers, +it is an important tool to understand the designed model +and accelerate the design cycle. In this section, we will +first introduce the overview of the explainable recommen- +dations, and then summarize the existing causal methods, +as well as some open problems related to causal inference. +8.1 +Problem Introduction +The research of explainable recommendation, as a sub-area +of explainable AI, was proposed and defined by [128]. +With the rapid development of deep neural networks, the +state-of-the art recommender systems widely adapt deep +models to improve the recommendation performance. How- +ever, these deep models are too complicated for users to +understand the decision made by the intelligent systems, +thus a deep model is usually considered as a black-box. +Recommender systems, serve as essential decision-making +systems in daily life, are required to provide accurate de- +cision results as well as underlying reasons. For example, +a stock investor needs to know which characteristics lead +to the recommendation before making the final decision. +A consumer hopes to understand why the recommended +items are worth buying before paying. +The explainable models can be either model-intrinsic +or model-agnostic. The former one refers to generating +explanations simultaneously with the recommendation re- +sults and the later one refers to generating explanations +after providing the recommendation results. Model-intrinsic +(also known as ad-hoc) explainable models usually de- +sign the explanation generation mechanism as a part of +decision-making process, and model-agnostic explanations +(also known as post-hoc) explainable models usually design +separate mechanisms for generating explanations. +The explanations can be presented in many different +ways, which usually depend on what kind of information +source is used for explanations. Typically, the explanations +can be presented as related users or items [89, 129], the +features of users or items [128, 130], generated textual +sentence [131, 132], visual explanations [133], graph [134], +etc. Existing works have made many successes in explain- +able recommendations with different information sources. +For example, Zhang et al. [128] propose Explicit Factor +Model (EFM) which extract explicit item features and user + +11 +opinions from user reviews to provide feature-level expla- +nations. Peake and Wang [129] extract association rules to +provide purchased items as an explanation in a model- +agnostic manner. Xian et al. [134] perform explicit reasoning +path with knowledge graph to provide recommendations +and explanations. In addition, Existing works have intro- +duced explainability into conversational recommendation. +Chen et al. [135] develop an Explainable Conversational +Recommendation (ECR) model to provide accurate recom- +mendations as well as high quality explanations by multi- +round conversations. Incorporating causal inference ideas +and techniques brings new opportunities for explainable +recommendations. In the following part, we will focus on +causal-related methods. Interested readers may refer other +surveys [29, 30] for more explainable recommendation ap- +proaches. +8.2 +Causal Methods +8.2.1 +Counterfactual +In recent years, counterfactual reasoning draws more and +more attention in explainable AI. For any AI system that +makes predictions based on machine learning models, no +matter white-box or black-box, counterfactual reasoning +looks for what input (e.g., aspects, features) should be +changed, and by how much, to acquire a different predic- +tion. Then, the altered input will comprise the explanation. +For instance, when generating explanations for a rejected +loan request, it could be something like: if your annual +income is 50, 000, instead of 30, 000, your request will not +be rejected. Some existing works have introduced the idea of +counterfactual reasoning into recommendation scenarios for +generating explanations, which looks for minimal changes +in the recommender system (e.g. item features, items in the +history, user’s behaviors, etc.) leading a different prediction +to identify the most essential part (e.g. item features, items in +the history, user’s behaviors, etc.) as the explanations. Ghaz- +imatin et al. [136] generate explanations for a recommender +system based on users’ actions in in the history. More specif- +ically, it introduces a searching algorithm on a knowledge +graph to look for the minimal set of user’s history to be cut +off, such that the user will receive different recommendation +results. Tan et al. [130] proposes a counterfactual explana- +tion framework for generating feature-level explanations. +It introduces two new concepts, explanation complexity +and explanation strength. These two concepts are used to +formulate a counterfactual optimization problem, as well as +an evaluation metric to evaluate the generated explanations. +Later in [137], a similar counterfactual explanation frame- +work is also used to explain which features are causing +fairness issues in recommender system. Tran et al. [138] +utilizes an influence function to analyze the training data. +Then, a counterfactual set of training data are used for +generating explanations. +8.2.2 +Causal Discovery +Explainable recommendation models based on causal dis- +covery are still in theirs infancy. Causal discovery methods +aim to extract causal relations among variables from the +data. Existing causal discovery based approaches in rec- +ommendation provide model-intrinsic explanations. More +specifically, through the extracted causal relations, the +causal discovery based recommendation models are able +to provide recommendations simultaneously with corre- +sponding causal relations as explanations. As we men- +tioned, causal discovery methods usually try to learn a +causal graph. In recommendation scenario, considering the +extremely large amount of items, the learned causal graphs +are typically based on item group level. For example, Wang +et al. [125] propose to learn a cluster-level causal graph +to guide sequential recommendation. Based on the learned +cluster-level causal graph and cluster assignment for each +item, the model is able to calculate the causal relations +between items. The item in interaction history with the +strongest causal relation with the recommended item is +identified as the explanation. Xu et al. [126] aim to learn a +causal graph on product type (PT) level for PT-level recom- +mendation. Particularly, the model takes collected feedback +data as the result of the mixture of two completing mech- +anisms: a causal mechanism based on user intention and +a intervention mechanism based on deployed recommen- +dation algorithm. The recommendation and corresponding +explanations are generated via the learned PT-level causal +graph. +8.3 +Open Problems +Despite the above successful usages in causal explainable +recommendation, there are open problems that expected to +be solved in the future. First, causal discovery based ex- +plainable recommendation models, are capable of generat- +ing model-intrinsic explanations, need further exploration. +Second, the current counterfactual explanation algorithms +are claimed to be model-agnostic because they are able to be +applied on any recommendation models (or at least a wide +range of recommendation models). However, the model +itself has to be reachable. It is not certain about how to +apply counterfactual explanation algorithms on an recom- +mendation model that are not accessible by the algorithm +user. Finally, there are currently no methods to leverage +other causal reasoning methods, such as the do-calculus, to +generate explanations. +9 +CAUSAL FAIRNESS IN RECOMMENDATION +Recommender system, as a powerful tool for business, +has been widely used to improve user engagement and +further create higher profit. Classical recommender systems +mainly care about how to precisely estimate user prefer- +ences. However, in recent years, concerns about fairness in +recommendation have attracted much attention from both +industry and academia [31, 32, 139, 140, 141]. With the +development of recommendation techniques, recommender +systems have been widely used to assist or even replace +human decision-making in several domains. Several studies +have shown that the unfairness may lead to negative conse- +quences [142, 143, 144], which in turn may have significant +social impacts. For example, in e-commerce, the unfairness +of exposure of items may hurt the benefits of the platform +and providers in long-term [145]; in educational recommen- +dation [146], an unfair system due to gender imbalance +[147] may discourage females from selecting STEM (i.e., + +12 +science, technology, engineering, and mathematics) topics, +which may affect society for generations; An unfair ad rec- +ommendation may even result in racial discrimination [148]. +Therefore, to increase the applications of recommender sys- +tems and maintain a healthy social impact, it is critical to +consider fairness in recommendations and build a reliable +decision-making system. +9.1 +Problem Introduction +Before achieving fairness in recommender systems, one +should first understand the reasons of unfairness. Bias and +discrimination are two commonly accepted causes of unfair- +ness [31, 32, 33, 149]. Biases in recommender systems mainly +consist of bias in data and bias in algorithm. The bias in +data may come from data generation, collection, sampling, +and storage. For example, in recommender systems, the +training data is collected from a deployed system, if the +algorithm underlying the deployed system makes biased +predictions, then the generated data may involve biases. The +bias in data may affect the algorithms, since most machine +learning algorithms rely on data to be trained and make pre- +dictions after training. If the training data contains biases, +the algorithms trained on them will learn biased knowl- +edge from these biases and further lead to unfairness. For +example, if the training data shows significant imbalance +between majority user/item group and minority user/item +group, it is high likely that the recommendation algorithm +learns much better on the majority group and results in +discrimination on the minority group. Except for the bias +in data, the recommendation algorithm itself may enhance +existing biases and cause unfairness, which is referred to +the bias in algorithm. For example, some recommendation +algorithms may enhance the popularity bias, where popular +items will get more recommendation than less popular +items with equal or similar quality. Discrimination, as a +multidisciplinary problem [150, 151, 152], is also a cause of +unfairness defined as an unjustified difference in treatment +on the basis of any physical or cultural characteristic (e.g., +race, gender, etc.) due to human prejudice and stereotyping. +It is worth mentioning that unfairness is not only caused +by bias and discrimination. For example, there may exist +conflicts or trade-offs between different kinds of fairness +[31, 33, 153], where achieving one fairness will hurt another +fairness. +To fight against unfairness, it is important to define fair- +ness. In general machine learning, fairness can be defined +on target level (i.e., to achieve fairness on group-level or +individual-level). Specifically, fairness can be categorized +into group fairness and individual fairness. +• Group Fairness: Group fairness defines the fairness on +group-level, which is based on the idea that different +groups should be treated equally. Here the groups can be +divided in many ways, where the most commonly used +way is to split the groups based on some explicit sensitive +attributes. +• Individual Fairness: Individual fairness defines fairness +on individual-level, which is based on the idea that similar +individuals should receive similar predictions. Moreover, +individual fairness can be theoretically considered as a +very special group fairness, which divides each individual +into different groups. +Since fairness in recommender systems relates to the +benefits from multiple stakeholders [144, 154, 155, 156, 157], +the request of fairness may come from different sides. There- +fore, the definition of fairness in recommendation can also +be divided into user-side fairness and item-side fairness. +• User-side Fairness: User-side fairness aims to satisfy the +fairness requirements from users (consumers). The re- +quest from the user side are mainly focusing on the recom- +mendation quality (i.e., recommendation performance). +The user-side fairness can be achieved on both group- +level and individual-level. User-side fairness on group- +level aims to reduce the discrepancy of recommendation +quality between different user groups, where the user +groups are divided by sensitive features, such as race +or gender [142, 158], or by assigned features (e.g., cold +users vs. heavy users [159], active users vs. inactive users +[140, 160]). For user-side fairness on individual-level, the +recommendation quality should be unchanged even an +individual’s sensitive features have changed. For exam- +ple, Li et al. [139] incorporate the idea of counterfactual +fairness [91] to design a recommendation model which +makes the recommendation performance unchanged even +the user’s sensitive features are flipped in the counterfac- +tual world. +• Item-side Fairness: Item-side fairness aims to satisfy the +fairness from items side, which mainly focuses on re- +questing equal exposure opportunity of items to maintain +market fairness. Here the items refer to “items” to be +ranked or recommended. For example, in e-commerce, +the items refer to products to be sold; in recruitment +system, the items refer to job seekers (item providers). +One branch of existing work focuses on achieving fairness +according to item attributes. For example, some works +[145, 161, 162, 163, 164, 165] achieve the fair exposure +between popular and unpopular items to prevent unpop- +ular items from being under-exposed. Moreover, another +branch of research work mainly focuses on achieving fair- +ness based on the sensitive attributes of item providers, +such as gender [166, 167, 168], geographic provenience +[169, 170, 171], etc. +It is worth noting that the user-side fairness and item- +side fairness may not exclusive to each other, where two- +sided fairness [172, 173, 174, 175, 176] approaches are pro- +posed to satisfy the fairness demands from both sides. +Besides the taxonomies mentioned above, there are also +some taxonomies [31, 33] that are used to classify fairness +in recommendation from other perspectives. For example, +static fairness vs. dynamic fairness [143, 143, 177, 178]; short- +term fairness vs. long-term fairness [145, 179]; populational +fairness vs. personalized fairness [139, 180, 181]; blackbox +fairness vs. explainable fairness [137], centralized fairness +vs. decentralized fairness [182, 183]. +Typically, the proposed approaches to achieve fairness +in recommendations can be roughly divided into three cate- +gories: pre-processing methods, in-processing methods and +post-processing methods [31, 33, 149, 184]. Pre-processing +methods usually aim to achieve fairness by minimizing +the bias in the data before the model training. Compared +with other types of methods, there are fewer works on pre- +processing methods. Some representative methods include + +13 +fairness-aware data sampling approach to cover items of +all groups, data balancing approach [185] to increase the +coverage of minority groups and data repairing approaches +to ensure label correctness and remove disparate impact +[186]. In-processing methods propose to incorporate fair- +ness requirements as a part of the objective function to +achieve fairness during the training. Typically, the fair- +ness requirement works as a regularizer or a constraint +[66, 140, 145, 158, 162, 187, 188, 189, 190, 191]. To minimize +the unfairness while minimizing the original loss function +(i.e., recommendation accuracy loss), it is also important +to find a trade-off between recommendation accuracy and +fairness [145, 192], which is also sometimes formulated as +a multi-objective learning problem [192]. Post-processing +methods aim to achieve fairness in inference stage after the +training, by techniques such as re-ranking [140, 193, 194, +195] or multi-armed bandit [196, 197, 198]. To measure the +unfairness, many different fairness metrics are proposed. +For example, Absolute Difference (AD) [66] measures the +absolute difference between the performance of protected +group and unprotected group; Normalized Discounted KL- +divergence [199] calculates a normalized discounted cumu- +lative value of KL-divergence for each position, etc. More +possible fairness metrics can be found in [32]. +Recently, researchers have noticed that fairness cannot +be well detected by solely correlation or association. Specif- +ically, fairness criteria are based on solely joint distribu- +tion of random variables [200], such as outcomes, features, +sensitive attributes, etc. However, recent work [201] shows +that any definition of fairness that depends merely on +the joint probability distribution is not necessarily capable +of detecting discrimination. Therefore, many approaches +[91, 93, 200, 202, 203] are proposed to address the problem +of unfairness through the lens of causality. +In general machine learning, causal-based fairness nota- +tions are mostly defined on intervention or counterfactual. +To measure the unfairness in causal-based fairness, one +challenge is understanding the causal relationships that +account for different outcomes. Causal graph, as a power- +ful tool for causal reasoning, is usually used to represent +the causal relationships among variables. Given the causal +graph capturing the causal relationships, many causal ef- +fects are used to measure the unfairness. For example, ATE +(as Eq.(3), also known as Total Effect [27]) is used to measure +the effect of changing sensitive attributes to the outcomes, +Kilbertus et al. [204] measure the indirect causal effects +[205] from sensitive attributes to outcomes and eliminate the +directed path from sensitive attributes to outcomes except +via a resolving variable, where resolving variables refer +to any variables in the causal graph that are influenced +by sensitive attributes in a non-discriminatory way. More +details of causal-based fairness notations can be found in +[76]. Counterfactual fairness is a commonly used definition +of fairness in causal-based fairness. Counterfactual fairness +is an individual-level causal-based fairness notion, which +requires that the predicted outcome should be the same +in the counterfactual world as in the real world for any +individual [91]. The basic idea is minimizing the ATT (as +Eq.(4), some references also name it as ETT [27, 68, 132]) +conditioned on all features to receive the same probability +distribution in the factual and counterfactual world. For +counterfactual fairness in recommendation, the definition is +given as follows [139]: +Definition 15. (Counterfactual Fairness in Recommendation) +The counterfactual fairness is satisfied for a recommendation +model if for any user u with sensitive attributes Z = z and +remaining features X = x: +P(Lz|X = x, Z = z) = P(Lz′|X = x, Z = z) +(21) +for all L and any value z′ attainable by Z, where L denotes the +top-k recommendation list for user u. +In the next section, we will introduce some causal meth- +ods for fairness in recommendation. +9.2 +Causal Methods +9.2.1 +Reweighting +As we mentioned before, bias is a widely accepted cause of +unfairness, thus some existing work adopts inverse propen- +sity scoring (IPS) methods to solve the bias in recommen- +dation. For example, popularity bias will lead to item-side +unfairness that popular items may obtain more exposure +opportunities. The IPS approaches the biases are caused +by non-randomly assigned treatment, thus use the inverse +propensity to reweight the samples to remove the bias. +For example, Schnabel et al. [90] consider recommendation +as treatment and apply an IPS estimator in an Empirical +Risk Minimization framework for learning to solve bias +in recommendation. Saito et al. [70] design an IPS-based +estimator for unbiased pairwise learning. Wang et al. [106] +use a small part of unbiased data to train a propensity +model and use biased data to train an IPS-based rating +model. The IPS-based approaches are easy to implement but +it requires an accurate propensity estimator and suffers from +high variance [206, 207]. +Although biases in data are commonly recognized as +the main causes of unfairness in recommendation, the re- +lationship between bias and fairness has not been clearly +understood or discussed. More specifically, the debiasing +methods are usually proposed to improve the recommen- +dation performance by removing bias, thus the models are +evaluated by recommendation metrics instead of fairness +metrics. Many works on fairness are not implemented by de- +biasing methods but directly designed by fairness require- +ments, which may result in a trade-off between accuracy +and fairness. In the following part, we focus on fairness +methods. More discussion of debiasing methods can be +found in Section 12. +9.2.2 +Counterfactual +Counterfactual fairness, as a causality-based definition of +fairness, requires the predicted outcomes to be the same in +the counterfactual world as in the factual world. To achieve +counterfactual fairness, it is important but challenging for +a fair model to predict the outcomes in the counterfactual +world (i.e., the sensitive attributes have been changed). +Ma et al. [208] propose a counterfactual data augmenta- +tion module, which is trained based on a variational auto- +encoder with a fairness constraint, to generate counterfac- +tual data with different sensitive attributes. By maximiz- +ing the similarity between the representation learned from + +14 +the original data and the different counterfactual data, the +designed model is able to achieve counterfactual fairness. +Mehrotra et al. [209] use counterfactual estimation to eval- +uate recommendation policies in terms of the trade-odd +beween relevance and fairness, and propose a recommen- +dation model considering user’s tolerance towards fairness. +The idea of counterfactual is used not only in fairness model +design but also in fairness diagnosis. Specifically, fairness +diagnosis aims to find out the reasons that cause model un- +fairness. Inspired by the idea of counterfactual explanation +[130, 210], Ge et al. [137] propose a counterfactual reasoning +approach to learn critical features that significantly influ- +ence the fairness-utility trade-off and use them as fairness +explanation for feature-based recommendation. +9.2.3 +Structural Equations +A Structural Causal Model consists of a causal graph, which +captures the direct causal relations among variables, and +a set of structural equations, which builds the quantitative +relations among variables. As we introduced in Section +6, if the structural equations are given, the interventional +or counterfactual outcomes can be obtained by replacing +the value of variables in structural equations. Inspired by +this idea, some works on fairness utilize the learned or +pre-defined structural equations. For example, some works +[92, 211, 212, 213, 214, 215] model different causal effects +from learned structural equations to discover discrimination +and further remove them. Kilbertus et al. [204] develop +a practical procedure to remove discrimination given the +structural equation model. +9.2.4 +Causal Graphs +Some other causality-based methods utilize causal graphs +to capture the underlying data generation mechanism and +apply other techniques to achieve fairness. For example, +Huang et al. [216] use the d-separation set identified from +the causal graph to design a fair upper confidence bound +bandit algorithm for online recommendation. Li et al. [139] +design a model based on a causal graph to generate feature +independent user representations via adversary learning. +Concretely, the model trains a predictor and an adversarial +classifier simultaneously, where the predictor learns the +representations for recommendation and the classifier mini- +mizes the predictor’s ability to predict the sensitive features. +9.3 +Open Problems +As we introduced above, researchers start to realize the +importance of considering causality-based fairness in rec- +ommendation [76, 201]. However, the foundation of causal +fairness in recommendation has not been well established. +Specifically, the fairness techniques are well explored in +classification tasks, however, those techniques may not be +directly migrated to the recommendation problem even if +the recommendation can be considered as a classification +task in some cases. For example, a straightforward method +[91] to achieve counterfactual fairness in classification is +removing sensitive attributes from the input to guarantee +the independence between the outcomes and the sensitive +features. However, in recommender systems, some existing +approaches do not use features for recommendation, such +as most collaborative filtering based models [217], but still +suffer from unfairness. The reason is that the interaction +information contains the hidden relationship between sensi- +tive features and user-item interaction, and this underlying +relationship will be captured by the model during the col- +laborative learning thus leading to unfairness. Therefore, it +is critical to have more explorations about the underlying +causal mechanism of unfairness. Additionally, it can help +the community to establish a connection between bias and +fairness. +10 +CAUSAL ROBUSTNESS IN RECOMMENDATION +Recently, the robustness of machine learning become in- +creasingly important. Because model time is very time- +consuming therefore, the recommender system models are +not re-trained frequently in practice. Traditionally, the rec- +ommender system assumes that the pattern of the training +dataset and the test dataset are the same. However, there +is a difference between the training dataset and the real- +world data. The difference might be caused by the naturally +distribution shift or intend attacking [218].Training on such +a training dataset will result in performance decreasing +when we apply the model to real-world data. In this case, +how to construct a robust model is very important. +10.1 +Problem Introduction +To begin with, we need to know which aspects harm the ro- +bustness of the recommender system. In general the dataset +will be split into three subset in the training progress (train- +ing set, validation set and test set). Most of the robustness +happen on training set and test set. For example, if the +training dataset if not big enough can this may cause the +overfitting or underfitting problem. In this case, we may +get a bad results on the test set. Specifically, the robustness +problem can be categorized as following: +• Distributional shift Many exist recommender systems +assume that the distribution for the training set and +test set are identical. However this assumption do not +meet the real-world scenarios, and this makes lots +of exist recommendation models cannot achieve the +performance we expect when we deploy them online +[219]. Many of the current recommender system models +are trained based on existing collected datasets. The +distribution of the data can be different when the new +model is deployed [220]. Even though the data is new +collected, there are still transformation risks [218]. Be- +cause there is a time period between training the model +and deploying the model. It is long enough for the +information of the collected be to attacked or changed +due to the processing error. +• Transformation Most of the recommender system are +training with the users’ and items’ feature. Based on the +feature, the recommender system can provide the users +a set of recommendation items. However, the users’ +and items’ feature can be corrupted or misleading. +For example, if users use VPN when they purchase a +item, then the location information could be wrong. +With such data, the recommender system may provide +wrong items to the users. + +15 +• Attack Except the transformation, attack may also +change the correctness of the training data. With the +development of e-commence, more people start to pur- +chase items online. There are huge benefits associated +with this. Therefore, the system are highly likely to be +attacked with the purpose to increase or decrease the +ranking of some items. For example, the users may +unwillingly changed their rating and reviews of the +purchased product. +• Sparsity Compared to the huge number of user and +item, most of the sequential recommendation train set +is sparsity. As we mentioned above, this may cause +overfitting or underfitting problem and lead to low +performance on test set. +10.2 +Method in Robustness +10.2.1 +Counterfactual +Recommendation model suffers from the data sparsity prob- +lem. For example, in the e-commerce application, compared +to the large number of user and items, the users purchase +history is quiet sparsity. Therefore, the model cannot get +enough data for training and result in the low predic- +tion performance. One approach to solve this problem is +counterfactual data augmentation. By using counterfactual +reasoning to generate new training data, and together with +the original data can enhance the performance of the rec- +ommendation model. Counterfactual Data-Augmentation +Sequential Recommendation (CASR) provides us a frame- +work to solve the problem [221]. For a training sample +({u, t1, t2, ...tl}, tl+1), the model will first indicate an index +d, and replace a td with an item ta. Suppose et ∈ RD is +the embedding of item t, D is the embedding size of the +item. For a given sample ({u, t1, t2, ...tl}, tl+1), the model +will optimize the following object: +min +ta∈C ∥eta − etd∥2 +2 +s.t. tl+1 ̸= arg max +t∈I S(t|u, t1, ..., td−1, ta, td+1, ..., tl) +(22) +where I is the set of all item.C is the item set for replace- +ment, which can be specified as I or other set to involve of +some prior knowledge. S is the sampler used to generate +new sequential data. In this function, the object tries to +minimize the distance between the original item and the +replace item. And the constraint make sure the changed item +is not the original one. In this case, we can generate data that +not the same as the original one but similar to the original +one. +10.2.2 +Causal Graph +Some existing works introduced the idea of causal represen- +tation learning to mitigate the distributional shift problem. +The model split the user features into the observed group +and unobserved group and set two types of preference +depending on whether it is affected by the observed feature +or not [222]. According to the casual graph, a framework +is created to model the interaction generation procedure. +And to deal with the unobserved feature, they design a new +Variational Auto-Encoder (VAE) to infer the unobserved +feature from the historical interaction and observed features. +10.2.3 +Reweighting +Moreover, reweighting methods have been introduced to +improve the robustness of recommendation, for example, +Li et al.[113] consider to enhance the robustness of rec- +ommendation when there are agnostic distributional shifts +between training data and testing data. To this end, the +paper introduces a personalized feature selection method +for Factorization Machines (FMs) through referring to the +confounder balancing approach to balance the confounders +of each feature. In specific, considering there is usually no +prior knowledge of the causal structure of input variables +in FMs, the paper considers to treat every feature as a +treatment variable and aims to estimate its causal effect on +the outcome. When one feature is treated as a treatment, +the other features are considered as confounders. The paper +refers to the idea of confounder balancing [223, 224] to +learn a weight matrix to reweight each sample through +balancing the distributions of confounders across different +treatment features, so that FMs will assign a weight to each +feature that implies its causal effect on the target variable, +and thus help to select causal features for achieving robust +recommendations. +10.3 +Open Problems +Existing works on robustness problem can only focus on +some specific problems. For example, using counterfactual +data augmentation problem to mitigate sparsity problem +and using causal representation learning to solve distri- +bution shift problem. And if we apply these methods on +other robustness problems, the experimental performance +might decrease a lot. And most of the current existing works +are unexplained. They might have good performance on +solving some problems, but they cannot explain which part +of the model improves the performance and which part of +the model can have more improvement. An explained ro- +bustness method that can be applied on multiple problems +is a great challenge. +11 +UPLIFT-BASED RECOMMENDATION +Modern recommender systems usually aim to recommend +items that users are most likely to interact with (e.g., click, +purchase, etc.). However, users may interact with some +items even without recommendation. Based on this fact, +some existing works propose to recommend items with +high interaction probability lift instead of high interaction +probability values. +A closely related area is uplift modeling, which refers +to the techniques used to estimate the incremental impact +of a treatment on the outcomes. Uplift modeling is both a +causal inference and a machine learning problem [225]. It +is a causal inference problem because the two required out- +comes (i.e., receive treatment or no treatment) for calculating +the incremental impact are exclusive for an individual. It is +also a machine learning problem because it needs models to +predict reliable uplift values for decision making. Theoreti- +cally, uplift modeling aims to estimate a treatment effect on +outcomes [225]. There are three main approaches in existing +literature: the Two-Model approach trains two models on +treated data and controlled data respectively, and uses the + +16 +difference between two predictions to calculate the uplift +value [226]; the class transformation approach builds the +connection between the treated group and the controlled +group based on some assumptions [227]; the direct estima- +tion approach designs a model to directly estimate the uplift +value [228, 229]. +11.1 +Problem Introduction +Recommender systems have been employed in several in- +dustrial domain to increase the profit of the business and +improve user engagement. To achieve this goal, most of +the recommendation models are designed to increase user +action (e.g., click, purchase, etc.) by recommending items +that have highest interaction probabilities. However, most +recommender systems neglect a fact that users may take +actions on some items regardless of whether the system +recommends them [75, 230]. For example, a user will pur- +chase bottled water with 95% probability and energy drink +with 50% probability if the system recommends them. In +the opinion of most traditional recommender systems, it +would be better to recommend bottled water since it is more +likely to be purchased by the user. However, if the system +does not recommend them, bottled water, as a product for +daily use, may still have 90% probability to be purchased, +while energy drink may only have 20% probability of being +purchased. Recommending energy drink seems to be a +better choice since it has higher lift of purchase probability +(i.e., 30% vs 5%), which in turn may be expected to lead +to more profit. Based on this motivation, there is a trend +to design uplift-based recommender systems which aim to +recommend items with high lift. +Some previous works have been aware of the impact +of recommendation but have not solve it from a causal +view. For example, Bodapati [231] proposes a two-stage +model which separately trains the awareness and satisfac- +tion stages for items. By training the model based on firm- +initiated purchase data (i.e., purchases as a consequence +of recommendation) and self-initiated purchase data (i.e., +purchases other than firm-initiated purchases), the model +aims to recommend items that maximize the expected in- +cremental number of purchase from recommendation. Sato +et al. [230] propose a purchase prediction model which +incorporates individual differences in recommendation re- +sponsiveness. More specifically, the model includes user- +specific and item-specific responsiveness to maximize the +impact of recommendation. +An uplift in recommender systems is defined as an in- +crease of user actions (e.g., click, purchase, like, etc.) caused +by recommendations. Considering the uplift is defined as +difference between situations with and without recommen- +dation, from the perspective of causal inference, the uplift +can be mathematically represented by potential outcomes. +More specifically, taking recommendation as the treatment, +let Y (1) be the potential outcome with a recommendation, +Y (0) be the potential outcome without a recommendation. +Considering the binary situation, Y (1) = 1 and Y (0) = 1 +imply that a user will take actions on the item with and +without recommendation, respectively. The uplift of an item +for a user is Y (1) − Y (0). In the following subsection, we +will introduce some existing works on uplift-based recom- +mendation models with causal inference. +11.2 +Causal Methods +11.2.1 +Data Processing +One challenge of estimating the uplift value is that each +individual cannot observe both the factual and counterfac- +tual outcomes (i.e., outcomes with and without recommen- +dations). Thus there is no observed ground truth for the +uplift value (i.e., the causal effect of recommendation). To +overcome this issue, one possible solution is regarding the +training data. Sato et al. [75] propose a sampling method +on the observational data for an uplift-based optimization. +Specifically, by observing purchase and recommendation +logs, for a given user, an item can be either purchased or not +purchased and either recommended or not recommended. +The proposed optimization samples positive and negative +instances that are specific to the uplift task from four classes +items (i.e., recommended and purchased, recommended +and not purchased, not recommended and purchased, not +recommended and not purchased). Therefore, by taking +the sampled labels for uplift task as the ground truth, +the proposed optimization is able to learn the uplift value +for user-item pairs. Except for the sampling methods on +observational data, training on experimental data is also an +available option. Shang et al. [86] propose a reinforcement +learning based approach, which incorporate a deep uplift +network to learn the causal effect of different actions as a +reward function. The uplift network learns from the training +data collected from a randomized experiment. +11.2.2 +Counterfactual +According to the calculation of the uplift value, a straight- +forward way is to estimate the counterfactual outcomes. +Although the randomized experiment is ideal for estimating +the causal effect, it is impractical to apply randomized +experiment in all recommendation scenarios since it is time- +consuming and expensive. Therefore, it is essential to esti- +mate the counterfactual outcomes only based on the obser- +vational data. Inspired by the idea of collaborative filtering, +Xie et al. [87] believe that similar users have both similar +tastes on items and similar treatment effect under recom- +mendations. The proposed approach is designed based on +tensor factorization with three dimensions as user, item and +treatment. More specifically, for a three dimentional tensor +with m users, n items and l treatments, the element yu,i,t +can be predicted as follows. +ˆyu,i,t = pT +u qi + pT +u dt + qT +i dt +(23) +where pu, qi, dt are latent representation of user u, item i +and treatment t, respectively. The predicted value of ˆyu,i,t +is used to infer the potential outcome for a user-item pair +(u, i) under treatment t. Taking binary treatment setting as +an example, the uplift value for a user-item pair (u, i) can be +estimated by ˆyu,i,t=1−ˆyu,i,t=0. Sato et al. [88] apply a match- +ing estimator [232] to estimate unobserved counterfactual +outcomes and further estimate the causal effect for recom- +mendation. More specifically, following the neighborhood +methods in recommender systems, the proposed approach +replaces the potential outcomes with the weighted average +over the observed outcome for a set of neighbors to calculate +the causal effect, where the neighbors can be neighborhood +users or neighborhood items. + +17 +11.2.3 +Reweighting +Estimating causal effect from the observational data only +is challenging, since the ground truth is unobservable and +the estimation is prone to the biases in the observational +data. To overcome this issue, some existing works design +IPS-based approaches to estimate unbiased causal effect for +recommendation or evaluation. The unbiased estimation of +the uplift value (i.e., the causal effect) can be formuted +by IPS [233]. In practice, IPS is prone to suffer from high +variance issue. To tackle this problem, Sato et al. [233] +apply capped inverse propensity scoring (CIPS) to train +an unbiased uplift-based model; Sato et al. [75] propose a +unbiased estimator for uplift-based evaluation using self- +normalized inverse propensity scoring (SNIPS) [234]; Xiao +and Wang [235] apply doubly robust technique [107, 236] +to train an unbiased and robust model for uplift-based +recommendation. +11.3 +Open Problems +Existing works on uplift-based recommendation mainly fo- +cus on representing uplift value and estimating the causal +effect using potential outcome framework. Structural causal +model, as a power tool for causal inference, has rarely +been used for uplift-based recommendation. Existing works +using structural causal models are trying to estimate user’s +preference if the system recommends a certain item, which +can be estimated by do-operations on designed causal +graph. However, it is still not clear how to estimate the pref- +erence without recommendation using do-operation. First, +the structural causal model requires the designed causal +graph. Existing works on causal recommendation using +structural causal models rarely explicitly involve the impact +of recommendation into the causal graph, however for +uplift-based recommendation, whether requiring a specific +causal graph that explicitly depict the impact of recommen- +dation still needs to be discussed. Secondly, mathematical +representation of the preference without recommendation +using do-operation is also a challenge. Finally, for uplift- +based recommendation, if the designed causal graph and +mathematical representation of preference without recom- +mendation are decided, applying causal techniques on pref- +erence without recommendation may differ from existing +works. +12 +CAUSAL UNBIASEDNESS +IN RECOMMENDA- +TION +Nowadays, recommendation algorithms have been widely +used in several applications to alleviate information over- +loading in our daily life. Although recommender systems +(RS) have obtained huge impacts in a wide range of real- +world applications, it still faces many bias issues which +are challenging, and if left unattended, will affect the long- +term benefits of the recommender systems. Bias issues are +common in RS since one nature of RS is the feedback loop. +Following a generally accepted understanding [35, 36], the +feedback loop in RS can be divided into three parts from a +bird’s-eye view: 1) the data collection part (user → data); +2) the model training part (data → model); 3) the model +serving part (model → user). Different definitions of bias +issues exist in each part and the whole feedback loop. We +will introduce more details in the following parts. +12.1 +Problem Introduction +As we mentioned, the bias issues exist in each part as well +as the whole of the feedback loop. We will introduce the +different definition of bias in the feedback loop of RS as +follows: +• Bias in Data refers to the distribution difference between +the collected data for training and the ideal test data. +Typically, the training data for RS is observational instead +of experimental. The user decision may be affected by sev- +eral factors such as exposure mechanism of RS, thus the +training distribution is different from the test distribution. +Additionally, the training data may not truly represent +user preference, misleading recommendation model to +inaccurate prediction. We will introduce four kinds of bias +in data as follows: +– Selection Bias: Selection bias stems from users’ explicit +feedback (i.e., ratings). Selection bias means the ob- +served ratings are not representative of all ratings due +to users’ selection. It is also referred as missing-not-at- +random (MNAR). +– Exposure Bias: Exposure bias usually happens in recom- +mendations with implicit feedback. Since the informa- +tion about which item the user dislikes is unavailable in +observed data, the learning process will use unobserved +interactions to represent negative preference. Exposure +bias means unobserved interactions do not necessarily +represent the user’s negative preference since the users +are merely exposed to a small portion of items. +– Conformity Bias: Conformity Bias means that users tend +to behave similarly to the others in the group, even if +their behavior goes against their own judgment, which +makes the feedback may not represent users’ true pref- +erence. +– Position Bias: Position bias is common in recommenda- +tion, especially the results are presented by a ranking +list. Position bias means that users tend to interact with +items in higher position in the recommendation list +even if the items in higher position may not be highly +relevant. +• Bias in Model refers to the bias in the model design. +Bias i not always harmful. In fact, the bias in model +empower the model to achieve the ability to generalize +the prediction to unobserved examples. +– Inductive Bias: Inductive bias represents the assump- +tions made the model designer to better learn the ob- +jective and to generalize beyond training data. +• Bias in Results refers to the phenomenon that the rec- +ommendation algorithms tend to exhibit bias in recom- +mendation results presented to users. Typically, the biases +in recommendation results are studied from two perspec- +tives, one is popularity bias and the other is unfairness. +We have introduced fairness and related methods in sec- +tion 9, thus in this section, we will limit the bias in results +to popularity bias. +– Popularity Bias: Popularity bias refers to the phe- +nomenon that popular items are recommended more +frequently than their popularity warrant. + +18 +• Feedback Loop Bias refers to the amplified bias intro- +duced by the whole RS feedback loop mechanism. Data +bias will lead to data imbalance and result in bias issues +in recommendation results, while the biased recommen- +dation will in turn impact the user’s behavior and further +amplify the bias in the future recommendation. Taking +popularity bias as an example, the popular items get +more exposure in the observed data, which in turn obtain +increase opportunity to be recommended, resulting in +amplified bias, where popular items become more pop- +ular and non-popular items become even less popular +[237, 238, 239]. These amplified bias caused by feedback +loop, if left unattended, will result in echo chambers +[115, 240] or filter bubble [241, 242, 243, 244], which will +decrease the diversity and increase the homogenization. +In general, there are two ways for debiasing in rec- +ommender systems, one is debiasing during training and +the other is debising during evaluation. Introducing causal +inference into debias recommendation makes a great success +in recent years. In the following part, we will introduce +existing works on debiased recommendation models based +on causal inference. +12.2 +Causal Methods +12.2.1 +Data Processing +To address the bias problem in recommender systems, +one straightforward solution is to leverage unbiased data +[82, 96, 97, 98, 99, 100, 101, 102]. As we mentioned in section, +combining the limited experimental data and observational +data is a possible solution under the relaxed ignorability +assumption. In recommender systems, the experimental +data, which is also called as unbiased data, intervene the +system by using a random recommendation policy instead +of a normal recommendation policy. More specifically, for +each user, they do not use recommendation models to +show items, but instead randomly select some items to +show. Leveraging unbiased data helps to achieve debiased +prediction because applying random recommendation will +break the feedback loop. The key challenge is how to +incorporate a small portion of unbiased data into model +design. For example, Rosenfeld et al. [96] and Bonner and +Vasile [82] apply two recommendation models for biased +data and unbiased data respectively and connect two mod- +els by regularization. Yuan et al. [97] learn a imputation +models with unbiased data for ad click prediction. Chen +et al. [101] leverage unbiased data by meta-learning. Despite +the effectiveness on handle biases by using unbiased data, +collecting unbiased data will randomly recommend items to +users instead of using personalized recommendation model, +which will inevitably hurt users’ experience and revenues of +the platform. +12.2.2 +Reweighting +Another commonly used method is based on reweight- +ing, which use inverse propensity scores to reweight the +data sample for different bias issues, such as selection +bias [90, 106], exposure bias [70, 72, 73, 245], position bias +[246, 247, 248, 249, 250, 251], etc. The key challenge is +how to estimate the propensity scores and how to apply +it into optimization. Some works [70, 252] use popularity- +based propensity estimator. Some works [73, 248, 250, 251] +propose a dual problem to both optimize a propensity +estimator and a recommendation model. Some works [249] +propose to learn propensity scores from the observational +data. Some works [207, 235] use doubly robust model to +handle inaccurate propensity estimators. Inverse propensity +scoring methods have some limitations, such as inaccurate +propensity scores and suffering from high variance problem +[206]. +12.2.3 +Causal Adjustment +Causal adjustment is another promising direction for ad- +dressing bias issues [77, 79, 81, 114, 115]. With the help of +do-operator, the designed models aim to estimate the causal +preference P(Y |U, do(V )) with intervening item exposure +rather than the pure associative preference P(Y |U, V ) esti- +mated by traditional recommendation models. Intuitively, it +can be understood as to answer a counterfactual question: +what would the preference be if we intervene to expose +the item to the user? Causal adjustment is used to estimate +the causal preference with observational data. More specif- +ically, causal adjustment includes back-door adjustment +[68], front-door adjustment [68], etc. Based on the designed +causal graph representing the underlying mechanism of +data generation in recommender systems, the first thing is +to identify a set of variables satisfying the corresponding +criterion (e.g., back-door criterion for back-door adjustment, +front-door criterion for front-door adjustment), then apply +causal adjustment on identified variable set to estimate the +causal preference. For example, Zhang et al. [79] apply back- +door adjustment to mitigate the exposure bias caused by the +item popularity; Xu et al. [77] leverage front-door adjust- +ment to remove the effect of unobserved confounders; Wang +et al. [114] utilize back-door adjustment to mitigate the effect +of popularity bias. Causal adjustment requires to identify +a set of variables satisfying the corresponding criterion, +however, given a reasonable causal graph for recommender +systems, it is not always find out a set of variables satisfying +such criterion. But the designed causal graph will guide the +model design from other ways. +12.2.4 +Causal Graph +Causal graph, as an effective and powerful tool for causal +modeling, is used to depict the data generation process in +recommender systems. Based on the designed causal graph, +researchers will take it as the guidance to design causal +models for debiasing [80, 253]. For example, Zhao et al. [253] +and Zheng et al. [80] disentangle the effect from bias and +user’s preference based on the designed causal graph and +recommend items solely based on user’s preference; Wei +et al. [95] and Wang et al. [94] represent the counterfactual +world based on the designed causal graph and perform +counterfactual reasoning for recommendation. +12.3 +Open Problems +Inverse propensity scoring (IPS) is a valuable method +for debiasing. However, the effectiveness of IPS methods +highly rely on the correctness of propensity scores. How +to obtain correct propensity scores is still an important + +19 +yet unsolved question. Existing work usually design sim- +ple propensity estimator based on some item character- +istics, such as popularity-based propensity [70], or learn +the propensity scores from data [73, 248, 249, 250, 251]. +Whether using correct propensity scores can be only esti- +mated indirectly through the improvement for recommen- +dation performance. Therefore, quantitative evaluation of +the correctness of propensity scores is still an open problem +and need further exploration. +13 +OPEN PROBLEMS AND FUTURE DIRECTIONS +13.1 +Underlying Causal Mechanisms +Recall the existing works we introduced above, most of +them are based on the underlying causal mechanisms of rec- +ommender systems, which are represented by pre-defined +causal relations. In general, there are three levels of pre- +defined causal relations. The first level is identifying cause +and effect only. For example, IPS methods only investigate +the quantitative relationship between two variable, one is +cause, the other is effect. For example, in some IPS based +models for debiasing in recommendation, the cause is item +exposure, and the effect is the probability of interactions. +The second one is defining causal graphs, which identify the +causal relationship between all variable pairs (i.e., whether +causal relation exists, the direction of causal relation if +exists). By pre-defining the causal graph, some existing +works design models with the guidance of causal graph. +For example, some works [80, 253] disentangle multiple +cause on the effect based on the defined causal graph to +achieve unbiasedness. The last level is structural causal +models, which define not only the causal relations but also +quantitative relations (i.e., structural equations take causes +as input and return the value of effect). The effectiveness +of proposed models are highly related to the correctness of +underlying causal mechanism. Currently, most of the exist- +ing works define the underlying causal mechanism through +expert knowledge. The correctness of the pre-defined causal +mechanism can only be indirectly reflected by the recom- +mendation performance. Therefore, a direct and quantita- +tive evaluation of the defined causal mechanisms deserves +for further exploration. Another observation is that different +models may have different pre-defined causal mechanisms +even under the same practical scenario. As such, we believe +that a universal causal mechanism should be proposed. +13.2 +Causal Discovery +Apart from concerns about the accuracy of pre-defined +causal mechanisms, another limitation of pre-defined causal +mechanism is that pre-defined causal mechanisms by expert +knowledge are usually quite simple, which only involve few +factors into consideration. However, in real-world scenario, +the decision-making process (i.e., the underlying causal +mechanism of recommender systems) may involve much +more factors, which beyond the comprehension of domain +experts. Therefore, learning causal relations from data is an +important yet unsolved problem in recommender systems. +There exist few works [125, 126] design causal discovery +methods based on continuous optimization [127, 254] for +recommendation. The learned causal mechanism can be +used for explainable recommendation or be leveraged to +improve recommendation. Therefore, it is a promising di- +rection to propose causal discovery methods for recom- +mendation. It is also a challenge to evaluate the proposed +causal discovery methods for recommendation. Since there +is no ground-truth causal mechanism in real-world data, the +causal discovery methods in recommendation are usually +indirectly evaluated by recommendation performance. To +directly evaluate causal discovery methods for recommen- +dation, one possible solution is using simulation (we will +introduce it in the next section). +13.3 +Causality Driven Simulations +Simulation is one of the most powerful approach to build +environments in which the recommender systems can be +measured and analyzed. Building simulation for recommen- +dation will benefit both industry and academia. For exam- +ple, for industry, simulation provide controllable environ- +ment for practitioners to analyze the objectives of interest, +such as some business purpose, to accelerate the pace of +application development without the ethical risks. For re- +searchers in academia, due to the restrictions of accessibility +of real-world recommender systems, some proposed meth- +ods cannot be evaluated. This issue can be addressed by +using simulations. Existing simulations leverage reinforce- +ment learning techniques to simulate the decision making +process under a designed environment. However, existing +simulations without underlying causal mechanisms may +lead to inaccurate and unstable decision-making. Leverag- +ing causal mechanisms into simulation will achieve more +stable system for long-term analysis and causal-related +analysis as well. For example, causality driven simulations +can be used to evaluate causal discovery methods in rec- +ommendation. Thus, causality driven simulation will play +an essential role in recommender systems, which deserves +further explorations. +14 +CONCLUSION +In this survey, we provide comprehensive review of causal +inference methods for recommendation. We first provide +the fundamental knowledge of recommender systems. We +then introduce existing work in perspective of both causal +inference and recommender systems. 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Drouin, “Differentiable causal discov- +ery from interventional data,” Advances in Neural In- +formation Processing Systems, vol. 33, pp. 21 865–21 877, +2020. + diff --git a/DNE2T4oBgHgl3EQfoQhP/content/tmp_files/load_file.txt b/DNE2T4oBgHgl3EQfoQhP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fce9a664a7ce626025c29d64e4f8787a52134107 --- /dev/null +++ b/DNE2T4oBgHgl3EQfoQhP/content/tmp_files/load_file.txt @@ -0,0 +1,2917 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf,len=2916 +page_content='1 Causal Inference for Recommendation: Foundations, Methods and Applications Shuyuan Xu, Jianchao Ji, Yunqi Li, Yingqiang Ge, Juntao Tan, Yongfeng Zhang Abstract—Recommender systems are important and powerful tools for various personalized services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Traditionally, these systems use data mining and machine learning techniques to make recommendations based on correlations found in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, relying solely on correlation without considering the underlying causal mechanism may lead to various practical issues such as fairness, explainability, robustness, bias, echo chamber and controllability problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, researchers in related area have begun incorporating causality into recommendation systems to address these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this survey, we review the existing literature on causal inference in recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We discuss the fundamental concepts of both recommender systems and causal inference as well as their relationship, and review the existing work on causal methods for different problems in recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Finally, we discuss open problems and future directions in the field of causal inference for recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Index Terms—Recommender Systems, Causal Inference !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 1 INTRODUCTION R ECOMMENDER systems have been recognized as one of the most effective tools to alleviate the information overloading, and have been widely deployed in many real- world systems, such as e-commerce platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Ama- zon, eBay), social networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Facebook, Twitter), video- sharing platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Youtube, TikTok) and streaming services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Netflix, Hulu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general, these systems use advanced techniques to learn users’ preferences from his- torical data, along with collected user, item, and content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' And the development of these techniques has advanced rapidly in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Generally speaking, recommendation algorithms can be categorized into three major types: collaborative filter- ing, content-based recommendation and hybrid methods [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Collaborative filtering (CF) models are based on a key idea that similar users may share similar interest and similar items may be liked by similar users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Early memory- based CF models, such as user-based CF [4, 5] and item- based CF [6, 7], take the row or column vectors of the user-item rating matrix as the user and item vector rep- resentations, and calculate the similarity between users or items for recommendation based on pre-defined similarity functions such as cosine similarity and Pearson correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To extract latent semantic meanings from the matrix, researchers later explored learned user and item vec- tor representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' This started with Latent Factor Models (LFM) such as matrix factorization [8], probabilistic matrix factorization [9] and factorization machines [10], which are widely adopted models in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In these models, each user and item is learned as a latent representation to calcu- late the matching score of each user-item pair, usually based on inner-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The development of deep learning and neural networks has further extended CF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For exam- ple, [11, 12, 13, 14] adopts simple user and item representa- tions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', one-hot vectors) and learns a complex matching The authors are with the Department of Computer Science, Rutgers Univer- sity, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Emails: {shuyuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='xu, jianchao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='ji, yunqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='li, yingqiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='ge, juntao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='tan, yongfeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='zhang}@rutgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='edu function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [15, 16, 17, 18, 19] learn complex user and item representations and adopt a simple matching function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', inner product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' User representations can also be directly calculated from historical interactions, such as in sequential recommendation [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Content-based recommendation will utilize rich information about users and items, or even context information, to enhance recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In order to learn the similarities among items based on the side information, the representation approaches applied by the content-based recommendations have been developed from simple models such as TF-IDF [22] to deep learning based models such as DNN [23], CNN [24], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Hybrid ap- proaches combine collaborative filtering and content-based methods, which exploit the benefit of both methods and avoid their certain limitations [1, 2, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The foundation of traditional recommendation algo- rithms is mining or learning the correlative pattern from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, many collaborative filtering models aim to learn the user-item correlative pattern, some content- based recommendation models aim to learn the feature- feature correlative pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the real-world appli- cations are driven by underlying causal mechanisms, pure correlative learning without considering the causation will lead to some practical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We take the classic “beer and diapers” problems as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Pure correlative learning will learn the strong correlation pattern between beer and diapers, thus recommend beer for customers bought diapers or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the underlying mechanism is that young fathers usually buy beer and diapers together, and recommending beer or diapers without considering the underlying mechanism will cause confusion and further hurt user’s satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, it is important to advance from correlative learning to causal learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Formally speaking, causal inference studies the causal relation between cause and effect, where cause takes respon- sible for the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Two famous and popular frameworks are the potential outcome framework (also known as the Neyman–Rubin Potential Outcomes or the Rubin Causal Model) [26] and the structural causal model (SCM) [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='04016v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='IR] 8 Jan 2023 2 Both causal frameworks contribute to the development of causal recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By leveraging the underlying causal mechanisms in recommender systems, causal rec- ommendation is able to handle different practical issues, including explainability, fairness, robustness, uplift, and unbiasedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Contribution of this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this survey, we aim to provide a comprehensive review of causal inference for rec- ommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We first introduce the fundamental knowl- edge of recommender systems and then discuss existing work of causal inference for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, we explore the causal inference in recommender systems in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The first dimension follows the pipeline of causal inference, including concepts, notations, and tech- niques in causal inference, and the connection between causal inference and recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The second dimension follows the practical problems in recommenda- tion, including problem introduction, causal methods, and open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, we include explainability, fairness, robustness, uplift-based, unbiasedness in recom- mendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Finally, we highlight several open problems in causal inference for recommendation that remain to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Difference with Existing Surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Several surveys in recommender systems or causal inference have been pub- lished in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [29] and Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [30] review explainable recommendation, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [31] and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [32] review fairness in recommendation, Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [33], Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [34] and Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [35] summarize trustworthy recommender systems, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [36] review bias in recommendation, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [2] review the deep learning based recommendation algorithms, Ko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [37] provide a comprehensive review of recommender systems, Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [38] provide a comprehensive review of causal inference methods, Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [39] and Vowels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [40] summarize existing methods on causal structural learning and causal discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [41] summarize existing work on causal inference in recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Unlike Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [41] mainly introduce existing work in perspective of recommender systems, our survey provide systematic review in perspective of both causal inference and recom- mender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' This survey is organized as follows: Sec- tion 2 introduces the preliminaries of recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' From Section 3 to 7, we introduce fundamental knowledge of causal inference and the connection with recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Section 8 to 12 introduce existing causal meth- ods on explainable recommendation, fairness in recommen- dation, uplift-based recommendation, robust recommenda- tion, unbiased recommendation, respective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In Section 13, we discuss some open problems and future directions in causal inference for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Section 14 concludes this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 2 PRELIMINARIES FOR RECOMMENDER SYSTEMS In general, recommender systems aim to model user prefer- ences based on collected information, including user profile, item profile, and user-item interactions, and further predict users’ future interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' User profile represents the reg- istered information of the user, which may include user id, user age, user gender, user income, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Recommender systems may only use partial information for recommen- dation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', using user id only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The term “items” rep- resents different objects in differnt recommender systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', product in e-commerce, other users in social networks, videos in online video platform, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' According to different definition of “item”, item profile may include different item features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, products in e-commerce may take brand, category, price, image , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' in item profile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' videos in online video platform item profile may take video length, content description, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' in video recommendation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' other users in social networks may take corresponding user pro- files as item profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Similarly, recommender systems may only partial information of item profile for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Interactions refer to possible user behaviors towards items according to defined task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', click, purchase, rate, add-to- cart, review for e-commerce recommendation, like, dislike, share for video recommendation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general recom- mender systems, interactions are typically represented in two ways, one is explicit feedback, the other is implicit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Explicit feedback, such as ratings and reviews, is the explicit representation of users’ preference (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', rating score as 5 means that user like this item), while implicit feedback, such as click, is collected during user-system in- teraction process and implicitly represent users’ preference (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', user’s click behavior means that it is likely that user likes the corresponding item).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Traditional recommendation algorithms can be roughly categorized into collaborative filtering, content-based rec- ommendation and hybrid models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The basic idea of col- laborative filtering (CF) is that similar users may share similar interests and similar items may be likede by similar users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' CF methods can be further divided into memory- based CF and model-based CF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' memory-based CF makes predictions by a simple similarity measurement over his- torical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, user-based CF [4, 5] or item- based CF [6, 7] takes the row or column vector of the user- item rating matrix as the representation of each user or item and calculate the similarity by a simple measurement such as cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Model-based CF leverage a model to learn the representation of users and items to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It starts from Latent Factor Models, such as matrix factorization [8], probabilistic matrix factorization [9], tensor factorization [42], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Deep learning and neural networks have further extend CF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Deep CF methods can be further divided into similarity learning approach and representation learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The similarity learning approaches [15, 16, 17, 18, 19] leverage simple representa- tion of users and items (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', one-hot vectors) and learns a complex matching function to make prediction on each user-item pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The representation learning approaches learn complex representation of users and items, and then apply a simple matching function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', inner product) to calcu- late the prediction scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Content-based recommendation [23, 24, 43, 44, 45, 46, 47], on the other hand, replies on rich user and item profile to recommend items similar to the ones the user prefered in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, in a movie recommender system, the model tries to understand the features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', actors, directors, genres, tags, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=') of movies that a user has rate highly in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Then, only the movies that match the preferred features of the user 3 would be recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Hybrid models combine collabora- tive filtering and content-based methods, which exploit the benefit of both methods and avoid their certain limitations [1, 2, 25, 48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Moreover, several works, such as [50, 51], have empirically demonstrated that the hybrid approaches are able to achieve more accurate recommendation than pure collaborative and content-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Besides above traditional recommendation algorithms, there are some other recommendation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Sequen- tial recommendation [52] (also related to session-based or session-aware recommendation), which leverage the times- tamp information of interactions to suggest items, have become increasingly popular in academic research and in- dustrial application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Traditional sequential recommendation models employ simple machine learning approaches to model sequential data, such as Markov chain [53], session- based KNN [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' With the development of deep learning techniques, many deep models obtain tremendous achieve- ments in sequential recommendation, including RNN [55], LSTM [56], CNN [57, 58], attention models [59] and memory networks[60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Moreover, with increasing success achieved by foundation models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Large Language Models) on natural language tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', T5 [61], GPT-3 [62], OPT [63], PaLM [64]), recommender system community, leverage the unique characteristic of recommender systems, has devel- oped the research on personalized foundation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, P5 [65], as a pretrain, personalized prompt,and predict paradigm for recommendation, formulates recom- mendation as a language understanding and generation task to serve as a foundation model for many recommen- dation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The recommendation models learn users’ preference based on collected information, and make recommendation based on learned preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, a recommender system will provide a personalized recommendation list along with possible explanations to a specific user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Rec- ommender systems will first predict user’s preference to- wards a set of candidate items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Then the system will rank candidate items to provide personalized recommendation list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is worth mentioning that the ranking process is not necessarily solely based on the predicted scores provided by the recommendation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is possible to re-rank the list based on different demands, such as diversity, fairness, some business purpose, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' After generating personalized recommendation list, some recommendation systems may provide explanations along with recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The ex- planations can be either generated simultaneously with the recommendation or after the recommendation, depending on the recommendation model is explainable or black-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To evaluate the performance of recommender systems, it is important to define the characteristics of a good rec- ommender system and quantify the characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For a recommendation model with ability of predicting rating scores, a excellent model should be able to predict accu- rate ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, RMSE or MSE is used to evaluate the recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By considering the accu- racy of ranking list and whether the user’s prefered items recommended by the list, some commonly used metrics include Precision, Recall, F-Measure, NDCG, ROC Curve, AUC, MRR, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Besides above metrics used to evaluate recommendation performance, some metrics are used to evaluate the recommendation model in perspective of other purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Absolute Difference (AD) [66] is used to evaluate the fairness of recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 3 CAUSAL NOTATIONS IN RECOMMENDATION Causal inference is a critical research topic stemmed from statistics [28, 67, 68], and has been widely used in many do- mains for decades, such as computer science, public policy, economic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this section, we introduce causal notations and demonstrate how to apply them in recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 What is Causation Causation (also refer to as causality) is a terminology that is usually compared to and discussed with correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Although both correlation and causation explore the rela- tionship between variables, it is well known that “corre- lation does not imply causation” [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Causation takes a step further than correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Intuitively, causation explicitly applies to the case that event A causes event B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' On the other hand, correlation is a much simple relation that event A is related to event B, but one event does not necessarily cause another event to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, a study has shown that the data of monthly ice cream sales is highly related to the number of monthly shark attacks across the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Although the two variables are highly correlated, it is impossible to conclude that consuming ice cream causes shark attacks (or vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is more likely that both ice cream sales and shark attacks increase in the summer due to other factors such as warm weather, which leads to both variables being correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Similar examples can be found in recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The beer-and-diapers story is a good example to illustrate the difference between causation and correlation in recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' There is an observation that beer and diapers sell well together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Based on pure correla- tive learning, beer should be recommended for customers who bought diapers or vice versa because of the strong correlation pattern between beers and diapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the underlying causal mechanism is that young fathers may pick up some diapers while buying beer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, directly recommending items without considering the underlying causation may lead to confusion and scarified recommen- dation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general, understanding causation helps us to better understand how the world works and can improve the performance of recommendation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To theoretically study the causation, it is required to understand the mathematical representation of causation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general, there are two commonly used frameworks for causal inference, one is the potential outcomes framework (also known as the Neyman–Rubin Potential Outcomes or the Rubin Causal Model) [26] and the other is the structural causal model framework [27, 28] proposed by Pearl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Existing works usually introduce two framework separately, how- ever, we think both frameworks are logically equivalent [28] and follow the similar intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In the following sections, we will introduce those two frameworks following the intuitive idea of causation, including the connections and differences of two frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Key mathematical notations of Causation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Causal inference refers to a process of drawing a conclusion that a specific treatment was the “cause” of the outcome that 4 was observed [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, the atomic goal is to estimate the outcome if any specific treatment has been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Both frameworks use mathematical notations to represent the desired value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For Rubin Causal Model, the basic element is called potential outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Potential Outcome) A potential outcome is the outcome for an individual under a possible treatment Let X (X ∈ {x1, x2, · · · , xn}) denotes the treatment, where n is the total number of possible treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Most of the literature considers the binary treatment, for example, taking medicine is denoted as X = 1 and not taking medicine is denoted as X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Under the binary treatment, the group of individuals with treatment X = 1 is named as the treated group, and the group of individuals with treatment X = 0 is called as the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Generally, the potential outcome of treatment with value xi is denoted as Y (X = xi), which can be simplified as Y (xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The average potential outcome of treatment with value xi can be denoted as E[Y (xi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For any individual, only one treatment can be applied while keeping other variables unchanged, thus only one potential outcome can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, potential outcomes can be further divided into two categories, the observed one is named as observed outcome while the remaining unobserved potential outcomes are named as counterfactual outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In recommendation, the outcome is usually defined as user behavior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', click, purchase) or user preference (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', rating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Unbiased recommendation models define the treat- ment as exposure, in which the observed feedback Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', observed outcome) can be modeled as the product of two unobserved variables exposure O and relevance R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Y = O · R) [70, 71, 72, 73, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, in recom- mender systems, Y = 0 can be either negative samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', R = 0) or potential positive samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', R = 1, O = 0), which lead to data bias in recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To achieve personalized recommendation, the models usually estimate the potential outcome Yu,v for a certain user-item pair (u, v) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Yu,v = Ou,v · Ru,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By correctly estimating potential outcome Yu,v(Ou,v = 1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Ru,v = Yu,v(Ou,v = 1)), the designed model is able to achieve unbiased recom- mendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Uplift-based recommendation models define the treatment as recommendation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', 1 for recommended, 0 for not recommended) [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For each observed user-item pair, only one treatment can be observed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', recommended or not recommended).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, it is a challenge of estimating the counterfactual outcome to calculate the uplift value for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To achieve fairness, the treatment can also be defined as the sensitive attribute [76](e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', 1 for privileged group and 0 for disadvantaged group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Besides the treatment variable and the outcome variable, some other variables can be observed, and they can be further categorized as pre-treatment variables and the post- treatment variables [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Pre-treatment Variables) Pre-treatment variables are the variables that are not affected by the treatment, which are also named as background variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Post-treatment Variables) Post-treatment vari- ables are the variables that are affected by the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Different recommendation scenario may include differ- ent information and causal mechanisms, thus the specific definition of pre-treatment variables and post-treatment variables may vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In addition to the potential outcome, Pearl uses another popular representation which distinguishes correlation and causation using do-operation [27, 68] from the perspective of probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Supposed that X denotes the treatment and Y denotes the outcome, correlation and causation pursue different probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, correlation estimates the conditional probability P(Y |X) from observational data to determine the correlative relation between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By contrast, causal inference estimates P(Y |do(X = xi)) rep- resenting the outcome under a possible treatment xi, where do-operation intuitively denotes applying the treatment in- stead of observing the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The average outcome of applying treatment xi can be represented as E[Y |do(X = xi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' A specific probability P(Y = y|do(X = x)) can be simplified as P(y|do(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As we mentioned before, existing causal frameworks follow the same intuition, therefore, the mathematical notations of do-operations and potential outcomes can be converted to each other in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, in unbiased recommendation model, the treat- ment is usually defined as exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The results for a user-item pair (u, v) under exposure can be expressed as P(Y |u, v, do(X = 1)) where Y is the outcome and X is the exposure variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' If we define variable V as exposed items, then it can also be represented by P(Y |u, do(V = v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Sim- ilarly, the causal notations in uplift-based recommendation and fairness for recommendation can also be expressed by do-operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By defining do-operations, intervention as a basic con- cept in causal inference can be formally defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As we mentioned above, the do-operation denotes applying the treatment, which can be also defined as the intervention on the treatment variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We will introduce more details in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Counterfactual is an important concepts in both the potential outcome framework and structural causal model, which represents the difference with factual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, counterfactual represents the scenario that the treatment variable had a different value compared with the observed value in the factual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, con- sidering the treatment as taking drugs and the outcome as recovery, a patient who took drugs and recovered may wonder if he would have been recovered if he hadn’t taken the drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, in the factual world, the patient took drugs and recovered, and in the counterfactual world, the patient did not take drugs and we wonder if he would recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Similar example can be observed in recommender systems, for uplift-based recommendation, the treatment is defined as recommendation, the outcome is defined as user behaviors, and the system aims to maximize the increment of user behavior caused by recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the item cannot be both recommended and not recommended in the factual world, therefore, it is necessary to apply coun- terfactual into recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Counterfactual has been widely applied into recommender systems to address prac- tical issues and made great success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We will demonstrate details in this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 5 4 CAUSAL ASSUMPTIONS IN RECOMMENDATION In this section, we will introduce commonly used assump- tions in causal inference [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Stable Unit Treatment Value Assumption (SUTVA)) The potential outcomes for any individual do not vary with the treatment assigned to other individual, and, for each individual, there are no difference forms or versions of each treatment level, which lead to different potential outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' This assumption emphasizes the independence of each individual, which means that there are no interconnections between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In recommendation, the individual usually represents the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The traditional recommendation implicitly assumes the independence between users, which satisfies the SUTVA assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, this assumption does not always hold in practical recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, in the recommendation for social networks, the users may connect with each other through the network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Some recommendation models do not have ex- plicit users, for example, in session-based recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, the individual may be considered as the ses- sions, which temporally connect with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Ignorability) Given the variables W, which are not affected by the treatment, treatment assignment X is independent to the potential outcomes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Y (1), Y (0) ⊥⊥ X|W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The ignorability assumption is also named as the un- confoundedness assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' This assumption defines the treatment assignment under certain condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, for individuals with the same variables W, the treatment assignment is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' This assumption is accepted by many recommendation algorithms, however, in real-world recommender systems, there may exists unobserved vari- ables affect both the treatment and outcome, which has been studied by existing works [77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Positivity) For any value of variables W, which are not affected by the treatment, the treatment assignment is not deterministic: P(X = x|W = w) > 0, ∀x and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (1) This assumption guarantees the feasibility and signifi- cance of estimating the treatment effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' If for some values of W, the treatment assignment is deterministic, then the outcomes of at least one treatment can not be observed for- ever for these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, estimating the treatment effect is impractical and meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' This assumptions hold in recommendation algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For each user, every items have the chance to be exposed to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Items that cannot be exposed are not within the research scope of recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' With above three assumptions, the connection between the observed outcomes and the potential outcomes can be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' E[Y (x)|W = w] =E[Y (x)|W = w, X = x] =E[Y |W = w, X = x] (2) Apart from above three commonly used assumptions, there is another way to represent the assumed mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (e) (d) (a) Chain (b) Fork (c) Collider Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' X, Y , Z represent three variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (a)-(c) show three funda- mental causal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (d) show and example of causal graph, and (e) represents the manipulated graph of (d) when intervene on variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Structural Causal Model (SCM)) A SCM consists of a set of endogenous (V ) and a set of exogenous (U) variables connected by a set of functions (F) that determine the values of the variables in V based on the values of the variables in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' SCM is the key concept in the Pearl’s causal framework, which provides stronger assumptions (than potential out- comes framework) about the mechanisms behind the sce- narios, which indicates the relationships between variables other than the treatment and the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Each SCM is as- sociated with a graphical model G, represented as a Directed Acyclic Graph (DAG), where each node is a variable in U or V and each edge is a function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Each edge corresponds to a causal assumption: If the variable Y is the child of a variable X, then it is assumed that X is the direct cause of Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' If the variable Y is the descendant of a variable X, then it is assumed that X is the potential cause of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The causal graph is the key difference between potential outcomes framework and structural causal model framework, where potential outcomes framework does not consider the causal graph to depict causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, we think that both frameworks are built on some assumptions, and the causal graph is just a stronger assumption, which cannot completely separate two frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We introduce three fundamental causal graph in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Causal graph is a straightforward way to represents the underlying mechanism of recommender systems, and three typical causal graphs in Figure 1 often appear in the mechanisms of recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, the chain structure in Figure 1(a) appears in [77], where item decides intrinsic item features and intrinsic item features further decides user preference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' the fork structure in Figure 1(b) appears in [79], where item popularity is considered as a common cause of both item exposure and interaction probability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' the collider structure in Figure 1(c) appears in [80], where user click is the common outcome of both user interest and conformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For SCM, an existing work [81] has shown that the traditional recommendation and causal recommendation can be unified through a causal view, where the recommendation models aim to estimate 6 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Many traditional recommendation and causal recommendation can be unified under different causal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In the graphs, U is user, V is item, X is user interaction history, Y is preference score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (a) Causal graph for non-personalized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (b) Causal graph for simi- larity matching-based CF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (c) Causal graph that considers the causality from user to item [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (d) Causal graph used in [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' P(Y |U, do(X)) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', Y represents the user preference, U denotes users, V denotes items) but with different causal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More details can be found in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As we mentioned before, the intervention on the treat- ment variable can be interpreted as applying do-operation on the treatment variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Intuitively, the do-operation means directly intervention, which cut off the influence from other variables to the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, considering two variable X and Y , the desired interventional proba- bility P(y|do(x)) can be intuitively calculated as Pm(y|x), which is the observed probability on the manipulated graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, the manipulated graph removes all the income edges to the treatment variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, considering a simple causal graph as Figure 1 (d), where X is the treat- ment, Y is the outcome, and Z is the confounder, P(y|do(x)) on the original causal graph G is the same as Pm(y|x) on the manipulated graph Gm shown as Figure 1 (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' An example in recommendation is taking intervention on item exposure, which generate the data of randomized experiments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', data generation process follows the manipulated causal graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We will introduce more details about randomized experiments in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Similar to the intervention on causal graphs, the intervention on structural equations take the intervened value as the input to calculate the output of the structural equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The introduced assumptions bridge the gap between the observed correlation and the estimated causation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We will introduce some commonly used methods based on introduced assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 5 CAUSAL EFFECTS IN RECOMMENDATION After introducing the basic representation of the causal representation, many different kinds of causal effects can be defined using basic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' A basic causal effect is called as the treatment effect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', the outcome change if another treatment has been applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, the treatment effect can be measured at the population, treated group, subgroup and individual levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Here we define the treatment effect under binary treatment to make it clear, and it can be extended to multiple treatments by comparing their potential outcomes [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We takes the potential outcome as an example, and the do- operation can be applied in similar ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The treatment effect at the population level is named as the Average Treatment Effect (ATE) (some reference also name it as the Average Causal Effect [68] or the Total Effect [83]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' which is defined as: ATE = E[Y (1)] − E[Y (0)] (3) The treatment effect at the treated group level is called Average Treatment effect on the Treated Group (ATT) (some reference also name it as Effect of Treatment on the Treated (ETT)[27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 68]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' which is defined as: ATT = E[Y (1)|X = 1] − E[Y (0)|X = 1] (4) where Y (1)|X = 1 and Y (0)|X = 1 represent the potential outcomes under both treatments of the treated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For the subgroup level, the treatment effect is named as Conditional Average Treatment Effect (CATE), which is defined as: CATE = E[Y (1)|W = w] − E[Y (0)|W = w] (5) where W denotes the variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', grouping by multiple variables) defining the subgroup which are not affected by the treatment, and Y (1)|W = w and Y (0)|W = w are the potential outcomes under both treatments within the subgroup with W = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' At the individual level, the treatment effect is called as Individual Treatment Effect (ITE), which can be rep- resented as: ITE = Yi(1) − Yi(0) (6) where Yi(1) and Yi(0) are the potential outcomes for treat- ment X = 1 and X = 0 of individual i respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The ITE is considered equivalent as the CATE [84, 85] if each subgroup represents an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The treatment effect on different level has been used as quantitative evaluation in recommender systems to handle many issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, ITE is used to estimate the uplift value of recommendation [75, 86, 87, 88];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' ATE can be used to evaluate explanations [89] and estimate unbiased preference [90];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' ATT is used to evaluate counterfactual fairness [91];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In addition to the treatment effect at different levels we introduced above, there are some causal effects for mediation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' A mediation model seeks to explain the process that underlines a causal relationship between the treatment and the outcome via the inclusion of a third variable, known as a mediator variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Let X, Y , and M denote treatment, outcome, and mediator respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We will introduce three types of effects under the binary treatment for mediation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' First, Controlled Direct Effect (CDE) measure the ex- pected increase in Y as the treatment changes, while the mediator is set to a specific value m for the entire popula- tion, which can be defined as: CDE(m) = E[Y |do(X = 1, M = m)]−E[Y |do(X = 0, M = m)] (7) Second, Natural Direct Effect (NDE) measures the ex- pected increase in the outcome as the treatment changes, while the mediator is set to whatever value it would have 7 attained prior to the change, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', X = 0, which can be defined as: NDE = E[Y |do(X = 1, M = M0)] − E[Y |do(X = 0, M = M0)] (8) where M0 represents the value of mediator under treatment as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Finally, Natural Indirect Effect (NIE) measures the ex- pected increase in outcome when the treatment is held constant at X = 0, while M changes to whatever value it would have attained under X = 1, which can be defined as: NIE = E[Y |do(X = 0, M = M1)] − E[Y |do(X = 0, M = M0)] (9) where M1 represents the value of mediator under treatment as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' NIE captures the portion of the effect which can be explained by mediation alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The above direct and indirect effects play an important role in recommendation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The direct and indirect effects help the models quantitatively evaluate path-specific effects to detect and remove undesired effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, they can be used to identify direct and indirect discrimina- tion to achieve or explain fairness [92, 93], they can be used to identify and remove some bias [94, 95], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 6 CAUSAL ESTIMATION METHODS IN RECOMMEN- DATION Having defined the causal effects, the next logical step is to ask how can we estimate those effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' One way is to perform a randomized experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Randomized Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To measure the average treatment effect, an ideal way is to apply different treatment to the same group of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the ideal solution is impractical in real-world situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It can only be approximate by a randomized ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, a randomized experiment randomly assigns individuals into the treated group or the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The estimated ATE can be obtained by the difference of the average outcomes of two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To understand why a randomized experiment is the golden standard for estimating the average treatment effect, it is necessary to understand how correlation is different from causation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' E[Y |X = 1] − E[Y |X = 0] 1=E[Y (1)|X = 1] − E[Y (0)|X = 0] 2= E[Y (1)|X = 1] − E[Y (0)|X = 1] � �� � ATT + E[Y (0)|X = 1] − E[Y (0)|X = 0] � �� � bias (10) Here, step 1 follows the fact that Y (1) is the observed outcome when conditioning on X = 1 and Y (0) is the observed outcome when conditioning on X = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' step 2 adds and subtracts E[Y (0)|X = 1] to construct the ATT term and the bias term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The bias term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (10) creates the gap between the correlation and causation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The randomized ex- periment eliminate the bias term by randomly assigning in- dividuals into the treated group or the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, the random assignment makes the potential out- comes are independent from the treatment Y (1), Y (0) ⊥⊥ X (it does not imply that outcomes are independent from the treatment), thus E[Y (0)|X = 1] = E[Y (0)|X = 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Given Y (1), Y (0) ⊥⊥ X, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (10) can be rewrite as: E[Y |X = 1] − E[Y |X = 0] = E[Y (1)] − E[Y (0)] (11) Therefore, a randomized experiment can simply estimate ATE as the difference of the average outcomes of the treated group and the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In recommendation, the ran- domized experiments are usually used to handle the bias [82, 96, 97, 98, 99, 100, 101, 102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, by taking item exposure as the treatment, the randomized experiments follow the random recommendation policy instead of the deployed policy, in which return the unbiased data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', also called as uniform data) for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' A randomized experiment is not a one-size-fits-all solu- tion for causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In reality, randomized experiments are always time-consuming and expensive, thus the study usually involve small number of individuals, which may not be representative of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Meanwhile, ethi- cal concerns largely limit the applications of the random- ized experiments such as environmental health studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In addition, the randomized experiments cannot explain the causation on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, given the wide availability of observational data, the observational study is a shortcut for causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Observational Data Although the observational study could be a shortcut for causal inference, there are some issues of the observational data should be carefully considered during designing the causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The existence of confounders is a critical problem in the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Confounders) Confounders are variables that affect both the treatment assignment and the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Due to the existence of confounders, some spurious effect may be observed (taking relationship between ice cream consumption and shark attacks as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Con- founders widely exists in recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The exis- tence of confounders often results in different bias based on the definition of confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, taking item pop- ularity as a confounder, it will lead to popularity bias [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In addition to some observed and measurable confounders, such as item popularity, some unobserved or immeasurable confounders (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', which violate the ignorability assumption in Section 4) exist in real-world recommendation and have been widely studied by the community [74, 78, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Simpson’s paradox is another phenomenon that could be observed in the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' From Table 1, it can be observed that in both male and female groups, taking the drug has a better recovery rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' but in the total population, not taking drug has a better recovery rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' This phenomenon is usually caused by confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The Simpson’s para- dox can be also observed in recommender systems [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 8 TABLE 1 Results of a study into a new drug, with gender being taken into account [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' a/b represents a out of b recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Drug No Drug Male 81/87 (93%) 234/270 (87%) Female 192/263 (73%) 55/80 (69%) Total 273/350 (78%) 289/350 (83%) Macdonald [103] observes the Simpson’s paradox in offline evaluation for recommendation, and propose a method to mitigate the paradox in offline evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Compared to the experimental data, observational data only provides the information about what has occurred, but the why a specific treatment is token is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Given that the treatment assignment mechanism is unknown, the bias term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (10) cannot be eliminated or quantitatively measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, the bias caused by unknown treatment assignment is also a critical issue that should be carefully handled in model design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Methods Relying on Assumptions In some complex scenarios, it is risky to assume the causal mechanism based on prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, SUTVA, ignorability and positivity assumptions support some meth- ods to estimate the potential outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' One commonly used method is based on the idea of reweighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As we mentioned before, due to the unknown treatment assignment mechanism, there may exists the bias problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By assigning appropriate weight to each sample in the observational data, a pseudo-population can be created on which the distributions of the treated group and the control group are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' There are two commonly used reweighting methods: inverse propensity scoring and con- founder balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Propensity Score) The propensity score is defined as the conditional probability of treatment given background variables: e(w) = P(X = 1|W = w) (12) Given the propensity scores defined above, inverse propensity scoring methods [104, 105] assign a weight based on propensity score to each observed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Thus the es- timated ATE based on the observed samples can be rewrite as: ATEIP S = 1 n1 � i,xi=1 yi e(wi) − 1 n0 � j,xj=0 yj 1 − e(wj) (13) Inverse propensity scoring (IPS) is often used to de- sign unbiased estimator for recommender systems [90, 106], where the propensity score can be pre-defined or learned from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Although the use of propensity score is effective to reduce the bias, there are some issues during applying IPS in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' First, the correctness of the IPS estimator highly relies on the correctness of the propensity score estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To handle this dilemma, some augmented IPS methods are proposed, such as doubly robust estimator [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Another drawback is that the IPS estimator has variance problem, that the estimator is unstable if the estimated propensity scores are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To overcome this drawback, some methods propose to clip the propensity score [70] or trim samples with small propensity scores [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Another reweighting method is confounder balancing [109, 110, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The motivation is that the confounders can be balanced by the moments, which uniquely determine the distribution of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Thus the sample weights can be learned to estimate the causal effect through reweighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The confounder balancing based methods are used for stable learning [112] and robust recommendation [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In addition to reweighting methods, stratification is an- other representative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The idea of stratification is to split the entire population into homogeneous subgroups, which makes the treated group and the control group are similar in each subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Ideally, in this case, the samples in the same subgroup can be viewed as sampled from the data under randomized experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Macdonald [103] adopt this idea to mitigate Simpson’s paradox in offline evaluation for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In some applications, the causal mechanism is safely assumed based on prior knowledge or expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, the causal mechanism can be represented as a SCM as we introduced before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Although structural causal model framework requires stronger assumptions than potential outcomes framework, it also enable reasoning through the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Using a SCM, the key difference between causation and correlation is do-operations, which is the basic element to estimate the causal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As we mentioned, the do- operation can estimated by manipulated graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the data from the manipulated graph is generated from randomized experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The approaches based on the data generated by the original causal graph are useful in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Applying backdoor adjustment is a popular approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Back-door Criterion) A set of variables Z sat- isfies the backdoor criterion related to an ordered pair of variables (X, Y ) in a causal graph G if Z satisfies both (1) No node in Z is a descendant of X and (2) Z blocks every path between X and Y that contains an arrow into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Through identifying a set of variables satisfying the back-door criterion, the causal effect can be estimated using back-door adjustment formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Back-door Adjustment) If a set of variables Z satisfy the back-door criterion related to an ordered pair of variables (X, Y ), and if P(x, z) > 0, then the causal effect of X on Y is identifiable and is given by P(y|do(x)) = � z P(y|x, z)P(z) (14) Given the population of the observed data, if we divide the subgroup based on value of Z, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (14) can be considered as calculating the causal effect by the weighted sum of each subgroup, which is very similar to the stratification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Additionally, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (14) can be rewrite as: P(y|do(x)) = � z P(y, x, z) P(x|z) (15) where P(x|z) is known as the “propensity score”, therefore, the back-door adjustment is also an alternative representa- tion of IPS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The back-door adjustment is widely 9 (b) (a) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (a) An example of applying back-door adjustment on the causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (b) An example of causal graph with an unobserved confounder, in which the causal values can be estimated by the front-door adjust- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' used to address issues in recommendation, such as bias issues [79, 114], echo chambers [115], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' When we consider the do-operations, the interventions are not limited to actions that force a variable or a group of variables to take on specific value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general, interventions may involve dynamic policies in which the treatment vari- able X is made to respond in a specified way to some set Z of other variables, which is denoted as x = g(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, the estimated causal effect P(Y = y|do(X = g(Z)) can be calculated as: P(Y = y|do(X = g(Z))) = � z P(Y = y|do(X = g(Z)), Z = z)P(Z = z|do(X = g(Z))) = � z P(Y = y|do(X = g(Z)), Z = z)P(Z = z) = � z P(Y = y|do(X = x), Z = z)|x=g(z)P(Z = z) (16) In recommendation, the feedback data is collected from a deployed recommendation algorithm, thus the recom- mendation policy exists in the data generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Considering the dynamic policy as the recommendation policy, conditional intervention can also be applied to design causal recommendation models [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In recommendation scenario, observing an interaction in the feedback data does not imply that the interaction is destined to happen, thus the causal adjustment methods is sometimes applied with counterfactual reasoning [77, 81, 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Apart from above adjustment formulas, there are some rules are valid for interventional probabilities, which are called as the rules of do-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Before introducing the specific rules, we first introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Let X, Y , Z, and W be arbitrary disjoint sets of nodes in a causal DAG G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' GX denotes the graph obtained by deleting from G all arrows pointing to nodes in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Likewise, GX denotes as the graph obtained by deleting from G all arrows emerging from nodes in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The rules of do-calculus can be represented using above notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (The rules of do-calculus) The following three rules are valid for every interventional distribution compatible with a causal graph G Rule 1 (Insertion/deletion of observations): P(y|do(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' w) = P(y|do(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' w) if (Y ⊥⊥ Z|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' W)GX (17) Rule 2 (Action/observation exchange): P(y|do(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' do(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' w) = P(y|do(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' w) if (Y ⊥⊥ Z|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' W)GXZ (18) Rule 3 (Insertion/deletion of actions): P(y|do(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' do(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' w) = P(y|do(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' w) if (Y ⊥⊥ Z|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' W)GXZ(W ) (19) where Z(W) is the set of Z-nodes that are not ancestors of any W-nodes in GX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' With the help of the rules of do-calculus and introduced adjustment formulas, the interventional probabilities can be estimated by the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='4 Methods with Relaxed Assumptions Although above methods relying on introduced assump- tions basically satisfy the requirement of estimating causal effect from the observational data, in practice, for some specific applications, the introduced assumptions may not always hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' There are some methods trying to estimate the causal effect with relaxed assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' SUTVA assumes that individuals are independent and identical distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, in some real-world applica- tions, such as social networks, SUTVA cannot hold anymore since individuals are inherently interconnected with each other through the network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To handle this issue in real applications, a commonly used approach is applying a model, which capture the interconnection, into a causal inference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For examples, applying graph convolu- tional networks into a causal inference model to handle the network data [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The ignorability assumption assumes that the treament assignment is independent to the potential outcomes given the background variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, it is impossible to identify and collect all the background variables in real world, thus the ignorability assumption is hard to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In other words, there may exist unobserved confounders as we mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Only using observational data to estimate the causal effect is difficult, an alternative way is to combine the limited experimental data and observational data together [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In recommendation, the unbiased data is collected from randomized experiments, using a small part of unbaised data and a large part of observed feedback is a popular way to design unbiased recommendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Another solution is based on the assumed SCM, which models the unobserved confounders into the causal graph (an example is shown in Figure 3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Similar to applying the back-door adjustment, we first identify a set of variables satisfying the front-door criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Front-door Criterion) Given an ordered pair of variables (X, Y ) in a causal graph G, a set of variables Z satisfies the front-door criterion with respect to (X, Y ) if Z satisfies the following conditions: – Z intercepts all directed paths from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' – There is no unblocked back-door path from X to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' – X blocks all back-door paths from Z to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Given a set of variables that satisfies the front-door criterion, we can identify the causal effect with unobserved confounders [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 10 Definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Front-door Adjustment) If a set of variables Z satisfy the front-door criterion related to an ordered pair of variables (X, Y ), and if P(x, z) > 0, then the causal effect of X on Y is identifiable and is given by P(y|do(x)) = � z P(z|x) � x′ P(y|x′, z)P(x′) (20) The existence of unobserved confounders is widely rec- ognized by the community [74, 77, 78, 118], there are some works [77, 118] that attempt to apply front-door adjustment in recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Using instrumental variables is a possible way to get around the ignorability assumption and conduct causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Instrumental variables are defined as variables that only affect the outcome via the treatment variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Typical instrumental variables methods [119, 120] adopt two-stage models: the first stage reconstructs the treatment variable based on the instrumental variable and the second stage reconstructs the outcome based on the treatment from the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In recommender systems, Si et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [121] adopt the instrumental variable to design a model-agnostic recommendation framework using search data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 7 CAUSAL DISCOVERY IN RECOMMENDATION The above methods aim to learn the causal effect, there is another branch of causal models targeting at learning causal relations, which is also known as causal discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Except for few works only aim to identify treatment and outcome [122], most of the works aim to discover causal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Following [39, 123], traditional methods can be divided into three categories: constraint-based, socre-based and those based on functional causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Constraint-based Algorithms learn a set of causal graphs that satisfy the conditional independence embedded in the data and statistical tests are utilized to verify if a candidate graph satisfies the independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Score-based algorithms learn causal graphs by maximizing the scoring function S(X, G), which returns the score of the causal graph G given data X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Algorithms based on Functional causal models (FCMs) usually define a variable as a function of its directed causes and some noise term (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', linearly weighted by the adjacency matrix of the causal graph [124]) and optimize the designed objective to learn the parameters of the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We only briefly introduce the causal discovery methods, interested readers may refer to [39, 123] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Most existing works in causal recommendation are based on pre-defined causal graph representing the underlying causal mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The pre-defined causal graphs are usu- ally defined based on expert knowledge, which may be inaccurate and quite simple (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', only involve few vari- ables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Leveraging causal discovery in recommendation will handle these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' There exist few works [125, 126] design recommender systems with causal discovery techniques based on continuous optimization [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The learned causal mechanism will increase the explainability of recommender systems and guide the model design for other aspects, such as fairness, unbiasedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 8 CAUSAL EXPLAINABILITY IN RECOMMENDATION With the development of machine learning, accuracy is no longer the only only pursuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Moreover, transparency and trustworthiness start to obtain increasing attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, heathcare AI is required to provide not only accurate diagnoses, but also supporting explanations to convince patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Recommender systems, with humans in the loop, also require transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Explainable recommen- dation, which emerged and developed with the pursuit of transparency and trustworthiness of recommender systems, has been increasing popular in both academia and indus- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It aims to provide explanations for the recommended items, which will benefit the community in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For consumers, the explainable recommendation is able to help them make better decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For the platform, it may improve the transparency, persuasiveness, trustworthiness and user’s satisfaction of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For model developers, it is an important tool to understand the designed model and accelerate the design cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this section, we will first introduce the overview of the explainable recommen- dations, and then summarize the existing causal methods, as well as some open problems related to causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Problem Introduction The research of explainable recommendation, as a sub-area of explainable AI, was proposed and defined by [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' With the rapid development of deep neural networks, the state-of-the art recommender systems widely adapt deep models to improve the recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' How- ever, these deep models are too complicated for users to understand the decision made by the intelligent systems, thus a deep model is usually considered as a black-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Recommender systems, serve as essential decision-making systems in daily life, are required to provide accurate de- cision results as well as underlying reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, a stock investor needs to know which characteristics lead to the recommendation before making the final decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' A consumer hopes to understand why the recommended items are worth buying before paying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The explainable models can be either model-intrinsic or model-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The former one refers to generating explanations simultaneously with the recommendation re- sults and the later one refers to generating explanations after providing the recommendation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Model-intrinsic (also known as ad-hoc) explainable models usually de- sign the explanation generation mechanism as a part of decision-making process, and model-agnostic explanations (also known as post-hoc) explainable models usually design separate mechanisms for generating explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The explanations can be presented in many different ways, which usually depend on what kind of information source is used for explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Typically, the explanations can be presented as related users or items [89, 129], the features of users or items [128, 130], generated textual sentence [131, 132], visual explanations [133], graph [134], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Existing works have made many successes in explain- able recommendations with different information sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [128] propose Explicit Factor Model (EFM) which extract explicit item features and user 11 opinions from user reviews to provide feature-level expla- nations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Peake and Wang [129] extract association rules to provide purchased items as an explanation in a model- agnostic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Xian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [134] perform explicit reasoning path with knowledge graph to provide recommendations and explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In addition, Existing works have intro- duced explainability into conversational recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [135] develop an Explainable Conversational Recommendation (ECR) model to provide accurate recom- mendations as well as high quality explanations by multi- round conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Incorporating causal inference ideas and techniques brings new opportunities for explainable recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In the following part, we will focus on causal-related methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Interested readers may refer other surveys [29, 30] for more explainable recommendation ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Causal Methods 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Counterfactual In recent years, counterfactual reasoning draws more and more attention in explainable AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For any AI system that makes predictions based on machine learning models, no matter white-box or black-box, counterfactual reasoning looks for what input (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', aspects, features) should be changed, and by how much, to acquire a different predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Then, the altered input will comprise the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For instance, when generating explanations for a rejected loan request, it could be something like: if your annual income is 50, 000, instead of 30, 000, your request will not be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Some existing works have introduced the idea of counterfactual reasoning into recommendation scenarios for generating explanations, which looks for minimal changes in the recommender system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' item features, items in the history, user’s behaviors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=') leading a different prediction to identify the most essential part (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' item features, items in the history, user’s behaviors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=') as the explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Ghaz- imatin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [136] generate explanations for a recommender system based on users’ actions in in the history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specif- ically, it introduces a searching algorithm on a knowledge graph to look for the minimal set of user’s history to be cut off, such that the user will receive different recommendation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [130] proposes a counterfactual explana- tion framework for generating feature-level explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It introduces two new concepts, explanation complexity and explanation strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' These two concepts are used to formulate a counterfactual optimization problem, as well as an evaluation metric to evaluate the generated explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Later in [137], a similar counterfactual explanation frame- work is also used to explain which features are causing fairness issues in recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [138] utilizes an influence function to analyze the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Then, a counterfactual set of training data are used for generating explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Causal Discovery Explainable recommendation models based on causal dis- covery are still in theirs infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Causal discovery methods aim to extract causal relations among variables from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Existing causal discovery based approaches in rec- ommendation provide model-intrinsic explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, through the extracted causal relations, the causal discovery based recommendation models are able to provide recommendations simultaneously with corre- sponding causal relations as explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As we men- tioned, causal discovery methods usually try to learn a causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In recommendation scenario, considering the extremely large amount of items, the learned causal graphs are typically based on item group level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [125] propose to learn a cluster-level causal graph to guide sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Based on the learned cluster-level causal graph and cluster assignment for each item, the model is able to calculate the causal relations between items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The item in interaction history with the strongest causal relation with the recommended item is identified as the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [126] aim to learn a causal graph on product type (PT) level for PT-level recom- mendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Particularly, the model takes collected feedback data as the result of the mixture of two completing mech- anisms: a causal mechanism based on user intention and a intervention mechanism based on deployed recommen- dation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The recommendation and corresponding explanations are generated via the learned PT-level causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Open Problems Despite the above successful usages in causal explainable recommendation, there are open problems that expected to be solved in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' First, causal discovery based ex- plainable recommendation models, are capable of generat- ing model-intrinsic explanations, need further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Second, the current counterfactual explanation algorithms are claimed to be model-agnostic because they are able to be applied on any recommendation models (or at least a wide range of recommendation models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the model itself has to be reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is not certain about how to apply counterfactual explanation algorithms on an recom- mendation model that are not accessible by the algorithm user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Finally, there are currently no methods to leverage other causal reasoning methods, such as the do-calculus, to generate explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 9 CAUSAL FAIRNESS IN RECOMMENDATION Recommender system, as a powerful tool for business, has been widely used to improve user engagement and further create higher profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Classical recommender systems mainly care about how to precisely estimate user prefer- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, in recent years, concerns about fairness in recommendation have attracted much attention from both industry and academia [31, 32, 139, 140, 141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' With the development of recommendation techniques, recommender systems have been widely used to assist or even replace human decision-making in several domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Several studies have shown that the unfairness may lead to negative conse- quences [142, 143, 144], which in turn may have significant social impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, in e-commerce, the unfairness of exposure of items may hurt the benefits of the platform and providers in long-term [145];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' in educational recommen- dation [146], an unfair system due to gender imbalance [147] may discourage females from selecting STEM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', 12 science, technology, engineering, and mathematics) topics, which may affect society for generations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' An unfair ad rec- ommendation may even result in racial discrimination [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, to increase the applications of recommender sys- tems and maintain a healthy social impact, it is critical to consider fairness in recommendations and build a reliable decision-making system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Problem Introduction Before achieving fairness in recommender systems, one should first understand the reasons of unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Bias and discrimination are two commonly accepted causes of unfair- ness [31, 32, 33, 149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Biases in recommender systems mainly consist of bias in data and bias in algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The bias in data may come from data generation, collection, sampling, and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, in recommender systems, the training data is collected from a deployed system, if the algorithm underlying the deployed system makes biased predictions, then the generated data may involve biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The bias in data may affect the algorithms, since most machine learning algorithms rely on data to be trained and make pre- dictions after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' If the training data contains biases, the algorithms trained on them will learn biased knowl- edge from these biases and further lead to unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, if the training data shows significant imbalance between majority user/item group and minority user/item group, it is high likely that the recommendation algorithm learns much better on the majority group and results in discrimination on the minority group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Except for the bias in data, the recommendation algorithm itself may enhance existing biases and cause unfairness, which is referred to the bias in algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, some recommendation algorithms may enhance the popularity bias, where popular items will get more recommendation than less popular items with equal or similar quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Discrimination, as a multidisciplinary problem [150, 151, 152], is also a cause of unfairness defined as an unjustified difference in treatment on the basis of any physical or cultural characteristic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', race, gender, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=') due to human prejudice and stereotyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is worth mentioning that unfairness is not only caused by bias and discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, there may exist conflicts or trade-offs between different kinds of fairness [31, 33, 153], where achieving one fairness will hurt another fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To fight against unfairness, it is important to define fair- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general machine learning, fairness can be defined on target level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', to achieve fairness on group-level or individual-level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, fairness can be categorized into group fairness and individual fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Group Fairness: Group fairness defines the fairness on group-level, which is based on the idea that different groups should be treated equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Here the groups can be divided in many ways, where the most commonly used way is to split the groups based on some explicit sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Individual Fairness: Individual fairness defines fairness on individual-level, which is based on the idea that similar individuals should receive similar predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Moreover, individual fairness can be theoretically considered as a very special group fairness, which divides each individual into different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Since fairness in recommender systems relates to the benefits from multiple stakeholders [144, 154, 155, 156, 157], the request of fairness may come from different sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' There- fore, the definition of fairness in recommendation can also be divided into user-side fairness and item-side fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' User-side Fairness: User-side fairness aims to satisfy the fairness requirements from users (consumers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The re- quest from the user side are mainly focusing on the recom- mendation quality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', recommendation performance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The user-side fairness can be achieved on both group- level and individual-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' User-side fairness on group- level aims to reduce the discrepancy of recommendation quality between different user groups, where the user groups are divided by sensitive features, such as race or gender [142, 158], or by assigned features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', cold users vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' heavy users [159], active users vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' inactive users [140, 160]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For user-side fairness on individual-level, the recommendation quality should be unchanged even an individual’s sensitive features have changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For exam- ple, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [139] incorporate the idea of counterfactual fairness [91] to design a recommendation model which makes the recommendation performance unchanged even the user’s sensitive features are flipped in the counterfac- tual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Item-side Fairness: Item-side fairness aims to satisfy the fairness from items side, which mainly focuses on re- questing equal exposure opportunity of items to maintain market fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Here the items refer to “items” to be ranked or recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, in e-commerce, the items refer to products to be sold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' in recruitment system, the items refer to job seekers (item providers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' One branch of existing work focuses on achieving fairness according to item attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, some works [145, 161, 162, 163, 164, 165] achieve the fair exposure between popular and unpopular items to prevent unpop- ular items from being under-exposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Moreover, another branch of research work mainly focuses on achieving fair- ness based on the sensitive attributes of item providers, such as gender [166, 167, 168], geographic provenience [169, 170, 171], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is worth noting that the user-side fairness and item- side fairness may not exclusive to each other, where two- sided fairness [172, 173, 174, 175, 176] approaches are pro- posed to satisfy the fairness demands from both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Besides the taxonomies mentioned above, there are also some taxonomies [31, 33] that are used to classify fairness in recommendation from other perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, static fairness vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' dynamic fairness [143, 143, 177, 178];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' short- term fairness vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' long-term fairness [145, 179];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' populational fairness vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' personalized fairness [139, 180, 181];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' blackbox fairness vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' explainable fairness [137], centralized fairness vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' decentralized fairness [182, 183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Typically, the proposed approaches to achieve fairness in recommendations can be roughly divided into three cate- gories: pre-processing methods, in-processing methods and post-processing methods [31, 33, 149, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Pre-processing methods usually aim to achieve fairness by minimizing the bias in the data before the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Compared with other types of methods, there are fewer works on pre- processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Some representative methods include 13 fairness-aware data sampling approach to cover items of all groups, data balancing approach [185] to increase the coverage of minority groups and data repairing approaches to ensure label correctness and remove disparate impact [186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In-processing methods propose to incorporate fair- ness requirements as a part of the objective function to achieve fairness during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Typically, the fair- ness requirement works as a regularizer or a constraint [66, 140, 145, 158, 162, 187, 188, 189, 190, 191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To minimize the unfairness while minimizing the original loss function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', recommendation accuracy loss), it is also important to find a trade-off between recommendation accuracy and fairness [145, 192], which is also sometimes formulated as a multi-objective learning problem [192].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Post-processing methods aim to achieve fairness in inference stage after the training, by techniques such as re-ranking [140, 193, 194, 195] or multi-armed bandit [196, 197, 198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To measure the unfairness, many different fairness metrics are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Absolute Difference (AD) [66] measures the absolute difference between the performance of protected group and unprotected group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Normalized Discounted KL- divergence [199] calculates a normalized discounted cumu- lative value of KL-divergence for each position, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More possible fairness metrics can be found in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Recently, researchers have noticed that fairness cannot be well detected by solely correlation or association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specif- ically, fairness criteria are based on solely joint distribu- tion of random variables [200], such as outcomes, features, sensitive attributes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, recent work [201] shows that any definition of fairness that depends merely on the joint probability distribution is not necessarily capable of detecting discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, many approaches [91, 93, 200, 202, 203] are proposed to address the problem of unfairness through the lens of causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general machine learning, causal-based fairness nota- tions are mostly defined on intervention or counterfactual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To measure the unfairness in causal-based fairness, one challenge is understanding the causal relationships that account for different outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Causal graph, as a power- ful tool for causal reasoning, is usually used to represent the causal relationships among variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Given the causal graph capturing the causal relationships, many causal ef- fects are used to measure the unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, ATE (as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (3), also known as Total Effect [27]) is used to measure the effect of changing sensitive attributes to the outcomes, Kilbertus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [204] measure the indirect causal effects [205] from sensitive attributes to outcomes and eliminate the directed path from sensitive attributes to outcomes except via a resolving variable, where resolving variables refer to any variables in the causal graph that are influenced by sensitive attributes in a non-discriminatory way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More details of causal-based fairness notations can be found in [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Counterfactual fairness is a commonly used definition of fairness in causal-based fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Counterfactual fairness is an individual-level causal-based fairness notion, which requires that the predicted outcome should be the same in the counterfactual world as in the real world for any individual [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The basic idea is minimizing the ATT (as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (4), some references also name it as ETT [27, 68, 132]) conditioned on all features to receive the same probability distribution in the factual and counterfactual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For counterfactual fairness in recommendation, the definition is given as follows [139]: Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' (Counterfactual Fairness in Recommendation) The counterfactual fairness is satisfied for a recommendation model if for any user u with sensitive attributes Z = z and remaining features X = x: P(Lz|X = x, Z = z) = P(Lz′|X = x, Z = z) (21) for all L and any value z′ attainable by Z, where L denotes the top-k recommendation list for user u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In the next section, we will introduce some causal meth- ods for fairness in recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Causal Methods 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Reweighting As we mentioned before, bias is a widely accepted cause of unfairness, thus some existing work adopts inverse propen- sity scoring (IPS) methods to solve the bias in recommen- dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, popularity bias will lead to item-side unfairness that popular items may obtain more exposure opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The IPS approaches the biases are caused by non-randomly assigned treatment, thus use the inverse propensity to reweight the samples to remove the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Schnabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [90] consider recommendation as treatment and apply an IPS estimator in an Empirical Risk Minimization framework for learning to solve bias in recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [70] design an IPS-based estimator for unbiased pairwise learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [106] use a small part of unbiased data to train a propensity model and use biased data to train an IPS-based rating model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The IPS-based approaches are easy to implement but it requires an accurate propensity estimator and suffers from high variance [206, 207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Although biases in data are commonly recognized as the main causes of unfairness in recommendation, the re- lationship between bias and fairness has not been clearly understood or discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, the debiasing methods are usually proposed to improve the recommen- dation performance by removing bias, thus the models are evaluated by recommendation metrics instead of fairness metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Many works on fairness are not implemented by de- biasing methods but directly designed by fairness require- ments, which may result in a trade-off between accuracy and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In the following part, we focus on fairness methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More discussion of debiasing methods can be found in Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Counterfactual Counterfactual fairness, as a causality-based definition of fairness, requires the predicted outcomes to be the same in the counterfactual world as in the factual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To achieve counterfactual fairness, it is important but challenging for a fair model to predict the outcomes in the counterfactual world (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', the sensitive attributes have been changed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [208] propose a counterfactual data augmenta- tion module, which is trained based on a variational auto- encoder with a fairness constraint, to generate counterfac- tual data with different sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By maximiz- ing the similarity between the representation learned from 14 the original data and the different counterfactual data, the designed model is able to achieve counterfactual fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Mehrotra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [209] use counterfactual estimation to eval- uate recommendation policies in terms of the trade-odd beween relevance and fairness, and propose a recommen- dation model considering user’s tolerance towards fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The idea of counterfactual is used not only in fairness model design but also in fairness diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, fairness diagnosis aims to find out the reasons that cause model un- fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Inspired by the idea of counterfactual explanation [130, 210], Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [137] propose a counterfactual reasoning approach to learn critical features that significantly influ- ence the fairness-utility trade-off and use them as fairness explanation for feature-based recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Structural Equations A Structural Causal Model consists of a causal graph, which captures the direct causal relations among variables, and a set of structural equations, which builds the quantitative relations among variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As we introduced in Section 6, if the structural equations are given, the interventional or counterfactual outcomes can be obtained by replacing the value of variables in structural equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Inspired by this idea, some works on fairness utilize the learned or pre-defined structural equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, some works [92, 211, 212, 213, 214, 215] model different causal effects from learned structural equations to discover discrimination and further remove them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Kilbertus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [204] develop a practical procedure to remove discrimination given the structural equation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='4 Causal Graphs Some other causality-based methods utilize causal graphs to capture the underlying data generation mechanism and apply other techniques to achieve fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [216] use the d-separation set identified from the causal graph to design a fair upper confidence bound bandit algorithm for online recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [139] design a model based on a causal graph to generate feature independent user representations via adversary learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Concretely, the model trains a predictor and an adversarial classifier simultaneously, where the predictor learns the representations for recommendation and the classifier mini- mizes the predictor’s ability to predict the sensitive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Open Problems As we introduced above, researchers start to realize the importance of considering causality-based fairness in rec- ommendation [76, 201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the foundation of causal fairness in recommendation has not been well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, the fairness techniques are well explored in classification tasks, however, those techniques may not be directly migrated to the recommendation problem even if the recommendation can be considered as a classification task in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, a straightforward method [91] to achieve counterfactual fairness in classification is removing sensitive attributes from the input to guarantee the independence between the outcomes and the sensitive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, in recommender systems, some existing approaches do not use features for recommendation, such as most collaborative filtering based models [217], but still suffer from unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The reason is that the interaction information contains the hidden relationship between sensi- tive features and user-item interaction, and this underlying relationship will be captured by the model during the col- laborative learning thus leading to unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, it is critical to have more explorations about the underlying causal mechanism of unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Additionally, it can help the community to establish a connection between bias and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 10 CAUSAL ROBUSTNESS IN RECOMMENDATION Recently, the robustness of machine learning become in- creasingly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Because model time is very time- consuming therefore, the recommender system models are not re-trained frequently in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Traditionally, the rec- ommender system assumes that the pattern of the training dataset and the test dataset are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, there is a difference between the training dataset and the real- world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The difference might be caused by the naturally distribution shift or intend attacking [218].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='Training on such a training dataset will result in performance decreasing when we apply the model to real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, how to construct a robust model is very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Problem Introduction To begin with, we need to know which aspects harm the ro- bustness of the recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general the dataset will be split into three subset in the training progress (train- ing set, validation set and test set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Most of the robustness happen on training set and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, if the training dataset if not big enough can this may cause the overfitting or underfitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, we may get a bad results on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, the robustness problem can be categorized as following: Distributional shift Many exist recommender systems assume that the distribution for the training set and test set are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However this assumption do not meet the real-world scenarios, and this makes lots of exist recommendation models cannot achieve the performance we expect when we deploy them online [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Many of the current recommender system models are trained based on existing collected datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The distribution of the data can be different when the new model is deployed [220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Even though the data is new collected, there are still transformation risks [218].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Be- cause there is a time period between training the model and deploying the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is long enough for the information of the collected be to attacked or changed due to the processing error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Transformation Most of the recommender system are training with the users’ and items’ feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Based on the feature, the recommender system can provide the users a set of recommendation items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the users’ and items’ feature can be corrupted or misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, if users use VPN when they purchase a item, then the location information could be wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' With such data, the recommender system may provide wrong items to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 15 Attack Except the transformation, attack may also change the correctness of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' With the development of e-commence, more people start to pur- chase items online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' There are huge benefits associated with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, the system are highly likely to be attacked with the purpose to increase or decrease the ranking of some items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, the users may unwillingly changed their rating and reviews of the purchased product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Sparsity Compared to the huge number of user and item, most of the sequential recommendation train set is sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As we mentioned above, this may cause overfitting or underfitting problem and lead to low performance on test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Method in Robustness 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Counterfactual Recommendation model suffers from the data sparsity prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, in the e-commerce application, compared to the large number of user and items, the users purchase history is quiet sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, the model cannot get enough data for training and result in the low predic- tion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' One approach to solve this problem is counterfactual data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By using counterfactual reasoning to generate new training data, and together with the original data can enhance the performance of the rec- ommendation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Counterfactual Data-Augmentation Sequential Recommendation (CASR) provides us a frame- work to solve the problem [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For a training sample ({u, t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='tl}, tl+1), the model will first indicate an index d, and replace a td with an item ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Suppose et ∈ RD is the embedding of item t, D is the embedding size of the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For a given sample ({u, t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='tl}, tl+1), the model will optimize the following object: min ta∈C ∥eta − etd∥2 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' tl+1 ̸= arg max t∈I S(t|u, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', td−1, ta, td+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', tl) (22) where I is the set of all item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='C is the item set for replace- ment, which can be specified as I or other set to involve of some prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' S is the sampler used to generate new sequential data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this function, the object tries to minimize the distance between the original item and the replace item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' And the constraint make sure the changed item is not the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In this case, we can generate data that not the same as the original one but similar to the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Causal Graph Some existing works introduced the idea of causal represen- tation learning to mitigate the distributional shift problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The model split the user features into the observed group and unobserved group and set two types of preference depending on whether it is affected by the observed feature or not [222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' According to the casual graph, a framework is created to model the interaction generation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' And to deal with the unobserved feature, they design a new Variational Auto-Encoder (VAE) to infer the unobserved feature from the historical interaction and observed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Reweighting Moreover, reweighting methods have been introduced to improve the robustness of recommendation, for example, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [113] consider to enhance the robustness of rec- ommendation when there are agnostic distributional shifts between training data and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To this end, the paper introduces a personalized feature selection method for Factorization Machines (FMs) through referring to the confounder balancing approach to balance the confounders of each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In specific, considering there is usually no prior knowledge of the causal structure of input variables in FMs, the paper considers to treat every feature as a treatment variable and aims to estimate its causal effect on the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' When one feature is treated as a treatment, the other features are considered as confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The paper refers to the idea of confounder balancing [223, 224] to learn a weight matrix to reweight each sample through balancing the distributions of confounders across different treatment features, so that FMs will assign a weight to each feature that implies its causal effect on the target variable, and thus help to select causal features for achieving robust recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Open Problems Existing works on robustness problem can only focus on some specific problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, using counterfactual data augmentation problem to mitigate sparsity problem and using causal representation learning to solve distri- bution shift problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' And if we apply these methods on other robustness problems, the experimental performance might decrease a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' And most of the current existing works are unexplained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' They might have good performance on solving some problems, but they cannot explain which part of the model improves the performance and which part of the model can have more improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' An explained ro- bustness method that can be applied on multiple problems is a great challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 11 UPLIFT-BASED RECOMMENDATION Modern recommender systems usually aim to recommend items that users are most likely to interact with (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', click, purchase, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, users may interact with some items even without recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Based on this fact, some existing works propose to recommend items with high interaction probability lift instead of high interaction probability values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' A closely related area is uplift modeling, which refers to the techniques used to estimate the incremental impact of a treatment on the outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Uplift modeling is both a causal inference and a machine learning problem [225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is a causal inference problem because the two required out- comes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', receive treatment or no treatment) for calculating the incremental impact are exclusive for an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is also a machine learning problem because it needs models to predict reliable uplift values for decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Theoreti- cally, uplift modeling aims to estimate a treatment effect on outcomes [225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' There are three main approaches in existing literature: the Two-Model approach trains two models on treated data and controlled data respectively, and uses the 16 difference between two predictions to calculate the uplift value [226];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' the class transformation approach builds the connection between the treated group and the controlled group based on some assumptions [227];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' the direct estima- tion approach designs a model to directly estimate the uplift value [228, 229].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Problem Introduction Recommender systems have been employed in several in- dustrial domain to increase the profit of the business and improve user engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To achieve this goal, most of the recommendation models are designed to increase user action (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', click, purchase, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=') by recommending items that have highest interaction probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, most recommender systems neglect a fact that users may take actions on some items regardless of whether the system recommends them [75, 230].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, a user will pur- chase bottled water with 95% probability and energy drink with 50% probability if the system recommends them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In the opinion of most traditional recommender systems, it would be better to recommend bottled water since it is more likely to be purchased by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, if the system does not recommend them, bottled water, as a product for daily use, may still have 90% probability to be purchased, while energy drink may only have 20% probability of being purchased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Recommending energy drink seems to be a better choice since it has higher lift of purchase probability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', 30% vs 5%), which in turn may be expected to lead to more profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Based on this motivation, there is a trend to design uplift-based recommender systems which aim to recommend items with high lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Some previous works have been aware of the impact of recommendation but have not solve it from a causal view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Bodapati [231] proposes a two-stage model which separately trains the awareness and satisfac- tion stages for items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By training the model based on firm- initiated purchase data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', purchases as a consequence of recommendation) and self-initiated purchase data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', purchases other than firm-initiated purchases), the model aims to recommend items that maximize the expected in- cremental number of purchase from recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [230] propose a purchase prediction model which incorporates individual differences in recommendation re- sponsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, the model includes user- specific and item-specific responsiveness to maximize the impact of recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' An uplift in recommender systems is defined as an in- crease of user actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', click, purchase, like, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=') caused by recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Considering the uplift is defined as difference between situations with and without recommen- dation, from the perspective of causal inference, the uplift can be mathematically represented by potential outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, taking recommendation as the treatment, let Y (1) be the potential outcome with a recommendation, Y (0) be the potential outcome without a recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Considering the binary situation, Y (1) = 1 and Y (0) = 1 imply that a user will take actions on the item with and without recommendation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The uplift of an item for a user is Y (1) − Y (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In the following subsection, we will introduce some existing works on uplift-based recom- mendation models with causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Causal Methods 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Data Processing One challenge of estimating the uplift value is that each individual cannot observe both the factual and counterfac- tual outcomes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', outcomes with and without recommen- dations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Thus there is no observed ground truth for the uplift value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', the causal effect of recommendation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To overcome this issue, one possible solution is regarding the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [75] propose a sampling method on the observational data for an uplift-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Specifically, by observing purchase and recommendation logs, for a given user, an item can be either purchased or not purchased and either recommended or not recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The proposed optimization samples positive and negative instances that are specific to the uplift task from four classes items (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', recommended and purchased, recommended and not purchased, not recommended and purchased, not recommended and not purchased).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, by taking the sampled labels for uplift task as the ground truth, the proposed optimization is able to learn the uplift value for user-item pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Except for the sampling methods on observational data, training on experimental data is also an available option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Shang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [86] propose a reinforcement learning based approach, which incorporate a deep uplift network to learn the causal effect of different actions as a reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The uplift network learns from the training data collected from a randomized experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Counterfactual According to the calculation of the uplift value, a straight- forward way is to estimate the counterfactual outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Although the randomized experiment is ideal for estimating the causal effect, it is impractical to apply randomized experiment in all recommendation scenarios since it is time- consuming and expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, it is essential to esti- mate the counterfactual outcomes only based on the obser- vational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Inspired by the idea of collaborative filtering, Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [87] believe that similar users have both similar tastes on items and similar treatment effect under recom- mendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The proposed approach is designed based on tensor factorization with three dimensions as user, item and treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, for a three dimentional tensor with m users, n items and l treatments, the element yu,i,t can be predicted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' ˆyu,i,t = pT u qi + pT u dt + qT i dt (23) where pu, qi, dt are latent representation of user u, item i and treatment t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The predicted value of ˆyu,i,t is used to infer the potential outcome for a user-item pair (u, i) under treatment t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Taking binary treatment setting as an example, the uplift value for a user-item pair (u, i) can be estimated by ˆyu,i,t=1−ˆyu,i,t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [88] apply a match- ing estimator [232] to estimate unobserved counterfactual outcomes and further estimate the causal effect for recom- mendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, following the neighborhood methods in recommender systems, the proposed approach replaces the potential outcomes with the weighted average over the observed outcome for a set of neighbors to calculate the causal effect, where the neighbors can be neighborhood users or neighborhood items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 17 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Reweighting Estimating causal effect from the observational data only is challenging, since the ground truth is unobservable and the estimation is prone to the biases in the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To overcome this issue, some existing works design IPS-based approaches to estimate unbiased causal effect for recommendation or evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The unbiased estimation of the uplift value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', the causal effect) can be formuted by IPS [233].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In practice, IPS is prone to suffer from high variance issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To tackle this problem, Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [233] apply capped inverse propensity scoring (CIPS) to train an unbiased uplift-based model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [75] propose a unbiased estimator for uplift-based evaluation using self- normalized inverse propensity scoring (SNIPS) [234];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Xiao and Wang [235] apply doubly robust technique [107, 236] to train an unbiased and robust model for uplift-based recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Open Problems Existing works on uplift-based recommendation mainly fo- cus on representing uplift value and estimating the causal effect using potential outcome framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Structural causal model, as a power tool for causal inference, has rarely been used for uplift-based recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Existing works using structural causal models are trying to estimate user’s preference if the system recommends a certain item, which can be estimated by do-operations on designed causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, it is still not clear how to estimate the pref- erence without recommendation using do-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' First, the structural causal model requires the designed causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Existing works on causal recommendation using structural causal models rarely explicitly involve the impact of recommendation into the causal graph, however for uplift-based recommendation, whether requiring a specific causal graph that explicitly depict the impact of recommen- dation still needs to be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Secondly, mathematical representation of the preference without recommendation using do-operation is also a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Finally, for uplift- based recommendation, if the designed causal graph and mathematical representation of preference without recom- mendation are decided, applying causal techniques on pref- erence without recommendation may differ from existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 12 CAUSAL UNBIASEDNESS IN RECOMMENDA- TION Nowadays, recommendation algorithms have been widely used in several applications to alleviate information over- loading in our daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Although recommender systems (RS) have obtained huge impacts in a wide range of real- world applications, it still faces many bias issues which are challenging, and if left unattended, will affect the long- term benefits of the recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Bias issues are common in RS since one nature of RS is the feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Following a generally accepted understanding [35, 36], the feedback loop in RS can be divided into three parts from a bird’s-eye view: 1) the data collection part (user → data);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 2) the model training part (data → model);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 3) the model serving part (model → user).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Different definitions of bias issues exist in each part and the whole feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We will introduce more details in the following parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Problem Introduction As we mentioned, the bias issues exist in each part as well as the whole of the feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We will introduce the different definition of bias in the feedback loop of RS as follows: Bias in Data refers to the distribution difference between the collected data for training and the ideal test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Typically, the training data for RS is observational instead of experimental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The user decision may be affected by sev- eral factors such as exposure mechanism of RS, thus the training distribution is different from the test distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Additionally, the training data may not truly represent user preference, misleading recommendation model to inaccurate prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We will introduce four kinds of bias in data as follows: – Selection Bias: Selection bias stems from users’ explicit feedback (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', ratings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Selection bias means the ob- served ratings are not representative of all ratings due to users’ selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is also referred as missing-not-at- random (MNAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' – Exposure Bias: Exposure bias usually happens in recom- mendations with implicit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Since the informa- tion about which item the user dislikes is unavailable in observed data, the learning process will use unobserved interactions to represent negative preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Exposure bias means unobserved interactions do not necessarily represent the user’s negative preference since the users are merely exposed to a small portion of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' – Conformity Bias: Conformity Bias means that users tend to behave similarly to the others in the group, even if their behavior goes against their own judgment, which makes the feedback may not represent users’ true pref- erence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' – Position Bias: Position bias is common in recommenda- tion, especially the results are presented by a ranking list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Position bias means that users tend to interact with items in higher position in the recommendation list even if the items in higher position may not be highly relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Bias in Model refers to the bias in the model design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Bias i not always harmful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In fact, the bias in model empower the model to achieve the ability to generalize the prediction to unobserved examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' – Inductive Bias: Inductive bias represents the assump- tions made the model designer to better learn the ob- jective and to generalize beyond training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Bias in Results refers to the phenomenon that the rec- ommendation algorithms tend to exhibit bias in recom- mendation results presented to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Typically, the biases in recommendation results are studied from two perspec- tives, one is popularity bias and the other is unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We have introduced fairness and related methods in sec- tion 9, thus in this section, we will limit the bias in results to popularity bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' – Popularity Bias: Popularity bias refers to the phe- nomenon that popular items are recommended more frequently than their popularity warrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 18 Feedback Loop Bias refers to the amplified bias intro- duced by the whole RS feedback loop mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Data bias will lead to data imbalance and result in bias issues in recommendation results, while the biased recommen- dation will in turn impact the user’s behavior and further amplify the bias in the future recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Taking popularity bias as an example, the popular items get more exposure in the observed data, which in turn obtain increase opportunity to be recommended, resulting in amplified bias, where popular items become more pop- ular and non-popular items become even less popular [237, 238, 239].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' These amplified bias caused by feedback loop, if left unattended, will result in echo chambers [115, 240] or filter bubble [241, 242, 243, 244], which will decrease the diversity and increase the homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general, there are two ways for debiasing in rec- ommender systems, one is debiasing during training and the other is debising during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Introducing causal inference into debias recommendation makes a great success in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In the following part, we will introduce existing works on debiased recommendation models based on causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Causal Methods 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Data Processing To address the bias problem in recommender systems, one straightforward solution is to leverage unbiased data [82, 96, 97, 98, 99, 100, 101, 102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As we mentioned in section, combining the limited experimental data and observational data is a possible solution under the relaxed ignorability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In recommender systems, the experimental data, which is also called as unbiased data, intervene the system by using a random recommendation policy instead of a normal recommendation policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, for each user, they do not use recommendation models to show items, but instead randomly select some items to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Leveraging unbiased data helps to achieve debiased prediction because applying random recommendation will break the feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The key challenge is how to incorporate a small portion of unbiased data into model design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Rosenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [96] and Bonner and Vasile [82] apply two recommendation models for biased data and unbiased data respectively and connect two mod- els by regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [97] learn a imputation models with unbiased data for ad click prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [101] leverage unbiased data by meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Despite the effectiveness on handle biases by using unbiased data, collecting unbiased data will randomly recommend items to users instead of using personalized recommendation model, which will inevitably hurt users’ experience and revenues of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Reweighting Another commonly used method is based on reweight- ing, which use inverse propensity scores to reweight the data sample for different bias issues, such as selection bias [90, 106], exposure bias [70, 72, 73, 245], position bias [246, 247, 248, 249, 250, 251], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The key challenge is how to estimate the propensity scores and how to apply it into optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Some works [70, 252] use popularity- based propensity estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Some works [73, 248, 250, 251] propose a dual problem to both optimize a propensity estimator and a recommendation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Some works [249] propose to learn propensity scores from the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Some works [207, 235] use doubly robust model to handle inaccurate propensity estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Inverse propensity scoring methods have some limitations, such as inaccurate propensity scores and suffering from high variance problem [206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Causal Adjustment Causal adjustment is another promising direction for ad- dressing bias issues [77, 79, 81, 114, 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' With the help of do-operator, the designed models aim to estimate the causal preference P(Y |U, do(V )) with intervening item exposure rather than the pure associative preference P(Y |U, V ) esti- mated by traditional recommendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Intuitively, it can be understood as to answer a counterfactual question: what would the preference be if we intervene to expose the item to the user?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Causal adjustment is used to estimate the causal preference with observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specif- ically, causal adjustment includes back-door adjustment [68], front-door adjustment [68], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Based on the designed causal graph representing the underlying mechanism of data generation in recommender systems, the first thing is to identify a set of variables satisfying the corresponding criterion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', back-door criterion for back-door adjustment, front-door criterion for front-door adjustment), then apply causal adjustment on identified variable set to estimate the causal preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [79] apply back- door adjustment to mitigate the exposure bias caused by the item popularity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [77] leverage front-door adjust- ment to remove the effect of unobserved confounders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [114] utilize back-door adjustment to mitigate the effect of popularity bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Causal adjustment requires to identify a set of variables satisfying the corresponding criterion, however, given a reasonable causal graph for recommender systems, it is not always find out a set of variables satisfying such criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' But the designed causal graph will guide the model design from other ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='4 Causal Graph Causal graph, as an effective and powerful tool for causal modeling, is used to depict the data generation process in recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Based on the designed causal graph, researchers will take it as the guidance to design causal models for debiasing [80, 253].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [253] and Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [80] disentangle the effect from bias and user’s preference based on the designed causal graph and recommend items solely based on user’s preference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [95] and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' [94] represent the counterfactual world based on the designed causal graph and perform counterfactual reasoning for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Open Problems Inverse propensity scoring (IPS) is a valuable method for debiasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, the effectiveness of IPS methods highly rely on the correctness of propensity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' How to obtain correct propensity scores is still an important 19 yet unsolved question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Existing work usually design sim- ple propensity estimator based on some item character- istics, such as popularity-based propensity [70], or learn the propensity scores from data [73, 248, 249, 250, 251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Whether using correct propensity scores can be only esti- mated indirectly through the improvement for recommen- dation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, quantitative evaluation of the correctness of propensity scores is still an open problem and need further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 13 OPEN PROBLEMS AND FUTURE DIRECTIONS 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='1 Underlying Causal Mechanisms Recall the existing works we introduced above, most of them are based on the underlying causal mechanisms of rec- ommender systems, which are represented by pre-defined causal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' In general, there are three levels of pre- defined causal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The first level is identifying cause and effect only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, IPS methods only investigate the quantitative relationship between two variable, one is cause, the other is effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, in some IPS based models for debiasing in recommendation, the cause is item exposure, and the effect is the probability of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The second one is defining causal graphs, which identify the causal relationship between all variable pairs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', whether causal relation exists, the direction of causal relation if exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' By pre-defining the causal graph, some existing works design models with the guidance of causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, some works [80, 253] disentangle multiple cause on the effect based on the defined causal graph to achieve unbiasedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The last level is structural causal models, which define not only the causal relations but also quantitative relations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', structural equations take causes as input and return the value of effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The effectiveness of proposed models are highly related to the correctness of underlying causal mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Currently, most of the exist- ing works define the underlying causal mechanism through expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The correctness of the pre-defined causal mechanism can only be indirectly reflected by the recom- mendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, a direct and quantita- tive evaluation of the defined causal mechanisms deserves for further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Another observation is that different models may have different pre-defined causal mechanisms even under the same practical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' As such, we believe that a universal causal mechanism should be proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='2 Causal Discovery Apart from concerns about the accuracy of pre-defined causal mechanisms, another limitation of pre-defined causal mechanism is that pre-defined causal mechanisms by expert knowledge are usually quite simple, which only involve few factors into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, in real-world scenario, the decision-making process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=', the underlying causal mechanism of recommender systems) may involve much more factors, which beyond the comprehension of domain experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, learning causal relations from data is an important yet unsolved problem in recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' There exist few works [125, 126] design causal discovery methods based on continuous optimization [127, 254] for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' The learned causal mechanism can be used for explainable recommendation or be leveraged to improve recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Therefore, it is a promising di- rection to propose causal discovery methods for recom- mendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' It is also a challenge to evaluate the proposed causal discovery methods for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Since there is no ground-truth causal mechanism in real-world data, the causal discovery methods in recommendation are usually indirectly evaluated by recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' To directly evaluate causal discovery methods for recommen- dation, one possible solution is using simulation (we will introduce it in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content='3 Causality Driven Simulations Simulation is one of the most powerful approach to build environments in which the recommender systems can be measured and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Building simulation for recommen- dation will benefit both industry and academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For exam- ple, for industry, simulation provide controllable environ- ment for practitioners to analyze the objectives of interest, such as some business purpose, to accelerate the pace of application development without the ethical risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For re- searchers in academia, due to the restrictions of accessibility of real-world recommender systems, some proposed meth- ods cannot be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' This issue can be addressed by using simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Existing simulations leverage reinforce- ment learning techniques to simulate the decision making process under a designed environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' However, existing simulations without underlying causal mechanisms may lead to inaccurate and unstable decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Leverag- ing causal mechanisms into simulation will achieve more stable system for long-term analysis and causal-related analysis as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' For example, causality driven simulations can be used to evaluate causal discovery methods in rec- ommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Thus, causality driven simulation will play an essential role in recommender systems, which deserves further explorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' 14 CONCLUSION In this survey, we provide comprehensive review of causal inference methods for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We first provide the fundamental knowledge of recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We then introduce existing work in perspective of both causal inference and recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' More specifically, on the one hand, we introduce knowledge about causal in- ference and demonstrate its connection with recommender systems, on the other hand, we introduce different problems in recommender systems and how causal inference applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Finally, we further list some open problems and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' We hope this survey can benefit researchers and practitioners in this area and inspire more research work in causal inference for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} +page_content=' Jannach, M.' metadata={'source': 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877, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE2T4oBgHgl3EQfoQhP/content/2301.04016v1.pdf'} diff --git a/EtE1T4oBgHgl3EQfEgOL/content/tmp_files/2301.02891v1.pdf.txt b/EtE1T4oBgHgl3EQfEgOL/content/tmp_files/2301.02891v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce54d5ad17c6ac428cf38ec703b4c60e29ef01b8 --- /dev/null +++ b/EtE1T4oBgHgl3EQfEgOL/content/tmp_files/2301.02891v1.pdf.txt @@ -0,0 +1,1149 @@ +Geometric quantum discord and coherence in a dipolar interacting magnetic system +Clebson Cruz,1, ∗ Maron F. Anka,2, † Hamid-Reza Rastegar-Sedehi,3, ‡ and Cleidson Castro4, § +1Grupo de Informa¸c˜ao Quˆantica e F´ısica Estat´ıstica, Centro de Ciˆencias Exatas e das Tecnologias, +Universidade Federal do Oeste da Bahia - Campus Reitor Edgard Santos. Rua Bertioga, +892, Morada Nobre I, 47810-059 Barreiras, Bahia, Brasil. +2Instituto de F´ısica, Universidade Federal Fluminense, +Av. Gal. Milton Tavares de Souza s/n, 24210-346 Niter´oi, Rio de Janeiro, Brasil. +3Department of Physics, College of Sciences, Jahrom University, Jahrom 74135-111, Iran +4Centro de Forma¸c˜ao de Professores, Universidade Federal do Recˆoncavo da Bahia, +Avenida Nestor de Mello Pita, 535 Amargosa, Bahia, Brazil. +(Dated: January 10, 2023) +The study of low-dimensional metal complexes has revealed fascinating characteristics regarding the ground- +state crossover shown by spin-gaped systems. In this context, this work explores the effect of the quantum-level +crossing, induced by the magnetic anisotropies of dipolar interacting systems, on the quantum discord and +coherence of the system. The analytical expressions for the quantum discord, based on Schatten 1-norm, and +the l1 trace-norm quantum coherence for dinuclear spin-1/2 systems, are provided in terms of the magnetic +anisotropies. The results show that, while the quantum discord has a clear signature of the quantum level- +crossing, the basis dependence of the quantum coherence hides the crossover regarding the measured basis. In +addition, the global quantum coherence is wholly stored within the correlations of the system, regardless of its +reference basis. +Keywords: Dipolar Interaction; Qunatum discord; Quantum Coherence; Quantum-level crossing. +I. +INTRODUCTION +The study of the quantum properties of composite systems has led to a revolution in the development of emerging quantum +technologies [1–3]. The new generation of quantum devices explores physical properties associated with quantum correlations +between particles [4, 5] and superposition principle for the system states [1, 6, 7]. In this scenario, the characterization of the +quantumness of the physical systems is of paramount importance since the existence of quantum correlations and coherence are +a valuable resource for several quantum tasks [4, 8, 9]. +However, the characterization of quantum correlations is a rather complicated task from the theoretical [10] and experimental +[11] point of view. This scenario is aggravated in Condensed Matter systems, where the number of interacting components in +the system is usually on the order of the Avogadro number [12]. Nevertheless, there are a few exceptions, like low-dimensional +metal complexes (LDMC), for which full knowledge about their quantum properties can be obtained through the corresponding +analytical solutions [4, 6, 13–20]. In such solid-state systems, intra-molecular interactions are strong enough to suppress ex- +trinsic and intermolecular interactions [4, 13, 14, 21]. Therefore, their quantum features exhibit high stability against external +perturbations such as temperatures [4, 13, 14, 16, 21], magnetic fields [4, 6, 16, 22], and pressures [6, 15]. These characteris- +tics make these systems promising platforms for the development of emerging quantum technologies [4, 23–26]. In regard to +these possible applications, the study of dipolar interacting magnetic systems has received considerable attention in the quantum +information literature [17, 27–31]. +In this work, we present a theoretical study of quantum correlations and coherence for a dipolar interacting magnetic system, +exploring the effects of magnetic anisotropies on the quantumness of the system. As a result, this study provides to the literature +analytical expressions, in terms of the magnetic anisotropies, for the quantum discord, based on Schatten 1-norm, and the l1 +trace-norm quantum coherence written in an arbitrary basis, defined by the co-latitude and longitude angles of the Bloch sphere +representation. According to the findings, the behavior of the quantum discord carries a noteworthy signature of the quantum +level-crossing, caused by population changes resulting from the alteration of Boltzmann weights arising from the change of the +magnetic anisotropies of the system. On the other hand, the basis dependency of quantum coherence is detrimental in terms of +recognizing this crossover. In this regard, the measurement of the average quantum coherence is numerically obtained in order +to obtain a basis-independent perspective for this quantum resource. The results not only demonstrate that the average coherence +∗Electronic address: clebson.cruz@ufob.edu.br +†Electronic address: maronanka@id.uff.br +‡Electronic address: h.rastegar@jahromu.ac.ir +§Electronic address: ccastro@ufrb.edu.br +arXiv:2301.02891v1 [quant-ph] 7 Jan 2023 + +2 +is completely stored within the correlations of the system, yet they also demonstrate that it is possible to retrieve the signature of +the energy-level crossover present on the quantum discord measurement. Furthermore, the findings show how dipolar interaction +coupling magnetic anisotropies impact quantum correlations and coherence in a dinuclear spin-1/2 system. Thus, the dipolar +interaction model is a viable foundation for quantum technologies based on quantum discord and coherence. +II. +DINUCLEAR METAL COMPLEX WITH DIPOLAR INTERACTION +The class of dinuclear metal complexes undergoes several types of magnetic coupling [12]. Among these are Heisenberg +exchange, which is isotropic under rotations in spin space [4, 6, 20, 21], and Dzyaloshinskii-Moriya (DM) interaction [32– +34], which accounts for weak ferromagnetism in some antiferromagnetic materials [12]. A ubiquitous example of anisotropic +coupling in LDMCs is the dipolar interaction [17, 27–31]. This coupling arises from the influence of a magnetic field yielded +by one of the magnetic moments in the other ones [12]. In particular, for dinuclear metal complexes, the Hamiltonian which +describes this interaction is given by: +H = −1 +3 +⃗S T +A · +↔D · ⃗S B , +(1) +where ⃗S j = {S x +j, S y +j, S z +j} are the spin operators and +↔D = diag(∆ − 3ϵ, ∆ + 3ϵ, −2∆) is a diagonal tensor, with ϵ and ∆ being the +rhombic and axial parameters, respectively, related to the magnetic anisotropies in the dipolar model [12]. In particular, ∆ is +related to the zero-field splitting of the energy levels [35]. Considering the Hamiltonian, Eq. (1), written in the S (z) eigenbasis, +∆ > 0 becomes a signature that the spins are on z-axis, while ∆ < 0 indicates that the spins will be on the x − y plane. +Considering a dinuclear metal complex in with d9 electronic configuration, Eq. (1) can describe two coupled spin 1/2 particles +in the corresponding S z eigenbasis {|00⟩, |01⟩, |10⟩, |11⟩} +H = 1 +6 +�������������� +∆ +3ϵ +−∆ −∆ +−∆ −∆ +3ϵ +∆ +�������������� +, +(2) +The energy levels of the system from the coupling parameters are composed by the +E1 = 0 , +E2 = −2∆ , +E3 = ∆ + 3ϵ , +E4 = ∆ − 3ϵ . +(3) +From the thermal equilibrium, the density matrix for the coupled system is described by the Gibbs form ρAB = Z−1e−H/kBT, +where +Z = Tr(e−H/kBT) = 2eβ∆/6 cosh +�β∆ +6 +� ++ 2e−β∆/6 cosh +�βϵ +2 +� +. +(4) +is the canonical the partition function, with kB representing the Boltzmann’s constant. Thus, the dinuclear density matrix at sites +labeled by A and B can be written in the S z eigenbasis as the so-called X-shaped mixed state +ρAB = e− β∆ +6 +Z +������������������� +cosh +� βϵ +2 +� +− sinh +� βϵ +2 +� +e +2β∆ +6 cosh +� β∆ +6 +� +e +2β∆ +6 sinh +� β∆ +6 +� +e +2β∆ +6 sinh +� β∆ +6 +� +e +2β∆ +6 cosh +� β∆ +6 +� +− sinh +� βϵ +2 +� +cosh +� βϵ +2 +� +������������������� +. +(5) +The density matrix eigenvalues (population) and their corresponding eigenvectors can be written as: +PΨ− = +1 +1 + e +β∆ +3 + 2e− β∆ +6 cosh +� βϵ +2 +� → |Ψ−⟩, +(6) +PΨ+ = +1 +1 + e− β∆ +3 + 2e− β∆ +6 cosh +� βϵ +2 +� → |Ψ+⟩, +(7) +PΦ+ = +1 +1 + eβϵ + eβ( ∆+ϵ +2 ) + eβ( ∆+3ϵ +6 ) → |Φ+⟩, +(8) +PΦ− = +eβϵ +1 + eβϵ + eβ( ∆+ϵ +2 ) + eβ( ∆+3ϵ +6 ) → |Φ−⟩, +(9) + +3 +where +|Ψ±⟩ = +1√ +2 +(|01⟩ ± |10⟩) , +|Φ±⟩ = +1√ +2 +(|00⟩ ± |11⟩) . +(10) +are the so-called Bell states, which represent the maximally entangled states for a bipartite system [36]. +The study of LDMC has attracted the attention of both theoretical and experimental condensed matter physics communities +due to the fascinating properties of their ground states [6, 37]. In the presence of an external magnetic field, this systems +typically show a quantum level-crossing between its ground state and the first excited one when the field reaches a critical value +since the external magnetic field splits its energy levels, changing their corresponding populations. However, since the dipolar +interaction arises from the influence of the magnetic field created by one of the magnetic moments in the other, the splitting in +energy levels is ruled by the axial (∆) and rhombic (ϵ) parameters, as can be seen in Eq. (3). In this regard, Fig. 1 shows the +populations as a function of the ratio between the magnetic anisotropies and the energy scale factor kBT. +- 20 +- 10 +0 +10 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ϵ/kBT +Populations +- 20 +- 10 +0 +10 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ϵ/kBT +Populations +(a) +(b) +- 20 +- 10 +0 +10 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Δ/kBT +Populations +- 20 +- 10 +0 +10 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Δ/kB +Δ/kB +T +Populations +PΨ- +PΨ+ +PΦ+ +PΦ- +ϵ/kB= 5K +ϵ/kB=-5K += 5K +Δ/kB=-5K +FIG. 1: (Color online) Populations, Eqs. (6)–(9), as a function of the ratio between the magnetic anisotropies and the energy scale factor kBT. +(a) Axial dependence considering the rhombic parameter ϵ/kB = 5 K (left) and ϵ/kB = −5 K (right). (b) Rhombic dependence considering the +axial parameter ∆/kB = 5 K (left) and ∆/kB = −5 K (right). The inset shows the magnetic anisotropy dependence on the energy levels. +As can be seen, in agreement with Eq. (3), when the spins are in the x − y plane (∆ < 0) with positive rhombic parameter +(ϵ > 0), the ground state is the given by the state |Φ−⟩ (with population PΦ−), and there is no quantum level crossing. Thus, the +system remains in the ground state. However, changing the signal of the rhombic parameter induces a quantum level crossing +between states |Φ−⟩ and |Φ+⟩ (with population PΦ+). Moreover, for the spins oriented in the z axis (∆ > 0), it is possible to +observe a quantum level crossing between the state |Φ−⟩ (if ϵ > 0) or |Φ+⟩ (if ϵ < 0) and the state |Ψ−⟩ (with population PΨ+) by +increasing the ratio ∆/kBT to the critical point ∆ = |ϵ|. +In reference [29], the authors study the effect of the magnetic anisotropies, described by the axial and rhombic parameters, +on the nonlocal correlations of a dipolar interacting system of two spins-1/2, identified by the Peres-Horodecki separability +criterion [38, 39]. In addition, they explore the change in the ground state on the thermal entanglement for the teleportation +process. However, although quantum entanglement provides one path toward the characterization of nonlocal correlations, it +does not encompass all quantum correlations in the system [21, 40–51]. Therefore, in order to expand this result, the following +section presents a study of the quantum correlations and coherence described by the Schatten 1-norm geometric quantum discord +and the l1 trace-norm quantum coherence. + +30 +20 +10 +10 +20 +0 +0 +5 +10 +E20 +10 +0 +10 +20 +30 +10 +0 +5 +10 +E20 +10 +0 +-10 +20 +0 +5 +0 +5 +1020 +10 +ler +0 +-10 +20 +0 +5 +0 +5 +1060 +40 +20 +20 +40 +60 +20 +-10 +0 +10 +2060 +40 +20 +0 +20 +40 +60 +20 +0 +10 +204 +III. +QUANTUM DISCORD +Quantum discord has been defined as a measurement of the quantumness of correlations in a quantum system. It has been first +introduced as an entropic measurement of genuinely quantum correlations in a quantum state, defined as the difference between +the total and the classical correlation [40] Q(ρAB) = I(ρA : ρB) − C(ρAB), where I(ρA : ρB) = S (ρA) + S (ρB) − S (ρAB) represents +the mutual information between the subsystems A and B, and C(ρAB) is the classical correlation of the composite system ρAB +defined as C(ρAB) = max{Bk} +�S (ρA) − � +k pkS (ρk)�, with the maximization taking over positive operator-valued measurements +(POVM’s) {Bk} performed locally only on subsystem B. However, this analytical maximization over POVMs is an arduous task +even for a two-qubit system [10, 11, 40, 41, 47, 51]. In this scenario, the class of entropic measurements of correlations, such as +the entropic quantum discord, is defined as nondeterministic polynomial time (NP-complete) problems [10]. Consequently, only +a few results for the analytical expression of entropic quantum discord, and only for certain classes of states are exact solutions +known [41, 43, 45, 46, 49, 50, 52]. Due to this fact, alternative measurements of quantum correlations have been proposed +[41, 43, 45–50, 52–60], especially quantifiers based on geometric arguments [47–49, 53, 57–60]. +Geometric approaches are widely used to characterize and quantify quantum resources in a wide variety of quantum systems +[61]. In particular, the Schatten 1-norm quantum discord [4, 21, 47, 48, 62], is a reliable geometric-based quantifier of the +amount of quantum correlations in metal complexes [4, 21, 62, 63]. The so-called geometric quantum discord can be defined in +terms of the minimal distance between a set ω of closest classical-quantum states ρc [21, 47, 48], given by: +ρc = +� +k +pkΠ{A} +k +⊗ ρ{B} +k , +(11) +where 0 ≤ pk ≤ 1 and � +k pk = 1; {Π{A} +k } define a set of orthogonal projectors for a given subsystem A and ρ{B} +k +the reduced +density matrix for the subsystem B [47, 48]. Therefore, the geometric quantum discord can be expressed as +QG(ρAB) = min +ω ∥ρAB − ρc∥ , +(12) +where ∥M∥ = Tr +� √ +M†M +� +is the so-called 1-norm, and ρAB is the given quantum state at thermal equilibrium, Eq. (5). +Therefore, considering the given dinuclear magnetic system of spins-1/2 in a quantum spin-lattice, ruled by a dipolar Hamil- +tonian H, Eq.(1), the invariance under π rotation around a given spin axis (Z2 symmetry) [12, 46] allow us to compute the +geometric quantum discord, based on Schatten 1-norm, for the two-qubit X state, Eq. (5), as [53, 64] +QG(ρAB) = 1 +2 +� +φ2 +1max{φ2 +2, φ2 +3} − φ2 +2min{φ2 +1, φ2 +3} +max{φ2 +2, φ2 +3} − min{φ2 +1, φ2 +3} + φ2 +1 − φ2 +2 +(13) +where +φ1 = +e +β∆ +6 ���−1 + eβ∆/3��� + 2 +����sinh +� βϵ +2 +����� +����2 cosh +� βϵ +2 +� ++ eβ∆/6 + eβ∆/2 +���� +, +(14) +φ2 = +e +∆ +6 ���−1 + eβ∆/3��� − 2 +����sinh +� βϵ +2 +����� +����2 cosh +� βϵ +2 +� ++ eβ∆/6 + eβ∆/2 +���� +, +(15) +φ3 = +2 +eβ∆/3 cosh +� β∆ +6 +� +sech +� βϵ +2 +� ++ 1 +− 1 . +(16) +Considering the dipolar magnetic system in thermal equilibrium described by Eq. (5), it is possible to examine how the +magnetic anisotropies, represented by the axial (∆) and rhombic (ϵ) coupling parameters, affects the thermal quantum discord +in the system. Fig. 2 shows the geometric quantum discord, based on Schatten 1-norm, Eq. (13), as a function of the ratio +∆/kBT and ϵ/kBT. As expected, the quantum discord reaches its maximum (saturated) value of 1/2 as T approaches zero. As +the temperature rises, the value of quantum discord decreases inexorably and goes to zero when T ≫ |∆| and T ≫ |ϵ|. On the +other hand, given the spins in the x − y plane (∆ < 0), it is sufficient that only T ≫ |ϵ| to the discord reaches its minimum value. +However, if the spins are in the z-axis (∆ > 0), one can increase the quantum discord by increasing the axial parameter ∆ even +when T ≫ |ϵ|. +Furthermore, regarding the magnetic anisotropies, the quantum discord presents a signature of the quantum level crossing in +the dipolar interacting system, highlighted on the solid white line in Fig. 2. Considering the spins oriented in the z-axis (∆ > 0), +the zero-field splitting leads the system to a quantum level crossing in the critical boundary ∆ = |ϵ|, where it is possible to +detect a crossover between the states |Ψ+⟩ and |Φ−⟩, if ϵ > 0, or |Φ+⟩, if ϵ < 0. Moreover, for the spins oriented in the x − y + +5 +plane (∆ < 0), it is possible to observe a quantum level crossing between the state |Φ−⟩ and |Φ−⟩ in the critical boundary ϵ = 0. +On the other hand, the degree of quantum discord in the system can be increased by gradient ascent of the function QG(ρAB), +perpendicularly to the crossing boundary, which occurs for values in which |ϵ| ≫ kBT (for ∆ < 0), ∆ ≫ kBT (for |ϵ| ≪ ∆), and +|ϵ| ≫ ∆), corresponding to the lightest region in Fig. 2.Therefore, by controlling the axial (∆) and rhombic (ϵ) anisotropies is +possible to manage the degree of quantum discord in the dipolar interacting system. +In addition, in order to compare quantum discord to the level of entanglement in the system under investigation, we use the +concurrence measure. Typically, concurrence is used to assess entanglement in bipartite systems, and it can be easily computed +for any two-qubit system. The thermal concurrence examines the resemblance between the considered quantum state in thermal +equilibrium and its bit-flipped density matrix, ¯ρ = ρAB(σy ⊗ σy)ρ∗ +AB(σy ⊗ σy). In particular, for the X- shaped density matrix, Eq. +(5), the concurrence is analytically defined as +C(ρAB) := max{0, A, B}, +(17) +where +A = e− β∆ +6 +Z +�������e +2β∆ +6 sinh +�β∆ +6 +������� − cosh +�βϵ +2 +�� +, +(18) +B = e− β∆ +6 +Z +������sinh +�βϵ +2 +������ − e +2β∆ +6 cosh +�β∆ +6 +�� +. +(19) +Dashed green line in Fig. 2 denotes the boundary given by C(ρAB) = 0. Inside this region, the concurrence is zero, and the +state of the system is separable. However, within the region where entanglement is absent, the quantum discord of the system +is still considerably more than zero, ensuring the presence of quantum-correlated states even when the system is in a separable +syaye. On the other hand, for low temperatures, the entanglement is zero in the quantum level crossing boundary alongside +the quantum discord at the quantum level crossing boundary. In this scenario, the existence or absence of entanglement and, +therefore, quantum correlations, is dependent on its ground state, which might vary in response to magnetic anisotropies. Thus, +the variation of Boltzmann’s weights, Eqs. (6)-(9), associated with the occupancy of the energy levels, is the physical mechanism +responsible for the abrupt change in the quantum correlations near the energy-level crossover. +ϵ +kB T +Δ +kB T +PΨ+ +PΦ+ +PΦ- +FIG. 2: (Color online) Quantum Discord, based on Schatten 1-norm, for a dipolar interacting magnetic system, Eq. (13), as a function of +the ratios ∆/kBT and ϵ/kBT. The solid white line denotes the boundary between the quantum level crossings. The dashed green line is the +boundary given by the concurrence, Eq. (17), C(ρAB) = 0, inside which the entanglement of the system is absent. +IV. +QUANTUM COHERENCE +Similar to the approach proposed for the entanglement theory, where the quantum entanglement can be characterized by the +distance between a state of interest (ρ) and a set of states closed under local operations, and classical communication (separable +states) [38, 61, 65], Baumgratz et al. [65] provided the mathematical tools for quantifying the amount of quantum coherence +in a quantum system. Considering a d-dimensional Hilbert space, quantum coherence can be obtained from the minimal value + +Outf· J=6 +of a distance measurement D(ρ, σ), between the considered quantum state ρ and a set {σ = �d +k |k⟩⟨k| ∈ I} of incoherent states, +where the reference basis {|k⟩}{k=1,...,d} can be adequately defined considering the physics of the problem under investigation or +the task that requires this quantum resource [6, 65, 66]. In this scenario, since the non-vanishing off-diagonal terms of the +density operator ρ, which characterizes the quantum state of the system of interest, constitute the superposition from the chosen +reference basis [61, 65], the authors established a reliable measurement of quantum coherence through the l1 trace norm as [65] +Cl1 = min +σ∈I ∥ρ − σ∥l1 = +� +i�j +|⟨i|ρ| j⟩| . +(20) +Since coherence is a quantity that is reliant on the basis on which it is measured, it is essential to choose a reference basis for +the system within a metrology setting [61, 65]. In this scenario, the basis of an arbitrary quantum state can be altered by means +of unitary operations [36, 61]. In particular, for two-level systems such as spin-1/2, any reference basis can be obtained from the +unitary transformation +U(θ, φ) = +������� +cos +� θ +2 +� +−eiφ sin +� θ +2 +� +e−iφ sin +� θ +2 +� +cos +� θ +2 +� +������� , +(21) +where the θ and φ angles are the spherical equivalents of the co-latitude with respect to the z-axis, and the longitude concerning +the x-axis in a Bloch sphere representation, respectively [36, 67]. In this regard, the unitary transformation for the bipartite state +given by Eq. (5) is given by ρ{θ,φ} +AB = ˆUAB(θ, φ)ρAB ˆUAB(θ, φ) [67], where +ˆUAB(θ, φ) = U(θ, φ) ⊗ U(θ, φ) = +������������������� +cos2 � θ +2 +� +−eiφ sin +� θ +2 +� +cos +� θ +2 +� +−eiφ sin +� θ +2 +� +cos +� θ +2 +� +e2iφ sin2 � θ +2 +� +e−iφ sin +� θ +2 +� +cos +� θ +2 +� +cos2 � θ +2 +� +− sin2 � θ +2 +� +−eiφ sin +� θ +2 +� +cos +� θ +2 +� +e−iφ sin +� θ +2 +� +cos +� θ +2 +� +− sin2 � θ +2 +� +cos2 � θ +2 +� +−eiφ sin +� θ +2 +� +cos +� θ +2 +� +e−2iφ sin2 � θ +2 +� +e−iφ sin +� θ +2 +� +cos +� θ +2 +� +e−iφ sin +� θ +2 +� +cos +� θ +2 +� +cos2 � θ +2 +� +������������������� +(22) +By varying the co-latitude and longitude angles {θ, φ}, one can obtain the bipartite state ρAB, Eq. (5), in any reference basis. +Using the unitary transformation for the bipartite states, Eq. (22), in Eq. (5), one can obtain the representation of the density +operator for the dipolar interacting magnetic system of two spins-1/2 written in an arbitrary basis as +ρ{θ,φ} +AB = e− β∆ +6 +4Z +�������������� +ϱ11 +ϱ12 +ϱ12 +ϱ14 +ϱ∗ +12 +ϱ22 +ϱ23 +−ϱ12 +ϱ∗ +12 +ϱ23 +ϱ22 +−ϱ12 +ϱ∗ +14 −ϱ∗ +12 −ϱ∗ +12 +ϱ11 +�������������� +, +(23) +where +ϱ11 = 2 sin2(θ) +� +sinh +�β∆ +2 +� ++ cosh +�β∆ +2 +� +− sinh +�βϵ +2 +� +cos(2φ) +� ++ cosh +�βϵ +2 +� +(cos(2θ) + 3) , +(24) +ϱ12 = e−3iφ sin(θ) +� +2e2iφ cos(θ) +� +cosh +�βϵ +2 +� +− eβ∆/2� ++ e4iφ sinh +�βϵ +2 +� +(cos(θ) + 1) + sinh +�βϵ +2 +� +(cos(θ) − 1) +� +, +(25) +ϱ14 = 2e−2iφ sin2(θ) +� +cosh +�βϵ +2 +� +− eβ∆/2� +− 4e−4iφ sinh +�βϵ +2 +� +sin4 �θ +2 +� +− 4 sinh +�βϵ +2 +� +cos4 �θ +2 +� +, +(26) +ϱ22 = 2 +� +eβ∆/2 cos2(θ) + eβ∆/6 + sin2(θ) +� +sinh +�βϵ +2 +� +cos(2φ) + cosh +�βϵ +2 +��� +, +(27) +ϱ23 = 2eβ∆/2 cos2(θ) − 2eβ∆/6 + 2 sin2(θ) +� +sinh +�βϵ +2 +� +cos(2φ) + cosh +�βϵ +2 +�� +. +(28) +The diagonal entries of Eq. (23) are real, and the trace is 1. In addition, to ensure real eigenvalues, hermiticity restricts +off-diagonal elements to two complex numbers, i.e., ϱi j is the complex conjugate of ϱji. +Thus, from Eqs. (20) and (23), it is possible to write an analytical expression for the normalized quantum coherence in an +arbitrary basis, defined by the co-latitude and longitude angles {θ, φ}, as: +C{θ,φ} +l1 += e− β∆ +6 +6 |Z| +�4 |ϱ12| + |ϱ14| + |ϱ23|� . +(29) +In order to examine the relationship between quantum coherence and quantum correlations, a new metric known as corre- +lated coherence was established recently [67–69]. Quantum correlated coherence is a measure of coherence in which all local + +7 +components have been eliminated, i.e., all coherence in the system is totally recorded in the quantum correlations. For any +given quantum state ρ, the correlated contribution to quantum coherence may be calculated by subtracting the local coherence +of subsystems ρA = TrB(ρ) and ρB = TrA(ρ) from the overall coherence [68, 69]. Thus, the definition of correlated coherence +according to the l1-norm of coherence is: +Ccc(ρ{θ,φ} +AB ) := Cl1(ρ) − Cl1(ρA) − Cl1(ρB). +(30) +Considering the density matrix of the dipolar interacting magnetic system written in an arbitrary basis, Eq. (23), the reduced +density matrices of local subsystems are ρA = ρB = I/2, the maximally mixed state. Thus, regardless of the basis, the local +subsystems will remain in the maximally mixed state, since it is basis invariant [36]. Consequently, the local contribution for +the quantum coherence in this dipolar interacting system is always null, and the global coherence of the system, Eq. (29), +is totally recorded in the quantum correlations of the system, regardless of its reference basis. Therefore, for a number of +different combinations of values for the co-latitude and longitude angles {θ, φ}, the unitary transformation, Eq. (22), gives a +direct connection between the overall and the correlated degrees of coherence. +A. +Axial Coherence +In particular, due to the rotation symmetry of the dipolar interaction, the density matrix will be invariant when rotated both +spins by an angle π along any given spin axis. Thus, choosing the co-latitude angle as θ = nπ (n = {0, 1, 2, ...}), regardless of +the longitude angle φ, one can obtain the density matrix in the X-shaped form as described in Eq. (5). On the other hand, by +applying the unitary transformation for the bipartite states, Eq. (22), for {θ = π/2; φ = nπ}, and {θ = π/2; φ = nπ/2}, in Eq. (5) +one can obtain the density matrix S (x) and S (y) eigenbasis, respectively. +ρ{X,Y} +AB += e− β∆ +6 +2Z +���������������� +eβ∆/2 + e∓βϵ/2 +0 +0 +∓ +� +eβ∆/2 − e∓βϵ/2� +0 +eβ∆/6 + e±βϵ/2 e±βϵ/2 − eβ∆/6 +0 +0 +e±βϵ/2 − eβ∆/6 eβ∆/6 + e±βϵ/2 +0 +∓ +� +eβ∆/2 − e∓βϵ/2� +0 +0 +eβ∆/2 + e∓βϵ/2 +���������������� +. +(31) +As can be seen, due to the symmetry of the X-shaped density matrices [70], the X-structure of the operator is preserved. +Therefore, from Eqs. (5), (20) and (31), one can obtain the analytical expressions for the normalized axial quantum coherences +as +C{Z} +l1 += +2 +3 |Z| +� +eβ∆/6 +������sinh +�β∆ +6 +������� + e−β∆/6 +�����sinh +�βϵ +2 +������ +� +(32) +C{X,Y} +l1 += +e− β∆ +6 +3 |Z| +����eβ∆/2 − e∓βϵ/2��� + +���eβ∆/6 − e±βϵ/2��� +� +(33) +Fig. 3 shows the axial quantum coherence in S (i) spin eigenbasis, where i = {x, y, z}. Different from the behavior observed +for quantum discord (see Fig. 2), the axial quantum coherence is not sensible to the quantum level crossing. The quantum +coherence in each axis {x, y, z} is minimized in only one energy-level crossover. As can be seen, considering the spins oriented +in the z-axis (∆ > 0), the axial coherence in the S (x) eigenbasis is minimized on the critical boundary ∆ = −ϵ (with ϵ < 0), where +it is possible to detect a crossover between the states |Ψ+⟩ and |Φ+⟩, while the coherence in S (y) eigenbasis is minimized on the +critical boundary ∆ = ϵ (with ϵ > 0), where it is possible to detect a crossover between the states |Ψ+⟩ and |Φ−⟩ (see Fig. 2). As +expected from Eq. 33, if the rhombic parameter is null (ϵ = 0), C{X} +l1 += C{Y} +l1 . On the other hand, for the spins oriented in the x − y +plane, (∆ < 0), it is possible to observe that C{Z} +l1 is minimized in the quantum level crossing between the state |Φ−⟩ and |Φ−⟩ in +the critical boundary ϵ = 0. +As shown in Fig. 3, the basis dependence of the quantum coherence hides the energy-level crossover in this dipolar interacting +system regarding the measured basis. Therefore, the basis dependence on the quantum coherence defined by Baumgratz et al. +[65], can be unfavorable to recognizing the quantum level crossing caused by population changes resulting from the alteration +of Boltzman weights, Eqs. (6)-(9), arising from the change of the magnetic anisotropies of the dipolar interacting system. +B. +Average Coherence +Since the coherence formulated in the quantum resource theory is a basis-dependent measurement [8, 61, 66], it is natural to +define a basis-independent measurement [71–75]. Recent research has shown, via the use of relative entropies, as distance mea- +surements of quantum correlations, that basis-independent measurements of entropic quantum coherence are precisely identical + +8 +(a) (b) (c) +- 10 +- 5 +0 +5 +10 +- 10 +- 5 +0 +5 +10 +Δ +kB T +ϵ +kB T +- 10 +- 5 +0 +5 +10 +- 10 +- 5 +0 +5 +10 +Δ +kB T +ϵ +kB T +- 10 +- 5 +0 +5 +10 +- 10 +- 5 +0 +5 +10 +Δ +kB T +ϵ +kB T +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +SX eigenbasis +SY +SZ eigenbasis +eigenbasis +FIG. 3: (Color online) Axial quantum coherence based on l1 trace norm, for a dipolar interacting magnetic system, as a function of the ratios +∆/kBT and ϵ/kBT. The dashed white line represents the minimum value for the axial quantum coherence. +to entropic discord [74]. On the other hand, a possible basis-free measurement of quantum coherence for a quantum system +can be obtained from a geometrical standpoint by averaging the coherence of a state across all reference bases [71–73, 75]. +From a theoretical point of view, this measurement corresponds to averaging the coherence on a standard basis across all equiv- +alent states ρ{θ,φ} +AB += ˆUAB(θ, φ)ρAB ˆUAB(θ, φ). Therefore, as any two-qubit reference base can be created by applying the unitary +operation described in Eq. (22), the average quantum coherence can be obtained from Eq. (29) as +⟨Cl1⟩ = 1 +4π +2π +� +0 +π +� +0 +sin (θ)C{θ,φ} +l1 +dθdφ . +(34) +It is worth mentioning that these integrals are not trivial to solve, and an analytical expression for the average coherence is not +presented. However, it can be numerically integrated by any quadrature method [76]. In this scenario, Eq. (34) is estimated by +using the Clenshaw-Curtis rule on adaptively refined subintervals of the integration area [76, 77] since the numerical integration +algorithms are often equally efficient and effective as conventional algorithms for well-behaved integrands such as Eqs. (29) and +(30) [76]. +Fig. 4 shows the average quantum coherence for the dipolar magnetic interacting system. The solid white line represents the +threshold at which the quantum-level crossing, described in previous sections, actually occurs. As expected, based on Fig. 3, +when the temperature rises reaching the threshold T ≫ |∆| and T ≫ |ϵ|, the value of coherence reaches its lowest point and +will be equal to zero. However, the behavior of the average coherence is completely different from that observed in the axial +(basis-dependent) coherence shown in Fig. 3. +Moreover, besides unified frameworks from relative entropic measurements has shown that basis-independent entropic quan- +tum coherence is equivalent to entropic discord [74], this is not true for this geometrical approach. However, although the +contour lines of the average coherence are quite different from that shown in the discord presented in Fig. 2, it is still able to +identify the signature of the energy-level crossing that was seen during the measurement of the quantum discord. This result is +due to the fact that the global coherence is totally stored within the correlations of the system, and its average behavior is affected +by the presence of genuine quantum correlations measured by the quantum discord. +In addition, the entanglement of the system is absent within the area shown by the dashed green line that denotes the boundary +supplied by the concurrence, which is denoted by Eq. (17), C(ρAB) = 0. Thus, as one would anticipate based on the observation +of the quantum discord in Fig. 2, even in the absence of entanglement, the average coherence that is completely stored on the +correlations of the system is noticeably distinct from zero. +V. +CONCLUSIONS +In summary, this paper explored the influence of magnetic anisotropies on the quantumness of a dipolar interacting magnetic +system via a theoretical examination of the geometric quantum discord, measured by Schatten 1-norm, and the l1 trace-norm +quantum coherence. The analytical formulations for these quantum information quantifiers were obtained in terms of magnetic +anisotropies. In this scenario, the effects of dipolar coupling constants on these quantifiers are highlighted. It is demonstrated +that the presence of dipolar anisotropies increases the degree to which the system possesses quantum correlation and coherence. + +9 +0 +5 +10 +0 +5 +10 +- 10 +- 5 +- 10 +- 5 +Δ +kB T +ϵ +kB T +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +PΨ+ +PΦ+ +PΦ- +FIG. 4: (Color online) Average quantum coherence based on l1 trace norm, for a dipolar interacting magnetic system, as a function of the ratios +∆/kBT and ϵ/kBT. The solid white line represents the boundary between the quantum level crossings. The dashed green line is the boundary +given by the concurrence, Eq. (17), C(ρAB) = 0, inside which the entanglement of the system is absent. +As another remarkable result, it is proved that the global coherence, expressed in an arbitrary reference basis, determined by +the co-latitude and longitude angles of the Bloch sphere representation, is totally stored within the correlations of the system. +Moreover, according to the results, the behavior of quantum discord contains a notable hallmark of quantum level-crossing in +the system, in contrast to the basis-dependent axial quantum coherence, which hides the energy-level crossover regarding the +measured basis. +Therefore, the dependency of the base on the quantum coherence specified by Baumgratz might be deleterious in identifying +the crossing of levels owing to population changes originating from the changing of Boltzman weights due to the modification +of the magnetic anisotropies of the studied system. In this regard, the average quantum coherence was measured numerically +obtained in order to gain a viewpoint independent of the reference basis, unraveling that the average coherence is able to extract +the signature of the energy-level crossover present in the measurement of quantum discord. +Finally, the findings that were given provide light on the ways in which magnetic anisotropies caused by the dipolar interaction +coupling of a dinuclear spin-1/2 system influence quantum correlations and coherence. Therefore, the dipolar interaction model +is an excellent option for usage as a platform for quantum technologies that are based on quantum resources such as quantum +coherence and quantum discord. +ACKNOWLEDGEMENTS +C. Cruz gratefully acknowledges Mario Reis for the valuable discussions. M. F. Anka thanks FAPERJ for financial support. +[1] M. Mohseni, P. Read, H. Neven, S. Boixo, V. Denchev, R. Babbush, A. Fowler, V. Smelyanskiy, and J. Martinis, Nature 543, 171 (2017). +[2] M. 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Xiang, Applied Mathematics and Computation 340, 251 (2019). + diff --git a/EtE1T4oBgHgl3EQfEgOL/content/tmp_files/load_file.txt b/EtE1T4oBgHgl3EQfEgOL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..51569c51343d74ab3aa6f4a7125671301371501b --- /dev/null +++ b/EtE1T4oBgHgl3EQfEgOL/content/tmp_files/load_file.txt @@ -0,0 +1,853 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf,len=852 +page_content='Geometric quantum discord and coherence in a dipolar interacting magnetic system Clebson Cruz,1, ∗ Maron F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Anka,2, † Hamid-Reza Rastegar-Sedehi,3, ‡ and Cleidson Castro4, § 1Grupo de Informa¸c˜ao Quˆantica e F´ısica Estat´ıstica, Centro de Ciˆencias Exatas e das Tecnologias, Universidade Federal do Oeste da Bahia - Campus Reitor Edgard Santos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Rua Bertioga, 892, Morada Nobre I, 47810-059 Barreiras, Bahia, Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2Instituto de F´ısica, Universidade Federal Fluminense, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Gal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Milton Tavares de Souza s/n, 24210-346 Niter´oi, Rio de Janeiro, Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 3Department of Physics, College of Sciences, Jahrom University, Jahrom 74135-111, Iran 4Centro de Forma¸c˜ao de Professores, Universidade Federal do Recˆoncavo da Bahia, Avenida Nestor de Mello Pita, 535 Amargosa, Bahia, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (Dated: January 10, 2023) The study of low-dimensional metal complexes has revealed fascinating characteristics regarding the ground- state crossover shown by spin-gaped systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this context, this work explores the effect of the quantum-level crossing, induced by the magnetic anisotropies of dipolar interacting systems, on the quantum discord and coherence of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The analytical expressions for the quantum discord, based on Schatten 1-norm, and the l1 trace-norm quantum coherence for dinuclear spin-1/2 systems, are provided in terms of the magnetic anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The results show that, while the quantum discord has a clear signature of the quantum level- crossing, the basis dependence of the quantum coherence hides the crossover regarding the measured basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In addition, the global quantum coherence is wholly stored within the correlations of the system, regardless of its reference basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Keywords: Dipolar Interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Qunatum discord;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Quantum Coherence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Quantum-level crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' INTRODUCTION The study of the quantum properties of composite systems has led to a revolution in the development of emerging quantum technologies [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The new generation of quantum devices explores physical properties associated with quantum correlations between particles [4, 5] and superposition principle for the system states [1, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this scenario, the characterization of the quantumness of the physical systems is of paramount importance since the existence of quantum correlations and coherence are a valuable resource for several quantum tasks [4, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, the characterization of quantum correlations is a rather complicated task from the theoretical [10] and experimental [11] point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' This scenario is aggravated in Condensed Matter systems, where the number of interacting components in the system is usually on the order of the Avogadro number [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Nevertheless, there are a few exceptions, like low-dimensional metal complexes (LDMC), for which full knowledge about their quantum properties can be obtained through the corresponding analytical solutions [4, 6, 13–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In such solid-state systems, intra-molecular interactions are strong enough to suppress ex- trinsic and intermolecular interactions [4, 13, 14, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, their quantum features exhibit high stability against external perturbations such as temperatures [4, 13, 14, 16, 21], magnetic fields [4, 6, 16, 22], and pressures [6, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' These characteris- tics make these systems promising platforms for the development of emerging quantum technologies [4, 23–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In regard to these possible applications, the study of dipolar interacting magnetic systems has received considerable attention in the quantum information literature [17, 27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this work, we present a theoretical study of quantum correlations and coherence for a dipolar interacting magnetic system, exploring the effects of magnetic anisotropies on the quantumness of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' As a result, this study provides to the literature analytical expressions, in terms of the magnetic anisotropies, for the quantum discord, based on Schatten 1-norm, and the l1 trace-norm quantum coherence written in an arbitrary basis, defined by the co-latitude and longitude angles of the Bloch sphere representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' According to the findings, the behavior of the quantum discord carries a noteworthy signature of the quantum level-crossing, caused by population changes resulting from the alteration of Boltzmann weights arising from the change of the magnetic anisotropies of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' On the other hand, the basis dependency of quantum coherence is detrimental in terms of recognizing this crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this regard, the measurement of the average quantum coherence is numerically obtained in order to obtain a basis-independent perspective for this quantum resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The results not only demonstrate that the average coherence ∗Electronic address: clebson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cruz@ufob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='br †Electronic address: maronanka@id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='uff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='br ‡Electronic address: h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='rastegar@jahromu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='ir §Electronic address: ccastro@ufrb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='br arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='02891v1 [quant-ph] 7 Jan 2023 2 is completely stored within the correlations of the system, yet they also demonstrate that it is possible to retrieve the signature of the energy-level crossover present on the quantum discord measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Furthermore, the findings show how dipolar interaction coupling magnetic anisotropies impact quantum correlations and coherence in a dinuclear spin-1/2 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Thus, the dipolar interaction model is a viable foundation for quantum technologies based on quantum discord and coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' DINUCLEAR METAL COMPLEX WITH DIPOLAR INTERACTION The class of dinuclear metal complexes undergoes several types of magnetic coupling [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Among these are Heisenberg exchange, which is isotropic under rotations in spin space [4, 6, 20, 21], and Dzyaloshinskii-Moriya (DM) interaction [32– 34], which accounts for weak ferromagnetism in some antiferromagnetic materials [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' A ubiquitous example of anisotropic coupling in LDMCs is the dipolar interaction [17, 27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' This coupling arises from the influence of a magnetic field yielded by one of the magnetic moments in the other ones [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In particular, for dinuclear metal complexes, the Hamiltonian which describes this interaction is given by: H = −1 3 ⃗S T A · ↔D · ⃗S B , (1) where ⃗S j = {S x j, S y j, S z j} are the spin operators and ↔D = diag(∆ − 3ϵ, ∆ + 3ϵ, −2∆) is a diagonal tensor, with ϵ and ∆ being the rhombic and axial parameters, respectively, related to the magnetic anisotropies in the dipolar model [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In particular, ∆ is related to the zero-field splitting of the energy levels [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Considering the Hamiltonian, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (1), written in the S (z) eigenbasis, ∆ > 0 becomes a signature that the spins are on z-axis, while ∆ < 0 indicates that the spins will be on the x − y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Considering a dinuclear metal complex in with d9 electronic configuration, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (1) can describe two coupled spin 1/2 particles in the corresponding S z eigenbasis {|00⟩, |01⟩, |10⟩, |11⟩} H = 1 6 �������������� ∆ 3ϵ −∆ −∆ −∆ −∆ 3ϵ ∆ �������������� , (2) The energy levels of the system from the coupling parameters are composed by the E1 = 0 , E2 = −2∆ , E3 = ∆ + 3ϵ , E4 = ∆ − 3ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (3) From the thermal equilibrium, the density matrix for the coupled system is described by the Gibbs form ρAB = Z−1e−H/kBT, where Z = Tr(e−H/kBT) = 2eβ∆/6 cosh �β∆ 6 � + 2e−β∆/6 cosh �βϵ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (4) is the canonical the partition function, with kB representing the Boltzmann’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Thus, the dinuclear density matrix at sites labeled by A and B can be written in the S z eigenbasis as the so-called X-shaped mixed state ρAB = e− β∆ 6 Z ������������������� cosh � βϵ 2 � − sinh � βϵ 2 � e 2β∆ 6 cosh � β∆ 6 � e 2β∆ 6 sinh � β∆ 6 � e 2β∆ 6 sinh � β∆ 6 � e 2β∆ 6 cosh � β∆ 6 � − sinh � βϵ 2 � cosh � βϵ 2 � ������������������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5) The density matrix eigenvalues (population) and their corresponding eigenvectors can be written as: PΨ− = 1 1 + e β∆ 3 + 2e− β∆ 6 cosh � βϵ 2 � → |Ψ−⟩, (6) PΨ+ = 1 1 + e− β∆ 3 + 2e− β∆ 6 cosh � βϵ 2 � → |Ψ+⟩, (7) PΦ+ = 1 1 + eβϵ + eβ( ∆+ϵ 2 ) + eβ( ∆+3ϵ 6 ) → |Φ+⟩, (8) PΦ− = eβϵ 1 + eβϵ + eβ( ∆+ϵ 2 ) + eβ( ∆+3ϵ 6 ) → |Φ−⟩, (9) 3 where |Ψ±⟩ = 1√ 2 (|01⟩ ± |10⟩) , |Φ±⟩ = 1√ 2 (|00⟩ ± |11⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (10) are the so-called Bell states, which represent the maximally entangled states for a bipartite system [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The study of LDMC has attracted the attention of both theoretical and experimental condensed matter physics communities due to the fascinating properties of their ground states [6, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In the presence of an external magnetic field, this systems typically show a quantum level-crossing between its ground state and the first excited one when the field reaches a critical value since the external magnetic field splits its energy levels, changing their corresponding populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, since the dipolar interaction arises from the influence of the magnetic field created by one of the magnetic moments in the other, the splitting in energy levels is ruled by the axial (∆) and rhombic (ϵ) parameters, as can be seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this regard, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 1 shows the populations as a function of the ratio between the magnetic anisotropies and the energy scale factor kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='0 ϵ/kBT Populations 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='0 ϵ/kBT Populations (a) (b) 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='0 Δ/kBT Populations 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='0 Δ/kB Δ/kB T Populations PΨ- PΨ+ PΦ+ PΦ- ϵ/kB= 5K ϵ/kB=-5K = 5K Δ/kB=-5K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 1: (Color online) Populations, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (6)–(9), as a function of the ratio between the magnetic anisotropies and the energy scale factor kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (a) Axial dependence considering the rhombic parameter ϵ/kB = 5 K (left) and ϵ/kB = −5 K (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (b) Rhombic dependence considering the axial parameter ∆/kB = 5 K (left) and ∆/kB = −5 K (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The inset shows the magnetic anisotropy dependence on the energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' As can be seen, in agreement with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (3), when the spins are in the x − y plane (∆ < 0) with positive rhombic parameter (ϵ > 0), the ground state is the given by the state |Φ−⟩ (with population PΦ−), and there is no quantum level crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Thus, the system remains in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, changing the signal of the rhombic parameter induces a quantum level crossing between states |Φ−⟩ and |Φ+⟩ (with population PΦ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Moreover, for the spins oriented in the z axis (∆ > 0), it is possible to observe a quantum level crossing between the state |Φ−⟩ (if ϵ > 0) or |Φ+⟩ (if ϵ < 0) and the state |Ψ−⟩ (with population PΨ+) by increasing the ratio ∆/kBT to the critical point ∆ = |ϵ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In reference [29], the authors study the effect of the magnetic anisotropies, described by the axial and rhombic parameters, on the nonlocal correlations of a dipolar interacting system of two spins-1/2, identified by the Peres-Horodecki separability criterion [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In addition, they explore the change in the ground state on the thermal entanglement for the teleportation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, although quantum entanglement provides one path toward the characterization of nonlocal correlations, it does not encompass all quantum correlations in the system [21, 40–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, in order to expand this result, the following section presents a study of the quantum correlations and coherence described by the Schatten 1-norm geometric quantum discord and the l1 trace-norm quantum coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 30 20 10 10 20 0 0 5 10 E20 10 0 10 20 30 10 0 5 10 E20 10 0 10 20 0 5 0 5 1020 10 ler 0 10 20 0 5 0 5 1060 40 20 20 40 60 20 10 0 10 2060 40 20 0 20 40 60 20 0 10 204 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' QUANTUM DISCORD Quantum discord has been defined as a measurement of the quantumness of correlations in a quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' It has been first introduced as an entropic measurement of genuinely quantum correlations in a quantum state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' defined as the difference between the total and the classical correlation [40] Q(ρAB) = I(ρA : ρB) − C(ρAB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' where I(ρA : ρB) = S (ρA) + S (ρB) − S (ρAB) represents the mutual information between the subsystems A and B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' and C(ρAB) is the classical correlation of the composite system ρAB defined as C(ρAB) = max{Bk} �S (ρA) − � k pkS (ρk)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' with the maximization taking over positive operator-valued measurements (POVM’s) {Bk} performed locally only on subsystem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, this analytical maximization over POVMs is an arduous task even for a two-qubit system [10, 11, 40, 41, 47, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this scenario, the class of entropic measurements of correlations, such as the entropic quantum discord, is defined as nondeterministic polynomial time (NP-complete) problems [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Consequently, only a few results for the analytical expression of entropic quantum discord, and only for certain classes of states are exact solutions known [41, 43, 45, 46, 49, 50, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Due to this fact, alternative measurements of quantum correlations have been proposed [41, 43, 45–50, 52–60], especially quantifiers based on geometric arguments [47–49, 53, 57–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Geometric approaches are widely used to characterize and quantify quantum resources in a wide variety of quantum systems [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In particular, the Schatten 1-norm quantum discord [4, 21, 47, 48, 62], is a reliable geometric-based quantifier of the amount of quantum correlations in metal complexes [4, 21, 62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The so-called geometric quantum discord can be defined in terms of the minimal distance between a set ω of closest classical-quantum states ρc [21, 47, 48], given by: ρc = � k pkΠ{A} k ⊗ ρ{B} k , (11) where 0 ≤ pk ≤ 1 and � k pk = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' {Π{A} k } define a set of orthogonal projectors for a given subsystem A and ρ{B} k the reduced density matrix for the subsystem B [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, the geometric quantum discord can be expressed as QG(ρAB) = min ω ∥ρAB − ρc∥ , (12) where ∥M∥ = Tr � √ M†M � is the so-called 1-norm, and ρAB is the given quantum state at thermal equilibrium, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, considering the given dinuclear magnetic system of spins-1/2 in a quantum spin-lattice, ruled by a dipolar Hamil- tonian H, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (1), the invariance under π rotation around a given spin axis (Z2 symmetry) [12, 46] allow us to compute the geometric quantum discord, based on Schatten 1-norm, for the two-qubit X state, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5), as [53, 64] QG(ρAB) = 1 2 � φ2 1max{φ2 2, φ2 3} − φ2 2min{φ2 1, φ2 3} max{φ2 2, φ2 3} − min{φ2 1, φ2 3} + φ2 1 − φ2 2 (13) where φ1 = e β∆ 6 ���−1 + eβ∆/3��� + 2 ����sinh � βϵ 2 ����� ����2 cosh � βϵ 2 � + eβ∆/6 + eβ∆/2 ���� , (14) φ2 = e ∆ 6 ���−1 + eβ∆/3��� − 2 ����sinh � βϵ 2 ����� ����2 cosh � βϵ 2 � + eβ∆/6 + eβ∆/2 ���� , (15) φ3 = 2 eβ∆/3 cosh � β∆ 6 � sech � βϵ 2 � + 1 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (16) Considering the dipolar magnetic system in thermal equilibrium described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5), it is possible to examine how the magnetic anisotropies, represented by the axial (∆) and rhombic (ϵ) coupling parameters, affects the thermal quantum discord in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2 shows the geometric quantum discord, based on Schatten 1-norm, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (13), as a function of the ratio ∆/kBT and ϵ/kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' As expected, the quantum discord reaches its maximum (saturated) value of 1/2 as T approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' As the temperature rises, the value of quantum discord decreases inexorably and goes to zero when T ≫ |∆| and T ≫ |ϵ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' On the other hand, given the spins in the x − y plane (∆ < 0), it is sufficient that only T ≫ |ϵ| to the discord reaches its minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, if the spins are in the z-axis (∆ > 0), one can increase the quantum discord by increasing the axial parameter ∆ even when T ≫ |ϵ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Furthermore, regarding the magnetic anisotropies, the quantum discord presents a signature of the quantum level crossing in the dipolar interacting system, highlighted on the solid white line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Considering the spins oriented in the z-axis (∆ > 0), the zero-field splitting leads the system to a quantum level crossing in the critical boundary ∆ = |ϵ|, where it is possible to detect a crossover between the states |Ψ+⟩ and |Φ−⟩, if ϵ > 0, or |Φ+⟩, if ϵ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Moreover, for the spins oriented in the x − y 5 plane (∆ < 0), it is possible to observe a quantum level crossing between the state |Φ−⟩ and |Φ−⟩ in the critical boundary ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' On the other hand, the degree of quantum discord in the system can be increased by gradient ascent of the function QG(ρAB), perpendicularly to the crossing boundary, which occurs for values in which |ϵ| ≫ kBT (for ∆ < 0), ∆ ≫ kBT (for |ϵ| ≪ ∆), and |ϵ| ≫ ∆), corresponding to the lightest region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='Therefore, by controlling the axial (∆) and rhombic (ϵ) anisotropies is possible to manage the degree of quantum discord in the dipolar interacting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In addition, in order to compare quantum discord to the level of entanglement in the system under investigation, we use the concurrence measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Typically, concurrence is used to assess entanglement in bipartite systems, and it can be easily computed for any two-qubit system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The thermal concurrence examines the resemblance between the considered quantum state in thermal equilibrium and its bit-flipped density matrix, ¯ρ = ρAB(σy ⊗ σy)ρ∗ AB(σy ⊗ σy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In particular, for the X- shaped density matrix, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5), the concurrence is analytically defined as C(ρAB) := max{0, A, B}, (17) where A = e− β∆ 6 Z �������e 2β∆ 6 sinh �β∆ 6 ������� − cosh �βϵ 2 �� , (18) B = e− β∆ 6 Z ������sinh �βϵ 2 ������ − e 2β∆ 6 cosh �β∆ 6 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (19) Dashed green line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2 denotes the boundary given by C(ρAB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Inside this region, the concurrence is zero, and the state of the system is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, within the region where entanglement is absent, the quantum discord of the system is still considerably more than zero, ensuring the presence of quantum-correlated states even when the system is in a separable syaye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' On the other hand, for low temperatures, the entanglement is zero in the quantum level crossing boundary alongside the quantum discord at the quantum level crossing boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this scenario, the existence or absence of entanglement and, therefore, quantum correlations, is dependent on its ground state, which might vary in response to magnetic anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Thus, the variation of Boltzmann’s weights, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (6)-(9), associated with the occupancy of the energy levels, is the physical mechanism responsible for the abrupt change in the quantum correlations near the energy-level crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' ϵ kB T Δ kB T PΨ+ PΦ+ PΦ- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2: (Color online) Quantum Discord, based on Schatten 1-norm, for a dipolar interacting magnetic system, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (13), as a function of the ratios ∆/kBT and ϵ/kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The solid white line denotes the boundary between the quantum level crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The dashed green line is the boundary given by the concurrence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (17), C(ρAB) = 0, inside which the entanglement of the system is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' QUANTUM COHERENCE Similar to the approach proposed for the entanglement theory, where the quantum entanglement can be characterized by the distance between a state of interest (ρ) and a set of states closed under local operations, and classical communication (separable states) [38, 61, 65], Baumgratz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' [65] provided the mathematical tools for quantifying the amount of quantum coherence in a quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Considering a d-dimensional Hilbert space, quantum coherence can be obtained from the minimal value Outf· J=6 of a distance measurement D(ρ, σ), between the considered quantum state ρ and a set {σ = �d k |k⟩⟨k| ∈ I} of incoherent states, where the reference basis {|k⟩}{k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=',d} can be adequately defined considering the physics of the problem under investigation or the task that requires this quantum resource [6, 65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this scenario, since the non-vanishing off-diagonal terms of the density operator ρ, which characterizes the quantum state of the system of interest, constitute the superposition from the chosen reference basis [61, 65], the authors established a reliable measurement of quantum coherence through the l1 trace norm as [65] Cl1 = min σ∈I ∥ρ − σ∥l1 = � i�j |⟨i|ρ| j⟩| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (20) Since coherence is a quantity that is reliant on the basis on which it is measured, it is essential to choose a reference basis for the system within a metrology setting [61, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this scenario, the basis of an arbitrary quantum state can be altered by means of unitary operations [36, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In particular, for two-level systems such as spin-1/2, any reference basis can be obtained from the unitary transformation U(θ, φ) = ������� cos � θ 2 � −eiφ sin � θ 2 � e−iφ sin � θ 2 � cos � θ 2 � ������� , (21) where the θ and φ angles are the spherical equivalents of the co-latitude with respect to the z-axis, and the longitude concerning the x-axis in a Bloch sphere representation, respectively [36, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this regard, the unitary transformation for the bipartite state given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5) is given by ρ{θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='φ} AB = ˆUAB(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' φ)ρAB ˆUAB(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' φ) [67],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' where ˆUAB(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' φ) = U(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cos2 � θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='− sin2 � θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='−eiφ sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='e−iφ sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='− sin2 � θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cos2 � θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='−eiφ sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='e−2iφ sin2 � θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='e−iφ sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='e−iφ sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='cos2 � θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='(22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='By varying the co-latitude and longitude angles {θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' φ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' one can obtain the bipartite state ρAB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5), in any reference basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Using the unitary transformation for the bipartite states, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (22), in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' one can obtain the representation of the density operator for the dipolar interacting magnetic system of two spins-1/2 written in an arbitrary basis as ρ{θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='φ} AB = e− β∆ 6 4Z �������������� ϱ11 ϱ12 ϱ12 ϱ14 ϱ∗ 12 ϱ22 ϱ23 −ϱ12 ϱ∗ 12 ϱ23 ϱ22 −ϱ12 ϱ∗ 14 −ϱ∗ 12 −ϱ∗ 12 ϱ11 �������������� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (23) where ϱ11 = 2 sin2(θ) � sinh �β∆ 2 � + cosh �β∆ 2 � − sinh �βϵ 2 � cos(2φ) � + cosh �βϵ 2 � (cos(2θ) + 3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (24) ϱ12 = e−3iφ sin(θ) � 2e2iφ cos(θ) � cosh �βϵ 2 � − eβ∆/2� + e4iφ sinh �βϵ 2 � (cos(θ) + 1) + sinh �βϵ 2 � (cos(θ) − 1) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (25) ϱ14 = 2e−2iφ sin2(θ) � cosh �βϵ 2 � − eβ∆/2� − 4e−4iφ sinh �βϵ 2 � sin4 �θ 2 � − 4 sinh �βϵ 2 � cos4 �θ 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (26) ϱ22 = 2 � eβ∆/2 cos2(θ) + eβ∆/6 + sin2(θ) � sinh �βϵ 2 � cos(2φ) + cosh �βϵ 2 ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (27) ϱ23 = 2eβ∆/2 cos2(θ) − 2eβ∆/6 + 2 sin2(θ) � sinh �βϵ 2 � cos(2φ) + cosh �βϵ 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (28) The diagonal entries of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (23) are real, and the trace is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In addition, to ensure real eigenvalues, hermiticity restricts off-diagonal elements to two complex numbers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=', ϱi j is the complex conjugate of ϱji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Thus, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (20) and (23), it is possible to write an analytical expression for the normalized quantum coherence in an arbitrary basis, defined by the co-latitude and longitude angles {θ, φ}, as: C{θ,φ} l1 = e− β∆ 6 6 |Z| �4 |ϱ12| + |ϱ14| + |ϱ23|� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (29) In order to examine the relationship between quantum coherence and quantum correlations, a new metric known as corre- lated coherence was established recently [67–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Quantum correlated coherence is a measure of coherence in which all local 7 components have been eliminated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=', all coherence in the system is totally recorded in the quantum correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' For any given quantum state ρ, the correlated contribution to quantum coherence may be calculated by subtracting the local coherence of subsystems ρA = TrB(ρ) and ρB = TrA(ρ) from the overall coherence [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Thus, the definition of correlated coherence according to the l1-norm of coherence is: Ccc(ρ{θ,φ} AB ) := Cl1(ρ) − Cl1(ρA) − Cl1(ρB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (30) Considering the density matrix of the dipolar interacting magnetic system written in an arbitrary basis, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (23), the reduced density matrices of local subsystems are ρA = ρB = I/2, the maximally mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Thus, regardless of the basis, the local subsystems will remain in the maximally mixed state, since it is basis invariant [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Consequently, the local contribution for the quantum coherence in this dipolar interacting system is always null, and the global coherence of the system, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (29), is totally recorded in the quantum correlations of the system, regardless of its reference basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, for a number of different combinations of values for the co-latitude and longitude angles {θ, φ}, the unitary transformation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (22), gives a direct connection between the overall and the correlated degrees of coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Axial Coherence In particular, due to the rotation symmetry of the dipolar interaction, the density matrix will be invariant when rotated both spins by an angle π along any given spin axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Thus, choosing the co-latitude angle as θ = nπ (n = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='}), regardless of the longitude angle φ, one can obtain the density matrix in the X-shaped form as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' On the other hand, by applying the unitary transformation for the bipartite states, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (22), for {θ = π/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' φ = nπ}, and {θ = π/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' φ = nπ/2}, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5) one can obtain the density matrix S (x) and S (y) eigenbasis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' ρ{X,Y} AB = e− β∆ 6 2Z ���������������� eβ∆/2 + e∓βϵ/2 0 0 ∓ � eβ∆/2 − e∓βϵ/2� 0 eβ∆/6 + e±βϵ/2 e±βϵ/2 − eβ∆/6 0 0 e±βϵ/2 − eβ∆/6 eβ∆/6 + e±βϵ/2 0 ∓ � eβ∆/2 − e∓βϵ/2� 0 0 eβ∆/2 + e∓βϵ/2 ���������������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (31) As can be seen, due to the symmetry of the X-shaped density matrices [70], the X-structure of the operator is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (5), (20) and (31), one can obtain the analytical expressions for the normalized axial quantum coherences as C{Z} l1 = 2 3 |Z| � eβ∆/6 ������sinh �β∆ 6 ������� + e−β∆/6 �����sinh �βϵ 2 ������ � (32) C{X,Y} l1 = e− β∆ 6 3 |Z| ����eβ∆/2 − e∓βϵ/2��� + ���eβ∆/6 − e±βϵ/2��� � (33) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 3 shows the axial quantum coherence in S (i) spin eigenbasis, where i = {x, y, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Different from the behavior observed for quantum discord (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2), the axial quantum coherence is not sensible to the quantum level crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The quantum coherence in each axis {x, y, z} is minimized in only one energy-level crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' As can be seen, considering the spins oriented in the z-axis (∆ > 0), the axial coherence in the S (x) eigenbasis is minimized on the critical boundary ∆ = −ϵ (with ϵ < 0), where it is possible to detect a crossover between the states |Ψ+⟩ and |Φ+⟩, while the coherence in S (y) eigenbasis is minimized on the critical boundary ∆ = ϵ (with ϵ > 0), where it is possible to detect a crossover between the states |Ψ+⟩ and |Φ−⟩ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' As expected from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 33, if the rhombic parameter is null (ϵ = 0), C{X} l1 = C{Y} l1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' On the other hand, for the spins oriented in the x − y plane, (∆ < 0), it is possible to observe that C{Z} l1 is minimized in the quantum level crossing between the state |Φ−⟩ and |Φ−⟩ in the critical boundary ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 3, the basis dependence of the quantum coherence hides the energy-level crossover in this dipolar interacting system regarding the measured basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, the basis dependence on the quantum coherence defined by Baumgratz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' [65], can be unfavorable to recognizing the quantum level crossing caused by population changes resulting from the alteration of Boltzman weights, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (6)-(9), arising from the change of the magnetic anisotropies of the dipolar interacting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Average Coherence Since the coherence formulated in the quantum resource theory is a basis-dependent measurement [8, 61, 66], it is natural to define a basis-independent measurement [71–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Recent research has shown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' via the use of relative entropies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' as distance mea- surements of quantum correlations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' that basis-independent measurements of entropic quantum coherence are precisely identical 8 (a) (b) (c) 10 5 0 5 10 10 5 0 5 10 Δ kB T ϵ kB T 10 5 0 5 10 10 5 0 5 10 Δ kB T ϵ kB T 10 5 0 5 10 10 5 0 5 10 Δ kB T ϵ kB T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='30 SX eigenbasis SY SZ eigenbasis eigenbasis FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 3: (Color online) Axial quantum coherence based on l1 trace norm, for a dipolar interacting magnetic system, as a function of the ratios ∆/kBT and ϵ/kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The dashed white line represents the minimum value for the axial quantum coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' to entropic discord [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' On the other hand, a possible basis-free measurement of quantum coherence for a quantum system can be obtained from a geometrical standpoint by averaging the coherence of a state across all reference bases [71–73, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' From a theoretical point of view, this measurement corresponds to averaging the coherence on a standard basis across all equiv- alent states ρ{θ,φ} AB = ˆUAB(θ, φ)ρAB ˆUAB(θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, as any two-qubit reference base can be created by applying the unitary operation described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (22), the average quantum coherence can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (29) as ⟨Cl1⟩ = 1 4π 2π � 0 π � 0 sin (θ)C{θ,φ} l1 dθdφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (34) It is worth mentioning that these integrals are not trivial to solve, and an analytical expression for the average coherence is not presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, it can be numerically integrated by any quadrature method [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this scenario, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (34) is estimated by using the Clenshaw-Curtis rule on adaptively refined subintervals of the integration area [76, 77] since the numerical integration algorithms are often equally efficient and effective as conventional algorithms for well-behaved integrands such as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (29) and (30) [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 4 shows the average quantum coherence for the dipolar magnetic interacting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The solid white line represents the threshold at which the quantum-level crossing, described in previous sections, actually occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' As expected, based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 3, when the temperature rises reaching the threshold T ≫ |∆| and T ≫ |ϵ|, the value of coherence reaches its lowest point and will be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, the behavior of the average coherence is completely different from that observed in the axial (basis-dependent) coherence shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Moreover, besides unified frameworks from relative entropic measurements has shown that basis-independent entropic quan- tum coherence is equivalent to entropic discord [74], this is not true for this geometrical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' However, although the contour lines of the average coherence are quite different from that shown in the discord presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2, it is still able to identify the signature of the energy-level crossing that was seen during the measurement of the quantum discord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' This result is due to the fact that the global coherence is totally stored within the correlations of the system, and its average behavior is affected by the presence of genuine quantum correlations measured by the quantum discord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In addition, the entanglement of the system is absent within the area shown by the dashed green line that denotes the boundary supplied by the concurrence, which is denoted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (17), C(ρAB) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Thus, as one would anticipate based on the observation of the quantum discord in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 2, even in the absence of entanglement, the average coherence that is completely stored on the correlations of the system is noticeably distinct from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' CONCLUSIONS In summary, this paper explored the influence of magnetic anisotropies on the quantumness of a dipolar interacting magnetic system via a theoretical examination of the geometric quantum discord, measured by Schatten 1-norm, and the l1 trace-norm quantum coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The analytical formulations for these quantum information quantifiers were obtained in terms of magnetic anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this scenario, the effects of dipolar coupling constants on these quantifiers are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' It is demonstrated that the presence of dipolar anisotropies increases the degree to which the system possesses quantum correlation and coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 9 0 5 10 0 5 10 10 5 10 5 Δ kB T ϵ kB T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content='7 PΨ+ PΦ+ PΦ- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' 4: (Color online) Average quantum coherence based on l1 trace norm, for a dipolar interacting magnetic system, as a function of the ratios ∆/kBT and ϵ/kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The solid white line represents the boundary between the quantum level crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' The dashed green line is the boundary given by the concurrence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' (17), C(ρAB) = 0, inside which the entanglement of the system is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' As another remarkable result, it is proved that the global coherence, expressed in an arbitrary reference basis, determined by the co-latitude and longitude angles of the Bloch sphere representation, is totally stored within the correlations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Moreover, according to the results, the behavior of quantum discord contains a notable hallmark of quantum level-crossing in the system, in contrast to the basis-dependent axial quantum coherence, which hides the energy-level crossover regarding the measured basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, the dependency of the base on the quantum coherence specified by Baumgratz might be deleterious in identifying the crossing of levels owing to population changes originating from the changing of Boltzman weights due to the modification of the magnetic anisotropies of the studied system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' In this regard, the average quantum coherence was measured numerically obtained in order to gain a viewpoint independent of the reference basis, unraveling that the average coherence is able to extract the signature of the energy-level crossover present in the measurement of quantum discord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Finally, the findings that were given provide light on the ways in which magnetic anisotropies caused by the dipolar interaction coupling of a dinuclear spin-1/2 system influence quantum correlations and coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Therefore, the dipolar interaction model is an excellent option for usage as a platform for quantum technologies that are based on quantum resources such as quantum coherence and quantum discord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' ACKNOWLEDGEMENTS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Cruz gratefully acknowledges Mario Reis for the valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' Anka thanks FAPERJ for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} +page_content=' [1] M.' metadata={'source': 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and Computation 340, 251 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfEgOL/content/2301.02891v1.pdf'} diff --git a/F9E1T4oBgHgl3EQfqwX2/content/tmp_files/2301.03348v1.pdf.txt b/F9E1T4oBgHgl3EQfqwX2/content/tmp_files/2301.03348v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5644daca8a5ad1298cb7719575edb16ad92d96d --- /dev/null +++ b/F9E1T4oBgHgl3EQfqwX2/content/tmp_files/2301.03348v1.pdf.txt @@ -0,0 +1,1616 @@ +MNRAS 000, 1–13 (2023) +Preprint 10 January 2023 +Compiled using MNRAS LATEX style file v3.0 +A rocky exoplanet classification method and its application to +calculating surface pressure and surface temperature +Sarah R.N. McIntyre,1,2★ Penelope L. King,2 and Franklin P. Mills3,4 +1Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia +2Research School of Earth Sciences, Australian National University, Canberra, ACT 2601, Australia +3Fenner School of Environment and Society, Australian National University, Canberra, ACT 2601, Australia +4Space Science Institute, Boulder, CO 80301, USA +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +With over 5,000 exoplanets currently detected, there is a need for a primary classification +method to prioritise candidates for biosignature observations. Here, we develop a classification +method to categorise rocky exoplanets based on their closest solar system analogue using +available data of observed stellar and planetary features, masses, and radii, to model non- +thermal atmospheric escape, thermal atmospheric escape, and stellar irradiation boundaries. +Applying this classification method to the 720 rocky exoplanets in our sample with uncertainties +in planetary masses, radii, stellar temperatures, and fluxes propagated via a Monte Carlo model +indicates that 22% ± 8% are Mercury analogues, 39% ± 4% are Mars analogues, 11% ± 1% +are Venus analogues, 2% ± 1% are Earth analogues, and 26% ± 12% are without a known +planetary counterpart in our solar system. Extrapolating to conditions on LHS 3844b and GJ +1252b, our classification method gives results reasonably consistent with current observations. +Subsequently, to demonstrate the functionality of this classification method, we plot our +catalogued sample of exoplanets on an adjusted surface pressure versus temperature phase +diagram, presenting more realistic estimates of the potential surface phases (gas, liquid or ice). +Our new classification method could help target selection for future exoplanet characterisation +missions. +Key words: planets and satellites: terrestrial planets - planets and satellites: surfaces - cata- +logues +1 +INTRODUCTION +Over the past 28 years, astronomers have observed over 5,0001 ex- +trasolar planets, providing us with basic information regarding their +orbits, masses, and radii. Research on exoplanets is currently fo- +cused on determining which of these worlds may be habitable; for +example, rocky bodies (with sufficient gravity to support an atmo- +sphere) orbiting their host star at a distance where stellar insolation +flux is suitable for the existence of liquid water on their surface +(Kaltenegger 2017). Due to the inability to conduct in-situ explo- +ration, near-term studies to further characterise exoplanets will focus +on remote detection of their atmospheres and spectral observations +of possible biosignatures (Schwieterman et al. 2018). However, the +significant observing time required to characterise rocky exoplanets +limits the number of targets where we can conduct such extensive +observations. +Defining how potential atmospheric biosignatures vary under +★ E-mail: sarah.mcintyre@anu.edu.au +1 https://exoplanetarchive.ipac.caltech.edu/ +(Accessed +18 +December 2022) +different conditions is important when characterising exoplanets +(Schwieterman et al. 2018). Two significant parameters when con- +sidering the climatic conditions of an exoplanet are surface pressure +and surface temperature (Keles et al. 2018). Water’s stability on a +planetary surface as a liquid depends on both the surface tempera- +ture and pressure (Seager 2013). While the freezing point of liquid +water is not strongly dependent on surface pressure, the boiling point +is significantly affected by it (Vladilo et al. 2013). Furthermore, at +surface pressures below the triple point, liquid water is not stable +at any temperature. Thus, reliable estimates of the surface pressure +and temperature are essential for characterising the environment and +habitability of an exoplanet. Research suggests that a rise in surface +pressure could lead to a rise in temperature due to the greenhouse +effect (Kopparapu et al. 2014). However, a high surface pressure +can enhance cooling through increased Rayleigh scattering (Kast- +ing et al. 1993; Keles et al. 2018), Mie scattering (Kitzmann et al. +2010), or reflection due to clouds (Marley et al. 2013). Additionally, +exoplanet general circulation model (GDM) simulations and solar +system observations suggest that high surface pressures could in- +crease the latitudinal heat transport, cancelling seasonal variations +in the planet’s surface temperature and resulting in smaller global +© 2023 RAS +arXiv:2301.03348v1 [astro-ph.EP] 9 Jan 2023 + +2 +S.R.N McIntyre et al. +temperature variations (Bullock & Grinspoon 1996; Hansen et al. +2010; Leovy 2001; Trenberth & Caron 2001; Vladilo et al. 2013). +Despite the importance of surface pressure, current proposed +methods for its measurement, using remote-sensing techniques, are +challenging and may not fall within the wavelength cut-off for the +James Webb Space Telescope (Chamberlain et al. 2013; Crow et al. +2011; Gardner et al. 2006; Kasting et al. 2009; Misra et al. 2014). +Three-dimensional general circulation models (3D GCMs) are be- +ginning to provide insights into exoplanet atmospheres (Boutle et al. +2017; Del Genio et al. 2019a; Galuzzo et al. 2021; Lewis & Ham- +mond 2022; Turbet et al. 2016, 2018; Way et al. 2018; Wolf 2017). +While there are a significant number of 3D GCM exoplanet simu- +lations published given the lack of observational data so far, such +models are computationally intensive and not generally accessible +by the broader scientific community, limiting the number of sim- +ulations conducted to date (Del Genio et al. 2019b). Furthermore, +many 3D GCM simulations of rocky exoplanets model “Earth-like” +atmospheres, assuming ∼1 bar of N2 as the predominant component +of the atmosphere. +An initial estimate of an exoplanet’s surface pressure (𝑃𝑠𝑢𝑟 𝑓 ) +can be obtained from a simple model based on hydrostatic equi- +librium (e.g. Hall 2020; Kippenhahn et al. 1990; Kopparapu et al. +2014; Mordasini et al. 2012; Silva et al. 2017), using available +observational data on an exoplanet’s mass (𝑀𝑝) and radius (𝑅𝑝): +𝑃𝑠𝑢𝑟 𝑓 +𝑃⊕ += +� 𝑀𝑝 +𝑀⊕ +�2� 𝑅⊕ +𝑅𝑝 +�4 +(1) +where 𝑃⊕, 𝑀⊕ and 𝑅⊕ are the surface pressure, mass, and ra- +dius of Earth, respectively. For many exoplanets, the radius or the +mass is unknown, resulting in the publication of several mass-radius +relations dependent on the planet’s type, allowing us to calculate +the missing measurement (e.g. Chen & Kipping 2016; Nikouravan +2021; Otegi et al. 2020; Seager et al. 2007; Swift et al. 2011; Turbet +et al. 2020; Weiss & Marcy 2014; Zeng et al. 2016). Here, we follow +the NASA exoplanet database2 and use the Chen & Kipping (2016) +M-R relationship to fill in the missing parameter: +𝑅𝑝 ∼ 𝑀𝑝0.279±0.009 +(2) +Using the relation from Equation 2, the surface pressure in +Equation 1 can be written as: +𝑃𝑠𝑢𝑟 𝑓 +𝑃⊕ += +� 𝑅𝑝 +𝑅⊕ +�3.168±0.232 +(3) +This Earth normalisation significantly limits the range of pos- +sible surface pressure values (Equations 4 and 5), as evident when +calculating the maximum and minimum 𝑃𝑠𝑢𝑟 𝑓 using Equation 3 +and taking the upper radius limit for rocky exoplanets as 1.23𝑅⊕, +from the Chen & Kipping (2016) definition of the boundary be- +tween terrestrial and Jovian planets, and a lower radius limit of +0.3𝑅⊕, which corresponds to the size of the smallest exoplanet dis- +covered around a main-sequence star – Kepler 37b (Haghighipour +2015). +𝑃𝑠𝑢𝑟 𝑓 𝑚𝑎𝑥 = 1.014(1.23)3.168 = 1.95𝑏𝑎𝑟 +(4) +𝑃𝑠𝑢𝑟 𝑓 𝑚𝑖𝑛 = 1.014(0.3)3.168 = 0.02𝑏𝑎𝑟 +(5) +2 https://exoplanetarchive.ipac.caltech.edu/ +(Accessed +18 +December 2022) +The benefit of Equation 1 is that it only requires an exoplanet’s +radius or mass value. However, we obtain an Earth-centric estimate +within the 0.02 − 1.95 bar range because we use Earth’s surface +pressure and radius in our calculations. Our solar system contains +four rocky planets, each with a unique surface, atmosphere, struc- +ture, and evolution. Extrasolar rocky planets will likely display a +similarly wide variety of surface characteristics and interior com- +positions. While the radius range of 0.3 − 1.23𝑅⊕ covers all four +rocky planets in the solar system (Chen & Kipping 2016; Haghigh- +ipour 2015), the resulting surface pressure range only encompasses +Earth’s. This simple model excludes the 5×10−15 − 92 bar range +of surface pressure measurements observed on Mercury, Mars, and +Venus (Rasool et al. 1966; Seiff et al. 1985). +Alternative methods, such as comparing to a more appropriate +solar system analogue, should be considered rather than contin- +uing with the current approach of assigning Earth-like character- +istics to all rocky exoplanets. A similar approach has been used +for modelling the atmospheric chemistry and climate of Venus-like +exoplanets (Kane et al. 2019; Schaefer & Fegley 2011; Way & +Del Genio 2020). Furthermore, the scientific community has been +studying, observing, and probing the planets of the solar system +with multiple satellites, in situ missions, and remote sensing ob- +servations using ground- and space-based telescopes (Gröller et al. +2018; Jakosky et al. 2015; Marcq et al. 2018; McClintock & Lank- +ton 2007; McNutt Jr et al. 2010; Mills et al. 2006; Von Zahn et al. +1979; Withers et al. 2015). Thus, we have significant knowledge +of the solar system planets’ atmospheric profiles and compositions. +We have values for the surface pressures of Mercury, Mars, and +Venus and could use these planets’ respective pressure and radii +values to normalise Equation 1 more appropriately, provided we +can classify which exoplanets are likely to be Mercury, Mars, or +Venus analogues. +Previous classification schemes from Forget & Leconte (2014) +and Wordsworth & Kreidberg (2022) suggest that climates on ter- +restrial exoplanets should depend primarily on atmospheric com- +position, incident stellar flux, and tidal evolution. Here, we develop +a classification method using available data of observed stellar and +planetary features, masses, and radii, to model non-thermal atmo- +spheric escape, thermal atmospheric escape, and stellar irradiation +boundaries. We compute the escape velocities for the known rocky +exoplanets and compare them to the insolation and thermal velocity +of likely gases to determine whether the exoplanets can maintain +their atmospheres and, if so, which gases are retained. Furthermore, +we quantify planetary temperature conditions based on the incident +stellar fluxes to determine the likelihood of rocky exoplanets resid- +ing in a temperate or runaway greenhouse zone. We utilise these +factors to classify the current list of rocky exoplanets and group +them into categories based on similarity to the most appropriate so- +lar system analogue. Subsequently, to demonstrate the functionality +of this classification method, we combine it with an extension to +the simple surface pressure model (Equation 3), plot the adjusted +surface pressure vs temperature phase diagram, and discuss the im- +plications for the habitability of exoplanets and the optimisation of +target selection for future atmospheric observations. +2 +METHOD +2.1 +Non-thermal atmospheric escape +An exoplanet must have an atmosphere to have a significant surface +pressure. Thus, an important feature in determining surface pressure +MNRAS 000, 1–13 (2023) + +A rocky exoplanet classification method +3 +is the escape velocity of an exoplanet, which can be calculated as: +𝑉𝑒𝑠𝑐 = +√︄ +2𝐺𝑀𝑝 +𝑅𝑝 +(6) +where +𝐺 +is +the +universal +gravitational +constant +6.6743 × +10−11𝑚3𝑘𝑔−1𝑠−2. The escape itself is a rapid process, unlikely +to be directly observable. According to Zahnle & Catling (2017), +the cumulative impact of escape should be evident in the statis- +tical analysis of exoplanets, with a division between planets with +and without atmospheres, that they define as the “cosmic shoreline” +(Catling & Zahnle 2009; Zahnle & Catling 2013, 2017). The solar +system planets are neatly divided around the cosmic shoreline: +𝑆∗ = 5 × 10−16𝑉4 +𝑒𝑠𝑐 +(7) +where 𝑆∗ is insolation and 𝑣𝑒𝑠𝑐 is the escape velocity as defined in +Equation 6. The cosmic shoreline is a simple approximation of non- +thermal atmospheric escape based on observable parameters. While +there are additional non-thermal processes that could increase the +amount of atmospheric mass-loss, for example ion pickup (Lammer +et al. 2006), sputtering (Terada et al. 2009), dissociation and disso- +ciative recombination (Geppert & Larsson 2008), photo-chemical +energising mechanisms (Vidotto 2013), and charge exchange (Dong +et al. 2017); these models require supplementary information for +which we currently have no observed values and no clear method +to obtain. +2.2 +Thermal atmospheric escape +The thermal escape rate is a function of the planet’s escape velocity +and the temperature of the exobase. There are two types of ther- +mal escape in atmospheres: hydrodynamic escape and slow thermal +escape (Jeans escape). Highly irradiated large-mass exoplanets are +more likely to lose atmospheric mass through hydrodynamic blow- +off (Owen 2019). However, Konatham et al. (2020) suggest that +rocky exoplanets are more inclined to undergo slow thermal escape +caused by atmospheric species’ thermal velocities. Using the basic +principles of the kinetic theory of gases, we follow Konatham et al. +(2020) to predict probable atmospheric compositions of exoplanets +by identifying the atmospheric species that can leave their atmo- +spheres via slow thermal escape. According to the Konatham et al. +(2020) model, the thermal escape rate is defined by the thermal +velocity of the atmospheric species: +𝑈 = +√︂ +3𝑘𝑏𝑇 +𝑚 +(8) +where 𝑚 is the mass of a gas, 𝑘𝑏 is Boltzmann’s constant, and 𝑇 is +the exobase temperature. As an observable value for 𝑇 is currently +unavailable, we follow the Konatham et al. (2020) approach for a +fast rotating exoplanet and utilise its equilibrium temperature, 𝑇𝑒𝑞 +with albedo A = 0, as a conservative approach for estimating the +slow thermal escape of species from the atmosphere: +𝑇𝑒𝑞 = +� (1 − 𝐴) 𝑆∗ +4𝜎 +� 1 +4 +(9) +where 𝜎 is the Stefan–Boltzmann constant. Konatham et al. (2020) +infer results for rocky exoplanets experiencing slow thermal escape +using data from observations of gases escaping from solar system +planets’ atmospheres. Equation 10 relates𝑈 to 𝑣𝑒𝑠𝑐 for atmospheric +species to escape an exoplanet’s atmosphere: +𝑈 > 1 +10𝑣𝑒𝑠𝑐 +(10) +2.3 +Circumstellar Habitable Zone +The search for potential Earth analogues begins by examining the +Circumstellar Habitable Zone (CHZ), defined as the region in which +a rocky planet, with favourable atmospheric conditions, can sustain +liquid water on its surface (Kasting et al. 1993; Selsis et al. 2007; +Kopparapu et al. 2013, 2014). Kopparapu et al. (2013, 2014) esti- +mate the CHZ around stars with stellar effective temperatures (𝑇∗) +in the range of 2600–7200 K, for planetary masses between 0.1- +5M⊕, and assuming H2O (inner boundary) and CO2 (outer bound- +ary) dominated atmospheres, with N2 as the background gas. This +model quantifies incident stellar radiation fluxes that would result +in planetary temperature conditions shifting to a runaway snowball +or a runaway greenhouse. Here, we use the Kopparapu et al. (2014) +optimistic definition of the CHZ “early Mars” outer (Equation 11) +and “recent Venus” inner (Equation 12) boundaries: +𝑆𝐸𝑀 = 0.32 + 5.547 × 10−5(𝑇∗ − 5780) + 1.526 × 10−9(𝑇∗ − 5780)2 +−2.874 × 10−12(𝑇∗ − 5780)3 − 5.011 × 10−16(𝑇∗ − 5780)4 +(11) +𝑆𝑅𝑉 = 1.776 + 2.136 × 10−4(𝑇∗ − 5780) + 2.533 × 10−8(𝑇∗ − 5780)2 +−1.332 × 10−11(𝑇∗ − 5780)3 − 3.097 × 10−15(𝑇∗ − 5780)4 +(12) +where 𝑆𝐸𝑀 is the stellar flux required for the early Mars outer CHZ +boundary, and 𝑆𝑅𝑉 is the stellar flux required for the recent Venus +inner CHZ boundary. These CHZ boundaries define the exoplanets +in our sample that most closely resemble Earth and are therefore +likely to have liquid water on their surfaces. +2.4 +Venus Zone +There is a clear distinction in atmospheric evolution between Earth +and Venus, probably due to the significant difference in solar irra- +diance (approximately a factor of two). Kane et al. (2014) define +a “Venus Zone” (VZ) where a planet is considered to be a Venus +analogue rather than an Earth analogue. +We use the optimistic CHZ “recent Venus” boundary from +Equation 12 to define the runaway greenhouse outer VZ boundary, +where oceans completely evaporate, resulting in the inability to +execute a carbon cycle and efficiently moderate atmospheric CO2 +levels, leading to the formation of a thick Venus-like atmosphere. +As distance from the host star decreases, the likelihood of +substantial atmospheric mass loss increases. Kane et al. (2014) +determine the insolation flux required for Venus to cross the Zahnle +& Catling (2017) cosmic shoreline and use this value to denote +the complete atmospheric erosion of Venus analogues (𝑆𝐴𝐸) as an +approximation for the VZ inner boundary: +𝑆𝐴𝐸 ≈ 25𝑆⊕ +(13) +Just as planets in the CHZ could be considered Earth analogues +until more spectroscopic information becomes available, planets +inside the VZ could be considered Venus analogues until further +characterisation observations are undertaken. +2.5 +Surface pressure +After cataloguing our sample of rocky exoplanets, we adjust pa- +rameters such as surface pressure to more appropriately normalise +simple calculations. Taking the solar system planet values for sur- +face pressure (𝑃𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡) and radius (𝑅𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡) into account, +we adjust Equation 3 to: +𝑃𝑆𝑆−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 𝑃𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡 +� +𝑅𝑝 +𝑅𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡 +�3.168 +(14) +MNRAS 000, 1–13 (2023) + +4 +S.R.N McIntyre et al. +𝑃𝑀𝑒𝑟𝑐𝑢𝑟 𝑦−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 1.05 × 10−13𝑅𝑝3.168 +(15) +𝑃𝑀 𝑎𝑟𝑠−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 0.0467𝑅𝑝3.168 +(16) +𝑃𝑉 𝑒𝑛𝑢𝑠−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 108.27𝑅𝑝3.168 +(17) +𝑃𝐸𝑎𝑟𝑡ℎ−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 1.014𝑅𝑝3.168 +(18) +2.6 +Surface temperature +After calculating an adjusted 𝑃𝑠𝑢𝑟 𝑓 by normalising to the most +appropriate solar system analogue, we apply our rocky exoplanet +classification system to surface temperature estimates to plot an +analogue adjusted surface pressure vs surface temperature phase +diagram. Unfortunately, like surface pressure, direct measurements +of surface temperature are not typically available (Weisfeiler et al. +2015) and when modelling using 3D GCMs, Earth-centric assump- +tions of atmospheric composition or convection depth are frequently +made (e.g. Biserud 2022; Chaverot et al. 2021; Way et al. 2017; +Yang & Abbot 2014), creating a bias in surface temperature mod- +els. Del Genio et al. (2019b) correlate surface temperature (𝑇𝑠𝑢𝑟 𝑓 ) +and equilibrium temperature (𝑇𝑒𝑞), by attributing the difference +between the two to a greenhouse effect: +𝑇𝑠𝑢𝑟 𝑓 = 𝑇𝑒𝑞 + 𝐺𝑎 +(19) +where 𝐺𝑎 is the atmospheric greenhouse effect. Substituting Equa- +tion 9 into Equation 19, we attain a surface temperature equation: +𝑇𝑠𝑢𝑟 𝑓 = +� (1 − 𝐴) 𝑆∗ +4𝜎 +� 1 +4 ++ 𝐺𝑎 +(20) +Substituting observed values of the rocky solar system planets’ +bond albedo, insolation flux, and average surface temperature into +Equation 20 allows us to calculate the atmospheric greenhouse effect +for our solar system analogues (Table 1). +Table 1. Solar System analogue surface temperature calculation values. +Planet +𝑇𝑠𝑢𝑟 𝑓 (K) +𝐴 +𝑆∗ (Wm−2) +𝐺𝑎 (K) +Mercury +440 +0.07 +9082.7 +0.85 +Mars +210 +0.25 +586.2 +-0.15 +Venus +737 +0.77 +2601.3 +510.85 +Earth +288 +0.3 +1361.0 +33.85 +While we do not take the time evolution factor into account +in this paper, it should be noted that the values in Table 1 have +changed over the lifetime of the solar system and similar variations +are expected across the lifetime of exoplanet systems. +This surface temperature model only accounts for a broad +greenhouse effect and not specific greenhouse gas abundances. +However, it enables us to apply solar system planet values for bond +albedo (𝐴𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡) and atmospheric greenhouse (𝐺𝑎𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡) +from Table 1 and refine our surface temperature calculations by +adjusting Equation 20 to: +𝑇𝑆𝑆−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = +� �1 − 𝐴𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡 +� 𝑆∗ +4𝜎 +� 1 +4 ++ 𝐺𝑎𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡 +(21) +𝑇𝑀𝑒𝑟𝑐𝑢𝑟 𝑦−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 45.0𝑆∗ +1 +4 + 0.85 +(22) +𝑇𝑀 𝑎𝑟𝑠−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 42.6𝑆∗ +1 +4 − 0.15 +(23) +𝑇𝑉 𝑒𝑛𝑢𝑠−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 31.7𝑆∗ +1 +4 + 510.85 +(24) +𝑇𝐸𝑎𝑟𝑡ℎ−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 41.9𝑆∗ +1 +4 + 33.85 +(25) +2.7 +Sample selection and Monte Carlo calculations +We utilise NASA’s composite planet database3, in combination with +additional information on the Kepler planets’ radii provided by +Berger et al. (2020b) and updated stellar properties by Berger et al. +(2020a) to compose a catalogue of rocky exoplanets. Furthermore, +we use the Chen & Kipping (2016) M-R relationship detailed in +Equation 2 to calculate unknown radii or masses and their uncer- +tainties. +To ensure that we have included only rocky exoplanets in our +database, we follow the Chen & Kipping (2016) definition of the +boundary between Jovian and terrestrial planets, limiting our selec- +tion to exoplanets with radii 𝑅𝑝 ≤ 1.23 R⊕. +As we need information for stellar temperature, insolation flux, +planetary mass, and planetary radius as provided by the NASA +composite planet database, Berger et al. (2020b), and Berger et al. +(2020a), our total sample contains 720 rocky exoplanets. For each +exoplanet in our sample, we execute 10,000 Monte Carlo simu- +lations using a Gaussian probability distribution for uncertainties +on the stellar temperatures, insolation fluxes, planetary masses, and +planetary radii as provided by the NASA composite planet database, +Berger et al. (2020b), and Berger et al. (2020a). Furthermore, for +the subset of exoplanets where the mass-radius relation was used, +Equation 2 was input directly into the Monte Carlo simulation to +ensure the uncertainties were appropriately correlated. +The Monte Carlo simulations allow us to determine the median +and 68% confidence intervals on escape velocity (𝑣𝑒𝑠𝑐), equilibrium +temperature (𝑇𝑒𝑞), surface temperature (𝑇𝑠𝑢𝑟 𝑓 ), and surface pres- +sure (𝑃𝑠𝑢𝑟 𝑓 ). +3 +DATA ANALYSIS & DISCUSSION +3.1 +Exoplanet classification +In Figure 1, we plot stellar insolation flux (𝑆∗) against the escape +velocity (𝑣𝑒𝑠𝑐) for the exoplanets in our sample as well as the four +rocky solar system planets. The trend seen in Figure 1 is what we +would expect to see if escape were the primary factor influencing +the volatile inventories of exoplanets. With the aid of the Zahnle +3 https://exoplanetarchive.ipac.caltech.edu/cgi-bin/ +TblView/nph-tblView?app=ExoTbls&config=compositepars +(accessed 18 December 2022) +MNRAS 000, 1–13 (2023) + +A rocky exoplanet classification method +5 +& Catling (2017) cosmic shoreline (Equation 7), we can see that +planetary atmospheres are thick when the influence of the central +star is weak (measured by insolation) or the gravitational well is +deep (measured by escape velocity). On the other hand, we have +planets with thin or no atmospheres when the star is too bright or +gravity is weak. +Figure 1. Insolation and escape velocity for the 720 exoplanets in our sam- +ple. Exoplanets likely to have thin or no atmosphere are plotted with open +circles. Exoplanets likely to have an atmosphere are solid dots. Labelled solid +coloured circles represent observed values for rocky solar system planets. +The solid black line represents the Zahnle & Catling (2017) cosmic shoreline +(Equation 7). The dashed vertical line indicates the Chen & Kipping (2016) +cut-off for rocky exoplanets, 𝑅𝑝 ≤ 1.23 R⊕. Median and 68% confidence +intervals on escape velocity values were calculated using the Monte Carlo +simulations. +From our sample of 720 rocky exoplanets, 12% ± 4% reside +below the cosmic shoreline and are consequently likely to maintain +a significant atmosphere. Conversely, 88% ± 5% of the exoplan- +ets in our sample reside above the cosmic shoreline and are likely +to have thin or negligible atmospheres, based on the rapid escape +velocity parameter alone. The uncertainty values arise from taking +into account the 68% confidence intervals on insolation and es- +cape velocity. Taking the upper insolation error and lower escape +velocity error into account, 4% of exoplanets located in the area +where rocky planets are likely to maintain a significant atmosphere, +cross the cosmic shoreline and thus could potentially have thin or +no atmosphere. Conversely, taking the lower insolation error and +upper escape velocity error into account, 5% of exoplanets located +in the area where rocky planets have a thin or negligible atmo- +sphere, cross the cosmic shoreline and thus could potentially have +a substantial atmosphere. Furthermore, Figure 1 indicates that most +rocky exoplanets lie above the cosmic shoreline, where gravity is +weak and stellar flux strong, implying that our current observational +exoplanet data has a bias towards close-orbiting rocky planets with +limited atmospheres. +All rocky exoplanets discovered thus far are on close, highly +irradiated orbits, bombarded by large amounts of ionising EUV and +X-ray radiation (Lammer et al. 2022). The upper atmosphere of +an exoplanet also may be exposed to coronal mass ejections and +stellar winds, inducing additional non-thermal loss processes of ion +pickup (Lammer et al. 2006), sputtering (Terada et al. 2009), disso- +ciation and dissociative recombination (Geppert & Larsson 2008), +photo-chemical energising mechanisms (Vidotto 2013), and charge +exchange (Dong et al. 2017). These non-thermal escape mecha- +nisms could shift the cosmic shoreline to lower insolation at a given +escape velocity and are particularly important in the early phases +of the host star’s evolution, when XUV flux and CME rates may be +orders of magnitude higher (Do Amaral et al. 2022; Gronoff et al. +2020). However, models for these non-thermal processes require +additional information for which we currently have no observed +values, so it is beyond the scope of this paper. Alternatively, de- +pending on the stellar wind pressures, a significant magnetic field +could mitigate the non-thermal escape processes, resulting in a shift +of the cosmic shoreline to higher escape velocities (McIntyre et al. +2019; Egan et al. 2019). +There are two types of thermal escape in atmospheres; hy- +drodynamic escape and slow thermal escape (Jeans escape). Most +models have been designed for hydrodynamic conditions primarily +for small semi-major axis, high-mass, highly irradiated exoplanets +(Owen 2019; Tian 2015; Madhusudhan 2019). Conversely, low- +irradiated, small-mass rocky exoplanets are more likely to expe- +rience slow thermal escape driven by thermal velocities of atmo- +spheric species when assuming that equilibrium temperature is a +good guide to thermospheric temperature (Konatham et al. 2020). +Figure 2. Escape velocity (Equation 6) versus equilibrium temperature +(Equation 9) of our sample of 720 rocky exoplanets. Exoplanets that re- +side above the cosmic shoreline in Figure 1 and are likely to have thin or +no atmosphere are plotted with open circles. Exoplanets that reside below +the cosmic shoreline in Figure 1 and are likely to have an atmosphere are +plotted with solid dots. Labelled solid coloured circles represent calculated +values for rocky solar system planets. Solid black lines represent the thermal +velocity of atmospheric gas species (defined by Equations 8-10) (Konatham +et al. 2020), where exoplanet atmospheres may retain the gas at lower escape +velocities. Red shaded region denotes Mercury analogue exoplanets. Orange +shaded region denotes Mars analogues exoplanets. The dashed horizontal +line indicates the cut-off for rocky exoplanets as 𝑅𝑝 ≤ 1.23 R⊕. Median and +68% confidence intervals on escape velocity and equilibrium temperature +values were calculated using the Monte Carlo simulations. +In Figure 2, we compute the thermal velocities of selected +gases for our sample of rocky exoplanets (parameterised by the +equilibrium temperature from Equation 9) and compare them with +the escape velocity to determine which gases could be preserved +in the planets’ atmospheres, in accordance with Equation 10. The +diagonal lines depict the thermal velocity of various atmospheric +species as a function of kinetic temperature (also known as veloc- +MNRAS 000, 1–13 (2023) + +105 +Rocky exoplanets with thin +0 +or negligible atmospheres +104 +Rocky exoplanets with atmospheres +103 +0 +102. +S +8 +8 +Insolation +0 +0 +101 +0 +Mercury +Venus +100. +·Earth +Mars +10-1 +IZ +IN +31 +10-2. +Cosmic shoreline +R: +1 +10-3. +104 +103 +Escape Velocity (ms-1)H +oad +H2 +8 +0 +096 +Earth +104. +8 +0 +Venus +0 +He +Escape Velocity (ms- +0 +080 +0 +0 +00 +C +0 +0 +0 +08 +00 +00 +Mars +0 +8 +0 +O,CH4 +0 +Mercury +0 +H20,NH3 +0 +02 +CO2 +502 +Rocky exoplanets with thin +0 +Xe +or negligible atmospheres +Rocky exoplanets with atmospheres +103. +103 +102 +Eguilibrium Temperature (K)6 +S.R.N McIntyre et al. +ity lines) collated from Konatham et al. (2020). An exoplanet can +retain a specific atmospheric species if its velocity line is below the +position of the exoplanet in Figure 2. Conversely, an atmospheric +species escapes the exoplanet’s atmosphere if its velocity line is +above the exoplanet’s position. This allows us to utilise the kinetic +theory of gases to estimate potential atmospheric constituents for +our sample of rocky exoplanets. +In Figure 2, we can see that 22% ± 8% of rocky exoplanets in +our sample lie below the CO2 line in the red shaded region. All the +exoplanets in this subset were also located substantially above the +cosmic shoreline (Figure 1). Based on their positioning in Figures +1 and 2, these exoplanets are unlikely to be able to sustain a signifi- +cant atmosphere and thus will have limited surface pressure. When +comparing to the four rocky solar system analogues, we can see that +these exoplanets most closely resemble Mercury with its negligi- +ble atmosphere (surface pressure ∼5 picobars). Consequently, this +subset of exoplanets could be classified as Mercury analogues. +The next group of exoplanets are those located above the CO2 +line and below the H2O line. The 39% ± 4% of rocky exoplan- +ets from our sample residing within the orange shaded region in +Figure 2 also lie above the cosmic shoreline in Figure 1. Based +on their position in Figures 1 and 2, these exoplanets are likely +to have thin water-less atmospheres. When comparing to the four +rocky solar system analogues, we can see that the atmospheres of +these exoplanets most closely resemble present-day Mars with its +thin, CO2-rich, dry atmosphere (surface pressure of 0.00636 bars). +Consequently, this subset of exoplanets could be classified as Mars +analogues. +Figure 3. Location within the CHZ and VZ of 279 rocky exoplanets that +reside above the H2O line in Figure 2. The blue shaded region denotes +the CHZ. The purple shaded region denotes the VZ. Exoplanets that reside +above the cosmic shoreline in Figure 1 and are likely to have thin or no +atmosphere are plotted with open circles. Exoplanets that reside below the +cosmic shoreline in Figure 1 and are likely to have an atmosphere are +plotted with solid dots. Trappist-1h, the exoplanet shown beyond the Early +Mars boundary, is discussed further in 3.2 +The final 39% ± 12% of rocky exoplanets presented in Figure 2 +reside above the H2O line and are likely to have water in their atmo- +spheres. When comparing to the four rocky solar system analogues, +both Venus and Earth fall within similar locations in Figures 1 and 2 +yet have substantially different atmospheres with surface pressures +differing by two orders of magnitude. To determine which of the +279 rocky exoplanets located above the H2O line in Figure 2 are +likely to be Earth analogues or Venus analogues, we examine their +location in relation to the CHZ and VZ in Figure 3. +Using Equations 11 - 12, we plot the optimistic boundaries +of the CHZ in Figure 3 to quantify the insolation flux thresholds +where planetary conditions transition to either a runaway snowball +or a runaway greenhouse. The 2% ± 1% of rocky exoplanets that +are likely to have an atmosphere (Figure 1), with H2O present +(Figure 2), and also reside within the optimistic CHZ (Figure 3) +where the insolation is suitable for the presence of liquid water on +the planets’ surfaces, are the planets most similar to Earth. Thus, +these exoplanets could be classified as Earth analogues and are the +most likely to retain a ∼1 bar atmosphere including H2O molecules, +indicating a high potential for retaining water in their atmospheres +and on their surfaces. +Additionally, using Equations 12 - 13, we plot the boundaries +of the VZ in Figure 3 to quantify the insolation flux thresholds where +planetary conditions transition between a runaway greenhouse and +complete atmospheric erosion. In Figure 3, we see that 11% ± 1% +of rocky exoplanets with H2O present in their atmospheres (Figure +2) reside within the VZ. Out of these 11% ± 1% rocky exoplanets, +fifteen are plotted with open circles denoting their location above the +cosmic shoreline in Figure 1, indicating they are unlikely to have an +atmosphere. However, taking into account 68% confidence intervals +on insolation and escape velocity, these exoplanets cross the cosmic +shoreline and could potentially have a substantial atmosphere. Thus, +the 11% ± 1% of rocky exoplanets within uncertainties are likely to +have an atmosphere (Figure 1), with H2O present (Figure 2). Their +position within the VZ (Figure 3), indicates that their atmospheres +would be unable to maintain radiation balance, resulting in runaway +heating of the surface and the formation of a thick Venus-like atmo- +sphere and surface pressures in the order of 102 bar. Therefore, this +subset of rocky exoplanets could be classified as Venus analogues. +The final subset of 26% ± 12% rocky exoplanets from our sam- +ple reside above the H2O line in Figure 2 and are not located within +the CHZ or VZ in Figure 3. Furthermore, the majority of planets +in this subset are likely to have thin or negligible atmospheres, ac- +cording to Figure 1. Out of these 26% ± 12% rocky exoplanets, +nine exoplanets are located past the inner VZ atmospheric erosion +boundary, yet they are plotted with solid circles, indicating they +are likely to have an atmosphere according to Figure 1. However, +taking into account 68% confidence intervals on insolation and es- +cape velocity, these exoplanets cross the cosmic shoreline and could +potentially have a thin or negligible atmosphere. Kane et al. (2014) +suggest that these highly irradiated rocky exoplanets located past +the VZ’s inner boundary have completely eroded atmospheres. No +solar system analogues exist for this subset of rocky exoplanets, +which have the potential for H2O to be present, yet too high levels +of stellar flux eroding their atmospheres. +Combining the information from Figures 1-3, we develop the +classification method outlined in Figure 4. Comparing to previous +classification schemes, Wordsworth & Kreidberg (2022) classify a +subset of rocky exoplanets that have an atmosphere and equilibrium +temperatures above 300K as having no direct analogue in the solar +system. However, here, we classify these exoplanets as Venus-like +as we have determined that they retain an atmosphere that could +have H2O present despite their high temperatures. This subset will +be interesting to explore in future observations to study the key tran- +sitions in atmospheric composition and determine how they differ +from Venus. Additionally, in Figure 3 we have classified a different +subset than Wordsworth & Kreidberg (2022) having no solar sys- +tem analogue as those rocky exoplanets that have the potential for +MNRAS 000, 1–13 (2023) + +Atmospheric +Recent +Early +0 +6500. +0 +Erosion +Venus +Mars +. +0 +0 +6000 +G +Temperature +0 +0 +5500. +0 +8 +0 +5000 +00 +00 +0 +4500. +Effective +K +8 +4000: +0 +0 +CHZ +VZ +0 +3500 +Rocky exoplanets with +thin or negligible +M +3000 +atmospheres +Rocky exoplanets with +atmospheres +2500 +102 +100 +103 +101 +Insolation (S)A rocky exoplanet classification method +7 +Figure 4. Classification of rocky exoplanets into closest solar system analogue. The dashed lines indicate potential for crossover in classification due to +uncertainties in the input parameters. None of the exoplanets that were likely to maintain an atmosphere were below the H2O line. +H2O to be present, yet due to high levels of stellar flux, have no or +negligible atmospheres. +3.2 +Application of exoplanet classification to surface pressure +and surface temperature +To demonstrate the effect that the new classification system has on +simple normalised calculations, the surface pressures for all 720 +rocky exoplanets in our sample were computed using the original +Equation 3 (Figure 5a), and using the new Equations 15-18 (Figure +5b). +Figure 5a demonstrates the clustering around ∼1 bar surface +pressure due to the fact that surface pressure was normalised by +Earth’s value of 1.014 bar. Therefore, relative to their observed +values, the calculated values for Mercury and Mars are higher, and +Venus is lower. Furthermore, as we have limited the radius of rocky +exoplanets in our sample to 0.3𝑅⊕ ≤ 𝑅𝑝 ≤ 1.23𝑅⊕, the surface +pressure range is limited to 0.02 − 1.95 bar. +Figure 5b highlights the effect that the solar system analogue +classification method has made to the surface pressure model. There +is now a broader spread of surface pressure values for rocky exo- +planets ranging from 2 × 10−15 − 210 bars. Additionally, Figure +5b illustrates the division between planets that have thin to no at- +mospheres being Mercury or Mars analogues and planets with at- +mospheres being Venus or Earth analogues. The exceptions to this +division are the Venus analogue exoplanets whose median escape +velocities indicate an absence of an atmosphere, however within +68% confidence intervals, these Venus analogues cross the cosmic +shoreline from Figure 1 and indicate the potential for a thick Venus +like atmosphere, based on their location in Figure 3. Exoplanets +that are classified as having “no solar system analogue” have no +analogue adjusted equations and are thus omitted from Figure 5b. +Reliable estimates of surface temperature and pressure are es- +sential for characterising the environment and habitability of an +exoplanet. To further demonstrate the functionality of our exoplanet +classification method, we plot our sample of rocky exoplanets over +the phase diagram of pure water. In Figure 6a we plot the equilib- +rium temperature, where 𝐴 = 0 and 𝐺𝑎 = 0 (Equation 9), against +the original surface pressure values (Equation 3). Subsequently, af- +ter applying our solar system analogue classification, in 6b we plot +the surface temperature, with analogue-defined 𝐴 and 𝐺𝑎 values +(Equations 22-25), against the analogue-adjusted surface pressure +values (Equations 15-18). +Figure 6a illustrates the 0.02-1.95 bar limitations and clustering +around ∼1 bar surface pressure due to the Earth-centric normali- +sation in Equation 3. Additionally, the equilibrium temperature of +Venus, without accounting for any atmospheric greenhouse effect, +falsely indicates the likelihood of liquid water on Venus’s surface. +Thus, our subset of Venus analogue exoplanets which reside within +the liquid water zone in Figure 6a may display similar false-positive +results. +Figure 6b highlights the effect that the solar system analogue +classification method has made to the surface temperature vs pres- +sure phase diagram. The Venus analogues have shifted outside of the +liquid water zone, representing the significant effect an atmospheric +greenhouse plays in the potential habitability of an exoplanet. In Fig- +ure 6a there are no rocky exoplanets with an atmosphere recording +equilibrium temperature above 620K, however with our analogue +adjusted surface temperatures in Figure 6b, we see some Venus-like +exoplanets with surface temperatures of up to 950K, which could be +classified as Atmosphere type I from Miguel et al. (2011). On these +hot rocky exoplanets, the major gases present are likely to be Na, +O2, O, and Fe, with the near-crust atmospheres mainly composed +of H2O, CO2, and SO2 (Herbort et al. 2020; Miguel et al. 2011; +Miguel & Kaltenegger 2013; Schaefer & Fegley 2009; Schaefer +et al. 2012). +Furthermore, in Figure 6b there are six Earth analogues that are +likely to have an atmosphere (Figure 1) with H2O present (Figure +MNRAS 000, 1–13 (2023) + +Sample of currently +detected rocky exoplanets +Fig 1: Non-thermal escape +Fig 1: Non-thermal escape +Thin or negligible + Likely to maintain +an atmosphere +atmosphere +2: The +Fig 2: Thermal +escape +Above the H,O line: +Below the H,O line: +Above the H,O line: +Unlikely to have water +Likely to have water +Likely to have water +in the atmosphere +in the atmosphere +in the atmosphere +Fig 3: Insolation +Venu + boundaries +2 +Fig 3 +Between the CO2 and H,O lines: +Located inside of the inner +Located within the VZ: +Below the CO2 line: +Located within the CHZ: +Thin atmosphere and unlikely to +boundary of the VZ: +Likely to have a thick atmosphere +Negligible atmosphere +No runaway greenhouse effect +have water in the atmosphere +Atmospheric erosion too high +due to runaway greenhouse + Mercury analogue + Mars analogue + No solar system analogue + Earth analogue +Venus analogue8 +S.R.N McIntyre et al. +Figure 5. a. Surface pressure calculation using original Equation 3 applied to 720 rocky exoplanets in our sample. Open coloured circles represent the calculated +values for rocky solar system planets. Labelled solid coloured circles represent observed values for rocky solar system planets. Exoplanets that reside above +the cosmic shoreline in Figure 1 and are likely to have thin or no atmosphere are plotted with open circles. Exoplanets that reside below the cosmic shoreline +in Figure 1 and are likely to have an atmosphere are plotted with solid dots. Median and 68% confidence intervals on surface pressure values were calculated +using the Monte Carlo simulations. b. Surface pressure calculation using Equations 15-18 applied to our sample categorised according to Figure 4. Labelled +solid coloured circles represent observed values for rocky solar system planets. Exoplanets that reside above the cosmic shoreline in Figure 1 and are likely to +have thin or no atmosphere are plotted with open circles. Exoplanets that reside below the cosmic shoreline in Figure 1 and are likely to have an atmosphere +are plotted with solid dots. Exoplanets that are classified as having “no solar system analogue” have no analogue adjusted equations and are thus omitted from +this graph. Median and 68% confidence intervals on surface pressure values were calculated using the Monte Carlo simulations. +Figure 6. a. Phase diagram of pure water (blue shaded region) with surface pressure vs equilibrium temperature, applying Equations 3 and 9 to the 720 rocky +exoplanets in our sample. Labelled solid coloured circles represent rocky solar system planets. Exoplanets that reside above the cosmic shoreline in Figure 1 +and are likely to have thin or no atmosphere are plotted with open circles. Exoplanets that reside below the cosmic shoreline in Figure 1 and are likely to have +an atmosphere are plotted with solid dots. Median and 68% confidence intervals on surface temperature and surface pressure values were calculated using +the Monte Carlo simulations. b. Phase diagram of pure water (blue shaded region) with adjusted surface pressure vs surface temperature, applying Equations +15-18 and 22-25 to our sample classified according to Figure 4. Labelled solid coloured circles represent rocky solar system planets. Exoplanets that reside +above the cosmic shoreline in Figure 1 and are likely to have thin or no atmosphere are plotted with open circles. Exoplanets that reside below the cosmic +shoreline in Figure 1 and are likely to have an atmosphere are plotted with solid dots. Exoplanets that are classified as having “no solar system analogue” have +no analogue adjusted equations and are thus omitted from this graph. Median and 68% confidence intervals on surface temperature and surface pressure values +were calculated using the Monte Carlo simulations. +MNRAS 000, 1–13 (2023) + +a: Original Psurf Earth centric estimation +b: Adjusted Psurf using solar system analogue classification +102 +Venus +102- +Venus +Earth +10-1. +101. +(Bars) +0 +(Bars) +Mars +Pvenus - analogue +Venus +10 +Pressure +Pressure +PEarth - analogue +Earth +calc +100. +PMars - analogue +0 +PMercury - analogue +Mars +calc +Rocky exoplanets with thin +face +ce +Mercury +or negligible atmospheres +10 +Surfa +calc +Rocky exoplanets with atmospheres +Surf +0 +Rocky exoplanets with thin +10-2. +Mars +10-13 +or negligible atmospheres +Mercury +Rocky exoplanets with atmospheres +Mercury +0.6 +0.8 +1.0 +1.2 +0.4 +0.6 +0.8 +0.4 +1.0 +1.2 +Radius (R) +Radius (R)a: Original Earth centric estimation +b: Adjusted Psurf and Tsurf using solar system analogue classification +102- +OVenus +Rocky exoplanets with thin +Rocky exoplanets with atmospheres +. +0 +or negligible atmospheres +102. +Venus +Rocky exoplanets with thin +0 +Rocky exoplanets with atmospheres +or negligible atmospheres +(Bars) +101. +(Bars) +101. +100 +Earth + Pressure + Pressure +b%.%0 +0 +° +0 +100. +0 +10 +)-1, +0 +8&0 +0o +0 +8 +0 +0 +00% +Surface l +000% +urface +0 +10 +Mars +0 +0 +0 +10' +0 +S10-13J +0 +0 +0 +@0 +0 +00 +0 +10-14↓ +Mercury +10-2↓ +Mars +0 +0 +0 +500 +1000 +2000 +2500 +3000 +3500 +0 +500 +1000 +1500 +1500 +2000 +2500 +3000 +3500 +Surface Temperature (K) +Equilibrium Temperature (K)A rocky exoplanet classification method +9 +2) and reside within the optimistic CHZ (Figure 3), yet are located +to the left of the liquid water zone in Figure 6b indicating a higher +potential for water molecules to be in ice form. +Figure 7. Analogue adjusted surface pressure and surface temperature of GJ +1061 d, Proxima Cen b, Teegarden’s Star c, TRAPPIST-1 e, TRAPPIST-1 f, +TRAPPIST-1 g, and TRAPPIST-1 h in the context of liquid stability for H2O +(light blue shading), NH3 (green shading), H2S (pink shading), CO2 (dark +blue shading), and CH4 (red shading) adapted from Cengel et al. (2012). The +dashed coloured lines for CH4 (red dashed line), H2S (pink dashed line), and +CO2 (dark blue dashed line), represent the solid-gas boundaries. Median and +68% confidence intervals on analogue adjusted surface pressure and surface +temperature values were calculated using the Monte Carlo simulations. +To determine the types of ices likely present on the surface +of the six Earth analogues located in the water ice zone in Figure +6b, we plot their analogue adjusted surface pressure and surface +temperature values in the context of phase diagrams for H2O, NH3, +H2S, CO2, and CH4 in Figure 7 (Cengel et al. 2012). From Figure +7 it is evident that all of the planets are likely to form H2O ices. The +potential addition of surface water ice on these exoplanets could +increase their albedos, resulting in a corresponding decrease to +their surface temperatures from Equation 20. As more atmospheric +gas is trapped in ice form, this could also result in a decrease of +atmospheric pressure. Additionally, Teegarden’s Star c, TRAPPIST- +1 f, and TRAPPIST-1 g fall within the liquid NH3 phase, indicating +that if ammonia is present on their surfaces it is likely to be in liquid +form. All of these exoplanets will still maintain H2S, CO2, and CH4 +in gas form in their atmospheres. +TRAPPIST-1 h is the only rocky exoplanet in our sample that +resides past the Early Mars CHZ outer boundary in Figure 3, and +is the left-most exoplanet in Figure 6a. Based on its location in +Figures 1 and 2, TRAPPIST-1 h is likely to retain an atmosphere +with H2O present. However, due to its location in Figures 3 and 6a, +TRAPPIST-1 h is likely to be cooler than the Earth analogues, and +there is a higher potential for the molecules to be in ice form. From +Figure 7 it is evident that TRAPPIST-1 h is likely to form H2O, +NH3, H2S, and CO2 ices, while CH4 is likely to remain in gas form. +As evident in Figures 5 and 6, our rocky exoplanet classifi- +cation, when applied to surface pressure and surface temperature +models, allows us to present a more appropriate picture of the cur- +rent rocky exoplanet sample and provides a further layer of context +in the characterisation of exoplanets. +Our model predictions can be tested using two rocky exoplan- +ets for which the upper limits on atmospheric features have been +determined using secondary eclipse and phase variation observa- +tions. First, we compared our results to the Kreidberg et al. (2019) +observations of LHS 3844b. While LHS 3844b’s radius measure- +ment of 1.3R⊕ is technically outside our rocky exoplanet cut-off +range (𝑅𝑝 ≤ 1.23R⊕), when run through our models we find results +consistent with Kreidberg et al. (2019) of a hot rocky planet unable +to retain a substantial atmosphere. Using our classification method, +we categorise LHS 3844b as a “Non-solar system analogue” exo- +planet as it has the potential for H2O to be present, yet high levels +of stellar flux have likely eroded the atmosphere. This is consistent +with the findings from Kreidberg et al. (2019) of a small atmosphere +susceptible to erosion by stellar winds and thus likely being bare- +rock with low bond-albedo. Additionally, we compared our results +to the Crossfield et al. (2022) observations of GJ 1252b. While our +model assumes a fast-rotating exoplanet, unlike the slow-rotating +GJ 1252b, when run through our models, the results agree with +Crossfield et al. (2022) that GJ 1252b is a hot rocky exoplanet with +no significant atmosphere. Using our classification method, we cat- +egorise GJ 1252b as a close-in “Mars-analogue” exoplanet likely +to have a thin water-less atmosphere. Comparing our analogue ad- +justed temperature of 1011 ± 107 K to the Crossfield et al. (2022) +dayside brightness temperature of 1410+91 +−125 K, we fall within 2𝜎 of +their dayside GJ 1252b surface temperature. Despite using different +assumptions, our primary classification model is consistent with +other work using different techniques, giving us confidence in our +approach. +3.3 +Model limitations +The modelling employed in this paper provides a general approx- +imation of atmospheric escape velocity, thermal escape of species +from the atmosphere, and stellar irradiation boundaries for known +exoplanets; however, it is important to acknowledge its simplicity. +The Zahnle & Catling (2017) cosmic shoreline definition as- +sumes an average molecular weight of the atmospheric material +to represent all atmospheric compositions. However, if the atmo- +spheric material is composed of significantly heavier or lighter el- +ements, or if photochemistry which could significantly reduce the +mean molecular mass is considered, then the cosmic shoreline’s +ideal gas assumption may be under- or over-estimating atmospheric +escape rates. Furthermore, the time-frame for atmospheric loss will +differ depending on luminosity and stellar mass for a given planet. +Thermal loss rates may also be influenced by atmospheric compo- +sition, as Parkinson et al. (2022) show that an O-CO2 thermosphere +can cool efficiently. Atmospheric species can be lost through several +additional non-thermal processes for example ion pickup (Lammer +et al. 2006), sputtering (Terada et al. 2009), dissociation and disso- +ciative recombination (Geppert & Larsson 2008), photo-chemical +energising mechanisms (Vidotto 2013), and charge exchange (Dong +et al. 2017). Conversely, these atmospheric species could be gained +through volcanic degassing (Oosterloo et al. 2021) or impact events +that liberate gas from the surface (Kuwahara & Sugita 2015). Fur- +thermore, depending on the stellar wind pressures, the presence of +a significant magnetic field could reduce non-thermal atmospheric +erosion (McIntyre et al. 2019; Egan et al. 2019). Future research +should be conducted to investigate which atmospheric mass-loss +processes dominate in different planetary scenarios and the degree +to which they are inhibited by potential planetary magnetism. +Planetary rotation is an essential factor that could affect the +modelling employed here; however, this parameter is not yet able to +be directly observed for a broad sample of exoplanets. Yang & Ab- +bot (2014) demonstrate the dependence of the inner CHZ boundary +MNRAS 000, 1–13 (2023) + +101 +(Bars) +GJ/106ld +surface Pressure ( +Teegarden's +Star c +tProxima Cen b +TRAPPIST:1g+ +Earth +TRAPPIST-1f +TRAPPIST-1e +TRAPPIST-1h+ +H20 liquid +S +NH3 liquid +H2S liquid +CO2 liquid +CH4 liquid +10- +50 +100 +150 +200 +250 +300 +350 +400 +450 +Surface Temperature (K)10 +S.R.N McIntyre et al. +on planetary rotation. Strongly irradiated, rapidly rotating exoplan- +ets could lose water and develop low albedos, entering a runaway +greenhouse even while residing within the CHZ (Del Genio et al. +2019b; Kopparapu et al. 2016). Furthermore, species in the atmo- +spheres of a rapidly rotating exoplanet could reach high velocities +due to the increased temperature, which would accelerate the atmo- +spheric escape process (Konatham et al. 2020). While we can infer +the rotation period for tidally locked exoplanets as equivalent to their +orbital period, such planets could escape synchronous rotation by +being captured in spin-orbit resonances (Goldreich & Soter 1966; +Makarov et al. 2012; Rodríguez et al. 2012) or through resonant +planet-planet interactions with their exterior planetary companions +(Delisle et al. 2017; Vinson & Hansen 2017; Zanazzi & Triaud +2019). Additionally, there is no way to constrain the rotation pe- +riod without direct observations of exoplanets past the tidal locking +radius. In the future, photometric variability techniques have been +postulated to facilitate measurements of an exoplanet’s rotation (Fu- +jii & Kawahara 2012; Snellen et al. 2014). +Here, we only focus on the impact of stellar flux on the run- +away greenhouse boundary, as defined by Kopparapu et al. (2014). +However, there are additional ways surface temperature conditions +could increase to levels resulting in a runaway greenhouse, such +as tidal heating (Barnes et al. 2013; McIntyre 2022), especially +for planets in eccentric orbits (Williams & Pollard 2002; Kane & +Gelino 2012), or sufficiently increased CO2 levels that could drive +an atmosphere into a runaway greenhouse at further distances from +the host star (Kane et al. 2014). Nonetheless, the likelihood of a +runaway greenhouse will decrease dramatically as the distance past +the current VZ increases. +The surface temperature estimate utilised here only accounts +for a broad greenhouse effect and not specific greenhouse gas abun- +dances. For synchronously rotating exoplanets, future observations +of thermal phase curves could help further quantify the difference +between surface temperature and equilibrium temperature and pro- +vide a more accurate picture of an exoplanet’s atmospheric green- +house effect (Del Genio et al. 2019a; Lacis et al. 2010; Yang & Ab- +bot 2014). It would also be useful to measure exoplanets’ obliquity, +as this can result in high to low temperatures distributed from the +equator to the poles (Nowajewski et al. 2018). For example, Wang +et al. (2016) suggests that with higher obliquities, the habitability +around M dwarfs narrows. Obliquity is currently unobservable for +exoplanets, although it could be extrapolated when seasonal cycle +information on reflected starlight becomes available (Kane & Torres +2017). Furthermore, Ahlers (2016) determine that exoplanets with +inclined orbits around fast-rotating stars display changes in equilib- +rium temperature of up to 15% due to the increased irradiance near +the stellar poles. +Through these simplified models, we are attempting to use the +limited observational data on the characteristics of rocky exoplanets +we currently have available. In the future, additional observations of +obliquity, inclination, planetary rotation rates, stellar rotation rates, +and thermal phase curves could help strengthen the classification +method detailed here and help determine optimum targets for future +atmospheric observations of rocky exoplanets. +4 +CONCLUSIONS +Here, we use non-thermal atmospheric escape, thermal atmospheric +escape, and stellar irradiation boundaries to develop a primary clas- +sification method for current rocky exoplanets (𝑅𝑝 ≤ 1.23R⊕) and +group them into categories relative to the most appropriate solar sys- +tem analogue. When applying this primary classification method to +the 720 rocky exoplanets in our sample, results suggest that 22% ± +8% are Mercury analogues, 39% ± 4% are Mars analogues, 11% ± +1% are Venus analogues, 2% ± 1% are Earth analogues, and 26% ± +12% are without a known planetary counterpart in our solar system. +Implementing this classification method will help further char- +acterise the detected exoplanets by comparing them to a more ap- +propriate solar system analogue, rather than continuing with the +common approach where we estimate the values for numerous un- +known parameters by normalising to Earth. To demonstrate the +functionality of this classification method, we compare it to a +simple model for surface pressure. Using Earth-centric measure- +ments the calculated surface pressure range for rocky exoplan- +ets (0.3𝑅⊕ ≤ 𝑅𝑝 ≤ 1.23𝑅⊕) with the simple model spanned +0.02 − 1.95 bars. After applying the new primary classification +method to the rocky exoplanet sample, the surface pressure range +now expands to 2×10−15−210 bars, accounting for the full variation +observed in our solar system. Furthermore, our new rocky exoplanet +classification method, when applied to calculating surface pressure +and surface temperature, allows us to present a more varied picture +of the current rocky exoplanet sample and provides a further layer +of context in the characterisation of exoplanets; for example, the +presence of liquid, gas or ice. +The use of our new primary classification method could im- +prove inferences of temperature, composition, interior structure, +evolution and dynamics of rocky exoplanets, which would aid our +ability to interpret and model exoplanet atmospheres (Kane et al. +2019). Additionally, this classification method could benefit tar- +get selection for exoplanet characterisation missions by providing a +more robust starting point for potential atmospheric properties and +composition for comparison to future observations. +ACKNOWLEDGEMENTS +S.R.N.McIntyre gratefully acknowledges an Australian Government +Research Training Program (RTP) Scholarship. P.L. King acknowl- +edges funding from an Australian Research Council Discovery Pro- +gram grant (DP200100406). Helpful comments from an anonymous +referee are gratefully acknowledged. +DATA AVAILABILITY +The data underlying this article are available in the article and +in its online supplementary material which can be downloaded +in electronic form from the Centre de Données astronomiques de +Strasbourg (CDS) service via anonymous ftp cdsarc.u-strasbg. +fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/ +viz-bin/cat/J/MNRAS. +REFERENCES +Ahlers J. 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The rest of the table can be downloaded in +electronic form from CDS service and the publisher’s website. +Planet Name +M𝑝 (M⊕) +R𝑝 (R⊕) +v𝑒𝑠𝑐 (ms−1) +S∗ (S⊕) +T𝑒𝑞 (K) +Original P𝑠𝑢𝑟 𝑓 (Bar) +Adjusted P𝑠𝑢𝑟 𝑓 (Bar) +Adjusted T𝑠𝑢𝑟 𝑓 (K) +Planet Classification +TRAPPIST-1 b +1.37 ± 0.07 +1.12 ± 0.01 +12411.61 ± 322.28 +4.15 ± 0.16 +397.26 ± 3.82 +1.23 ± 0.14 +136.63 ± 08.53 +785.66 ± 2.65 +Venus analogue +TRAPPIST-1 c +1.31 ± 0.06 +1.10 ± 0.01 +12214.27 ± 270.09 +2.21 ± 0.09 +339.36 ± 3.44 +1.20 ± 0.12 +132.63 ± 08.43 +745.61 ± 2.38 +Venus analogue +TRAPPIST-1 d +0.39 ± 0.01 +0.79 ± 0.01 +7849.09 ± 131.55 +1.11 ± 0.04 +285.69 ± 2.60 +0.40 ± 0.03 +0.40 ± 0.03 +295.07 ± 2.35 +Earth analogue +TRAPPIST-1 e +0.69 ± 0.02 +0.92 ± 0.01 +9701.21 ± 166.83 +0.65 ± 0.03 +249.91 ± 2.86 +0.68 ± 0.06 +0.68 ± 0.06 +262.36 ± 2.63 +Earth analogue +TRAPPIST-1 f +1.04 ± 0.03 +1.05 ± 0.01 +11153.63 ± 180.93 +0.37 ± 0.01 +217.08 ± 1.46 +0.92 ± 0.07 +0.92 ± 0.07 +232.34 ± 1.35 +Earth analogue +TRAPPIST-1 g +1.32 ± 0.04 +1.13 ± 0.01 +12099.61 ± 186.13 +0.25 ± 0.01 +196.81 ± 1.99 +1.09 ± 0.08 +1.09 ± 0.08 +213.80 ± 1.83 +Earth analogue +TRAPPIST-1 h +0.33 ± 0.02 +0.76 ± 0.01 +7350.25 ± 234.95 +0.14 ± 0.01 +170.25 ± 3.03 +0.33 ± 0.05 +No solar system analogue +No solar system analogue +No solar system analogue +MNRAS 000, 1–13 (2023) + diff --git a/F9E1T4oBgHgl3EQfqwX2/content/tmp_files/load_file.txt b/F9E1T4oBgHgl3EQfqwX2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2276b1bc7c5b51dfdc3cc334fceff38e8a90928e --- /dev/null +++ b/F9E1T4oBgHgl3EQfqwX2/content/tmp_files/load_file.txt @@ -0,0 +1,1251 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf,len=1250 +page_content='MNRAS 000, 1–13 (2023) Preprint 10 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='0 A rocky exoplanet classification method and its application to calculating surface pressure and surface temperature Sarah R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' McIntyre,1,2★ Penelope L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' King,2 and Franklin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Mills3,4 1Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia 2Research School of Earth Sciences, Australian National University, Canberra, ACT 2601, Australia 3Fenner School of Environment and Society, Australian National University, Canberra, ACT 2601, Australia 4Space Science Institute, Boulder, CO 80301, USA Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' in original form ZZZ ABSTRACT With over 5,000 exoplanets currently detected, there is a need for a primary classification method to prioritise candidates for biosignature observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Here, we develop a classification method to categorise rocky exoplanets based on their closest solar system analogue using available data of observed stellar and planetary features, masses, and radii, to model non- thermal atmospheric escape, thermal atmospheric escape, and stellar irradiation boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Applying this classification method to the 720 rocky exoplanets in our sample with uncertainties in planetary masses, radii, stellar temperatures, and fluxes propagated via a Monte Carlo model indicates that 22% ± 8% are Mercury analogues, 39% ± 4% are Mars analogues, 11% ± 1% are Venus analogues, 2% ± 1% are Earth analogues, and 26% ± 12% are without a known planetary counterpart in our solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Extrapolating to conditions on LHS 3844b and GJ 1252b, our classification method gives results reasonably consistent with current observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Subsequently, to demonstrate the functionality of this classification method, we plot our catalogued sample of exoplanets on an adjusted surface pressure versus temperature phase diagram, presenting more realistic estimates of the potential surface phases (gas, liquid or ice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Our new classification method could help target selection for future exoplanet characterisation missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Key words: planets and satellites: terrestrial planets - planets and satellites: surfaces - cata- logues 1 INTRODUCTION Over the past 28 years, astronomers have observed over 5,0001 ex- trasolar planets, providing us with basic information regarding their orbits, masses, and radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Research on exoplanets is currently fo- cused on determining which of these worlds may be habitable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' for example, rocky bodies (with sufficient gravity to support an atmo- sphere) orbiting their host star at a distance where stellar insolation flux is suitable for the existence of liquid water on their surface (Kaltenegger 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Due to the inability to conduct in-situ explo- ration, near-term studies to further characterise exoplanets will focus on remote detection of their atmospheres and spectral observations of possible biosignatures (Schwieterman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, the significant observing time required to characterise rocky exoplanets limits the number of targets where we can conduct such extensive observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Defining how potential atmospheric biosignatures vary under ★ E-mail: sarah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='mcintyre@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='au 1 https://exoplanetarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='edu/ (Accessed 18 December 2022) different conditions is important when characterising exoplanets (Schwieterman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Two significant parameters when con- sidering the climatic conditions of an exoplanet are surface pressure and surface temperature (Keles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Water’s stability on a planetary surface as a liquid depends on both the surface tempera- ture and pressure (Seager 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' While the freezing point of liquid water is not strongly dependent on surface pressure, the boiling point is significantly affected by it (Vladilo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, at surface pressures below the triple point, liquid water is not stable at any temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Thus, reliable estimates of the surface pressure and temperature are essential for characterising the environment and habitability of an exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Research suggests that a rise in surface pressure could lead to a rise in temperature due to the greenhouse effect (Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, a high surface pressure can enhance cooling through increased Rayleigh scattering (Kast- ing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Keles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2018), Mie scattering (Kitzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2010), or reflection due to clouds (Marley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Additionally, exoplanet general circulation model (GDM) simulations and solar system observations suggest that high surface pressures could in- crease the latitudinal heat transport, cancelling seasonal variations in the planet’s surface temperature and resulting in smaller global © 2023 RAS arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='03348v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='EP] 9 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='N McIntyre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' temperature variations (Bullock & Grinspoon 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Leovy 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Trenberth & Caron 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Vladilo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Despite the importance of surface pressure, current proposed methods for its measurement, using remote-sensing techniques, are challenging and may not fall within the wavelength cut-off for the James Webb Space Telescope (Chamberlain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Crow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kasting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Misra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Three-dimensional general circulation models (3D GCMs) are be- ginning to provide insights into exoplanet atmospheres (Boutle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Del Genio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Galuzzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Lewis & Ham- mond 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Turbet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2016, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Way et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Wolf 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' While there are a significant number of 3D GCM exoplanet simu- lations published given the lack of observational data so far, such models are computationally intensive and not generally accessible by the broader scientific community, limiting the number of sim- ulations conducted to date (Del Genio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, many 3D GCM simulations of rocky exoplanets model “Earth-like” atmospheres, assuming ∼1 bar of N2 as the predominant component of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' An initial estimate of an exoplanet’s surface pressure (𝑃𝑠𝑢𝑟 𝑓 ) can be obtained from a simple model based on hydrostatic equi- librium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Hall 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kippenhahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2017), using available observational data on an exoplanet’s mass (𝑀𝑝) and radius (𝑅𝑝): 𝑃𝑠𝑢𝑟 𝑓 𝑃⊕ = � 𝑀𝑝 𝑀⊕ �2� 𝑅⊕ 𝑅𝑝 �4 (1) where 𝑃⊕, 𝑀⊕ and 𝑅⊕ are the surface pressure, mass, and ra- dius of Earth, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' For many exoplanets, the radius or the mass is unknown, resulting in the publication of several mass-radius relations dependent on the planet’s type, allowing us to calculate the missing measurement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Chen & Kipping 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Nikouravan 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Otegi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Seager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Swift et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Turbet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Weiss & Marcy 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Here, we follow the NASA exoplanet database2 and use the Chen & Kipping (2016) M-R relationship to fill in the missing parameter: 𝑅𝑝 ∼ 𝑀𝑝0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='279±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='009 (2) Using the relation from Equation 2, the surface pressure in Equation 1 can be written as: 𝑃𝑠𝑢𝑟 𝑓 𝑃⊕ = � 𝑅𝑝 𝑅⊕ �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='168±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='232 (3) This Earth normalisation significantly limits the range of pos- sible surface pressure values (Equations 4 and 5), as evident when calculating the maximum and minimum 𝑃𝑠𝑢𝑟 𝑓 using Equation 3 and taking the upper radius limit for rocky exoplanets as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23𝑅⊕, from the Chen & Kipping (2016) definition of the boundary be- tween terrestrial and Jovian planets, and a lower radius limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3𝑅⊕, which corresponds to the size of the smallest exoplanet dis- covered around a main-sequence star – Kepler 37b (Haghighipour 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 𝑃𝑠𝑢𝑟 𝑓 𝑚𝑎𝑥 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='014(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='168 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='95𝑏𝑎𝑟 (4) 𝑃𝑠𝑢𝑟 𝑓 𝑚𝑖𝑛 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='014(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='168 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='02𝑏𝑎𝑟 (5) 2 https://exoplanetarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='edu/ (Accessed 18 December 2022) The benefit of Equation 1 is that it only requires an exoplanet’s radius or mass value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, we obtain an Earth-centric estimate within the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='02 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='95 bar range because we use Earth’s surface pressure and radius in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Our solar system contains four rocky planets, each with a unique surface, atmosphere, struc- ture, and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Extrasolar rocky planets will likely display a similarly wide variety of surface characteristics and interior com- positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' While the radius range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23𝑅⊕ covers all four rocky planets in the solar system (Chen & Kipping 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Haghigh- ipour 2015), the resulting surface pressure range only encompasses Earth’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' This simple model excludes the 5×10−15 − 92 bar range of surface pressure measurements observed on Mercury, Mars, and Venus (Rasool et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Seiff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Alternative methods, such as comparing to a more appropriate solar system analogue, should be considered rather than contin- uing with the current approach of assigning Earth-like character- istics to all rocky exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' A similar approach has been used for modelling the atmospheric chemistry and climate of Venus-like exoplanets (Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Schaefer & Fegley 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Way & Del Genio 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, the scientific community has been studying, observing, and probing the planets of the solar system with multiple satellites, in situ missions, and remote sensing ob- servations using ground- and space-based telescopes (Gröller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Jakosky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Marcq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' McClintock & Lank- ton 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' McNutt Jr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Mills et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Von Zahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Withers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Thus, we have significant knowledge of the solar system planets’ atmospheric profiles and compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' We have values for the surface pressures of Mercury, Mars, and Venus and could use these planets’ respective pressure and radii values to normalise Equation 1 more appropriately, provided we can classify which exoplanets are likely to be Mercury, Mars, or Venus analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Previous classification schemes from Forget & Leconte (2014) and Wordsworth & Kreidberg (2022) suggest that climates on ter- restrial exoplanets should depend primarily on atmospheric com- position, incident stellar flux, and tidal evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Here, we develop a classification method using available data of observed stellar and planetary features, masses, and radii, to model non-thermal atmo- spheric escape, thermal atmospheric escape, and stellar irradiation boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' We compute the escape velocities for the known rocky exoplanets and compare them to the insolation and thermal velocity of likely gases to determine whether the exoplanets can maintain their atmospheres and, if so, which gases are retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, we quantify planetary temperature conditions based on the incident stellar fluxes to determine the likelihood of rocky exoplanets resid- ing in a temperate or runaway greenhouse zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' We utilise these factors to classify the current list of rocky exoplanets and group them into categories based on similarity to the most appropriate so- lar system analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Subsequently, to demonstrate the functionality of this classification method, we combine it with an extension to the simple surface pressure model (Equation 3), plot the adjusted surface pressure vs temperature phase diagram, and discuss the im- plications for the habitability of exoplanets and the optimisation of target selection for future atmospheric observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2 METHOD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='1 Non-thermal atmospheric escape An exoplanet must have an atmosphere to have a significant surface pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Thus, an important feature in determining surface pressure MNRAS 000, 1–13 (2023) A rocky exoplanet classification method 3 is the escape velocity of an exoplanet, which can be calculated as: 𝑉𝑒𝑠𝑐 = √︄ 2𝐺𝑀𝑝 𝑅𝑝 (6) where 𝐺 is the universal gravitational constant 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='6743 × 10−11𝑚3𝑘𝑔−1𝑠−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The escape itself is a rapid process, unlikely to be directly observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' According to Zahnle & Catling (2017), the cumulative impact of escape should be evident in the statis- tical analysis of exoplanets, with a division between planets with and without atmospheres, that they define as the “cosmic shoreline” (Catling & Zahnle 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Zahnle & Catling 2013, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The solar system planets are neatly divided around the cosmic shoreline: 𝑆∗ = 5 × 10−16𝑉4 𝑒𝑠𝑐 (7) where 𝑆∗ is insolation and 𝑣𝑒𝑠𝑐 is the escape velocity as defined in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The cosmic shoreline is a simple approximation of non- thermal atmospheric escape based on observable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' While there are additional non-thermal processes that could increase the amount of atmospheric mass-loss, for example ion pickup (Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2006), sputtering (Terada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2009), dissociation and disso- ciative recombination (Geppert & Larsson 2008), photo-chemical energising mechanisms (Vidotto 2013), and charge exchange (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' these models require supplementary information for which we currently have no observed values and no clear method to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='2 Thermal atmospheric escape The thermal escape rate is a function of the planet’s escape velocity and the temperature of the exobase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' There are two types of ther- mal escape in atmospheres: hydrodynamic escape and slow thermal escape (Jeans escape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Highly irradiated large-mass exoplanets are more likely to lose atmospheric mass through hydrodynamic blow- off (Owen 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, Konatham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020) suggest that rocky exoplanets are more inclined to undergo slow thermal escape caused by atmospheric species’ thermal velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Using the basic principles of the kinetic theory of gases, we follow Konatham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020) to predict probable atmospheric compositions of exoplanets by identifying the atmospheric species that can leave their atmo- spheres via slow thermal escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' According to the Konatham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020) model, the thermal escape rate is defined by the thermal velocity of the atmospheric species: 𝑈 = √︂ 3𝑘𝑏𝑇 𝑚 (8) where 𝑚 is the mass of a gas, 𝑘𝑏 is Boltzmann’s constant, and 𝑇 is the exobase temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' As an observable value for 𝑇 is currently unavailable, we follow the Konatham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020) approach for a fast rotating exoplanet and utilise its equilibrium temperature, 𝑇𝑒𝑞 with albedo A = 0, as a conservative approach for estimating the slow thermal escape of species from the atmosphere: 𝑇𝑒𝑞 = � (1 − 𝐴) 𝑆∗ 4𝜎 � 1 4 (9) where 𝜎 is the Stefan–Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Konatham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020) infer results for rocky exoplanets experiencing slow thermal escape using data from observations of gases escaping from solar system planets’ atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Equation 10 relates𝑈 to 𝑣𝑒𝑠𝑐 for atmospheric species to escape an exoplanet’s atmosphere: 𝑈 > 1 10𝑣𝑒𝑠𝑐 (10) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3 Circumstellar Habitable Zone The search for potential Earth analogues begins by examining the Circumstellar Habitable Zone (CHZ), defined as the region in which a rocky planet, with favourable atmospheric conditions, can sustain liquid water on its surface (Kasting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Selsis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2013, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2013, 2014) esti- mate the CHZ around stars with stellar effective temperatures (𝑇∗) in the range of 2600–7200 K, for planetary masses between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='1- 5M⊕, and assuming H2O (inner boundary) and CO2 (outer bound- ary) dominated atmospheres, with N2 as the background gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' This model quantifies incident stellar radiation fluxes that would result in planetary temperature conditions shifting to a runaway snowball or a runaway greenhouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Here, we use the Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2014) optimistic definition of the CHZ “early Mars” outer (Equation 11) and “recent Venus” inner (Equation 12) boundaries: 𝑆𝐸𝑀 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='32 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='547 × 10−5(𝑇∗ − 5780) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='526 × 10−9(𝑇∗ − 5780)2 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='874 × 10−12(𝑇∗ − 5780)3 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='011 × 10−16(𝑇∗ − 5780)4 (11) 𝑆𝑅𝑉 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='776 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='136 × 10−4(𝑇∗ − 5780) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='533 × 10−8(𝑇∗ − 5780)2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='332 × 10−11(𝑇∗ − 5780)3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='097 × 10−15(𝑇∗ − 5780)4 (12) where 𝑆𝐸𝑀 is the stellar flux required for the early Mars outer CHZ boundary, and 𝑆𝑅𝑉 is the stellar flux required for the recent Venus inner CHZ boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' These CHZ boundaries define the exoplanets in our sample that most closely resemble Earth and are therefore likely to have liquid water on their surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='4 Venus Zone There is a clear distinction in atmospheric evolution between Earth and Venus, probably due to the significant difference in solar irra- diance (approximately a factor of two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2014) define a “Venus Zone” (VZ) where a planet is considered to be a Venus analogue rather than an Earth analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' We use the optimistic CHZ “recent Venus” boundary from Equation 12 to define the runaway greenhouse outer VZ boundary, where oceans completely evaporate, resulting in the inability to execute a carbon cycle and efficiently moderate atmospheric CO2 levels, leading to the formation of a thick Venus-like atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' As distance from the host star decreases, the likelihood of substantial atmospheric mass loss increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2014) determine the insolation flux required for Venus to cross the Zahnle & Catling (2017) cosmic shoreline and use this value to denote the complete atmospheric erosion of Venus analogues (𝑆𝐴𝐸) as an approximation for the VZ inner boundary: 𝑆𝐴𝐸 ≈ 25𝑆⊕ (13) Just as planets in the CHZ could be considered Earth analogues until more spectroscopic information becomes available,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' planets inside the VZ could be considered Venus analogues until further characterisation observations are undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='5 Surface pressure After cataloguing our sample of rocky exoplanets, we adjust pa- rameters such as surface pressure to more appropriately normalise simple calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Taking the solar system planet values for sur- face pressure (𝑃𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡) and radius (𝑅𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡) into account, we adjust Equation 3 to: 𝑃𝑆𝑆−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 𝑃𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡 � 𝑅𝑝 𝑅𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡 �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='168 (14) MNRAS 000, 1–13 (2023) 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='N McIntyre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 𝑃𝑀𝑒𝑟𝑐𝑢𝑟 𝑦−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='05 × 10−13𝑅𝑝3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='168 (15) 𝑃𝑀 𝑎𝑟𝑠−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='0467𝑅𝑝3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='168 (16) 𝑃𝑉 𝑒𝑛𝑢𝑠−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='27𝑅𝑝3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='168 (17) 𝑃𝐸𝑎𝑟𝑡ℎ−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='014𝑅𝑝3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='168 (18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='6 Surface temperature After calculating an adjusted 𝑃𝑠𝑢𝑟 𝑓 by normalising to the most appropriate solar system analogue, we apply our rocky exoplanet classification system to surface temperature estimates to plot an analogue adjusted surface pressure vs surface temperature phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Unfortunately, like surface pressure, direct measurements of surface temperature are not typically available (Weisfeiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2015) and when modelling using 3D GCMs, Earth-centric assump- tions of atmospheric composition or convection depth are frequently made (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Biserud 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Chaverot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Way et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Yang & Abbot 2014), creating a bias in surface temperature mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Del Genio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2019b) correlate surface temperature (𝑇𝑠𝑢𝑟 𝑓 ) and equilibrium temperature (𝑇𝑒𝑞), by attributing the difference between the two to a greenhouse effect: 𝑇𝑠𝑢𝑟 𝑓 = 𝑇𝑒𝑞 + 𝐺𝑎 (19) where 𝐺𝑎 is the atmospheric greenhouse effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Substituting Equa- tion 9 into Equation 19, we attain a surface temperature equation: 𝑇𝑠𝑢𝑟 𝑓 = � (1 − 𝐴) 𝑆∗ 4𝜎 � 1 4 + 𝐺𝑎 (20) Substituting observed values of the rocky solar system planets’ bond albedo, insolation flux, and average surface temperature into Equation 20 allows us to calculate the atmospheric greenhouse effect for our solar system analogues (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Solar System analogue surface temperature calculation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Planet 𝑇𝑠𝑢𝑟 𝑓 (K) 𝐴 𝑆∗ (Wm−2) 𝐺𝑎 (K) Mercury 440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='07 9082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='85 Mars 210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='25 586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='15 Venus 737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='77 2601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='85 Earth 288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3 1361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='85 While we do not take the time evolution factor into account in this paper, it should be noted that the values in Table 1 have changed over the lifetime of the solar system and similar variations are expected across the lifetime of exoplanet systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' This surface temperature model only accounts for a broad greenhouse effect and not specific greenhouse gas abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, it enables us to apply solar system planet values for bond albedo (𝐴𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡) and atmospheric greenhouse (𝐺𝑎𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡) from Table 1 and refine our surface temperature calculations by adjusting Equation 20 to: 𝑇𝑆𝑆−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = � �1 − 𝐴𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡 � 𝑆∗ 4𝜎 � 1 4 + 𝐺𝑎𝑆𝑆−𝑝𝑙𝑎𝑛𝑒𝑡 (21) 𝑇𝑀𝑒𝑟𝑐𝑢𝑟 𝑦−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='0𝑆∗ 1 4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='85 (22) 𝑇𝑀 𝑎𝑟𝑠−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='6𝑆∗ 1 4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='15 (23) 𝑇𝑉 𝑒𝑛𝑢𝑠−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='7𝑆∗ 1 4 + 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='85 (24) 𝑇𝐸𝑎𝑟𝑡ℎ−𝑎𝑛𝑎𝑙𝑜𝑔𝑢𝑒 = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='9𝑆∗ 1 4 + 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='85 (25) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='7 Sample selection and Monte Carlo calculations We utilise NASA’s composite planet database3, in combination with additional information on the Kepler planets’ radii provided by Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020b) and updated stellar properties by Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020a) to compose a catalogue of rocky exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, we use the Chen & Kipping (2016) M-R relationship detailed in Equation 2 to calculate unknown radii or masses and their uncer- tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' To ensure that we have included only rocky exoplanets in our database, we follow the Chen & Kipping (2016) definition of the boundary between Jovian and terrestrial planets, limiting our selec- tion to exoplanets with radii 𝑅𝑝 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23 R⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' As we need information for stellar temperature, insolation flux, planetary mass, and planetary radius as provided by the NASA composite planet database, Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020b), and Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020a), our total sample contains 720 rocky exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' For each exoplanet in our sample, we execute 10,000 Monte Carlo simu- lations using a Gaussian probability distribution for uncertainties on the stellar temperatures, insolation fluxes, planetary masses, and planetary radii as provided by the NASA composite planet database, Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020b), and Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, for the subset of exoplanets where the mass-radius relation was used, Equation 2 was input directly into the Monte Carlo simulation to ensure the uncertainties were appropriately correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The Monte Carlo simulations allow us to determine the median and 68% confidence intervals on escape velocity (𝑣𝑒𝑠𝑐), equilibrium temperature (𝑇𝑒𝑞), surface temperature (𝑇𝑠𝑢𝑟 𝑓 ), and surface pres- sure (𝑃𝑠𝑢𝑟 𝑓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 3 DATA ANALYSIS & DISCUSSION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='1 Exoplanet classification In Figure 1, we plot stellar insolation flux (𝑆∗) against the escape velocity (𝑣𝑒𝑠𝑐) for the exoplanets in our sample as well as the four rocky solar system planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The trend seen in Figure 1 is what we would expect to see if escape were the primary factor influencing the volatile inventories of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' With the aid of the Zahnle 3 https://exoplanetarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='edu/cgi-bin/ TblView/nph-tblView?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='app=ExoTbls&config=compositepars (accessed 18 December 2022) MNRAS 000, 1–13 (2023) A rocky exoplanet classification method 5 & Catling (2017) cosmic shoreline (Equation 7), we can see that planetary atmospheres are thick when the influence of the central star is weak (measured by insolation) or the gravitational well is deep (measured by escape velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' On the other hand, we have planets with thin or no atmospheres when the star is too bright or gravity is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Insolation and escape velocity for the 720 exoplanets in our sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets likely to have thin or no atmosphere are plotted with open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets likely to have an atmosphere are solid dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Labelled solid coloured circles represent observed values for rocky solar system planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The solid black line represents the Zahnle & Catling (2017) cosmic shoreline (Equation 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The dashed vertical line indicates the Chen & Kipping (2016) cut-off for rocky exoplanets, 𝑅𝑝 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23 R⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Median and 68% confidence intervals on escape velocity values were calculated using the Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' From our sample of 720 rocky exoplanets, 12% ± 4% reside below the cosmic shoreline and are consequently likely to maintain a significant atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Conversely, 88% ± 5% of the exoplan- ets in our sample reside above the cosmic shoreline and are likely to have thin or negligible atmospheres, based on the rapid escape velocity parameter alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The uncertainty values arise from taking into account the 68% confidence intervals on insolation and es- cape velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Taking the upper insolation error and lower escape velocity error into account, 4% of exoplanets located in the area where rocky planets are likely to maintain a significant atmosphere, cross the cosmic shoreline and thus could potentially have thin or no atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Conversely, taking the lower insolation error and upper escape velocity error into account, 5% of exoplanets located in the area where rocky planets have a thin or negligible atmo- sphere, cross the cosmic shoreline and thus could potentially have a substantial atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, Figure 1 indicates that most rocky exoplanets lie above the cosmic shoreline, where gravity is weak and stellar flux strong, implying that our current observational exoplanet data has a bias towards close-orbiting rocky planets with limited atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' All rocky exoplanets discovered thus far are on close, highly irradiated orbits, bombarded by large amounts of ionising EUV and X-ray radiation (Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The upper atmosphere of an exoplanet also may be exposed to coronal mass ejections and stellar winds, inducing additional non-thermal loss processes of ion pickup (Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2006), sputtering (Terada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2009), disso- ciation and dissociative recombination (Geppert & Larsson 2008), photo-chemical energising mechanisms (Vidotto 2013), and charge exchange (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' These non-thermal escape mecha- nisms could shift the cosmic shoreline to lower insolation at a given escape velocity and are particularly important in the early phases of the host star’s evolution, when XUV flux and CME rates may be orders of magnitude higher (Do Amaral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Gronoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, models for these non-thermal processes require additional information for which we currently have no observed values, so it is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Alternatively, de- pending on the stellar wind pressures, a significant magnetic field could mitigate the non-thermal escape processes, resulting in a shift of the cosmic shoreline to higher escape velocities (McIntyre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' There are two types of thermal escape in atmospheres;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' hy- drodynamic escape and slow thermal escape (Jeans escape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Most models have been designed for hydrodynamic conditions primarily for small semi-major axis, high-mass, highly irradiated exoplanets (Owen 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Tian 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Madhusudhan 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Conversely, low- irradiated, small-mass rocky exoplanets are more likely to expe- rience slow thermal escape driven by thermal velocities of atmo- spheric species when assuming that equilibrium temperature is a good guide to thermospheric temperature (Konatham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Escape velocity (Equation 6) versus equilibrium temperature (Equation 9) of our sample of 720 rocky exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that re- side above the cosmic shoreline in Figure 1 and are likely to have thin or no atmosphere are plotted with open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside below the cosmic shoreline in Figure 1 and are likely to have an atmosphere are plotted with solid dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Labelled solid coloured circles represent calculated values for rocky solar system planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Solid black lines represent the thermal velocity of atmospheric gas species (defined by Equations 8-10) (Konatham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2020), where exoplanet atmospheres may retain the gas at lower escape velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Red shaded region denotes Mercury analogue exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Orange shaded region denotes Mars analogues exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The dashed horizontal line indicates the cut-off for rocky exoplanets as 𝑅𝑝 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23 R⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Median and 68% confidence intervals on escape velocity and equilibrium temperature values were calculated using the Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' In Figure 2, we compute the thermal velocities of selected gases for our sample of rocky exoplanets (parameterised by the equilibrium temperature from Equation 9) and compare them with the escape velocity to determine which gases could be preserved in the planets’ atmospheres, in accordance with Equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The diagonal lines depict the thermal velocity of various atmospheric species as a function of kinetic temperature (also known as veloc- MNRAS 000, 1–13 (2023) 105 Rocky exoplanets with thin 0 or negligible atmospheres 104 Rocky exoplanets with atmospheres 103 0 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' S 8 8 Insolation 0 0 101 0 Mercury Venus 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Earth Mars 10-1 IZ IN 31 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Cosmic shoreline R: 1 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 104 103 Escape Velocity (ms-1)H oad H2 8 0 096 Earth 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 8 0 Venus 0 He Escape Velocity (ms- 0 080 0 0 00 C 0 0 0 08 00 00 Mars 0 8 0 O,CH4 0 Mercury 0 H20,NH3 0 02 CO2 502 Rocky exoplanets with thin 0 Xe or negligible atmospheres Rocky exoplanets with atmospheres 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 103 102 Eguilibrium Temperature (K)6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='N McIntyre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' ity lines) collated from Konatham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' An exoplanet can retain a specific atmospheric species if its velocity line is below the position of the exoplanet in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Conversely, an atmospheric species escapes the exoplanet’s atmosphere if its velocity line is above the exoplanet’s position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' This allows us to utilise the kinetic theory of gases to estimate potential atmospheric constituents for our sample of rocky exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' In Figure 2, we can see that 22% ± 8% of rocky exoplanets in our sample lie below the CO2 line in the red shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' All the exoplanets in this subset were also located substantially above the cosmic shoreline (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Based on their positioning in Figures 1 and 2, these exoplanets are unlikely to be able to sustain a signifi- cant atmosphere and thus will have limited surface pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' When comparing to the four rocky solar system analogues, we can see that these exoplanets most closely resemble Mercury with its negligi- ble atmosphere (surface pressure ∼5 picobars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Consequently, this subset of exoplanets could be classified as Mercury analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The next group of exoplanets are those located above the CO2 line and below the H2O line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The 39% ± 4% of rocky exoplan- ets from our sample residing within the orange shaded region in Figure 2 also lie above the cosmic shoreline in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Based on their position in Figures 1 and 2, these exoplanets are likely to have thin water-less atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' When comparing to the four rocky solar system analogues, we can see that the atmospheres of these exoplanets most closely resemble present-day Mars with its thin, CO2-rich, dry atmosphere (surface pressure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='00636 bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Consequently, this subset of exoplanets could be classified as Mars analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Location within the CHZ and VZ of 279 rocky exoplanets that reside above the H2O line in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The blue shaded region denotes the CHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The purple shaded region denotes the VZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside above the cosmic shoreline in Figure 1 and are likely to have thin or no atmosphere are plotted with open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside below the cosmic shoreline in Figure 1 and are likely to have an atmosphere are plotted with solid dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Trappist-1h, the exoplanet shown beyond the Early Mars boundary, is discussed further in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='2 The final 39% ± 12% of rocky exoplanets presented in Figure 2 reside above the H2O line and are likely to have water in their atmo- spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' When comparing to the four rocky solar system analogues, both Venus and Earth fall within similar locations in Figures 1 and 2 yet have substantially different atmospheres with surface pressures differing by two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' To determine which of the 279 rocky exoplanets located above the H2O line in Figure 2 are likely to be Earth analogues or Venus analogues, we examine their location in relation to the CHZ and VZ in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Using Equations 11 - 12, we plot the optimistic boundaries of the CHZ in Figure 3 to quantify the insolation flux thresholds where planetary conditions transition to either a runaway snowball or a runaway greenhouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The 2% ± 1% of rocky exoplanets that are likely to have an atmosphere (Figure 1), with H2O present (Figure 2), and also reside within the optimistic CHZ (Figure 3) where the insolation is suitable for the presence of liquid water on the planets’ surfaces, are the planets most similar to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Thus, these exoplanets could be classified as Earth analogues and are the most likely to retain a ∼1 bar atmosphere including H2O molecules, indicating a high potential for retaining water in their atmospheres and on their surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Additionally, using Equations 12 - 13, we plot the boundaries of the VZ in Figure 3 to quantify the insolation flux thresholds where planetary conditions transition between a runaway greenhouse and complete atmospheric erosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' In Figure 3, we see that 11% ± 1% of rocky exoplanets with H2O present in their atmospheres (Figure 2) reside within the VZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Out of these 11% ± 1% rocky exoplanets, fifteen are plotted with open circles denoting their location above the cosmic shoreline in Figure 1, indicating they are unlikely to have an atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, taking into account 68% confidence intervals on insolation and escape velocity, these exoplanets cross the cosmic shoreline and could potentially have a substantial atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Thus, the 11% ± 1% of rocky exoplanets within uncertainties are likely to have an atmosphere (Figure 1), with H2O present (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Their position within the VZ (Figure 3), indicates that their atmospheres would be unable to maintain radiation balance, resulting in runaway heating of the surface and the formation of a thick Venus-like atmo- sphere and surface pressures in the order of 102 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Therefore, this subset of rocky exoplanets could be classified as Venus analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The final subset of 26% ± 12% rocky exoplanets from our sam- ple reside above the H2O line in Figure 2 and are not located within the CHZ or VZ in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, the majority of planets in this subset are likely to have thin or negligible atmospheres, ac- cording to Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Out of these 26% ± 12% rocky exoplanets, nine exoplanets are located past the inner VZ atmospheric erosion boundary, yet they are plotted with solid circles, indicating they are likely to have an atmosphere according to Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, taking into account 68% confidence intervals on insolation and es- cape velocity, these exoplanets cross the cosmic shoreline and could potentially have a thin or negligible atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2014) suggest that these highly irradiated rocky exoplanets located past the VZ’s inner boundary have completely eroded atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' No solar system analogues exist for this subset of rocky exoplanets, which have the potential for H2O to be present, yet too high levels of stellar flux eroding their atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Combining the information from Figures 1-3, we develop the classification method outlined in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Comparing to previous classification schemes, Wordsworth & Kreidberg (2022) classify a subset of rocky exoplanets that have an atmosphere and equilibrium temperatures above 300K as having no direct analogue in the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, here, we classify these exoplanets as Venus-like as we have determined that they retain an atmosphere that could have H2O present despite their high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' This subset will be interesting to explore in future observations to study the key tran- sitions in atmospheric composition and determine how they differ from Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Additionally, in Figure 3 we have classified a different subset than Wordsworth & Kreidberg (2022) having no solar sys- tem analogue as those rocky exoplanets that have the potential for MNRAS 000, 1–13 (2023) Atmospheric Recent Early 0 6500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 0 Erosion Venus Mars .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 0 0 6000 G Temperature 0 0 5500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 0 8 0 5000 00 00 0 4500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Effective K 8 4000: 0 0 CHZ VZ 0 3500 Rocky exoplanets with thin or negligible M 3000 atmospheres Rocky exoplanets with atmospheres 2500 102 100 103 101 Insolation (S)A rocky exoplanet classification method 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Classification of rocky exoplanets into closest solar system analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The dashed lines indicate potential for crossover in classification due to uncertainties in the input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' None of the exoplanets that were likely to maintain an atmosphere were below the H2O line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' H2O to be present, yet due to high levels of stellar flux, have no or negligible atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='2 Application of exoplanet classification to surface pressure and surface temperature To demonstrate the effect that the new classification system has on simple normalised calculations, the surface pressures for all 720 rocky exoplanets in our sample were computed using the original Equation 3 (Figure 5a), and using the new Equations 15-18 (Figure 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 5a demonstrates the clustering around ∼1 bar surface pressure due to the fact that surface pressure was normalised by Earth’s value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='014 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Therefore, relative to their observed values, the calculated values for Mercury and Mars are higher, and Venus is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, as we have limited the radius of rocky exoplanets in our sample to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3𝑅⊕ ≤ 𝑅𝑝 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23𝑅⊕, the surface pressure range is limited to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='02 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='95 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 5b highlights the effect that the solar system analogue classification method has made to the surface pressure model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' There is now a broader spread of surface pressure values for rocky exo- planets ranging from 2 × 10−15 − 210 bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Additionally, Figure 5b illustrates the division between planets that have thin to no at- mospheres being Mercury or Mars analogues and planets with at- mospheres being Venus or Earth analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The exceptions to this division are the Venus analogue exoplanets whose median escape velocities indicate an absence of an atmosphere, however within 68% confidence intervals, these Venus analogues cross the cosmic shoreline from Figure 1 and indicate the potential for a thick Venus like atmosphere, based on their location in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that are classified as having “no solar system analogue” have no analogue adjusted equations and are thus omitted from Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Reliable estimates of surface temperature and pressure are es- sential for characterising the environment and habitability of an exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' To further demonstrate the functionality of our exoplanet classification method, we plot our sample of rocky exoplanets over the phase diagram of pure water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' In Figure 6a we plot the equilib- rium temperature, where 𝐴 = 0 and 𝐺𝑎 = 0 (Equation 9), against the original surface pressure values (Equation 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Subsequently, af- ter applying our solar system analogue classification, in 6b we plot the surface temperature, with analogue-defined 𝐴 and 𝐺𝑎 values (Equations 22-25), against the analogue-adjusted surface pressure values (Equations 15-18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 6a illustrates the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='02-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='95 bar limitations and clustering around ∼1 bar surface pressure due to the Earth-centric normali- sation in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Additionally, the equilibrium temperature of Venus, without accounting for any atmospheric greenhouse effect, falsely indicates the likelihood of liquid water on Venus’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Thus, our subset of Venus analogue exoplanets which reside within the liquid water zone in Figure 6a may display similar false-positive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 6b highlights the effect that the solar system analogue classification method has made to the surface temperature vs pres- sure phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The Venus analogues have shifted outside of the liquid water zone, representing the significant effect an atmospheric greenhouse plays in the potential habitability of an exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' In Fig- ure 6a there are no rocky exoplanets with an atmosphere recording equilibrium temperature above 620K, however with our analogue adjusted surface temperatures in Figure 6b, we see some Venus-like exoplanets with surface temperatures of up to 950K, which could be classified as Atmosphere type I from Miguel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' On these hot rocky exoplanets, the major gases present are likely to be Na, O2, O, and Fe, with the near-crust atmospheres mainly composed of H2O, CO2, and SO2 (Herbort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Miguel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Miguel & Kaltenegger 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Schaefer & Fegley 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Schaefer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' in Figure 6b there are six Earth analogues that are likely to have an atmosphere (Figure 1) with H2O present (Figure MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 1–13 (2023) Sample of currently detected rocky exoplanets Fig 1: Non-thermal escape Fig 1: Non-thermal escape Thin or negligible Likely to maintain an atmosphere atmosphere 2: The Fig 2: Thermal escape Above the H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='O line: Below the H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='O line: Above the H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='O line: Unlikely to have water Likely to have water Likely to have water in the atmosphere in the atmosphere in the atmosphere Fig 3: Insolation Venu boundaries 2 Fig 3 Between the CO2 and H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='O lines: Located inside of the inner Located within the VZ: Below the CO2 line: Located within the CHZ: Thin atmosphere and unlikely to boundary of the VZ: Likely to have a thick atmosphere Negligible atmosphere No runaway greenhouse effect have water in the atmosphere Atmospheric erosion too high due to runaway greenhouse Mercury analogue Mars analogue No solar system analogue Earth analogue Venus analogue8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='N McIntyre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Surface pressure calculation using original Equation 3 applied to 720 rocky exoplanets in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Open coloured circles represent the calculated values for rocky solar system planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Labelled solid coloured circles represent observed values for rocky solar system planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside above the cosmic shoreline in Figure 1 and are likely to have thin or no atmosphere are plotted with open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside below the cosmic shoreline in Figure 1 and are likely to have an atmosphere are plotted with solid dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Median and 68% confidence intervals on surface pressure values were calculated using the Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Surface pressure calculation using Equations 15-18 applied to our sample categorised according to Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Labelled solid coloured circles represent observed values for rocky solar system planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside above the cosmic shoreline in Figure 1 and are likely to have thin or no atmosphere are plotted with open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside below the cosmic shoreline in Figure 1 and are likely to have an atmosphere are plotted with solid dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that are classified as having “no solar system analogue” have no analogue adjusted equations and are thus omitted from this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Median and 68% confidence intervals on surface pressure values were calculated using the Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Phase diagram of pure water (blue shaded region) with surface pressure vs equilibrium temperature, applying Equations 3 and 9 to the 720 rocky exoplanets in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Labelled solid coloured circles represent rocky solar system planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside above the cosmic shoreline in Figure 1 and are likely to have thin or no atmosphere are plotted with open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside below the cosmic shoreline in Figure 1 and are likely to have an atmosphere are plotted with solid dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Median and 68% confidence intervals on surface temperature and surface pressure values were calculated using the Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Phase diagram of pure water (blue shaded region) with adjusted surface pressure vs surface temperature, applying Equations 15-18 and 22-25 to our sample classified according to Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Labelled solid coloured circles represent rocky solar system planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside above the cosmic shoreline in Figure 1 and are likely to have thin or no atmosphere are plotted with open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that reside below the cosmic shoreline in Figure 1 and are likely to have an atmosphere are plotted with solid dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Exoplanets that are classified as having “no solar system analogue” have no analogue adjusted equations and are thus omitted from this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Median and 68% confidence intervals on surface temperature and surface pressure values were calculated using the Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) a: Original Psurf Earth centric estimation b: Adjusted Psurf using solar system analogue classification 102 Venus 102- Venus Earth 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (Bars) 0 (Bars) Mars Pvenus - analogue Venus 10 Pressure Pressure PEarth - analogue Earth calc 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' PMars - analogue 0 PMercury - analogue Mars calc Rocky exoplanets with thin face ce Mercury or negligible atmospheres 10 Surfa calc Rocky exoplanets with atmospheres Surf 0 Rocky exoplanets with thin 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Mars 10-13 or negligible atmospheres Mercury Rocky exoplanets with atmospheres Mercury 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='2 Radius (R) Radius (R)a: Original Earth centric estimation b: Adjusted Psurf and Tsurf using solar system analogue classification 102- OVenus Rocky exoplanets with thin Rocky exoplanets with atmospheres .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 0 or negligible atmospheres 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Venus Rocky exoplanets with thin 0 Rocky exoplanets with atmospheres or negligible atmospheres (Bars) 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (Bars) 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 100 Earth Pressure Pressure b%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='%0 0 ° 0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 0 10 )-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=" 0 8&0 0o 0 8 0 0 00% Surface l 000% urface 0 10 Mars 0 0 0 10' 0 S10-13J 0 0 0 @0 0 00 0 10-14↓ Mercury 10-2↓ Mars 0 0 0 500 1000 2000 2500 3000 3500 0 500 1000 1500 1500 2000 2500 3000 3500 Surface Temperature (K) Equilibrium Temperature (K)A rocky exoplanet classification method 9 2) and reside within the optimistic CHZ (Figure 3)," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' yet are located to the left of the liquid water zone in Figure 6b indicating a higher potential for water molecules to be in ice form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Analogue adjusted surface pressure and surface temperature of GJ 1061 d, Proxima Cen b, Teegarden’s Star c, TRAPPIST-1 e, TRAPPIST-1 f, TRAPPIST-1 g, and TRAPPIST-1 h in the context of liquid stability for H2O (light blue shading), NH3 (green shading), H2S (pink shading), CO2 (dark blue shading), and CH4 (red shading) adapted from Cengel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The dashed coloured lines for CH4 (red dashed line), H2S (pink dashed line), and CO2 (dark blue dashed line), represent the solid-gas boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Median and 68% confidence intervals on analogue adjusted surface pressure and surface temperature values were calculated using the Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' To determine the types of ices likely present on the surface of the six Earth analogues located in the water ice zone in Figure 6b, we plot their analogue adjusted surface pressure and surface temperature values in the context of phase diagrams for H2O, NH3, H2S, CO2, and CH4 in Figure 7 (Cengel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' From Figure 7 it is evident that all of the planets are likely to form H2O ices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The potential addition of surface water ice on these exoplanets could increase their albedos, resulting in a corresponding decrease to their surface temperatures from Equation 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' As more atmospheric gas is trapped in ice form, this could also result in a decrease of atmospheric pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Additionally, Teegarden’s Star c, TRAPPIST- 1 f, and TRAPPIST-1 g fall within the liquid NH3 phase, indicating that if ammonia is present on their surfaces it is likely to be in liquid form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' All of these exoplanets will still maintain H2S, CO2, and CH4 in gas form in their atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' TRAPPIST-1 h is the only rocky exoplanet in our sample that resides past the Early Mars CHZ outer boundary in Figure 3, and is the left-most exoplanet in Figure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Based on its location in Figures 1 and 2, TRAPPIST-1 h is likely to retain an atmosphere with H2O present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, due to its location in Figures 3 and 6a, TRAPPIST-1 h is likely to be cooler than the Earth analogues, and there is a higher potential for the molecules to be in ice form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' From Figure 7 it is evident that TRAPPIST-1 h is likely to form H2O, NH3, H2S, and CO2 ices, while CH4 is likely to remain in gas form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' As evident in Figures 5 and 6, our rocky exoplanet classifi- cation, when applied to surface pressure and surface temperature models, allows us to present a more appropriate picture of the cur- rent rocky exoplanet sample and provides a further layer of context in the characterisation of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Our model predictions can be tested using two rocky exoplan- ets for which the upper limits on atmospheric features have been determined using secondary eclipse and phase variation observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' First, we compared our results to the Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2019) observations of LHS 3844b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' While LHS 3844b’s radius measure- ment of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3R⊕ is technically outside our rocky exoplanet cut-off range (𝑅𝑝 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23R⊕), when run through our models we find results consistent with Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2019) of a hot rocky planet unable to retain a substantial atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Using our classification method, we categorise LHS 3844b as a “Non-solar system analogue” exo- planet as it has the potential for H2O to be present, yet high levels of stellar flux have likely eroded the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' This is consistent with the findings from Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2019) of a small atmosphere susceptible to erosion by stellar winds and thus likely being bare- rock with low bond-albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Additionally, we compared our results to the Crossfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2022) observations of GJ 1252b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' While our model assumes a fast-rotating exoplanet, unlike the slow-rotating GJ 1252b, when run through our models, the results agree with Crossfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2022) that GJ 1252b is a hot rocky exoplanet with no significant atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Using our classification method, we cat- egorise GJ 1252b as a close-in “Mars-analogue” exoplanet likely to have a thin water-less atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Comparing our analogue ad- justed temperature of 1011 ± 107 K to the Crossfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2022) dayside brightness temperature of 1410+91 −125 K, we fall within 2𝜎 of their dayside GJ 1252b surface temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Despite using different assumptions, our primary classification model is consistent with other work using different techniques, giving us confidence in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3 Model limitations The modelling employed in this paper provides a general approx- imation of atmospheric escape velocity, thermal escape of species from the atmosphere, and stellar irradiation boundaries for known exoplanets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' however, it is important to acknowledge its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The Zahnle & Catling (2017) cosmic shoreline definition as- sumes an average molecular weight of the atmospheric material to represent all atmospheric compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, if the atmo- spheric material is composed of significantly heavier or lighter el- ements, or if photochemistry which could significantly reduce the mean molecular mass is considered, then the cosmic shoreline’s ideal gas assumption may be under- or over-estimating atmospheric escape rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, the time-frame for atmospheric loss will differ depending on luminosity and stellar mass for a given planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Thermal loss rates may also be influenced by atmospheric compo- sition, as Parkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2022) show that an O-CO2 thermosphere can cool efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Atmospheric species can be lost through several additional non-thermal processes for example ion pickup (Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2006), sputtering (Terada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2009), dissociation and disso- ciative recombination (Geppert & Larsson 2008), photo-chemical energising mechanisms (Vidotto 2013), and charge exchange (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Conversely, these atmospheric species could be gained through volcanic degassing (Oosterloo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2021) or impact events that liberate gas from the surface (Kuwahara & Sugita 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Fur- thermore, depending on the stellar wind pressures, the presence of a significant magnetic field could reduce non-thermal atmospheric erosion (McIntyre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Future research should be conducted to investigate which atmospheric mass-loss processes dominate in different planetary scenarios and the degree to which they are inhibited by potential planetary magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Planetary rotation is an essential factor that could affect the modelling employed here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' however, this parameter is not yet able to be directly observed for a broad sample of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=" Yang & Ab- bot (2014) demonstrate the dependence of the inner CHZ boundary MNRAS 000, 1–13 (2023) 101 (Bars) GJ/106ld surface Pressure ( Teegarden's Star c tProxima Cen b TRAPPIST:1g+ Earth TRAPPIST-1f TRAPPIST-1e TRAPPIST-1h+ H20 liquid S NH3 liquid H2S liquid CO2 liquid CH4 liquid 10- 50 100 150 200 250 300 350 400 450 Surface Temperature (K)10 S." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='N McIntyre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' on planetary rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Strongly irradiated, rapidly rotating exoplan- ets could lose water and develop low albedos, entering a runaway greenhouse even while residing within the CHZ (Del Genio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, species in the atmo- spheres of a rapidly rotating exoplanet could reach high velocities due to the increased temperature, which would accelerate the atmo- spheric escape process (Konatham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' While we can infer the rotation period for tidally locked exoplanets as equivalent to their orbital period, such planets could escape synchronous rotation by being captured in spin-orbit resonances (Goldreich & Soter 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Makarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2012) or through resonant planet-planet interactions with their exterior planetary companions (Delisle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Vinson & Hansen 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Zanazzi & Triaud 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Additionally, there is no way to constrain the rotation pe- riod without direct observations of exoplanets past the tidal locking radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' In the future, photometric variability techniques have been postulated to facilitate measurements of an exoplanet’s rotation (Fu- jii & Kawahara 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Snellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Here, we only focus on the impact of stellar flux on the run- away greenhouse boundary, as defined by Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' However, there are additional ways surface temperature conditions could increase to levels resulting in a runaway greenhouse, such as tidal heating (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' McIntyre 2022), especially for planets in eccentric orbits (Williams & Pollard 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Kane & Gelino 2012), or sufficiently increased CO2 levels that could drive an atmosphere into a runaway greenhouse at further distances from the host star (Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Nonetheless, the likelihood of a runaway greenhouse will decrease dramatically as the distance past the current VZ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The surface temperature estimate utilised here only accounts for a broad greenhouse effect and not specific greenhouse gas abun- dances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' For synchronously rotating exoplanets, future observations of thermal phase curves could help further quantify the difference between surface temperature and equilibrium temperature and pro- vide a more accurate picture of an exoplanet’s atmospheric green- house effect (Del Genio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Lacis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Yang & Ab- bot 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' It would also be useful to measure exoplanets’ obliquity, as this can result in high to low temperatures distributed from the equator to the poles (Nowajewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' For example, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' (2016) suggests that with higher obliquities, the habitability around M dwarfs narrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Obliquity is currently unobservable for exoplanets, although it could be extrapolated when seasonal cycle information on reflected starlight becomes available (Kane & Torres 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, Ahlers (2016) determine that exoplanets with inclined orbits around fast-rotating stars display changes in equilib- rium temperature of up to 15% due to the increased irradiance near the stellar poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Through these simplified models, we are attempting to use the limited observational data on the characteristics of rocky exoplanets we currently have available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' In the future, additional observations of obliquity, inclination, planetary rotation rates, stellar rotation rates, and thermal phase curves could help strengthen the classification method detailed here and help determine optimum targets for future atmospheric observations of rocky exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 4 CONCLUSIONS Here, we use non-thermal atmospheric escape, thermal atmospheric escape, and stellar irradiation boundaries to develop a primary clas- sification method for current rocky exoplanets (𝑅𝑝 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23R⊕) and group them into categories relative to the most appropriate solar sys- tem analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' When applying this primary classification method to the 720 rocky exoplanets in our sample, results suggest that 22% ± 8% are Mercury analogues, 39% ± 4% are Mars analogues, 11% ± 1% are Venus analogues, 2% ± 1% are Earth analogues, and 26% ± 12% are without a known planetary counterpart in our solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Implementing this classification method will help further char- acterise the detected exoplanets by comparing them to a more ap- propriate solar system analogue, rather than continuing with the common approach where we estimate the values for numerous un- known parameters by normalising to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' To demonstrate the functionality of this classification method, we compare it to a simple model for surface pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Using Earth-centric measure- ments the calculated surface pressure range for rocky exoplan- ets (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='3𝑅⊕ ≤ 𝑅𝑝 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23𝑅⊕) with the simple model spanned 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='02 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='95 bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' After applying the new primary classification method to the rocky exoplanet sample, the surface pressure range now expands to 2×10−15−210 bars, accounting for the full variation observed in our solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Furthermore, our new rocky exoplanet classification method, when applied to calculating surface pressure and surface temperature, allows us to present a more varied picture of the current rocky exoplanet sample and provides a further layer of context in the characterisation of exoplanets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' for example, the presence of liquid, gas or ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The use of our new primary classification method could im- prove inferences of temperature, composition, interior structure, evolution and dynamics of rocky exoplanets, which would aid our ability to interpret and model exoplanet atmospheres (Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Additionally, this classification method could benefit tar- get selection for exoplanet characterisation missions by providing a more robust starting point for potential atmospheric properties and composition for comparison to future observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' ACKNOWLEDGEMENTS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='McIntyre gratefully acknowledges an Australian Government Research Training Program (RTP) Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' King acknowl- edges funding from an Australian Research Council Discovery Pro- gram grant (DP200100406).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Helpful comments from an anonymous referee are gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article are available in the article and in its online supplementary material which can be downloaded in electronic form from the Centre de Données astronomiques de Strasbourg (CDS) service via anonymous ftp cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='u-strasbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' fr (130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='5) or via https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='fr/ viz-bin/cat/J/MNRAS.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' The rest of the table can be downloaded in electronic form from CDS service and the publisher’s website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content=' Planet Name M𝑝 (M⊕) R𝑝 (R⊕) v𝑒𝑠𝑐 (ms−1) S∗ (S⊕) T𝑒𝑞 (K) Original P𝑠𝑢𝑟 𝑓 (Bar) Adjusted P𝑠𝑢𝑟 𝑓 (Bar) Adjusted T𝑠𝑢𝑟 𝑓 (K) Planet Classification TRAPPIST-1 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='01 12411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='61 ± 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='16 397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='26 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='14 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='63 ± 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='53 785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='66 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='65 Venus analogue TRAPPIST-1 c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='01 12214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='27 ± 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='09 339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='36 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='12 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='63 ± 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='43 745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='61 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='38 Venus analogue TRAPPIST-1 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='01 7849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='09 ± 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='04 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='69 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='03 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='07 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='35 Earth analogue TRAPPIST-1 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='01 9701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='21 ± 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='03 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='91 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='06 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='36 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='63 Earth analogue TRAPPIST-1 f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='01 11153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='63 ± 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='01 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='08 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='07 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='34 ± 1.' metadata={'source': 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+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} +page_content='05 No solar system analogue No solar system analogue No solar system analogue MNRAS 000, 1–13 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfqwX2/content/2301.03348v1.pdf'} diff --git a/FNFRT4oBgHgl3EQfyzjn/content/tmp_files/2301.13648v1.pdf.txt b/FNFRT4oBgHgl3EQfyzjn/content/tmp_files/2301.13648v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b193ccb64f06b764093198d0ba4c456d72311615 --- /dev/null +++ b/FNFRT4oBgHgl3EQfyzjn/content/tmp_files/2301.13648v1.pdf.txt @@ -0,0 +1,587 @@ +CSDN: COMBINING SHALLOW AND DEEP NETWORKS FOR ACCURATE REAL-TIME +SEGMENTATION OF HIGH-DEFINITION INTRAVASCULAR ULTRASOUND IMAGES +Shaofeng Yuan1∗ +Feng Yang2∗ +1 Institute of Artificial Intelligence, Insight Lifetech, Shenzhen, China +2 School of Biomedical Engineering, Southern Medical University, Guangzhou, China +ABSTRACT +Intravascular ultrasound (IVUS) is the preferred modality for +capturing real-time and high resolution cross-sectional im- +ages of the coronary arteries, and evaluating the stenosis. Ac- +curate and real-time segmentation of IVUS images involves +the delineation of lumen and external elastic membrane bor- +ders. +In this paper, we propose a two-stream framework +for efficient segmentation of 60 MHz high resolution IVUS +images. +It combines shallow and deep networks, namely, +CSDN. The shallow network with thick channels focuses to +extract low-level details. The deep network with thin chan- +nels takes charge of learning high-level semantics. Treating +the above information separately enables learning a model +to achieve high accuracy and high efficiency for accurate +real-time segmentation. To further improve the segmentation +performance, mutual guided fusion module is used to en- +hance and fuse both different types of feature representation. +The experimental results show that our CSDN accomplishes +a good trade-off between analysis speed and segmentation +accuracy. +Index Terms— Shallow network, Deep network, Real- +time segmentation, Intravascular ultrasound images, Medical +image segmentation +1. INTRODUCTION +Atherosclerosis is a disease of the vessel wall, and responsible +for many cardiovascular diseases. Compared with the in vitro +screening, the widespread application of the intravascular ul- +trasound (IVUS) imaging relies on its capability to visualize +the inner structure of vessels in real-time to diagnose the arte- +riosclerotic disease of the coronary artery. It can assess quan- +titative clinical measurements. However, accurate delineation +of the lumen and external elastic membrane (EEM) borders +is essential for the assessment of plaque burden and stenosis +degree. The current clinical practice relies on manual annota- +tion in the IVUS images, which is time-consuming and user- +dependent. +* Shaofeng Yuan and Feng Yang are the corresponding authors +(shaofeng.yuan.smu@gmail.com; yangf@smu.edu.cn). +In recent years, many deep learning methods based on +convolutional networks (ConvNets) have been widely used +in computer vision and image processing tasks due to their +excellent capacity of automatic feature extraction [1, 2]. +Considerable progress has been made in medical image com- +puting community [3]. Many ConvNets-based methods have +been developed to segment lumen and/or EEM regions. For +example, Ji et al. [4] proposed IVUS-Net series based on +U-shaped fully convolutional networks [5]. Different from +simple encoding and decoding layers in U-net, IVUS-Net +uses aggregated, multi-branch architectures in these layers. +Cao et al. +[6] selected DeepLabv3+ [7] as the segmenta- +tion network in their work. It combines the advantages of +spatial pyramid pooling module and encoder-decoder struc- +ture. Xia et al. [8] proposed a multi-scale feature aggregated +U-net (MFAUNet) to extract two membrane borders simul- +taneously. The feature aggregated module in skip connec- +tions utilizes the bi-directional convolutional long short-term +memory unit to extract the context information from the +spatial-temporal perspective. Ziemer et al. [9] proposed a +multi-frame ConvNet for lumen segmentation. Adding in- +formation about neighbouring frames surrounding the frame +of interest improved the segmentation performance. Inspied +by this work, we adopt three-frame IVUS images as input of +our CSDN. Li et al. [10] used two modified U-net for lumen +and media-adventitia borders, respectively. The results of two +segmentations are combined in the end. Szarski et al. [11] +proposed a real-time modified U-net augmented with learned +translation dependence (coordinate-aware FCN, CoordFCN). +However, the above methods don’t accomplish a good +trade-off between processing speed and segmentation accu- +racy. In this paper, a novel framework CSDN is proposed to +timely and accurately extract two important borders in IVUS +images by considering that low-level details and high-level +semantics are crucial to the real-time segmentation task, and +treated separately can achieve the trade-off between the accu- +racy and inference speed. +2. METHODS +The overview of the proposed CSDN architecture is shown in +Fig. 1 (a). +arXiv:2301.13648v1 [eess.IV] 30 Jan 2023 + +Multi-frame +IVUS Images +×2 +Deep Network +Mask of +Intermediate Frame +Downsample +×3 +×4 +Shallow Network +Upsample +Conv-BN-PReLU +Stem Block +Gather-Expansion Block +Context Block +Mutual Guided Fusion +(a) Architecture of CSDN +(b) Stem Block +3×3 Conv, +Stride=2 +3×3 MP, +Stride=2 +1×1 +Conv +3×3 Conv, +Stride=2 +C +3×3 Conv, +Stride=1 +Pooling Branch +Conv Branch +1×1 +Conv +Low-level +features from SN +High-level +features from DN +3×3 +DWConv +3×3 Conv, +Stride=2 +3×3 Conv, +Stride=1 +3×3 +DWConv +1×1 +Conv +3×3 AP, +Stride=2 +1×1 +Conv +4×4 +upsample +× +× ++ +3×3 Conv, +Stride=1 +(c) Mutual Guided Fusion +Fig. 1. (a) The architecture of the proposed CSDN. On the top side, Deep Network with thin channels is adopted to extract +high-level semantic information. On the bottom side, Shallow Network with thick channels is used to extract low-level detailed +information. At last, the output features from the above networks are input to Mutual Guided Fusion to generate final output +feature and make the final prediction. Before feature extraction and label assignment, Downsample and Upsample modules +are used to decrease and increase image size for accelerating segmentation. (b) Components of the Stem Block. (c) Compo- +nents of the Mutual Guided Fusion. MP is max pooling, AP is average pooling. DWConv is depth-wise convolution. Batch +normalization and PReLU are omitted. +2.1. Downsampling with pixel unshuffling +Directly extracting visual features in high resolution space, +ConvNets require more computational cost and memory foot- +print. However, during percutaneous coronary intervention +(PCI), accurate and real-time IVUS segmentation for lumen +and EEM area is a requirement. Shi et al. [12] proposed +to use pixel shuffling as an upsampling operation, alternative +to deconvolution layer. ConvNets with pixel shuffling, first +gets wide but low resolution feature maps, then arranges in- +put channels to produce a feature map with higher resolution. +Contrary to pixel shuffling, pixel unshuffling is a downsam- +pling operation, exchanging channel information with spatial +information. +As shown in Fig. 1, CSDN uses Downsam- +ple and Upsample modules to accelerate feature extraction of +neural networks. In practice, Downsample module include +bicubic interpolation and pixel unshuffling. After Downsam- +ple module, the width size of multi-frame IVUS images is +reduced by a factor of 4. +2.2. Shallow network +As shown on the bottom side in Fig. 1(a), the shallow network +with thick channels focuses to extract low-level details. This +network has a shallow structure with thick channels in build- +ing blocks, because we enforce it to encode enough spatial de- +tailed information. The shallow network has three feature ex- +traction blocks. Each block has three standard 2d convolution +layers. Each convolution layer is followed by a 2d batch nor- +malization and a PReLU activation function. In each block, +the first convolution layer has stride of 2 for downsampling +the size of feature maps. Obviously, the width size of output +from the shallow network is reduced by a factor of 8. +2.3. Deep network +As shown on the top side in Fig. 1(a), the deep network +with thin channels takes charge of learning high-level seman- +tics. This network has a deep architecture with thin channels +in building blocks for better segmentation performance. In + +dense prediction task, e.g., semantic segmentation, sufficient +receptive field is of importance for the good performance. +Therefor, we design deep network to provide sizeable re- +ceptive field using a fast down-sampling strategy. The deep +network has five context and semantic information feature +extraction stages. The first stage is the Stem Block, shown in +Fig. 1(b). This block has two branches to downsample input +feature maps in different manners. In the end of two branches, +both output feature maps are concatenated, then followed by +a convolution layer with batch normalization and PReLU ac- +tivation function. The last stage is the Context Block [13, 14]. +This block uses residual connection and global average pool- +ing to embed the global contextual information, providing +the maximum receptive field. The remaining three stages are +Gather-Expansion Block [14]. Each Gather-Expansion Block +has at least 2 gather-expansion (GE) layers. For example, the +first Gather-Expansion Block has 2 GE layers, the second has +3, and the third has 4, shown in Fig. 1(a) gold boxes. The +GE layer has 2 types, GE-Stride1 and GE-Stride2, details in +[14], and both are residual connections. The first GE layer +in each GE block uses GE-Stride2 for downsampling feature +maps to enlarge receptive field. The other GE layer in each +GE block uses GE-Stride1. +In GE-Stride1, a 3 × 3 con- +volution is used to gather local feature values and expand to +higher-dimensional space. Then an efficient 3 × 3 depth-wise +convolution is used independently over each channel from the +above step. In the end, a 1 × 1 convolution is used to project +higher-dimensional feature maps into a low channel capacity +space. +2.4. Feature fusion with mutual guided fusion +The feature representation of the shallow network and the +deep network is different and complementary. Thus, we use +mutual guided fusion block to merge both types of feature +representation, illustrated in Fig. 1(c). This block use contex- +tual information from deep network to guide the feature value +from shallow network. Meanwhile, this block use detail in- +formation from shallow network to guide the feature response +from deep network. Comparing to the simple feature fusion +like element-wise summation or feature concatenation, mu- +tual guided fusion strategy enables efficient communication +between both two neural networks. +3. EXPERIMENTS AND RESULTS +3.1. Dataset +In this work, we evaluate the proposed CSDN framework on +High-definition Intravascular Ultrasound (HDIVUS) database, +which is a partial and in-house dataset for IVUS image seg- +mentation created by Insight Lifetech. The dataset consists +of 2098 sets of B-mode IVUS images, including a complete +annotation of two important borders by expert cardiologists. +In total, the HDIVUS dataset contains IVUS images from 14 +patients. For each pullback, about 150 IVUS images were +sampled. All the original images have a size of 900 × 900 +pixels. +3.2. Evaluation metrics +The evaluation metrics used in this work are two types, based +on region and border. The region-based metrics are Dice- +Sørensen Coefficient (DSC), Intersection over Union(IoU) of +EEM and lumen. The border-based metrics is Hausdorff dis- +tance (HD) in the 95th percentage of EEM and lumen. In sce- +nario of real-time semantic segmentation, frame per second +(FPS) is also considered. +Table 1. Average performance in validation set +Lumen +Method +Params +FPS +DSC +IoU +HD95 +PE-UNetv1a +40863K +205 +0.953 +0.913 +0.198 +PE-UNetv2b +40871K +187 +0.952 +0.912 +0.193 +CoodFCN +475K +222 +0.937 +0.885 +0.292 +MFAUNet +646166K +98 +0.947 +0.903 +0.204 +IVUSNetv1 +16952K +179 +0.914 +0.846 +0.610 +IVUSNetv2 +39175K +69 +0.931 +0.877 +0.329 +DMUNet-Dc +7489K +223 +0.934 +0.880 +0.306 +DMUNet-Td +7489K +223 +0.940 +0.890 +0.347 +DMUNet-Fe +7489K +223 +0.941 +0.892 +0.280 +DLv3p-Rf +57949K +65 +0.953 +0.913 +0.183 +DLv3p-Xg +53419K +60 +0.957 +0.919 +0.177 +CSDN +1706K +151 +0.953 +0.913 +0.189 +EEM +Method +Params +FPS +DSC +IoU +HD95 +PE-UNetv1a +40863K +205 +0.960 +0.925 +0.261 +PE-UNetv2b +40871K +187 +0.962 +0.928 +0.236 +CoodFCN +475K +222 +0.945 +0.899 +0.515 +MFAUNet +646166K +98 +0.963 +0.930 +0.246 +IVUSNetv1 +16952K +179 +0.888 +0.803 +1.243 +IVUSNetv2 +39175K +69 +0.951 +0.911 +0.326 +DMUNet-Dc +7489K +223 +0.945 +0.899 +0.550 +DMUNet-Td +7489K +223 +0.937 +0.885 +0.577 +DMUNet-Fe +7489K +223 +0.940 +0.890 +0.582 +DLv3p-Rf +57949K +65 +0.962 +0.929 +0.224 +DLv3p-Xg +53419K +60 +0.964 +0.932 +0.213 +CSDN +1706K +151 +0.960 +0.925 +0.246 +a,bPretrained VGG16-Encoder without BN or with BN. +c,d,eLoss is Dice, Tversky and Focal loss. +f,gBackbone is ResNet-101 or Xception. +3.3. Implementation details +In all experiments, batch size in training is 16. The size of in- +put images is 900 × 900 pixels. The Adam optimizer is used + +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Fig. 2. Segmentation results from different methods. (a) PE-UNetv1 (b) CoordFCN (c) MFAUNet (d) IVUSNetv2 (e) DMUNet- +F (f) DLv3p-X (g) Our CSDN (h) Input. Red contour is ground truth of EEM, and green is lumen. Orange contour is prediction +of EEM, and gold is lumen. +with an initial learning rate of 1 × 10−3 and with a weight +decay of 1 × 10−4. The learning rate schedule is used with a +step size of 100, and learning rate is reduced by half in each +step. The total epoch we set is 300. The models are imple- +mented based on the PyTorch, and trained with a NVIDIA +GeFore GTX 3090 GPU. A hybrid loss combining Focal loss +and Dice loss is used. It is defined as: +L = LF ocal + LDice +For data augmentation, a combination of the following meth- +ods is used in training: translation, rotation, scaling, shearing, +left-right and up-down flipping. Because the channel size of +input images is 3, channel swapping is also used and centeral +frame is keep fixed. Deep supervision mechanism is further +used to boost segmentation performance. +3.4. Quantitative results +Tab. 1 reports the quantitative results and parameter size for +U-net with pretrained VGG16 encoder (PE-UNet series), +CoordFCN, MFAUNet, IVUS-Netv1, IVUS-Netv2, double +modified U-net with three different losses, DeepLabv3+ with +different backbones and our CSDN. Comparing to Coord- +FCN with 475K learnable parameters, CSDN imporves 1.6% +in DSC, 3.4% in IoU, and 0.103 mm in HD95, for lumen +segmentation, with only small increase in parameter size. For +EEM segmentation, CSDN imporves 1.5% in DSC, 2.6% +in IoU, and 0.269 mm in HD95. +Comparing to PE-UNet +series, CSDN is six times less than in parameter size, but +has similar or equal segmentation performance. +Although +CSDN’s FPS performance is less than PE-UNet series, the +GPU and CPU memory usage of PE-UNet series is large than +CSDN’s. It’s impractical to directly deploy PE-UNet series +in GPU with small memory, e.g., GTX 1050ti with 4GB. +Although MFAUNet is slightly worse than CSDN, the pa- +rameter size of MFAUNet is huge. The size of model weights +is about 2.6 GB. It’s impractical to deploy it into medical +devices with small main memory. Comparing to IVUS-Net +series and DMUNet series, our CSDN surpasses in term of +parameter size, region-based and border-based metrics with a +large margin. DeepLabv3+ is a semantic segmentation model +for general-purpose image segmentation with large latence. +Although it processes image in real-time with GTX 3090, it’s +very slow or even impractical with GTX 1050ti. Our pro- +posed CSDN is slightly worse than DeepLabv3+, however, +it’s easy to deploy it into different GPU and it’s smooth to +segment input high-definition images in real-time. +3.5. Qualitative results +We visualize manual and predictive results for both lumen +and EEM cases in Fig. 2. CSDN’s prediction tends to be +more close to boundary of lumen and EEM, while other meth- +ods except strong but slow DeepLabv3+ with Xception as the +backbone and MFAUNet with huge parameters have cases +with small disconnected prediction regions or uneven con- +tours. To demonstate the challenging task of IVUS image +segmentation, the last row of Fig. 2 shows one of the most +challenging cases where CSDN’s prediction can be a reason- +able segmentation from non-expert perspective. Shadow ar- +tifacts and peculiar bloodstream not only oblige learners to +have capacity of modeling the global dependence, but also + +OOexpect these algorithms to focus on local details. CSDN with +dual streams is suitable for handling with the above dilemma. +4. CONCLUSIONS +We have presented CSDN, a two-stream framework for effi- +cient and real-time segmentation of high-definition IVUS im- +ages. It combines shallow and deep networks. Treating low- +level details and high-level semantics information separately +enables learning a model to achieve high accuracy and high +efficiency for accurate real-time segmentation. In the future, +more experiments can be carried out on CSDN. +5. COMPLIANCE WITH ETHICAL STANDARDS +Informed consent was obtained from all individual partici- +pants involved in the study. +6. ACKNOWLEDGMENTS +This work is supported, in part, by the National Natural Sci- +ence Foundation of China (Nos. 61771233). +7. REFERENCES +[1] K. He, X. Zhang, S. Ren, and J. 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Wang, “Real-time sin- +gle image and video super-resolution using an efficient +sub-pixel convolutional neural network,” in Proceedings +of the IEEE conference on computer vision and pattern +recognition. IEEE, 2016, pp. 1874–1883. +[13] C. Yu, J. Wang, C. Peng, C. Gao, G. Yu, and N. Sang, +“Bisenet: Bilateral segmentation network for real-time +semantic segmentation,” +in Proceedings of the Euro- +pean conference on computer vision, 2018, pp. 325– +341. +[14] C. Yu, C. Gao, J. Wang, G. Yu, C. Shen, and N. Sang, +“Bisenet v2: Bilateral network with guided aggrega- +tion for real-time semantic segmentation,” International +journal of computer vision, vol. 129, no. 11, pp. 305– +3068, 2021. + diff --git a/FNFRT4oBgHgl3EQfyzjn/content/tmp_files/load_file.txt b/FNFRT4oBgHgl3EQfyzjn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..55319b94360c68b6fc3a4330f912baec5e504242 --- /dev/null +++ b/FNFRT4oBgHgl3EQfyzjn/content/tmp_files/load_file.txt @@ -0,0 +1,428 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf,len=427 +page_content='CSDN: COMBINING SHALLOW AND DEEP NETWORKS FOR ACCURATE REAL-TIME SEGMENTATION OF HIGH-DEFINITION INTRAVASCULAR ULTRASOUND IMAGES Shaofeng Yuan1∗ Feng Yang2∗ 1 Institute of Artificial Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Insight Lifetech,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Shenzhen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' China 2 School of Biomedical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Southern Medical University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' China ABSTRACT Intravascular ultrasound (IVUS) is the preferred modality for capturing real-time and high resolution cross-sectional im- ages of the coronary arteries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' and evaluating the stenosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Ac- curate and real-time segmentation of IVUS images involves the delineation of lumen and external elastic membrane bor- ders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In this paper, we propose a two-stream framework for efficient segmentation of 60 MHz high resolution IVUS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' It combines shallow and deep networks, namely, CSDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The shallow network with thick channels focuses to extract low-level details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The deep network with thin chan- nels takes charge of learning high-level semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Treating the above information separately enables learning a model to achieve high accuracy and high efficiency for accurate real-time segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' To further improve the segmentation performance, mutual guided fusion module is used to en- hance and fuse both different types of feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The experimental results show that our CSDN accomplishes a good trade-off between analysis speed and segmentation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Index Terms— Shallow network, Deep network, Real- time segmentation, Intravascular ultrasound images, Medical image segmentation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' INTRODUCTION Atherosclerosis is a disease of the vessel wall, and responsible for many cardiovascular diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Compared with the in vitro screening, the widespread application of the intravascular ul- trasound (IVUS) imaging relies on its capability to visualize the inner structure of vessels in real-time to diagnose the arte- riosclerotic disease of the coronary artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' It can assess quan- titative clinical measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' However, accurate delineation of the lumen and external elastic membrane (EEM) borders is essential for the assessment of plaque burden and stenosis degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The current clinical practice relies on manual annota- tion in the IVUS images, which is time-consuming and user- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Shaofeng Yuan and Feng Yang are the corresponding authors (shaofeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='yuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='smu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' yangf@smu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In recent years, many deep learning methods based on convolutional networks (ConvNets) have been widely used in computer vision and image processing tasks due to their excellent capacity of automatic feature extraction [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Considerable progress has been made in medical image com- puting community [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Many ConvNets-based methods have been developed to segment lumen and/or EEM regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' For example, Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' [4] proposed IVUS-Net series based on U-shaped fully convolutional networks [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Different from simple encoding and decoding layers in U-net, IVUS-Net uses aggregated, multi-branch architectures in these layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' [6] selected DeepLabv3+ [7] as the segmenta- tion network in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' It combines the advantages of spatial pyramid pooling module and encoder-decoder struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' [8] proposed a multi-scale feature aggregated U-net (MFAUNet) to extract two membrane borders simul- taneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The feature aggregated module in skip connec- tions utilizes the bi-directional convolutional long short-term memory unit to extract the context information from the spatial-temporal perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Ziemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' [9] proposed a multi-frame ConvNet for lumen segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Adding in- formation about neighbouring frames surrounding the frame of interest improved the segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Inspied by this work, we adopt three-frame IVUS images as input of our CSDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' [10] used two modified U-net for lumen and media-adventitia borders, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The results of two segmentations are combined in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Szarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' [11] proposed a real-time modified U-net augmented with learned translation dependence (coordinate-aware FCN, CoordFCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' However, the above methods don’t accomplish a good trade-off between processing speed and segmentation accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In this paper, a novel framework CSDN is proposed to timely and accurately extract two important borders in IVUS images by considering that low-level details and high-level semantics are crucial to the real-time segmentation task, and treated separately can achieve the trade-off between the accu- racy and inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' METHODS The overview of the proposed CSDN architecture is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='13648v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='IV] 30 Jan 2023 Multi-frame IVUS Images ×2 Deep Network Mask of Intermediate Frame Downsample ×3 ×4 Shallow Network Upsample Conv-BN-PReLU Stem Block Gather-Expansion Block Context Block Mutual Guided Fusion (a) Architecture of CSDN (b) Stem Block 3×3 Conv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Stride=2 3×3 MP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Stride=2 1×1 Conv 3×3 Conv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Stride=2 C 3×3 Conv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Stride=1 Pooling Branch Conv Branch 1×1 Conv Low-level features from SN High-level features from DN 3×3 DWConv 3×3 Conv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Stride=2 3×3 Conv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Stride=1 3×3 DWConv 1×1 Conv 3×3 AP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Stride=2 1×1 Conv 4×4 upsample × × + 3×3 Conv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Stride=1 (c) Mutual Guided Fusion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' (a) The architecture of the proposed CSDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' On the top side, Deep Network with thin channels is adopted to extract high-level semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' On the bottom side, Shallow Network with thick channels is used to extract low-level detailed information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' At last, the output features from the above networks are input to Mutual Guided Fusion to generate final output feature and make the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Before feature extraction and label assignment, Downsample and Upsample modules are used to decrease and increase image size for accelerating segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' (b) Components of the Stem Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' (c) Compo- nents of the Mutual Guided Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' MP is max pooling, AP is average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' DWConv is depth-wise convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Batch normalization and PReLU are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Downsampling with pixel unshuffling Directly extracting visual features in high resolution space, ConvNets require more computational cost and memory foot- print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' However, during percutaneous coronary intervention (PCI), accurate and real-time IVUS segmentation for lumen and EEM area is a requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' [12] proposed to use pixel shuffling as an upsampling operation, alternative to deconvolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' ConvNets with pixel shuffling, first gets wide but low resolution feature maps, then arranges in- put channels to produce a feature map with higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Contrary to pixel shuffling, pixel unshuffling is a downsam- pling operation, exchanging channel information with spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 1, CSDN uses Downsam- ple and Upsample modules to accelerate feature extraction of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In practice, Downsample module include bicubic interpolation and pixel unshuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' After Downsam- ple module, the width size of multi-frame IVUS images is reduced by a factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Shallow network As shown on the bottom side in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 1(a), the shallow network with thick channels focuses to extract low-level details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' This network has a shallow structure with thick channels in build- ing blocks, because we enforce it to encode enough spatial de- tailed information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The shallow network has three feature ex- traction blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Each block has three standard 2d convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Each convolution layer is followed by a 2d batch nor- malization and a PReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In each block, the first convolution layer has stride of 2 for downsampling the size of feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Obviously, the width size of output from the shallow network is reduced by a factor of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Deep network As shown on the top side in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 1(a), the deep network with thin channels takes charge of learning high-level seman- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' This network has a deep architecture with thin channels in building blocks for better segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In dense prediction task, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=', semantic segmentation, sufficient receptive field is of importance for the good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Therefor, we design deep network to provide sizeable re- ceptive field using a fast down-sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The deep network has five context and semantic information feature extraction stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The first stage is the Stem Block, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' This block has two branches to downsample input feature maps in different manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In the end of two branches, both output feature maps are concatenated, then followed by a convolution layer with batch normalization and PReLU ac- tivation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The last stage is the Context Block [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' This block uses residual connection and global average pool- ing to embed the global contextual information, providing the maximum receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The remaining three stages are Gather-Expansion Block [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Each Gather-Expansion Block has at least 2 gather-expansion (GE) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' For example, the first Gather-Expansion Block has 2 GE layers, the second has 3, and the third has 4, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 1(a) gold boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The GE layer has 2 types, GE-Stride1 and GE-Stride2, details in [14], and both are residual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The first GE layer in each GE block uses GE-Stride2 for downsampling feature maps to enlarge receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The other GE layer in each GE block uses GE-Stride1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In GE-Stride1, a 3 × 3 con- volution is used to gather local feature values and expand to higher-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Then an efficient 3 × 3 depth-wise convolution is used independently over each channel from the above step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In the end, a 1 × 1 convolution is used to project higher-dimensional feature maps into a low channel capacity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Feature fusion with mutual guided fusion The feature representation of the shallow network and the deep network is different and complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Thus, we use mutual guided fusion block to merge both types of feature representation, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' This block use contex- tual information from deep network to guide the feature value from shallow network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Meanwhile, this block use detail in- formation from shallow network to guide the feature response from deep network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Comparing to the simple feature fusion like element-wise summation or feature concatenation, mu- tual guided fusion strategy enables efficient communication between both two neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' EXPERIMENTS AND RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Dataset In this work, we evaluate the proposed CSDN framework on High-definition Intravascular Ultrasound (HDIVUS) database, which is a partial and in-house dataset for IVUS image seg- mentation created by Insight Lifetech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The dataset consists of 2098 sets of B-mode IVUS images, including a complete annotation of two important borders by expert cardiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In total, the HDIVUS dataset contains IVUS images from 14 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' For each pullback, about 150 IVUS images were sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' All the original images have a size of 900 × 900 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Evaluation metrics The evaluation metrics used in this work are two types, based on region and border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The region-based metrics are Dice- Sørensen Coefficient (DSC), Intersection over Union(IoU) of EEM and lumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The border-based metrics is Hausdorff dis- tance (HD) in the 95th percentage of EEM and lumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In sce- nario of real-time semantic segmentation, frame per second (FPS) is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Average performance in validation set Lumen Method Params FPS DSC IoU HD95 PE-UNetv1a 40863K 205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='198 PE-UNetv2b 40871K 187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='193 CoodFCN 475K 222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='292 MFAUNet 646166K 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='204 IVUSNetv1 16952K 179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='610 IVUSNetv2 39175K 69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='329 DMUNet-Dc 7489K 223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='880 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='306 DMUNet-Td 7489K 223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='940 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='890 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='347 DMUNet-Fe 7489K 223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='280 DLv3p-Rf 57949K 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='183 DLv3p-Xg 53419K 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='177 CSDN 1706K 151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='189 EEM Method 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='515 MFAUNet 646166K 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='930 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='246 IVUSNetv1 16952K 179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='803 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='243 IVUSNetv2 39175K 69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='326 DMUNet-Dc 7489K 223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='550 DMUNet-Td 7489K 223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='577 DMUNet-Fe 7489K 223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='940 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='890 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='582 DLv3p-Rf 57949K 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='224 DLv3p-Xg 53419K 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='213 CSDN 1706K 151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='246 a,bPretrained VGG16-Encoder without BN or with BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' c,d,eLoss is Dice, Tversky and Focal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' f,gBackbone is ResNet-101 or Xception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Implementation details In all experiments, batch size in training is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The size of in- put images is 900 × 900 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The Adam optimizer is used (a) (b) (c) (d) (e) (f) (g) (h) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Segmentation results from different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' (a) PE-UNetv1 (b) CoordFCN (c) MFAUNet (d) IVUSNetv2 (e) DMUNet- F (f) DLv3p-X (g) Our CSDN (h) Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Red contour is ground truth of EEM, and green is lumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Orange contour is prediction of EEM, and gold is lumen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' with an initial learning rate of 1 × 10−3 and with a weight decay of 1 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The learning rate schedule is used with a step size of 100, and learning rate is reduced by half in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The total epoch we set is 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The models are imple- mented based on the PyTorch, and trained with a NVIDIA GeFore GTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' A hybrid loss combining Focal loss and Dice loss is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' It is defined as: L = LF ocal + LDice For data augmentation, a combination of the following meth- ods is used in training: translation, rotation, scaling, shearing, left-right and up-down flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Because the channel size of input images is 3, channel swapping is also used and centeral frame is keep fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Deep supervision mechanism is further used to boost segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Quantitative results Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 1 reports the quantitative results and parameter size for U-net with pretrained VGG16 encoder (PE-UNet series), CoordFCN, MFAUNet, IVUS-Netv1, IVUS-Netv2, double modified U-net with three different losses, DeepLabv3+ with different backbones and our CSDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Comparing to Coord- FCN with 475K learnable parameters, CSDN imporves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='6% in DSC, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='4% in IoU, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='103 mm in HD95, for lumen segmentation, with only small increase in parameter size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' For EEM segmentation, CSDN imporves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='5% in DSC, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='6% in IoU, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='269 mm in HD95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Comparing to PE-UNet series, CSDN is six times less than in parameter size, but has similar or equal segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Although CSDN’s FPS performance is less than PE-UNet series, the GPU and CPU memory usage of PE-UNet series is large than CSDN’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' It’s impractical to directly deploy PE-UNet series in GPU with small memory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=', GTX 1050ti with 4GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Although MFAUNet is slightly worse than CSDN, the pa- rameter size of MFAUNet is huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' The size of model weights is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='6 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' It’s impractical to deploy it into medical devices with small main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Comparing to IVUS-Net series and DMUNet series, our CSDN surpasses in term of parameter size, region-based and border-based metrics with a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' DeepLabv3+ is a semantic segmentation model for general-purpose image segmentation with large latence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Although it processes image in real-time with GTX 3090, it’s very slow or even impractical with GTX 1050ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Our pro- posed CSDN is slightly worse than DeepLabv3+, however, it’s easy to deploy it into different GPU and it’s smooth to segment input high-definition images in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Qualitative results We visualize manual and predictive results for both lumen and EEM cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' CSDN’s prediction tends to be more close to boundary of lumen and EEM, while other meth- ods except strong but slow DeepLabv3+ with Xception as the backbone and MFAUNet with huge parameters have cases with small disconnected prediction regions or uneven con- tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' To demonstate the challenging task of IVUS image segmentation, the last row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 2 shows one of the most challenging cases where CSDN’s prediction can be a reason- able segmentation from non-expert perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Shadow ar- tifacts and peculiar bloodstream not only oblige learners to have capacity of modeling the global dependence, but also OOexpect these algorithms to focus on local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' CSDN with dual streams is suitable for handling with the above dilemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' CONCLUSIONS We have presented CSDN, a two-stream framework for effi- cient and real-time segmentation of high-definition IVUS im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' It combines shallow and deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' Treating low- level details and high-level semantics information separately enables learning a model to achieve high accuracy and high efficiency for accurate real-time segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' In the future, more experiments can be carried out on CSDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' COMPLIANCE WITH ETHICAL STANDARDS Informed consent was obtained from all individual partici- pants involved in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is supported, in part, by the National Natural Sci- ence Foundation of China (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 61771233).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFRT4oBgHgl3EQfyzjn/content/2301.13648v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': 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b/H9AyT4oBgHgl3EQfTPcY/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f6470637bd195ffef95063e44ae8ed5ac2bb5409280dc16ad1e36686b25bfecb +size 98231 diff --git a/INA0T4oBgHgl3EQfB__n/content/tmp_files/2301.01985v1.pdf.txt b/INA0T4oBgHgl3EQfB__n/content/tmp_files/2301.01985v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..474fac5c53404406fd7778bd822f8414b8b9c0ef --- /dev/null +++ b/INA0T4oBgHgl3EQfB__n/content/tmp_files/2301.01985v1.pdf.txt @@ -0,0 +1,755 @@ +arXiv:2301.01985v1 [math.CO] 5 Jan 2023 +Power Reducibility and Congruences +Rong-Hua Wang1 and Michael X.X. Zhong2 +1School of Mathematical Sciences +Tiangong University +Tianjin 300387, P.R. China +wangronghua@tiangong.edu.cn +2School of Science +Tianjin University of Technology +Tianjin 300384, P.R. China +zhong.m@tjut.edu.cn +Abstract. In this paper, a criterion on the power reducibility of holonomic +sequences is presented. As applications, we show Ap´ery numbers Ak and the +central Delannoy polynomials Dk(z) are both power reducible and present +series of congruences. For example, when p > 3 is a prime, we find that for +each r ∈ N, there is a p-adic integer cr such that +p−1 +� +k=0 +(2k + 1)2r+1Ak ≡ crp +(mod p3). +Keywords: power reducibility; congruence; Ap´ery number; Delannoy poly- +nomial. +1 +Introduction +In the 1990s, Wilf and Zeilberger [16–19] developed the WZ theory for han- +dling definite summations mechanically. Since then the mechanical proof +of combinatorial identities had received special attention. Zeilberger’s algo- +rithm, also known as the method of creative telescoping, is the core algorithm +in the WZ theory. Over the past three decades, extensive work has been +done around the generalizations and applications of Zeilberger’s algorithm. +The reduction-based approach is the one which gained much attention as it +separates the computation of telescopers and the corresponding certificates +and is thus more efficient compared to the original algorithm. +1 + +In the case of discrete functions, polynomial reduction was first intro- +duced in 2015 by Chen et al. [2] to modify the Abramov–Petkovˇsek reduc- +tion. The modified algorithm is more efficient and can be used to compute +minimal telescopers for bivariate hypergeometric terms. The first reduction- +based creative telescoping algorithm for more than two variables was pre- +sented by Chen et al. [1]. +In 2021, Hou, Mu and Zeilberger [5] presented another polynomial reduc- +tion process, which avoids the multiplicative decomposition needed in [2]. +This polynomial reduction was employed by Hou and Li [11] to derive new +hypergeometric identities, and was introduced into the q-rational case by +the authors [14] to prove and discover q-identities automatically. Recently, +the authors [15] generalized the polynomial reduction to the holonomic case. +This provides an algorithmic way to prove and discover new multi-sum iden- +tities. Especially, series for π involving the Domb numbers and the Franel +numbers were obtained. +The polynomial reduction in [5] was designed to derive infinite families +of supercongruences. More precisely, Hou et al. focused on hypergeometric +terms tk satisfying tk+1 +tk += a(k) +b(k), where a(k) = ±b(k + α) and b(β + k) = +±b(β − k) for some α, β ∈ K. Here K is a field of characteristic 0. Let +γ = β − α−1 +2 . When a(k) = b(k+α) (resp. a(k) = −b(k+α)), they provided +a criterion Theroem 4.2 (resp. Theorem 3.2) to show that after polynomial +reduction to (k − γ)m, one will obtain a linear combination of (k − γ)i with +0 ≤ i < deg a(k) , i and m are nonnegative integers sharing the same parity. +This means the polynomial reduction reduce the power and keep the form +of monomial at the same time. Such tk will be referred as power reducible +with respect to γ. Once this is the case, summations of (k + γ)mtk can be +simplified to summations of (k + γ)itk, which can be used to deduce new +identities from known ones. +In this paper, we will discuss the power reducibility of holonomic se- +quences. We provide a criterion (theorem 2.4) on deciding the power re- +ducibility utilizing the annihilators of a holonomic sequence, which can be +seen as a generalization of Hou et al.’s result for hypergeometric terms. +What’s more, our criterion uniforms the criteria in [5] and does not need +to introduce the constant α, β. Utilizing this power reducibility method, we +can derive new series of congruences . This paper was organized as follows. +In Section 2, we recall the process of polynomial reduction and presented +a criterion on the power reducibility for holonomic sequences. As applica- +tions, Section 3 is devoted to congruences involving the Ap´ery numbers or +2 + +the central Delannoy polynomials. +2 +Polynomial reduction and power reducibility +Let K be a field of characteristic 0. The annihilator of a sequence F(k) is +defined by +ann F(k) := +� +L = +J +� +i=0 +ai(k)σi ∈ K[k][σ] | L(F(k)) = 0 +� +, +(2.1) +where σ is the shift operator (that is, σF(k) = F(k + 1)). +A sequence +(F(k))∞ +k=0 is said to be holonomic (or, P-recursive) if ann F(k) ̸= {0}. We +call J in (2.1) the order of L, and the minimum order of L ∈ ann is called +the order of F(k). +For any operator L = �J +i=0 ai(k)σi with ai(k) ∈ K[k], the adjoint of L +is defined by +L∗ = +J +� +i=0 +σ−iai(k). +(2.2) +Then for any polynomial x(k) ∈ K[k], +L∗(x(k)) = +J +� +i=0 +ai(k − i)x(k − i). +From [10,15], we know once a holonomic sequence F(k) is given, one can +construct polynomials q(k) such that q(k)F(k) is summable. +Theorem 2.1. Suppose that (F(k))∞ +k=0 is a holonomic sequence and that +L = �J +i=0 ai(k)σi ∈ ann F(k) \ {0}. Then for any x(k) ∈ K[k], +L∗(x(k))F(k) = ∆ +� +− +J−1 +� +i=0 +ui(k)F(k + i) +� +, +(2.3) +where +ui(k) = +J−i +� +j=1 +ai+j(k − j)x(k − j), +i = 0, 1, 2, . . . , J − 1. +(2.4) +3 + +Note that, if identity (2.3) holds, summing over k from 0 to n − 1 on +both sides, we obtain +n−1 +� +k=0 +L∗(x(k))F(k) = +�J−1 +� +i=0 +ui(0)F(i) +� +− +�J−1 +� +i=0 +ui(n)F(n + i) +� +, +(2.5) +where ui is defined in (2.4). +Let the difference space corresponding to L defined as +SL = {L∗(x(k)) | x(k) ∈ K[k]}, +(2.6) +and denote by [p(k)]L = p(k) + SL the coset of a polynomial p(k). +Next, we try to characterize the dimension of the quotient space K[k]/SL. +Given a nonzero operator L = +J� +i=0 +ai(k)σi ∈ K[k][σ], let +bℓ(k) = +J +� +j=ℓ +�j +k +� +aJ−j(k + j − J) and d = max +0≤ℓ≤J{deg bℓ(k) − ℓ}. +(2.7) +For simplicity, we will call d in (2.7) the degree of L, written as d = deg(L) +when there is no confusion. Note that +f(s) = +J +� +ℓ=0 +[kd+ℓ](bℓ(k))sℓ +is a nonzero polynomial in s. Here [kd+ℓ](bℓ(k)) denotes the coefficient of +kd+ℓ in bℓ(k) and sℓ denotes the falling factorial defined by sℓ = s(s − +1) · · · (s − ℓ + 1). Let +RL = {s ∈ N | f(s) = 0}. +(2.8) +Then L is called degenerated if RL ̸= ∅. We may also say F(k) is degenerated +when L(F(k)) = 0 and RL ̸= ∅. +The degrees of L∗(x(k)) can be given as follows. +Lemma 2.2. +[15, Lemma 2.5] Let L = +J� +i=0 +ai(k)σi ∈ K[k][σ] \ {0} and +d = deg(L) as given by (2.7). Then for any nonzero polynomial x(k) ∈ K[k], +we have +deg L∗(x(k)) +� < d + deg x(k), +if L is degenerated and deg x(k) ∈ RL, += d + deg x(k), +otherwise. +4 + +The following theorem characterises the quotient space K[k]/SL. +Theorem 2.3. Let L = �J +i=0 ai(k)σi ∈ K[k][σ] \ {0}, d = deg(L) and RL +defined by (2.8). Then +K[k]/SL = ⟨[ki]L | i ∈ {0, 1, 2, . . . , d − 1} ∪ RL⟩ +(2.9) +Proof. We first consider the case when L is not degenerated, namely, RL = ∅. +By Lemma 2.2 and the Euclidean division algorithm, it is easy to see that +any polynomial Q(k) ∈ K[k] with m ≥ d can be decomposed as +Q(k) = +m−d +� +s=0 +usqs(k) + ˜q(k), +(2.10) +where us ∈ K, qs(k) ∈ SL, for 0 ≤ s ≤ m − d and ˜q(k) is a polynomial of +degree less than d. This proves (2.9). When L is degenerated, we have +Q(k) = +� +0≤s≤m−d +s/∈RL +usqs(k) + +� +0≤s≤m−d +s∈RL +vskd+s + ˜q(k), +(2.11) +where us, vs ∈ K, qs(k) ∈ SL, for 0 ≤ s ≤ m − d, and ˜q(k) is a polynomial +with deg ˜q(n) < d. This completes the proof. +Equality (2.10) (or (2.11)) is called the polynomial reduction on Q(k) +with respect to L. In general, one can characterize the degree but not the +structure of ˜q(k). By the following theorem, we will see that when Q(k) = +(k −β)m for some β ∈ K, m ∈ N, and the coefficients of L satisfy additional +conditions, the corresponding ˜q(k) is a linear combination of (k − β)j with +j has the same parity of m and j < d. +Theorem 2.4. Let L = �J +i=0 ai(k)σi ∈ K[k][σ] \ {0} and d = deg(L). +Suppose L is not degenerated and there exists γ ∈ K such that +ai(γ + k) = (−1)daJ−i(γ − k − J), +i = 0, 1, . . . , ⌊J +2 ⌋. +(2.12) +Then for any positive integer m, we have +[(k − γ)m]L ∈ ⟨[(k − γ)i]L | i ≡ m +(mod 2), 0 ≤ i < d⟩. +When conditions in Theorem 2.4 are satisfied, we say L is power reducible +with respect to γ. If L ∈ ann F(k) for some holonomic sequence F(k), one +may also say F(k) is power reducible with respect to γ when there is no +confusion. To prove the theorem, we first recall the following observation, +one can see [5, Lemma 3.1] for a simple proof. +5 + +Lemma 2.5. Let p(k) ∈ K[k] and γ ∈ K. Then the following two statements +are equivalent. +(1) p(γ + k) = p(γ − k) (p(γ + k) = −p(γ − k), respectively). +(2) p(k) is the linear combination of (k − γ)2i, i = 0, 1, . . . ((k − γ)2i+1, +i = 0, 1, . . ., respectively). +Proof of theorem 2.4 : For s ∈ N = {0, 1, 2, . . .}, take +xs(k) = (k − γ + J +2 )s and ps(k) = L∗(xs(k)) = +J +� +i=0 +ai(k − i)xs(k − i) (2.13) +Then it is easily checked that +xs(γ + k) = (−1)sxs(γ − k − J). +Thus +ps(γ + k) = +J +� +i=0 +ai(γ + k − i)xs(γ + k − i) += +J +� +i=0 +(−1)daJ−i(γ − k + i − J)(−1)sxs(γ − k + i − J) += (−1)d+s +J +� +i=0 +aJ−i(γ − k − (J − i))xs(γ − k − (J − i)) += (−1)d+s +J +� +i=0 +ai(γ − k − i)xs(γ − k − i) += (−1)d+sps(γ − k). +By Lemma 2.5, ps(k) is a linear combination of (k − γ)2i+1(resp. (k − γ)2i) +when d + s is odd(resp. even), i ∈ N. Since L is not degenerated, we know +deg ps = d + s. That is, if d is even, then +p2s(k) = +s+d/2 +� +i=0 +c2s,i(k − γ)2i, +p2s+1(k) = +s+d/2 +� +i=0 +c2s+1,i(k − γ)2i+1, +6 + +with constants cj,i ∈ K and c2s,s+d/2, c2s+1,s+d/2 ̸= 0; When m is even(resp. +odd), the polynomial reduction of (k − γ)m using p2s(k)(resp. +p2s+1(k)) +clearly leads to the conclusion. +If d is odd, then +p2s(k) = +s+(d−1)/2 +� +i=0 +˜c2s,i(k − γ)2i+1, +p2s+1(k) = +s+(d+1)/2 +� +i=0 +˜c2s+1,i(k − γ)2i, +with constants ˜cj,i ∈ K and ˜c2s,s+(d−1)/2, ˜c2s+1,s+(d+1)/2 ̸= 0. When m is +even(resp. odd), the polynomial reduction of (k − γ)m using p2s+1(k)(resp. +p2s(k)) also leads to the conclusion. +From the proof of Theorem 2.4, one can see apparently that if we multiple +xs(k) in (2.13) with a nonzero constant αs ∈ K∗, the conclusion still holds. +This fact is useful in the applications. +Theorem 2.6. When L = �J +i=0 ai(k)σi is power reducible with respect to +γ and d = deg(L). Let +xs(k) = αs · (k − γ + J +2 )s for any αs ∈ K∗. +Then for any positive integer m, there exist some ui, vj ∈ K such that +(k − γ)m = +� +0≤i 3 be prime. Then for each r ∈ N, there is a p-adic +integer cr only depending on r such that +p−1 +� +k=0 +(2k + 1)2r+1Ak ≡ crp +(mod p3). +(3.2) +To prove theorem 3.1, we fisrt need to determine the congruence prop- +ertities of L∗(x(k))Ak. +Lemma 3.2. Let L be given as in (3.1) and n a positive integer. Then +n−1 +� +k=0 +L∗(x(k))Ak = n3 (x(n − 1)An−1 − x(n − 2)An) . +(3.3) +for any polynomial x(k) ∈ K[k]. Here L∗ is the adjoint of L. +8 + +Proof. By Equality (2.5) and the fact u0(0)A0 + u1(0)A1 = 0, we have +n−1 +� +k=0 +L∗(x(k))A(k) = − (u0(n)An + u1(n)An+1) , +(3.4) +where u0(n) = n3x(n − 2) − (2n + 1)(17n2 + 17n + 5)x(n − 1) and u1(n) = +(n + 1)3x(n − 1). As L ∈ ann Ak, it is straightforward to check that for any +n ≥ 1 +(n + 1)3An+1 = (2n + 1)(17n2 + 17n + 5)An − n3An−1. +(3.5) +Substituting (3.5) into (3.4), we derive (3.3). +Corollary 3.3. Let L be as in (3.1). Then +n−1 +� +k=0 +L∗(x(k))Ak ≡ 0 +(mod n3) +(3.6) +for any polynomial x(k) ∈ Z[k]. +Since L in (3.1) is power reducible with respect to γ = − 1 +2. By the proof +of Theorem 2.6, if we set +xs(k) = 2s+1(k − γ + 2 +2)s = 2(2k + 3)s, +(3.7) +then L∗(xs(k)) is a linear combination of (2k + 1)i with i ≡ s + 1 (mod 2). +In fact, we can even prove the coefficients in the combination are all integers. +Lemma 3.4. Let L be as in (3.1) and xs(k) satisfy (3.7). Then +L∗(xs(k)) = −8(2k + 1)s+3 + +⌊(s+3)/2⌋ +� +j=1 +cj(2k + 1)s+3−2j, +(3.8) +where cj ∈ Z for all j = 1, 2, . . . , ⌊(s + 3)/2⌋. +9 + +Proof. For simplicity, let ℓ = 2k + 1. By the definition of L∗, we have +L∗(xs(k)) = +2 +� +i=0 +ai(k − i)xs(k − i) +=2(k + 1)3(2k + 3)s − 2(17k2 + 17k + 5)(2k + 1)s+1 + 2k3(2k − 1)s +=(ℓ + 1)3 +4 +(ℓ + 2)s − 1 +2(17ℓ2 + 3)ℓs+1 + (ℓ − 1)3 +4 +(ℓ − 2)s +=(ℓ + 1)3 +4 +s +� +j=0 +�s +j +� +2jℓs−j + (ℓ − 1)3 +4 +s +� +j=0 +�s +j +� +(−2)jℓs−j − ℓs(17ℓ3 + 3ℓ) +2 +=(ℓ3 + 3ℓ) +2 +s +� +j=0 +j even +�s +j +� +2jℓs−j + (3ℓ2 + 1) +2 +s +� +j=0 +j odd +�s +j +� +2jℓs−j − ℓs(17ℓ3 + 3ℓ) +2 += − 8ℓs+3 + +s +� +j=1 +j even +�s +j +� +2j−1ℓs−j(ℓ3 + 3ℓ) + +s +� +j=1 +j odd +�s +j +� +2j−1ℓs−j(3ℓ2 + 1) += − 8ℓs+3 + +⌊(s+3)/2⌋ +� +j=1 +cjℓs+3−2j +where cj ∈ Z for all j = 1, 2, . . . , s. +Proof of theorem 3.1: Firstly one can do the polynomial reduction on +(2k + 1)2r+1 with respect to the L given by (3.1). Notice that if we take +xs(k) = 2(2k + 3)s as defined in (3.7), by the expression for L∗(xs(k)) in +(3.8), we have +(2k + 1)2r+1 = +2r−2 +� +s=0 +vs +2us L∗(xs(k)) + νr +2µr (2k + 1), +(3.9) +for some µr, us ∈ N, νr, vs ∈ Z. Multiply both sides of the above identity +with Ak and then summing over k from 0 to p−1. Since p > 3, congruences +(3.13) follows from Corollary 3.3 and the fact that +p−1 +� +k=0 +(2k + 1)Ak ≡ p +(mod p3) +which was presented by Z.-W. Sun [7]. +Let F(k) = (−1)kAk. By Zeilberger’s algorithm, we find that +L = a2(k)σ2 + a1(k)σ + a0(k) ∈ ann F(k), +(3.10) +10 + +where +a2(k) = (k + 2)3, a1(k) = (2k + 3)(17k2 + 51k + 39), a0(k) = (k + 1)3. +It is easy to check that L in (3.10) is also power reducible with respect to +γ = − 1 +2. Then by similar discussions, we will derive that +n−1 +� +k=0 +L∗(x(k))(−1)kAk = n3 (x(n − 1)F(n − 1) − x(n − 2)F(n)) . +for any polynomial x(k) ∈ K[k] and that +L∗(xs(k)) = 9(2k + 1)s+3 + +⌊(s+3)/2⌋ +� +j=1 +cj(2k + 1)s+3−2j, +where cj ∈ Z for all j = 1, 2, . . . , ⌊(s + 3)/2⌋. +At this stage, it is straightforward to prove that for each r ∈ N, there is +a p-adic integer cr only depending on r such that +p−1 +� +k=0 +(2k + 1)2r+1Ak ≡ crp +�p +3 +� +(mod p3) +(3.11) +with the fact that for any prime p > 3 +p−1 +� +k=0 +(2k + 1)(−1)kAk ≡ p +�p +3 +� +(mod p3), +which was conjectured by Z.-W. Sun [7] and confirmed by Guo and Zeng [3]. +We remark that congruence (3.11) was conjectured by Z.-W. Sun [9] +and firstly proved by W. Xia and Z.-W. Sun [13] recently. +In fact, the +proofs in [13] also employ the method of polynomial reduction by taking +xs(k) = ks. However after a step of polynomial reduction to (2k + 1)2r+1 +with these xs(k), the obtained polynomial with lower degree is no-longer a +linear combination of (2k + 1)2i+1, 0 ≤ i < r, and hence it is complicated to +show that [(2k + 1)2r+1] = [(2k + 1)]. +3.2 +Congruences involving the central Delannoy numbers +The central Delannoy polynomials Dk(z), k ∈ N, are defined by +Dk(z) = +k +� +i=0 +�k +i +��k + i +i +� +zi. +11 + +One can consult [8] for several interesting congruences involving Dk(z). Note +that Dk = Dk(1) is the central Delannoy numbers which arise in many +enumeration problems in combinatorics. One can see that Dk(0) = 1 for any +k ∈ N, we will assume x ̸= 0 in the following text. Zeilberger’s algorithm +leads to +L = (k + 2)σ2 − (2k + 3)(2z + 1)σ + (k + 1) ∈ ann Dk(z). +(3.12) +One can check that L in (3.12) is not degenerated and d = deg(L) = 1. It is +straight forward to check that L is power reducible with respect to γ = − 1 +2. +Then we know [(2k + 1)2r+2]L = [k0]L by theorem 2.4. +Theorem 3.5. Let p be an odd prime. Then for each r ∈ N, there is a +p-adic integer cr such that +p−1 +� +k=0 +(2k + 1)2r+2Dk(z) ≡ cr +p−1 +� +k=0 +Dk(z) +(mod p). +(3.13) +Proof. The conclusion is clearly true when p divides z. We only need to +consider the case when gcd(p, z) = 1. Firstly, one can show that +n−1 +� +k=0 +L∗(xs(k))Dk(z) = n(xs(n − 1)Dn−1(z) − xs(n − 2)Dn(z)). +for any polynomial xs(k) ∈ K[k]. Then for any z ∈ Z \ {0, −1} and any +polynomial xs(k) ∈ Z[k], we have +n−1 +� +k=0 +L∗(xs(k))Dk(z) ≡ 0 +(mod n). +(3.14) +Similar to the proof of Lemma 3.4, we will arrive at +L∗(xs(k)) = −4z(2k + 1)s+1 + +s +� +j=1 +�s +j +� +2j+1(2k + 1)s−j, +where xs(k) is given by (3.7). Then +(2k + 1)2r+2 = +2r+1 +� +s=0 +vs +(−4z)us L∗(xs(k)) + +νr +(−4z)µr (2k + 1)0, +for some µr, us ∈ N, νr, vs ∈ Z. Multiply both sides of the above identity +with Dk(z) and then summing over k from 0 to p − 1, Congruences (3.13) +follows immediately from (3.14). +12 + +Corollary 3.6. Let p > 3 be a prime. Then for each r ∈ N, there is a +p-adic integer cr such that +p−1 +� +k=0 +(2k + 1)2r+2Dk ≡ cr +�−1 +p +� +(mod p). +Proof. This can be derived from Theorem 3.5 with x = 1 and the congruence +derived in [6] +p−1 +� +k=0 +Dk ≡ +�−1 +p +� +(mod p). +References +[1] S. Chen, Q.-H. Hou, H. Huang, G. Labahn and R.-H. Wang. Construct- +ing minimal telescopers for rational functions in three discrete variables. +Adv. in Appl. Math., 141 (2022), 102389. +[2] S. Chen, H. Huang, M. Kauers and Z. Li. +A modified Abramov- +Petkovsek reduction and creative telescoping for hypergeometric terms. +In ISSAC ’15, pages 117–124, 2015. ACM. +[3] V.J.W. Guo and J. Zeng. Proof of some conjectures of Z.-W. Sun on +congruences for Ap´ery polynomials. +J. Number Theory, 132 (2012), +1731–1740. +[4] Q.-H. Hou and K. Liu. Congurences and telescopings of P-recursive +sequences. J. Difference Equ. Appl., 27 (2021), 686–697. +[5] Q.-H. Hou, Y.-P. Mu and D. Zeilberger. +Polynomial reduction and +supercongruences. J. Symbolic Comput., 103 (2021), 127–140. +[6] Z.-W. Sun. On Delannoy numbers and Schr¨oder numbers. J. Number +Theory, 131 (2011), 2387–2397. +[7] Z.-W. Sun. On sums of Ap´ery polynomials and related congruences. J. +Number Theory, 132 (2012), 2673–2699. +[8] Z.-W. Sun. Congruences involving generalized central trinomial coeffi- +cients. Sci. China Math., 57(2014), 1375–1400. +13 + +[9] Z.-W. Sun. Congruences involving gn(x) = �n +k=0 +�n +k +�2�2k +k +� +xk. Ramanu- +jan J., 40 (2016), 511–533. +[10] J. van der Hoeven. Creative telescoping using reductions. Preprint:hal- +01773137v2, June 2018. +[11] Q.-H. Hou and G.-J. Li. Gosper summability of rational multiples of +hypergeometric terms. J. Difference Equ. Appl., 27 (2021), 1723–1733. +[12] A. van der Poorten. A proof that Euler missed · · · Ap´ery’s proof of the +irrationality of ζ(3). Math. Intelligencer, 1 (1979), 195–203. +[13] W. Xia and Z.-W Sun. +On congruences involving Ap´ery numbers. +arXiv:2212.09455. +[14] R.-H. Wang and M.X.X. Zhong. q-Rational reduction and q-analogues +of series for π. J. Symbolic Comput., 116 (2023), 58–71. +[15] R.-H. Wang and M.X.X. Zhong. Polynomial reduction for holonomic se- +quences and applications in π-series and congruences. arXiv:2205.11129. +[16] H.S. Wilf and D. Zeilberger. An algorithmic proof theory for hypergeo- +metric (ordinary and “q”) multisum/integral identities. Invent. Math., +108 (1992), 575–633. +[17] D. Zeilberger. A holonomic systems approach to special function iden- +tities. Int. J. Comput. Appl. Math., 32 (1990), 321–368. +[18] D. Zeilberger. A fast algorithm for proving terminating hypergeometric +identities. Discrete Math., 80 (1990), 207–211. +[19] D. Zeilberger. The method of creative telescoping. J. Symbolic Comput., +11 (1991), 195–204. +14 + diff --git a/INA0T4oBgHgl3EQfB__n/content/tmp_files/load_file.txt b/INA0T4oBgHgl3EQfB__n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3cad739171f0a53edb49f496392b45a9f7cc5718 --- /dev/null +++ b/INA0T4oBgHgl3EQfB__n/content/tmp_files/load_file.txt @@ -0,0 +1,556 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf,len=555 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='01985v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='CO] 5 Jan 2023 Power Reducibility and Congruences Rong-Hua Wang1 and Michael X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Zhong2 1School of Mathematical Sciences Tiangong University Tianjin 300387, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' China wangronghua@tiangong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='cn 2School of Science Tianjin University of Technology Tianjin 300384, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' China zhong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='m@tjut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='cn Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' In this paper, a criterion on the power reducibility of holonomic sequences is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' As applications, we show Ap´ery numbers Ak and the central Delannoy polynomials Dk(z) are both power reducible and present series of congruences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' For example, when p > 3 is a prime, we find that for each r ∈ N, there is a p-adic integer cr such that p−1 � k=0 (2k + 1)2r+1Ak ≡ crp (mod p3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Keywords: power reducibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' congruence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Ap´ery number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Delannoy poly- nomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' 1 Introduction In the 1990s, Wilf and Zeilberger [16–19] developed the WZ theory for han- dling definite summations mechanically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Since then the mechanical proof of combinatorial identities had received special attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Zeilberger’s algo- rithm, also known as the method of creative telescoping, is the core algorithm in the WZ theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Over the past three decades, extensive work has been done around the generalizations and applications of Zeilberger’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' The reduction-based approach is the one which gained much attention as it separates the computation of telescopers and the corresponding certificates and is thus more efficient compared to the original algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' 1 In the case of discrete functions, polynomial reduction was first intro- duced in 2015 by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' [2] to modify the Abramov–Petkovˇsek reduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' The modified algorithm is more efficient and can be used to compute minimal telescopers for bivariate hypergeometric terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' The first reduction- based creative telescoping algorithm for more than two variables was pre- sented by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' In 2021, Hou, Mu and Zeilberger [5] presented another polynomial reduc- tion process, which avoids the multiplicative decomposition needed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' This polynomial reduction was employed by Hou and Li [11] to derive new hypergeometric identities, and was introduced into the q-rational case by the authors [14] to prove and discover q-identities automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Recently, the authors [15] generalized the polynomial reduction to the holonomic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' This provides an algorithmic way to prove and discover new multi-sum iden- tities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Especially, series for π involving the Domb numbers and the Franel numbers were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' The polynomial reduction in [5] was designed to derive infinite families of supercongruences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' More precisely, Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' focused on hypergeometric terms tk satisfying tk+1 tk = a(k) b(k), where a(k) = ±b(k + α) and b(β + k) = ±b(β − k) for some α, β ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Here K is a field of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Let γ = β − α−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' When a(k) = b(k+α) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' a(k) = −b(k+α)), they provided a criterion Theroem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='2) to show that after polynomial reduction to (k − γ)m, one will obtain a linear combination of (k − γ)i with 0 ≤ i < deg a(k) , i and m are nonnegative integers sharing the same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' This means the polynomial reduction reduce the power and keep the form of monomial at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Such tk will be referred as power reducible with respect to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Once this is the case, summations of (k + γ)mtk can be simplified to summations of (k + γ)itk, which can be used to deduce new identities from known ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' In this paper, we will discuss the power reducibility of holonomic se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' We provide a criterion (theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='4) on deciding the power re- ducibility utilizing the annihilators of a holonomic sequence, which can be seen as a generalization of Hou et al.’s result for hypergeometric terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' What’s more, our criterion uniforms the criteria in [5] and does not need to introduce the constant α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Utilizing this power reducibility method, we can derive new series of congruences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' This paper was organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' In Section 2, we recall the process of polynomial reduction and presented a criterion on the power reducibility for holonomic sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' As applica- tions, Section 3 is devoted to congruences involving the Ap´ery numbers or 2 the central Delannoy polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' 2 Polynomial reduction and power reducibility Let K be a field of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' The annihilator of a sequence F(k) is defined by ann F(k) := � L = J � i=0 ai(k)σi ∈ K[k][σ] | L(F(k)) = 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='1) where σ is the shift operator (that is, σF(k) = F(k + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' A sequence (F(k))∞ k=0 is said to be holonomic (or, P-recursive) if ann F(k) ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' We call J in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='1) the order of L, and the minimum order of L ∈ ann is called the order of F(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' For any operator L = �J i=0 ai(k)σi with ai(k) ∈ K[k], the adjoint of L is defined by L∗ = J � i=0 σ−iai(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='2) Then for any polynomial x(k) ∈ K[k], L∗(x(k)) = J � i=0 ai(k − i)x(k − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' From [10,15], we know once a holonomic sequence F(k) is given, one can construct polynomials q(k) such that q(k)F(k) is summable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Suppose that (F(k))∞ k=0 is a holonomic sequence and that L = �J i=0 ai(k)σi ∈ ann F(k) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Then for any x(k) ∈ K[k], L∗(x(k))F(k) = ∆ � − J−1 � i=0 ui(k)F(k + i) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='3) where ui(k) = J−i � j=1 ai+j(k − j)x(k − j), i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='4) 3 Note that, if identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='3) holds, summing over k from 0 to n − 1 on both sides, we obtain n−1 � k=0 L∗(x(k))F(k) = �J−1 � i=0 ui(0)F(i) � − �J−1 � i=0 ui(n)F(n + i) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='5) where ui is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Let the difference space corresponding to L defined as SL = {L∗(x(k)) | x(k) ∈ K[k]}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='6) and denote by [p(k)]L = p(k) + SL the coset of a polynomial p(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Next, we try to characterize the dimension of the quotient space K[k]/SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Given a nonzero operator L = J� i=0 ai(k)σi ∈ K[k][σ], let bℓ(k) = J � j=ℓ �j k � aJ−j(k + j − J) and d = max 0≤ℓ≤J{deg bℓ(k) − ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='7) For simplicity, we will call d in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='7) the degree of L, written as d = deg(L) when there is no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Note that f(s) = J � ℓ=0 [kd+ℓ](bℓ(k))sℓ is a nonzero polynomial in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Here [kd+ℓ](bℓ(k)) denotes the coefficient of kd+ℓ in bℓ(k) and sℓ denotes the falling factorial defined by sℓ = s(s − 1) · · · (s − ℓ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Let RL = {s ∈ N | f(s) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='8) Then L is called degenerated if RL ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' We may also say F(k) is degenerated when L(F(k)) = 0 and RL ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' The degrees of L∗(x(k)) can be given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' [15, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='5] Let L = J� i=0 ai(k)σi ∈ K[k][σ] \\ {0} and d = deg(L) as given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Then for any nonzero polynomial x(k) ∈ K[k], we have deg L∗(x(k)) � < d + deg x(k), if L is degenerated and deg x(k) ∈ RL, = d + deg x(k), otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' 4 The following theorem characterises the quotient space K[k]/SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Let L = �J i=0 ai(k)σi ∈ K[k][σ] \\ {0}, d = deg(L) and RL defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Then K[k]/SL = ⟨[ki]L | i ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' , d − 1} ∪ RL⟩ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' We first consider the case when L is not degenerated, namely, RL = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='2 and the Euclidean division algorithm, it is easy to see that any polynomial Q(k) ∈ K[k] with m ≥ d can be decomposed as Q(k) = m−d � s=0 usqs(k) + ˜q(k), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='10) where us ∈ K, qs(k) ∈ SL, for 0 ≤ s ≤ m − d and ˜q(k) is a polynomial of degree less than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' This proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' When L is degenerated, we have Q(k) = � 0≤s≤m−d s/∈RL usqs(k) + � 0≤s≤m−d s∈RL vskd+s + ˜q(k), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='11) where us, vs ∈ K, qs(k) ∈ SL, for 0 ≤ s ≤ m − d, and ˜q(k) is a polynomial with deg ˜q(n) < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='10) (or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='11)) is called the polynomial reduction on Q(k) with respect to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' In general, one can characterize the degree but not the structure of ˜q(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' By the following theorem, we will see that when Q(k) = (k −β)m for some β ∈ K, m ∈ N, and the coefficients of L satisfy additional conditions, the corresponding ˜q(k) is a linear combination of (k − β)j with j has the same parity of m and j < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Let L = �J i=0 ai(k)σi ∈ K[k][σ] \\ {0} and d = deg(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Suppose L is not degenerated and there exists γ ∈ K such that ai(γ + k) = (−1)daJ−i(γ − k − J), i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' , ⌊J 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='12) Then for any positive integer m, we have [(k − γ)m]L ∈ ⟨[(k − γ)i]L | i ≡ m (mod 2), 0 ≤ i < d⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' When conditions in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='4 are satisfied, we say L is power reducible with respect to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' If L ∈ ann F(k) for some holonomic sequence F(k), one may also say F(k) is power reducible with respect to γ when there is no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' To prove the theorem, we first recall the following observation, one can see [5, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='1] for a simple proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Let p(k) ∈ K[k] and γ ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Then the following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' (1) p(γ + k) = p(γ − k) (p(γ + k) = −p(γ − k), respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' (2) p(k) is the linear combination of (k − γ)2i, i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' ((k − γ)2i+1, i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=', respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Proof of theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='4 : For s ∈ N = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' }, take xs(k) = (k − γ + J 2 )s and ps(k) = L∗(xs(k)) = J � i=0 ai(k − i)xs(k − i) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='13) Then it is easily checked that xs(γ + k) = (−1)sxs(γ − k − J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Thus ps(γ + k) = J � i=0 ai(γ + k − i)xs(γ + k − i) = J � i=0 (−1)daJ−i(γ − k + i − J)(−1)sxs(γ − k + i − J) = (−1)d+s J � i=0 aJ−i(γ − k − (J − i))xs(γ − k − (J − i)) = (−1)d+s J � i=0 ai(γ − k − i)xs(γ − k − i) = (−1)d+sps(γ − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='5, ps(k) is a linear combination of (k − γ)2i+1(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' (k − γ)2i) when d + s is odd(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' even), i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Since L is not degenerated, we know deg ps = d + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' That is, if d is even, then p2s(k) = s+d/2 � i=0 c2s,i(k − γ)2i, p2s+1(k) = s+d/2 � i=0 c2s+1,i(k − γ)2i+1, 6 with constants cj,i ∈ K and c2s,s+d/2, c2s+1,s+d/2 ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' When m is even(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' odd), the polynomial reduction of (k − γ)m using p2s(k)(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' p2s+1(k)) clearly leads to the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' If d is odd, then p2s(k) = s+(d−1)/2 � i=0 ˜c2s,i(k − γ)2i+1, p2s+1(k) = s+(d+1)/2 � i=0 ˜c2s+1,i(k − γ)2i, with constants ˜cj,i ∈ K and ˜c2s,s+(d−1)/2, ˜c2s+1,s+(d+1)/2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' When m is even(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' odd), the polynomial reduction of (k − γ)m using p2s+1(k)(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' p2s(k)) also leads to the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' From the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='4, one can see apparently that if we multiple xs(k) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='13) with a nonzero constant αs ∈ K∗, the conclusion still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' This fact is useful in the applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' When L = �J i=0 ai(k)σi is power reducible with respect to γ and d = deg(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Let xs(k) = αs · (k − γ + J 2 )s for any αs ∈ K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INA0T4oBgHgl3EQfB__n/content/2301.01985v1.pdf'} +page_content=' Then for any positive integer m, there exist some ui, vj ∈ K such that (k − γ)m = � 0≤i