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1
+ XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE
2
+ FIPS Compliant Quantum Secure Communication
3
+ using Quantum Permutation Pad
4
+ Abstract—Quantum computing has entered fast development
5
+ track since Shor’s algorithm was proposed in 1994. Multi-cloud
6
+ services of quantum computing farms are currently available. One
7
+ of which, IBM quantum computing, presented a road map
8
+ showing their Kookaburra system with over 4158 qubits will be
9
+ available in 2025. For the standardization of Post-Quantum
10
+ Cryptography or PQC, the National Institute of Standards and
11
+ Technology or NIST recently announced the first candidates for
12
+ standardization with one algorithm for key encapsulation
13
+ mechanism (KEM), Kyber, and three algorithms for digital
14
+ signatures. NIST has also issued a new call for quantum-safe
15
+ digital signature algorithms due June 1, 2023. This timeline shows
16
+ that FIPS-certified quantum-safe TLS protocol would take a
17
+ predictably long time. However, ‘steal now, crack later’ tactic
18
+ requires protecting data against future quantum threat actors
19
+ today. NIST recommended the use of a hybrid mode of TLS 1.3
20
+ with its extensions to support PQC. The hybrid mode works for
21
+ certain cases but FIPS certification for the hybridized
22
+ cryptomodule might still be required. This paper proposes to take
23
+ a nested mode to enable TLS 1.3 protocol with quantum-safe data,
24
+ which can be made available today and is FIPS compliant. We
25
+ discussed the performance impacts of the handshaking phase of
26
+ the nested TLS 1.3 with PQC and the symmetric encryption phase.
27
+ The major impact on performance using the nested mode is in the
28
+ data symmetric encryption with AES. To overcome this
29
+ performance reduction, we suggest using quantum encryption
30
+ with a quantum permutation pad for the data encryption with a
31
+ minor performance reduction of less than 10%.
32
+ Keywords—quantum communication, quantum encryption,
33
+ quantum decryption, quantum security, secure communication,
34
+ QPP, FIPS, TLS 1.3.
35
+ I. INTRODUCTION
36
+ Peter Shor proposed his celebrated quantum algorithm in
37
+ 1994 [1], which solves the NP-hard problem of prime integer
38
+ factorization in polynomial time. At its beginning, quantum
39
+ computing,
40
+ especially
41
+ universal
42
+ gate-based
43
+ quantum
44
+ computing, experienced a slow development phase for about
45
+ two decades. In 2019, Arute et al. from Google claimed
46
+ Quantum Supremacy with their 53-qubits Sycamore processor
47
+ [2]. This marked the start of global quantum computing race.
48
+ Since IBM released their 5-qubit quantum computer for public
49
+ access with Qiskit tool in 2017, IBM recently announced their
50
+ 433-qubit quantum computer and plan to double its qubits every
51
+ year for 2023 to reach over 4,000-qubits by 2025, outlined in
52
+ their development roadmap [3].
53
+ The fundamental shift from classical computing to quantum
54
+ computing is the shift in computing algebra from the classical
55
+ Boolean algebra to linear algebra, used in quantum computing.
56
+ That is, from classical logic gates, implemented in CPU, to
57
+ quantum logic gates to be implemented in QPU. That means,
58
+ quantum computing indicates a quadratic speedup of classical
59
+ computers or �2��2, where � denotes the number of qubits of a
60
+ QPU. This exponential computing power would break today’s
61
+ RSA-2048 in 10 seconds if a fault tolerant quantum computer
62
+ reaches 4099 qubits, while a classical computer would take 300
63
+ trillion years. Mosca predicted that there is a “1/2 chance of
64
+ breaking RSA-2048 by 2031” [4]. The year from quantum
65
+ computing to break classical public key RSA has been called
66
+ the Year to Quantum threat or Y2Q.
67
+ Symmetric cryptography such as the well-known Advanced
68
+ Encryption Standard or AES also suffered quadratic speedup of
69
+ the best quantum attack on the key space using Grover’s
70
+ algorithm proposed by Grover in 1996 [5]. That requires the
71
+ key length to be doubled in comparison with the equivalent
72
+ classical security level. For example, the classical AES-128
73
+ would be replaced by AES-256 for quantum security.
74
+ It has been well-understood that the upcoming quantum
75
+ computing systems will destroy the foundation of classical
76
+ public key infrastructure or PKI for both key establishment
77
+ such as RSA, Diffie-Hellman or elliptical curve Diffie-Hellman
78
+ and digital signature such as Digital Signature Algorithm or
79
+ DSA. National Institute of Standards and Technology or NIST
80
+ has announced its standardization process of Post-Quantum
81
+ Cryptography or PQC in November of 2017. In July 2022 [6],
82
+ NIST announced the lattice-based Kyber [7] to be its first
83
+ standardized key encapsulation mechanism or KEM. And
84
+ lattice-based Dillithium [8] Falcon [9], as well as hash-based
85
+ SPHINCS+ [10] to be first standardized digital signature
86
+ schemes. Other KEM candidates BIKE [11], Classic McEliece
87
+ [12], HQC [13], and SIKE [14] moved to the 4th round to be
88
+ considered further. NIST has also announced its reopening
89
+ Alex He
90
+ Quantropi Inc.
91
+ Ottawa, Canada
92
+ alex.he@quantropi.com
93
+ Dafu Lou
94
+ Quantropi Inc.
95
+ Ottawa, Canada
96
+ dafu.lou@quantropi.com
97
+ Eric She
98
+ DLS Technology Corporation
99
+ Ottawa, Canada
100
+ eshe@dlstech.com
101
+ Shangjie Guo
102
+ FinQ Tech Inc.
103
+ College Park, Maryland
104
+ sguo@finq.tech
105
+ Hareesh Watson
106
+ DLS Technology Corporation
107
+ Ottawa, Canada
108
+ hwatson@dlstech.com
109
+ Sibyl Weng
110
+ DLS Technology Corporation
111
+ Ottawa, Canada
112
+ sweng@dlstech.com
113
+ Maria Perepechaenko
114
+ Quantropi Inc.
115
+ Ottawa, Canada
116
+ maria.perepechaenko@quantropi.com
117
+
118
+ Randy Kuang
119
+ Quantropi Inc.
120
+ Ottawa, Canada
121
+ randy.kuang@quantropi.com
122
+ ORCID: 000-0002-5567-2192-
123
+ 0002-5567-2192
124
+ OORCID: RCID iD: 0000-0002-
125
+
126
+ submissions for digital signature standardization due Jaune
127
+ 2023 [15].
128
+ In 2022 some PQC algorithms such as Rainbow (a digital
129
+ signature scheme based on Multivariate Public Key
130
+ Cryptography or MPKC) [16] and Supersingular Isogeny
131
+ Diffie-Hellman protocol or SIDH [17, 18] proved to be
132
+ vulnerable to classical attacks. Another very interesting
133
+ cryptoanalysis was reported in the late 2022 by Wenger et al.
134
+ [19]. The team used transformers (a deep learning model) to
135
+ develop an attack on certain lattice-based schemes. They
136
+ noticed that the basic equation system for Learning With Error
137
+ or LWE can be expressed as linear regression used in machine
138
+ learning. If trained transformers, especially combined with
139
+ quantum computational advantage, can solve the Short Vector
140
+ Problem, then PQC algorithms face an enormous challenge.
141
+ Recall that most of the PQC standard schemes are lattice-based.
142
+ This issue is especially concerning since estimation on Y2Q has
143
+ not taken the power of quantum machine learning into account.
144
+ Some recent novel PQC algorithms were proposed by
145
+ Kuang, Perepechaenko, and Barbeau for KEM [20, 21] and
146
+ digital signature [22, 23, 24, 25], based on NP-complete
147
+ Modular Diophantine Equation Problem or MDEP. These
148
+ novel schemes share a foundation which we call Multivariate
149
+ Polynomial Public Key or MPPK. MPPK is built on two vector
150
+ spaces: a linear multivariate vector space { ��, … , �� }
151
+ containing noise variables used for obscurity and polynomial
152
+ vector space { 1, ��, … , ���� } containing secret message
153
+ variable. MPPK offers small parameter sizes at the level of
154
+ hundreds of bytes for key, ciphertext, and signature and
155
+ outperforms NIST finalists in key generation, encryption,
156
+ decryption, as well as signing and signature verification. They
157
+ could be considered as generic PQC algorithms for a wide range
158
+ of devices and systems.
159
+ NIST recommended using a hybrid scheme for quantum
160
+ resistant TLS 1.3 [26]. By leveraging the keyShare extension of
161
+ TLS 1.3, one can combine a NIST-approved classical key
162
+ establishment algorithm such as ECCDH in TLS 1.3 with one
163
+ or more PQC KEM algorithms and a NIST-approved digital
164
+ signature algorithm such as DSA with a list of PQC digital
165
+ signature algorithms for a chain signatures. This hybrid scheme
166
+ offers crypto agility as the ongoing process of PQC
167
+ standardization and cryptanalysis. However, this hybrid
168
+ scheme brings two potential limitations: 1) the hybrid TLS 1.3
169
+ may still require FIPS certification although its core
170
+ cryptomodule is certified in classical TLS 1.3. In general, the
171
+ FIPS certification comes at a cost for both time and money,
172
+ although it would be possible to receive the certificate; 2) The
173
+ certified hybrid TLS 1.3 must be integrated with an application
174
+ for a quantum resistant service. But in some cases, this
175
+ integration may be difficult for some applications running
176
+ inside web browsers.
177
+ This paper proposes a new hybrid scheme by nesting PQC
178
+ inside classical TLS 1.3, creating nested TLS 1.3, to overcome
179
+ the limitations in the above hybrid scheme. The nested TLS 1.3
180
+ does not require a new FIPS certification because it does not
181
+ change the existing certified TLS. Moreover, it supports TLS
182
+ 1.3 as well as any certified TLS such as TLS 1.2 or even earlier.
183
+ One drawback of the nested scheme is that it may reduce the
184
+ performance of the encryption and decryption operations
185
+ because the transmitted data would be encrypted twice if AES
186
+ is used for both encryptions. To minimize the performance
187
+ impact, we propose to use quantum encryption with a Quantum
188
+ Permutation Pad algorithm or QPP [27, 28]. The QPP will be
189
+ used to encrypt the raw data first, producing a quantum-
190
+ encrypted message used as a TLS 1.3 message to be encrypted
191
+ with AES.
192
+ II. FIPS COMPLIANT QUANTUM ENCRYPTION WITH QPP
193
+ In this section, we first introduce the concept of a nested TLS
194
+ 1.3 protocol, that uses quantum-safe cryptosystems. The nested
195
+ TLS 1.3 allows for a smooth transition from classical era to
196
+ quantum era while maintaining FIPS compliance. We then
197
+ discuss the nested TLS 1.3 handshake process. Next, we
198
+ consider the symmetric data encryption with QPP for
199
+ considerations of both performance and security.
200
+
201
+ A. Nested TLS 1.3 with PQC in TLS Handshaking Proccess
202
+ The proposed nested TLS 1.3 is illustrated in Fig. 1. The
203
+ figure illustrates an Open Systems’ Interconnection model
204
+ (OSI-model) consisting of 7 layers: physical, data link, internet,
205
+ transport,
206
+ session,
207
+ presentation,
208
+ and
209
+ application.
210
+ The
211
+ functionality of each layer is illustrated in [29]. On the right-
212
+ hand side, we mark the corresponding layers in TCP/IP model.
213
+ The existing TLS cryptomodule is in the transport layer, while
214
+ the nested TLS is in the application layer. White arrows denote
215
+ a FIPS certified TLS 1.3 for clientHello request from the client
216
+ and serverHello response from the server to establish a shared
217
+ session for session data encryption and decryption. The NIST-
218
+ Approved key agreement protocol, ECCDH, is used for forward
219
+ secrecy in the existing TLS. The RSA algorithm is excluded
220
+ from the key agreement protocol. RSA, DSA, and ECDSA are
221
+ paired with hash functions for digital signature in the existing
222
+ TLS. The current conventional FIPS certified TLS
223
+ cryptomodule will be vulnerable to quantum computing attacks
224
+ once the fault tolerable quantum computers are available.
225
+ However, the "steal now crack later" tactics are already in use,
226
+ meaning all encrypted data today is at risk. Immediate action is
227
+ imperative to protect data against future quantum threat actors.
228
+ If sensitive information requires to remain secret for over 10
229
+ years, then it would not be wise to wait for the FIPS certified
230
+ TLS 1.3 with quantum resistance.
231
+ The proposed nested TLS 1.3 with PQC cryptographic
232
+ modules is independent from the FIPS certified TLS
233
+ cryptomodule in a sense that packets from the nested TLS 1.3
234
+ become data packets for the FIPS certified TLS. Since the outer
235
+ classical TLS cryptomodule is not altered, the nested TLS 1.3
236
+ does not violate the FIPS certification. This solution can be
237
+ considered as a promising FIPS compliant TLS 1.3 for quantum
238
+ security. The nested TLS 1.3 can be used to turn “steal now,
239
+ crack later” into “steal now, safe forever”.
240
+ Nested TLS 1.3 is based on the Open Quantum Safe or OQS
241
+ OpenSSL to support PQC KEM algorithms such as the NIST
242
+ finalists Kyber and Saber, as well as MPPK [20] and
243
+ Homomorphic Polynomial Public Key further evolved from
244
+ MPPK [30]. For the digital signatures, nested TLS 1.3 supports
245
+ PQC digital signature algorithms such as NIST finalists Falcon,
246
+
247
+ Dilithium, Rainbow, as well as MPPK/DS to be submitted to
248
+ NIST for standardization in 2023 [22].
249
+
250
+ B. Performance of a Nested TLS 1.3 Handshake
251
+ We tested the performance of a nested TLS1.3 handshake
252
+ in a local machine on a 16-core Intel®Core™ i7-10700 CPU at
253
+ 2.90 GHz system for all the measured primitives. Fig. 2
254
+ illustrates the performance of TLS 1.3 handShake for each pair
255
+ of KEM and digital signature schemes in terms of NIST
256
+ security levels. In general, Rainbow digital signature
257
+ demonstrates the worst performances for TLS handshake with
258
+ about 500 connections/second at all NIST security level I, 65
259
+ connections/second at all NIST security level III, and 30
260
+ connections/second at all NIST security level V.
261
+
262
+ By pairing MPPK/DS with Saber, Kyber, MPPK KEM, and
263
+ HPPK, the MPPK/DS scheme outperforms digital signature
264
+ schemes Falcon and Dilithium over 30% at security level I, over
265
+ 35% at security level III, and 40% at security level V,
266
+ respectively. On the other hand, pairing MPPK KEM and
267
+ HPPK with NIST digital signature finalists Falcon and
268
+ Dilithium, as well as MPPK/DS, the pairs of MPPK KEM with
269
+ MPPK/DS and HPPK with MPPK/DS outperform NIST
270
+ finalists Falcon and Dilithium. HPPK pairing with MPPK/DS
271
+ demonstrates a slightly better performance than MPPK KEM
272
+ pairing with MPPK/DS, about 10% for all three security levels.
273
+ Fig. 2 also demonstrates that the average TLS 1.3
274
+ handShake can be completed at sub-million second. For
275
+ example, MPPK/DS paired with MPPK KEM and HPPK would
276
+ establish about 5000 TLS 1.3 connections per second which
277
+ gives 0.2 ms/connection. In an actual cloud environment, the
278
+ performance would be reduced due to the network latency. A
279
+ typical network latency is at 10 ms level, so a sub-million
280
+ second processing time contribute no impact on a practical TLS
281
+ 1.3 handShake. That means, a nested TLS 1.3 inside the
282
+ existing TLS cryptomodule would not impact the overall
283
+ performance if we consider the fact that there is only one
284
+ handshaking per session.
285
+
286
+ C. Nested TLS 1.3 with PQC in Symmetric Encryption
287
+ After the handshake process of a nested TLS is complete,
288
+ communication peers establish a shared session key for
289
+ symmetric encryption during the session. Undoubtedly, the
290
+ NIST-Approved AES-256 can be used for data encryption in
291
+ the nested TLS, and the produced ciphertext would be
292
+ Quantum Encryption
293
+ Encryption
294
+ 1. ClientHello 
295
+
296
+
297
+
298
+
299
+
300
+
301
+
302
+
303
+
304
+
305
+ 
306
+
307
+
308
+ 
309
+
310
+
311
+
312
+
313
+ 2. ServerHello 
314
+
315
+
316
+
317
+ Nested TLS 1.3
318
+ 1. ClientHello 
319
+
320
+
321
+
322
+ 2. ServerHello 
323
+
324
+
325
+
326
+ Existing TLS
327
+ Figure 21. Illustration of nested TLS with PQC and quantum
328
+ encryption in OSI and TCP/IP models. The colorful OSI model is
329
+ taken from literature [29]. The TCP/IP model is indicated on the right-
330
+ hand side. The white arrows refer to the existing certified TLS 1.x and
331
+ green arrows denote the nested TLS 1.3 with quantum resistant
332
+ cryptographic modules.
333
+ Figure 12. TLS 1.3 handshake performances (connections/second) are
334
+ illustrated in terms of PQC KEM algorithms paired with PQC digital
335
+ signature algorithms. The x-axis is marked with KEM schemes and y-
336
+ axis represents handshaking connections per second. Digital signature
337
+ schemes are listed in the legends. NIST security level I, III, and V are
338
+ associated with the top, middle, and bottom graphs respectively.
339
+
340
+ Connectior
341
+ 2000
342
+ 1000
343
+ 66
344
+ 65
345
+ 66
346
+ 65
347
+ 0
348
+ Saber
349
+ Kyber768
350
+ MPPKL 416
351
+ HPPKL 214
352
+ Key ExchangeMechanism
353
+ TLs 1.3 handshake performance of different asymmetric algorithms - security level 5
354
+ Signature algorithm
355
+ second
356
+ 4188
357
+ 4000
358
+ Falcon-1024
359
+ 3584
360
+ Dilithium5
361
+ 3025
362
+ per
363
+ 3000
364
+ 2893
365
+ Rainbow-V-Classic
366
+ 2506
367
+ Connections
368
+ 2432
369
+ 2239
370
+ 2225
371
+ MPPKDS 623
372
+ 2089
373
+ 2000
374
+ 1000
375
+ 30
376
+ 30
377
+ 31
378
+ 30
379
+ 0
380
+ FireSaber
381
+ Kyber1024
382
+ MPPKL 417
383
+ HPPKL_215
384
+ KeyExchangeMechanismTLS1.3handshakeperformanceofdifferentasymmetricalgorithms-securitylevel1
385
+ 5219
386
+ Signaturealgorithm
387
+ 5000
388
+ 4762
389
+ 4709
390
+ Falcon-512
391
+ Dilithium2
392
+ persecond
393
+ 4000
394
+ 3852
395
+ 3983
396
+ 3943
397
+ 3849
398
+ Rainbow-l-Classic
399
+ 3782
400
+ MPPKDS_223
401
+ 3000
402
+ 3047
403
+ Connections
404
+ 2000
405
+ 1000
406
+ 529
407
+ 477
408
+ 480
409
+ 508
410
+ 0
411
+ LightSaber
412
+ Kyber512
413
+ MPPKL415
414
+ HPPKL_213
415
+ KeyExchangeMechanism
416
+ TLs 1.3 handshake performance of different asymmetric algorithms - security level 3
417
+ 5000
418
+ 5037
419
+ persecond
420
+ Signature algorithm
421
+ 4591
422
+ 4277
423
+ Dilithium3
424
+ 4000
425
+ Rainbow-lll-Classic
426
+ 3319
427
+ 3140
428
+ MPPKDS 423
429
+ 3000
430
+ 2778
431
+ 2776
432
+ SUP
433
+ P
434
+ Et
435
+ Frgn
436
+ da
437
+ ho
438
+ on
439
+ on
440
+ abl
441
+ CO
442
+ UD
443
+ CT
444
+ ep
445
+ min
446
+ RP,
447
+ ca
448
+ dr
449
+ ne
450
+ 2
451
+ e
452
+ .HnSession7.Application
453
+ tion4.Transport6:Data-
454
+ Google
455
+ Networkprocessto application
456
+ 7.Application
457
+ DNS,WWW/HTTP,P2P,EMAIL/POP,SMTP,Telnet,FTP
458
+ Presentation
459
+ Datarepresentationand encryption
460
+ Recognizingdata:HTML,DOC,JPEG,MP3,AvI,Sockets
461
+ Session
462
+ Interhostcommunication
463
+ SessionestablishmentinTCP,SiP,RTP,RPC-Namedpipes
464
+ End-to-endconnectionsandreliability
465
+ 4.Transport
466
+ TCP,UDP,SCTP,SSL,TLS
467
+ Pathdeterminationandlogicaladdressing
468
+ 3.Network
469
+ IP,ARP,IPseCICMP.IGMP.OSPF
470
+ Router
471
+ MAC
472
+ MAC
473
+ Physicaladdressing
474
+ 2.Data Link
475
+ Ethernet,8o2.11,MAC/LLC,VLAN,ATM,HDP,FibreChannel
476
+ FrameRelay,HDLC,PPP,Q.921,TokenRing
477
+ Media,signal,and binarytransmission
478
+ 1.Physical
479
+ RS-232,RI45.V.34.100BASE-TX,SDH.DSL,802.11encrypted again by the outer FIPS certified AES-256 with its
480
+ own session key. In this case, the session keys may be
481
+ potentially obtained by attackers using “Steal now, crack later”
482
+ strategy, and later decrypted using quantum attacking
483
+ mechanisms. However, if the nested TLS uses quantum-safe
484
+ algorithms for encryption, then vulnerability of the outer
485
+ session does not influence the security of transmitted data to a
486
+ great extent. That is, if the nested TLS 1.3 with PQC establishes
487
+ a secure session key for session data encryption, then the
488
+ attacker with quantum resources will not be able to decrypt the
489
+ data encrypted in the nested TLS layer even if they were able
490
+ to decrypt the data encrypted in the outer classical layer. This
491
+ nested TLS 1.3 with PQC stops the “steal now, crack later” and
492
+ offers the “steal now, safe forever” services.
493
+ If AES-256 is used for encryption in conventional outer and
494
+ nested layer, that would cause the overall performance to drop
495
+ by 50%. However, there is no requirement to use AES-256 for
496
+ the nested layer encryption. Under the consideration of the FIPS
497
+ compliance, the inner data encryption does not need to be the
498
+ NIST-Approved and FIPS certified, we can choose quantum
499
+ encryption with Quantum Permutation Pad or QPP [28],
500
+ implemented classically with permutation matrices. Unlike
501
+ AES-256 encryption with 14 rounds, QPP encryption follows
502
+ the same way as quantum gate operations or matrix vector
503
+ multiplications. QPP encryption is bijective transformation so
504
+ typical pre-randomization and dispatch techniques would be
505
+ applied before the gate operations to avoid statistic patterns
506
+ appearing in the ciphertext. Quantum encryption with QPP
507
+ demonstrates excellent performance in both encryption and
508
+ decryption, being over 10x faster than AES-256 [31, 32, 33, 34]
509
+ Using QPP algorithm for encryption in the inner layer does not
510
+ impact the performance as greatly as AES-256. The drop in
511
+ performance with QPP is less than 10%. At this cost we can
512
+ offer “steal now, safe forever” services. In addition, quantum
513
+ encryption with QPP has been implemented into IBM quantum
514
+ computers and compiled into 2-qubit and 3-qubit quantum
515
+ circuits [35, 36, 37].
516
+ For the detailed discission of quantum encryption with QPP,
517
+ please refer to [27-34]. Here we briefly summarize classical or
518
+ quantum implementation of QPP as follows shown in Fig. 3:
519
+ 1. Choose the number of n-bits or n-qubits used to
520
+ generate the permutation gates, for example, � �
521
+ 4, 8.
522
+ 2. Choose the number of gates to be used, M. M=64
523
+ for � � 8 in the digital QKD implementation
524
+ [38]. It can be reduced to 8 permutation gates with
525
+ � � 4.
526
+ 3. Session key is first expanded for M permutation
527
+ gates and then mapped to a set of permutation
528
+ gates through the Init module where the Fisher-
529
+ Yates shuffling algorithm.
530
+ 4. The session key is also used to seed a
531
+ cryptographic pseudo random number generator or
532
+ PRNG
533
+ 5. The pseudo random number generated by PRNG
534
+ is used to pre-randomize the plaintext m with XOR
535
+ and then dispatch the randomized data to the
536
+ indexed permutation gate for encryption and then
537
+ the ciphertext c is output accordingly.
538
+ 6. At the receiving side, the process is symmetric to
539
+ the encryption side, but the permutation gate must
540
+ be reversed or transposed. The pseudo random
541
+ number is first used to dispatch the ciphertext to
542
+ the indexed permutation for decryption and after
543
+ then derandomize with the pseudo random
544
+ number, the original plaintext m is obtained.
545
+ From Fig. 3, we can see that quantum encryption with QPP
546
+ only consists of three steps to complete its encryption:
547
+ randomization to remove any statistic bias, dispatching to the
548
+ indexed permutation gate/matrix, and then gate operation
549
+ applied to the randomized plaintext. Overall, the entire
550
+ encryption process may take at most the process time of a single
551
+ round in AES encryption. That is why QPP would be 10x faster
552
+ than AES. Therefore, the nested TLS 1.3 with PQC only has a
553
+ minor impact on the communication performance, less than
554
+ 10%.
555
+ If the attackers apply the “steal now, crack later” strategy
556
+ and wait for the quantum computers to break the public key for
557
+ the session key, then they can decrypt the outer AES encryption
558
+ and obtain the quantum encrypted ciphertext. The quantum
559
+ encrypted ciphertext is then requires to be cracked. However,
560
+ QPP as well as PQC algorithms and any other quantum-safe
561
+ algorithms are designed to withstand classical and quantum
562
+ attacks.
563
+ We have performed randomness analysis on ciphertext
564
+ produced by QPP. Table 1 illustrates the randomness analysis
565
+ of very biased English character files and ciphertext encrypted
566
+ with QPP using ENT test tool. ENT randomness test tool is very
567
+ sensitive to bit and byte level bias, especially detected in the
568
+ Chi Square values. ENT outputs six reports on their entropy
569
+ per 8 bits, Chi Square value, p-value, arithmetic mean, Monte
570
+ Carlo �, and serial correlation. Table 1 shows that the total
571
+ biased English plaintext is encrypted with QPP into ciphertexts
572
+ which demonstrate excellent randomness, especially the Chi
573
+ Square value 233.2 with a p-value 0.83. The acceptable p-value
574
+ a good randomness is from 0.01 to 9.99. The ciphertext also
575
+ demonstrated excellent value for arithmetic mean 127.49,
576
+ Monte Carlo � = 3.14198164, and finally the serial correlation
577
+ at 9.3�10��.
578
+ Init
579
+ PRNG
580
+
581
+
582
+
583
+
584
+
585
+
586
+
587
+
588
+
589
+
590
+
591
+
592
+
593
+
594
+
595
+
596
+
597
+
598
+
599
+
600
+
601
+
602
+ ……..
603
+
604
+
605
+
606
+
607
+
608
+
609
+
610
+
611
+
612
+
613
+
614
+
615
+
616
+
617
+
618
+
619
+
620
+
621
+
622
+
623
+
624
+
625
+ Dispatcher
626
+ QPP
627
+ Key
628
+ m
629
+ c
630
+ Figure 3. Quantum encryption with QPP is illustrated with the session
631
+ key, plaintext data, and ciphertext together with related modules.
632
+
633
+ Indeed, the nested TLS 1.3 with PQC offers FIPS
634
+ compliant solution with a quantum encryption component. We
635
+ this this solution as a good strategy for transition from classical
636
+ security to quantum security without waiting for NIST
637
+ standardization and FIPS certification to complete necessary
638
+ processes. Crypto agility is essential nowadays, since PQC and
639
+ other algorithms are novel and might have undiscovered
640
+ attacks, especially as quantum computing matures. So,
641
+ whenever a specific algorithm is found to be vulnerable, the
642
+ algorithm can be easily removed from the nested cryptomodule
643
+ and replaced with a new one in a convenient and quick manner.
644
+ When all PQC KEMs and digital signature algorithm are
645
+ standardized and the FIPS certification is required, then the
646
+ whole TLS 1.3 cryptomodule would be used for FIPS
647
+ certification. Once it is FIPS certified, the nested TLS 1.3
648
+ automatically becomes certified quantum resistant TLS 1.3
649
+ cryptomodule to replace the classical TLS 1.3. With this, we
650
+ feel confident to turn the “steal now, crack later” into “Steal
651
+ now, safe forever.”
652
+ Table 1. ENT testing is tabulated for statistically biased plaintext
653
+ inputs and ciphertext encrypted with QPP, together with their ideal
654
+ values.
655
+
656
+ III. CONCLUSION
657
+ In this work, we propose a FIPS compliant TLS 1.3 solution
658
+ with a nested quantum-secure TLS component. This solution
659
+ makes seamless transition and mitigation from classical
660
+ security to quantum security, that does not require any wait time
661
+ for standardization and certification. Given that the
662
+ standardization and FIPS certification process might take 10
663
+ years, adversaries can use this time to take advantage of the
664
+ “steal now, crack later” strategy. The proposal of the nested
665
+ TLS 1.3 with quantum-safe component could turn the “steal
666
+ now, crack later” into “steal now, safe forever”, while
667
+ preserving FIPS certified outer TLS layer. Therefore, this
668
+ proposed work is critical for protecting sensitive data today
669
+ with long shelf life against future quantum threats, which may
670
+ impact sectors including public health, insurance, genetics,
671
+ retirement. Any symmetric algorithm can be used in the nested
672
+ TLS layer. However, to overcome performance impact in
673
+ symmetric encryption, we suggest using quantum encryption
674
+ with QPP to further enhance the data security even with the
675
+ successful crack the outer public key with quantum computer,
676
+ the inner TLS 1.3 is still secure. In the future, we plan to build
677
+ the nested TLS 1.3 with PQC and test its real performance in a
678
+ cloud environment in comparison with normal TLS 1.3 with
679
+ PQC.
680
+ ACKNOWLEDGMENT
681
+ We acknowledge that Ryan Toth provided the OQS
682
+ OpenSSL diagram shown in Fig. 2. The overall performance of
683
+ TLS 1.3 hand shaking with MPPK/DS and MPPK KEM, as
684
+ well as HPPK would be published separately.
685
+
686
+ We acknowledge that the image in Fig.1 has been originally
687
+ taken from [29]. The image is licensed under the Creative
688
+ Commons Attribution 4.0 International [39]. We have made
689
+ modifications to the original image available in its original form
690
+ in [29].
691
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+ Entropy (bits)
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+ 4.224280
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+ p-Value
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1
+ 1
2
+ Sample-Efficient Unsupervised Domain Adaptation
3
+ of Speech Recognition Systems:
4
+ A case study for Modern Greek
5
+ Georgios Paraskevopoulos Student Member, IEEE, Theodoros Kouzelis, Georgios Rouvalis, Athanasios
6
+ Katsamanis Member, IEEE, Vassilis Katsouros Member, IEEE, Alexandros Potamianos Fellow, IEEE
7
+ Abstract—Modern speech recognition systems exhibits rapid
8
+ performance degradation under domain shift. This issue is
9
+ especially prevalent in data-scarce settings, such as low-resource
10
+ languages, where diversity of training data is limited. In this work
11
+ we propose M2DS2, a simple and sample-efficient finetuning
12
+ strategy for large pretrained speech models, based on mixed
13
+ source and target domain self-supervision. We find that including
14
+ source domain self-supervision stabilizes training and avoids
15
+ mode collapse of the latent representations. For evaluation, we
16
+ collect HParl, a 120 hour speech corpus for Greek, consisting
17
+ of plenary sessions in the Greek Parliament. We merge HParl
18
+ with two popular Greek corpora to create GREC-MD, a test-
19
+ bed for multi-domain evaluation of Greek ASR systems. In our
20
+ experiments we find that, while other Unsupervised Domain
21
+ Adaptation baselines fail in this resource-constrained environ-
22
+ ment, M2DS2 yields significant improvements for cross-domain
23
+ adaptation, even when a only a few hours of in-domain audio
24
+ are available. When we relax the problem in a weakly supervised
25
+ setting, we find that independent adaptation for audio using
26
+ M2DS2 and language using simple LM augmentation techniques
27
+ is particularly effective, yielding word error rates comparable to
28
+ the fully supervised baselines.
29
+ Index Terms—Unsupervised Domain Adaptation, Automatic
30
+ Speech Recognition, Multi-Domain Evaluation, Greek Speech
31
+ I. INTRODUCTION
32
+ Automatic Speech recognition (ASR) models have matured
33
+ to the point where they can enable commercial, real-world
34
+ applications, e.g., voice assistants, dictation systems, etc., thus
35
+ being one of machine learning’s success stories. However,
36
+ the performance of ASR systems rapidly deteriorates when
37
+ the test data domain differs significantly from the training
38
+ data. Domain mismatches can be caused by differences in
39
+ the recording conditions, such as environmental noise, room
40
+ reverberation, speaker and accent variability, or shifts in the
41
+ target vocabulary. These issues are extenuated in the case of
42
+ low-resource languages, where diversity in the training data
43
+ is limited due to poor availability of high-quality transcribed
44
+ audio. Therefore, specialized domain adaptation approaches
45
+ need to be employed when operating under domain-shift.
46
+ Unsupervised Domain Adaptation (UDA) methods are of
47
+ special interest, as they do not rely on expensive annotation
48
+ G. Paraskevopoulos is with the Graduate School of ECE, National Technical
49
+ University of Athens, Athens, Greece
50
+ G. Paraskevopoulos, T. Kouzelis, G. Rouvalis, A. Katsamanis, V. Katsouros
51
+ are with the Institute for Speech and Language Processing, Athena Research
52
+ Center, Athens, Greece
53
+ A. Potamianos is with the Faculty of ECE, National Technical University
54
+ of Athens, Athens, Greece
55
+ of domain-specific data for supervised in-domain training.
56
+ In contrast to supervised approaches, where the existence of
57
+ labeled data would allow to train domain-specific models,
58
+ UDA methods aim to leverage data in the absense of labels
59
+ to improve system performance in the domain of interest [1],
60
+ [2]. In the context of speech recognition the importance of
61
+ UDA is extenuated, as the transcription and alignment pro-
62
+ cess is especially expensive and time-consuming. Adaptation
63
+ methods have been explored since the early days of ASR,
64
+ at different levels of the system and different deployment
65
+ settings [3]. UDA has been used to improve the robustness
66
+ of ASR on a variety of recording conditions including far-
67
+ field speech, environmental noise and reverberation [4], [5],
68
+ [6]. Furthermore, UDA has been used for speaker adaptation,
69
+ and to improve performance under speaker, gender and accent
70
+ variability [7], [8]. UDA has also been employed for multilin-
71
+ gual and cross-lingual ASR, in order to improve ASR models
72
+ for low-resource languages [9], adapt to different dialects [10],
73
+ and even train speech recognition systems for endangered
74
+ languages [11].
75
+ Classical speech adaptation techniques involve feature-
76
+ based techniques, e.g., speaker normalization [12], feature-
77
+ based approaches [13]–[15], or multi-condition training [16].
78
+ Generally, traditional approaches require some knowledge
79
+ about the target domain, and the domain mismatch, e.g.,
80
+ regarding the noise and reverberation variability [17], and
81
+ require specific engineering for each adaptation scenario.
82
+ Modern ASR pipelines, increasingly rely on end-to-end
83
+ neural networks, e.g., [18], [19], or large pretrained models
84
+ with self-supervised objectives [20], [21]. The key approaches
85
+ employed for UDA of end-to-end ASR models can be grouped
86
+ in three categories, namely, teacher-student learning [10],
87
+ domain adversarial training [22], and target domain self-
88
+ supervision [23]. The benefit of these techniques is that they
89
+ do not require any special knowledge about the source or
90
+ the target domain. This makes end-to-end UDA approaches
91
+ versatile and able to be utilized in a larger array of adaptation
92
+ scenarios. In particular, adaptation through self-supervision
93
+ has been shown to be a robust, simple and efficient technique
94
+ for adaptation of state-of-the-art speech models [24].
95
+ Here, we leverage in-domain self-supervision to propose
96
+ the Mixed Multi-Domain Self-Supervision (M2DS2) finetun-
97
+ ing strategy, enabling sample-efficient domain adaptation of
98
+ wav2vec2 [20] based speech recognition models, even when
99
+ available in-domain data are scarce. Our key contributions are
100
+ arXiv:2301.00304v1 [cs.CL] 31 Dec 2022
101
+
102
+ 2
103
+ TABLE I
104
+ SUMMARY OF RELATED WORKS ON UNSUPERVISED DOMAIN ADAPTATION FOR ASR.
105
+ Work
106
+ Method
107
+ Model
108
+ Adaptation Setting
109
+ Language
110
+ [23], [25], [26]
111
+ Teacher-Student
112
+ Hard and soft labels
113
+ Conformer RNN-T [27]
114
+ Transformer CTC
115
+ RNN-T [19]
116
+ News speech, Voice search, Far-field,
117
+ Telephony, YouTube
118
+ English
119
+ [4], [5]
120
+ Teacher-Student
121
+ Soft labels
122
+ TDNN-LSTM [28]
123
+ Noise, Far-field
124
+ English
125
+ [29]
126
+ Teacher-Student
127
+ Hard and soft labels
128
+ NiN-CNN [30]
129
+ Dialects
130
+ Children speech
131
+ Japanese
132
+ [31]
133
+ Teacher-Student
134
+ Soft labels
135
+ Streaming RNN-T [32]
136
+ Multilingual
137
+ English,
138
+ Brazilian Portuguese,
139
+ Russian,
140
+ , Turkish,
141
+ Nordic/Germanic
142
+ [6], [33], [34]
143
+ Domain Adversarial Training
144
+ TDNN Kaldi [35], [36]
145
+ DNN-HMM
146
+ DNN-HMM
147
+ Noise, Channel
148
+ English
149
+ [37]
150
+ Domain Adversarial Training
151
+ RNN-CTC [38]
152
+ Far-field
153
+ English
154
+ [8], [39]
155
+ Domain Adversarial Training
156
+ TDNN Kaldi
157
+ RNN-T
158
+ Accent
159
+ Mandarin
160
+ [7], [40]
161
+ Domain Adversarial Training
162
+ DNN-HMM
163
+ CNN-DNN
164
+ Speaker, Gender,
165
+ Accent
166
+ English
167
+ [9]
168
+ Domain Adversarial Training
169
+ DSN [41]
170
+ Multilingual
171
+ Hindi, Sanskri
172
+ [24], [42]
173
+ Continual Pre-Training
174
+ wav2vec2 [20]
175
+ Audiobooks, Accents,
176
+ Ted Talks, Telephony,
177
+ Crowd-sourced, Parlamentary speech
178
+ English
179
+ [43]
180
+ Continual Pre-Training
181
+ wav2vec2
182
+ Cross-lingual
183
+ Korean
184
+ [11], [44]
185
+ Continual Pre-Training
186
+ XLSR-53 [21]
187
+ wav2vec2
188
+ Low resource languages
189
+ Ainu
190
+ Georgian, Somali,
191
+ Tagalog, Farsi
192
+ organized as follows:
193
+ 1) Inspired by recent advances on UDA for Natural Lan-
194
+ guage Processing systems [45], we propose a finetuning
195
+ strategy for speech models, where the self-supervised
196
+ objective is based on a contrastive loss in Section III.
197
+ Contrary to prior works, who leverage only in-domain
198
+ self-supervision, we find that in this contrastive setting
199
+ this leads to mode-collapse of the latent representations,
200
+ and mixed source and target domain self-supervision
201
+ is essential. We demonstrate this empirically in Sec-
202
+ tion VII-B.
203
+ 2) We collect and curate HParl, the largest publicly avail-
204
+ able1 speech corpus for Greek, collected from plenary
205
+ sessions in the Greek Parliament between 2018 and
206
+ 2022. We establish a data collection, pre-processing
207
+ and alignment pipeline that can be used for continuous
208
+ data integration, as the parliamentary proceedings get
209
+ regularly uploaded. We provide a detailed description of
210
+ our data collection process and the dataset statistics in
211
+ Section IV-A. HParl is merged in Section IV with two
212
+ popular Greek corpora (Logotypografia and Common-
213
+ Voice) to create GREC-MD, a testbed for multi-domain
214
+ evaluation of ASR systems in Greek.
215
+ 3) We demonstrate that, while other baselines fail at UDA
216
+ in our resource-constrained setting, M2DS2 can improve
217
+ model performance in the target domain in multiple
218
+ adaptation scenarios in Section VII. Specifical emphasis
219
+ is given in the sample efficiency of our approach in Sec-
220
+ 1We plan to release this version of HParl under the CC BY-NC 4.0 license
221
+ upon publication. The other corpora used in this work are available through
222
+ their respective distributors.
223
+ tion VII-A, where we demonstrate successful adaptation
224
+ even when we reduce the available in-domain data.
225
+ 4) When we relax the problem to a weakly supervised
226
+ adaptation setting, where some in-domain text is avail-
227
+ able but the pairing between audio and text is unknown,
228
+ we find that M2DS2 can be effectively combined with
229
+ simple N-gram adaptation techniques to get compara-
230
+ ble performance with the fully supervised baseline in
231
+ Section VIII. Furthermore we find that a simple text
232
+ augmentation approach, based on perplexity filtering of a
233
+ large corpus can produce strong adaptation results, even
234
+ for small amounts of in-domain text.
235
+ Additionally, we provide a formulation of the UDA problem
236
+ for ASR in Section II-A and link prior works to this formu-
237
+ lation in Sections II-B, II-C and II-D. We provide detailed
238
+ experimental settings for reproducibility in Section V, and
239
+ an upper-bound estimation for UDA performance with fully
240
+ supervised finetuning in Section VI.
241
+ II. BACKGROUND
242
+ We start by formally defining the Unsupervised Domain
243
+ Adaptation (UDA) problem. Initially, we formulate the prob-
244
+ lem in a classification setting and then we extend it for
245
+ speech recognition. We then provide an overview of different
246
+ adaptation approaches in the literature, and link each approach
247
+ to the UDA problem formulation. Table I presents a summary
248
+ of the key adaptation settings and applications that are ex-
249
+ plored in the literature. We see, that a relatively small amount
250
+ of methods, and their variants, is used to address multiple
251
+ real-world ASR problems, for example, cross-lingual, accent,
252
+ speaker and noise adaptation. Furthermore, while the majority
253
+
254
+ 3
255
+ of the works focus on the English language, there is an effort
256
+ to explore other popular languages, e.g., Mandarin, and under-
257
+ resourced languages, e.g., Ainu, Somali etc.
258
+ A. Problem Definition
259
+ Formally, the problem of UDA can be defined as follows.
260
+ Let X ⊆ Rn be a real-valued space that consists of n-
261
+ dimentional feature vectors x ∈ X, and Y a finite set of
262
+ labels y ∈ Y , i.e., Y = {1, 2, . . . , L}. Furthermore, assume
263
+ two different distributions, i.e., the source domain distribution
264
+ S(x, y) and the target domain distribution T (x, y), defined on
265
+ the cartesian product X × Y .
266
+ The goal is to train a model that learns a mapping between
267
+ feature vectors xT to their respective labels yT for samples
268
+ drawn from the target distribution (xT , yT ) ∼ T .
269
+ At training time we have access to samples from the source
270
+ distribution S(x, y) and the marginalized target distribution
271
+ T (x), i.e., no target labels are provided. We define the training
272
+ dataset D as the concatenation of the source and target training
273
+ sets, D = (DS, DT ). DS and DT are defined as sequences of
274
+ tuples, i.e.,
275
+ DS = {(xi, yi) | (xi, yi) ∼ S(x, y), 1 ≤ i ≤ N}
276
+ DT = {(xi, ∅) | xi ∼ T (x), 1 ≤ i ≤ M},
277
+ (1)
278
+ where we draw N samples from S(x, y) and M samples
279
+ from T (x). Finally, we augment tuples in D with a domain
280
+ indicator function:
281
+ D = {(xi, y′
282
+ i, 1i) | 1 ≤ i ≤ N + M}
283
+ 1i =
284
+
285
+ 0
286
+ if xi ∼ S(x),
287
+ 1
288
+ if xi ∼ T (x).
289
+ y′
290
+ i =
291
+
292
+ yi
293
+ if xi ∼ S(x),
294
+
295
+ if xi ∼ T (x).
296
+ (2)
297
+ 1) Unsupervised (Acoustic) Adaptation for ASR: The above
298
+ definition can be directly extended in the case of speech
299
+ recognition, with some modifications. In detail, we modify
300
+ the feature space X, to be the set of (finite) sequences of
301
+ real-valued feature vectors (xk)k∈N\{∞} ∈ X ⊆ (Rn)∗.
302
+ Furthermore, the label space Y is modified to be the set
303
+ of sequences (ym)m∈N\{∞}, where Y
304
+ = ({1, 2, . . . , L})∗
305
+ contains finite-length sequences over a finite lexicon. For
306
+ CTC training we make the assumption that k > m for any
307
+ sample (xk, ym), i.e., feature sequences are longer than their
308
+ respective label sequences [46]. The rest of the definitions need
309
+ no modifications.
310
+ 2) Unsupervised (Language) Adaptation for ASR: Adapta-
311
+ tion for ASR systems can also be performed at the language
312
+ level, i.e., the label space. In this setting, we assume that
313
+ the target domain samples are drawn from the marginalized
314
+ target distribution T (y). The target dataset DT now consists of
315
+ tuples in the form (∅, yi), where yi is the label word sequence
316
+ (ym)m∈N\{∞} for the i-th sample.
317
+ 3) Weakly supervised Adaptation for ASR: The last setting
318
+ we explore is the case were both audio and language in-
319
+ domain samples are available, but the mapping between them
320
+ is unknown. This situation can be encountered in real-world
321
+ settings, e.g., in the case in-domain audio and text are collected
322
+ independently. For example consider the case where audio
323
+ clips from news casts are collected, along with contemporary
324
+ newspaper articles. Another example is the case where long
325
+ audio clips alongside with transcriptions are available, but no
326
+ fine-grained time alignments2. In this case the target domain
327
+ samples are drawn independently from the marginalized dis-
328
+ tributions T (x) and T (y), and the target dataset DT consists
329
+ of tuples in the form (xi, ∅) and (∅, yi).
330
+ B. Teacher-Student Models
331
+ Teacher-Student learning or self-training, is one of the
332
+ earliest methods in semi-supervised learning [47]–[49]. The
333
+ key idea is to reduce the problem of unsupervised learning
334
+ of the task at hand in the target domain to a supervised one.
335
+ The general methodology is to train a teacher model gS using
336
+ the labeled data in the source domain DS, and then use this
337
+ for inference on the target domain to produce pseudolabels
338
+ ˆyi = gS(xi), xi ∼ T (x). The target domain dataset DT is
339
+ augmented with these silver labels, to contain tuples (xi, ˆyi).
340
+ Finally, a student model gT is trained in a supervised fashion,
341
+ using the augmented DT or a combination of DS and DT .
342
+ This process is usually repeated, with the student model
343
+ serving as the teacher model for the next iteration, until no
344
+ further improvement is observed. More recently, soft target
345
+ Teacher-Student learning has been explored for ASR [26],
346
+ [31], [50], where the KL divergence between the teacher and
347
+ student output label distributions is used as the loss function.
348
+ Being trained only on the source domain data the teacher
349
+ model is susceptible to error propagation. Filtering is a com-
350
+ monly used technique to achieve the right balance between
351
+ the size of the target domain used for training the student
352
+ model and the noise in the pseudolabels. Confidence scoring
353
+ based on the likelihood is usually applied, discarding those
354
+ utterances for which the hypothesized labels are untrustworthy
355
+ [51]. In [25] dropout is used to measure the model uncertainty.
356
+ The agreement between model predictions with and without
357
+ dropout are used for confidence scoring. In [23] a multi-task
358
+ training objective with a confidence loss is applied to minimise
359
+ the binary cross entropy between the estimated confidence and
360
+ the binary target sequence. In order to learn more robust and
361
+ generalizable features from the teacher model, Noisy Student
362
+ Training (NST) has been proposed in [52]. The teacher models
363
+ generates pseudolabels for DT while the student models are
364
+ trained on a heavily augmented version of DT [52]. In [52],
365
+ [53] the augmentation of the input target data is performed
366
+ with SpecAugment [54], while in [29] a spectrum frequency
367
+ augmentation is performed.
368
+ In [4] Teacher-Student learning with soft labels is introduced
369
+ for ASR to tackle noisy, far-field, and children speech. In
370
+ 2While a fully supervised in-domain dataset can be constructed in this
371
+ case using long / forced alignment methods, this is not a focal point for the
372
+ experimental part of this work.
373
+
374
+ 4
375
+ [5], this approach is extended for LF-MMI based models and
376
+ used for noisy, far-field and bandwidth adaptation. In [29] a
377
+ weighted sum of hard and soft target cross entropy losses
378
+ is used for Japanese dialects and children speech adaptation.
379
+ Ramabhadran et al. [31] propose a self-adaptive distillation,
380
+ and a method for distilling from multiple teachers that is
381
+ applied across several multilingual ASR systems for different
382
+ language groups. A comparison between soft and hard targets
383
+ for RNN-T models [19] showed that soft targets perform better
384
+ when both the teacher and student models have the same
385
+ architecture. Otherwise, hard targets are superior [50].
386
+ C. Domain Adversarial Training
387
+ Domain Adversarial Training (DAT) was initially introduced
388
+ for image classification [55]. The key idea is to train a
389
+ model that learns deep features that solve the task at hand
390
+ in the source domain, while being invariant with respect
391
+ to the domain shift. Concretely, the model is trained end-
392
+ to-end using a combination of the supervised task loss Lt,
393
+ learned on DS, and the domain discrimination loss La, i.e.,
394
+ L = Lt − αLa. The loss La is binary cross-entropy, trained
395
+ for domain discrimination using the tuples (xi, 1i). Notice
396
+ the − sign in the loss indicates adversarial learning, i.e., the
397
+ model should learn features that cannot discriminate between
398
+ domains, while solving the task.
399
+ In [6] DAT is employed for noise adaptation on a noise
400
+ corrupted version of WSJ [56] as the target dataset. Using the
401
+ Aurora-4 [57] dataset which has labels associated to the noise
402
+ type, Serdyuk et al. [33] train an adversarial noise classifier. In
403
+ [8] and [39] DAT is utilized for accent adaptation for Mandarin
404
+ and English respectively. Anoop C.S. et al. [9] propose DAT,
405
+ to address the scarcity of data in low-resource languages which
406
+ share a common acoustic space with a high-resource language,
407
+ namely Sanskrit and Hindi. They empirically demonstrate the
408
+ effectiveness of adversarial training, presenting experiments
409
+ with and without the reversal of the domain classification loss.
410
+ D. Leveraging In-domain Self-supervision
411
+ These lines of work have roots in Natural Language Pro-
412
+ cessing tasks [45], [58], and explore domain adaptation by
413
+ leveraging the in-domain data DT for self-supervised learning.
414
+ The core focus is domain adaptation of large pre-trained
415
+ models, e.g., [59], and self-supervision is achieved by use
416
+ of the pre-training self-supervised loss Ls. This process can
417
+ either take part in stages, via continual pre-training [58], or by
418
+ constructing a multitask objective L = Lt + αLs, as in [45].
419
+ Continual Pre-Training (CPT) has been explored for adap-
420
+ tation of ASR models. Robust wav2vec2 [24] explores the
421
+ effectiveness of CPT for domain adaptation, indicating the
422
+ importance of utilizing unlabeled in-domain data. In CASTLE
423
+ [42], CPT is combined with an online pseudolabeling strategy
424
+ for domain adaptation of wav2vec2. Cross-dataset evaluation
425
+ for popular English speech corpora indicates that CPT helps
426
+ to reduce the error rate in the target domain. In [43] and [11]
427
+ CPT is utilized for cross-lingual adaptation of wav2vec2 for
428
+ Korean and Ainu respectively. Notably for Ainu, which is an
429
+ endagered language, CPT has resulted in significant system
430
+ Fig. 1. Target-domain adaptation through self-supervision. In the left we see
431
+ the general pre-training stage of XLSR-53 using the self-supervised loss Ls.
432
+ General pre-training is performed on 56, 000 hours of audio in 53 languages.
433
+ In the right, we see the proposed domain-adaptive finetuning stage, where the
434
+ speech recognition task is learned using transcribed source domain data, while
435
+ adaptation to the target domain is performed by including the self-supervised
436
+ loss over (audio-only) source and target domain data
437
+ improvement. DeHaven and Jayadev [44] compare CPT and
438
+ pseudolabeling for adapting XLSR-53 to four under-resourced
439
+ languages, i.e., Georgian, Somali, Tagalog and Farsi. They find
440
+ that both approaches yield similar improvements, with CPT
441
+ being the more computationally efficient approach.
442
+ While CPT yields significant improvements in a variety of
443
+ tasks, one common theme in these works is the assumption
444
+ of hundreds or thousands of hours of available in-domain
445
+ data, mostly from online resources, e.g., YouTube. This can be
446
+ infeasible when we consider more niche adaptation settings,
447
+ or possible privacy concerns, e.g., how would one collect
448
+ 1000 hours of psychotherapy sessions in Greek? In this work,
449
+ we explore domain adaptation methods in a more resource-
450
+ constrained environment.
451
+ III. DOMAIN ADAPTATION THROUGH MULTI-DOMAIN
452
+ SELF-SUPERVISION
453
+ The proposed approach is based on end-to-end adaptation of
454
+ a large pre-trained speech model during the finetuning phase,
455
+ by including in-domain self-supervision. We extend UDALM
456
+ [45], that has shown promise for NLP tasks, for adaptation of
457
+ wav2vec2 based acoustic models, and specifically XLSR. We
458
+ focus on the problem of UDA in the context of a low-resource
459
+ language, i.e., Greek. The key finding of our exploration is
460
+ that straight-forward extension of UDALM, i.e., by using only
461
+ target domain self-supervision, underperforms in this setting,
462
+ and use of both source and target domain data is essential for
463
+ successful adaptation. In this section, first, we will present
464
+ a quick overview of the XLSR-53 training procedure, and
465
+ then we are going to outline the proposed domain adaptation
466
+ approach, which is shown in Fig. 1.
467
+ A. XLSR-53
468
+ XLSR-53 [21] is a massively pre-trained speech model,
469
+ trained on 56, 000 hours of multilingual speech, covering 53
470
+ languages. The model is based on wav2vec2 [20], which is
471
+ composed of a multi-layer convolutional feature encoder, that
472
+
473
+ General Pretraining
474
+ Finetuning
475
+ Ls
476
+ LCTC
477
+ Ls
478
+ Masked Transformer
479
+ Masked Transformer
480
+ XLSR
481
+ XLSR
482
+ MLS, CommonVoice and BABEL
483
+ Source Domain
484
+ Target Domain
485
+ 56.0000 of speech data from 53 languages5
486
+ TABLE II
487
+ THE GREC-MD CORPUS. WE CAN SEE THE DURATION OF EACH SPLIT IN H O U R S:M I N U T E S:S E C O N D S FORMAT, AS WELL AS THE NUMBER OF
488
+ SPEAKERS FOR EACH OF THE SUB-CORPORA.
489
+ Dataset
490
+ Domain
491
+ Speakers
492
+ Train
493
+ Dev
494
+ Test
495
+ Total Duration
496
+ HParl
497
+ Public (political) speech
498
+ 387
499
+ 99:31:41
500
+ 9:03:33
501
+ 11:12:28
502
+ 119:47:42
503
+ CV
504
+ Crowd-sourced speech
505
+ 325
506
+ 12:16:17
507
+ 1:57:44
508
+ 1:59:19
509
+ 16:13:20
510
+ Logotypografia
511
+ News casts
512
+ 125
513
+ 51:58:45
514
+ 9:08:35
515
+ 8:59:22
516
+ 70:06:42
517
+ Total
518
+ -
519
+ 713
520
+ 163:46:43
521
+ 20:09:52
522
+ 22:11:44
523
+ 206:08:19
524
+ extracts audio features zt from the raw audio, and a trans-
525
+ former context encoder that maps the latent audio features to
526
+ the output hidden states ct. Each latent feature zt corresponds
527
+ to 25 ms of audio with stride 20 ms. A contrastive objective Lc
528
+ is used for pre-training. For this, product quantization [60] is
529
+ applied to the features zt, and then a discrete approximation of
530
+ zt is obtained by sampling from a Gumbel-softmax distribution
531
+ [61], to obtain discrete code vectors qt, organized into G = 2
532
+ codebooks with V
533
+ = 320 vocabulary entries each. The
534
+ contrastive loss aims to identify the correct code vector for
535
+ a given time step, among a set of distractors Qt, obtained
536
+ through negative sampling from other timesteps. To avoid
537
+ mode collapse, a diversity loss Ld is included by maximizing
538
+ the entropy over the averaged softmax distribution over the
539
+ code vector entries ¯pg. The total loss is:
540
+ Ls = −log
541
+ es(zt,qt)
542
+
543
+ ˜q∼Qt es(zt,˜q)
544
+
545
+ ��
546
+
547
+ Contrastive Loss
548
+ Diversity Loss
549
+
550
+ ��
551
+
552
+ − 1
553
+ GV
554
+ G
555
+
556
+ g=1
557
+ V
558
+
559
+ v=1
560
+ ¯pg,vlog(¯pg,v)
561
+ (3)
562
+ B. Domain Adaptive finetuning for Contrastive Learning of
563
+ Speech Representations
564
+ Fig. 1 shows the proposed finetuning process. The key
565
+ intuition is that we want the model to synergistically learn
566
+ the task at hand (in our case ASR), while being adapted to
567
+ the target domain by in-domain self-supervision. In the left
568
+ we see the general pre-training stage of XLSR-53, which is
569
+ pre-trained on 56K hours of multilingual audio corpora using
570
+ the contrastive pre-training objective. In the right we see the
571
+ proposed finetuning stage, which is inspired by [45].
572
+ During finetuning we form a mixed objective function:
573
+ L = LCT C(xs, ys) + αLs(xs) + βLs(xt),
574
+ (4)
575
+ where (xs, ys) ∼ S(x, y), xt ∼ T (x), LCT C is the CTC
576
+ objective function, optimized using transcribed source domain
577
+ data, and Ls is the contrastive loss from Eq. (3). We scale the
578
+ contribution of each term using hyper-parameters α and β.
579
+ Note that contrary to [45], who use only in-domain self-
580
+ supervision, we leverage both source and target domain sam-
581
+ ples for the mixed self-supervision. We find that this is essen-
582
+ tial in our case to avoid mode collapse, i.e., the model using
583
+ only a few of the available discrete code vectors. Simultaneous
584
+ self-supervision on both the source and target data alleviates
585
+ mode collapse by anchoring the target code vector space to
586
+ have a similar structure as the source code vectors.
587
+ Hence we refer to this approach as Mixed Multi-Domain
588
+ Self-Supervision (M2DS2).
589
+ IV. THE GREC-MD CORPUS
590
+ For our experiments we compose a speech corpus for the
591
+ Greek language, that is suitable for multi- and cross-domain
592
+ evaluation. The GREC-MD corpus contains 206 hours of
593
+ Greek speech. Audio is segmented into individual utterances
594
+ and each utterance is paired with its corresponding tran-
595
+ scription. Table II summarizes the included sub-corpora, as
596
+ well as the train, development and test splits. The dataset is
597
+ constructed with three core principles in mind:
598
+ 1) Data Volume: We collect the largest publicly available
599
+ speech recognition corpus for the Greek language, able
600
+ to scale to hundreds of hours of transcribed audio.
601
+ 2) Temporal Relevance: Language changes over time. We
602
+ aim at an up-to-date corpus that encompasses the latest
603
+ terms and topics that appear in daily speech.
604
+ 3) Multi-Domain Evaluation: Single domain evaluation
605
+ can lead to misleading estimations of the expected
606
+ performance for ASR models. For example, state-of-
607
+ the-art ASR models [27] achieve under 5% Word Error
608
+ Rate (WER) on Librispeech [62] test sets, but this is
609
+ an over-estimation of system performance in the field.
610
+ This is extenuated when considering different acoustic
611
+ conditions or terminology. We consider multi-domain
612
+ evaluation essential when developing and deploying
613
+ real-world ASR models.
614
+ To satisfy the first two points, we collect data from a public,
615
+ continuously updated resource, i.e., the Hellenic Parliament
616
+ Proceedings, where recordings of the parliamentary sessions
617
+ are regularly uploaded. The benefit of using this resource is the
618
+ straight-forward collection of a continuously growing, multi-
619
+ speaker corpus of transcribed audio that is always up-to-date,
620
+ as the parliamentary discussions revolve around current affairs.
621
+ We refer to this corpus as HParl. For the multi-domain evalua-
622
+ tion, we merge HParl with two publicly available corpora, that
623
+ have different acoustic and language characteristics. We refer
624
+ to the merged, multi-domain corpus as GREC-MD. In this
625
+ Section, we will describe the collection and curation process
626
+ of HParl, and present the relevant statistics for the experiments.
627
+ TABLE III
628
+ PLENARY SESSIONS INCLUDED IN HPARL. THE HOURS COLUMN REFERS
629
+ TO THE RAW (UNSEGMENTED) HOURS OF COLLECTED AUDIO.
630
+ Start date
631
+ End date
632
+ #Sessions
633
+ Hours
634
+ 15-02-2022
635
+ 01-03-2022
636
+ 10
637
+ 55
638
+ 18-01-2019
639
+ 01-02-2019
640
+ 10
641
+ 52
642
+ 28-03-2019
643
+ 10-05-2019
644
+ 20
645
+ 108
646
+ 10-12-2018
647
+ 21-12-2018
648
+ 10
649
+ 88
650
+
651
+ 6
652
+ Fig. 2. Overview of the Hellenic Parliament Chamber. The chamber has an
653
+ amphitheatrical shape and can accomodate approximately 400 − 450 people.
654
+ The positions of the key speakers, i.e., current speaker and the parliament
655
+ president are annotated in the image.
656
+ A. Collection and Curation of HParl
657
+ Modern technological advances allow for more direct gov-
658
+ ernment transparency, through the commodification of storage
659
+ and internet speeds. In this spirit, the records of plenary ses-
660
+ sions of the Hellenic Parliament are made publicly available,
661
+ for direct access through a webpage3. The available video
662
+ recordings date back to 2015. For each plenary session, a
663
+ video recording is uploaded, along with a full transcription
664
+ that is recorded verbatim, and in real time by the parlia-
665
+ ment secretaries. For the creation of HParl, we build a web-
666
+ crawler that can traverse and download the video recordings,
667
+ along with the transcriptions from the official website. The
668
+ collection process is parallelized over multiple threads, and
669
+ parameterized by a range of dates and, optionally, a target
670
+ corpus size in GB or in hours. For this version of HParl, we
671
+ collect the plenary sessions in four date ranges, as described in
672
+ Table III. The majority of the collected sessions are from 2019,
673
+ but we also include sessions from 2018 and 2022 to include
674
+ coverage of different topics. The individual components of the
675
+ HParl curation pipeline are: Audio Pre-processing, Text Pre-
676
+ processing, Alignment, Post-processing, and dataset Splitting.
677
+ 1) Audio Pre-processing: Fig. 2 shows the layout of the
678
+ Hellenic Parliament Chamber. Plenary sessions mainly take
679
+ place in this room, or in the secondary House Chamber that
680
+ has similar setup but is smaller in size. Because of the room
681
+ and microphone characteristics, the captured audio in the
682
+ video streams contains reverberation, due to sound reflections.
683
+ We employ a light preprocessing pipeline, by passing the
684
+ input video streams through FFmpeg, and converting them to
685
+ monophonic, lossless audio format at 16000 Hz sampling rate.
686
+ The resulting audio is not passed through any de-reverberation
687
+ or speech enhancement software. The resulting audio files have
688
+ a minimum, average and maximum duration of 6 minutes, 6
689
+ hours and 16 hours respectively.
690
+ 2) Text Pre-processing: The text files contain full, word-
691
+ by-word transcription of the speeches and questions asked by
692
+ members of the audience, as well as extra annotations made
693
+ by the parliament secretaries. Some annotations are relevant,
694
+ 3https://www.hellenicparliament.gr/en/
695
+ i.e., the speaker name, while others are plain text descriptions
696
+ of events happening during the session and need to be filtered
697
+ out (e.g., “The session is interrupted for a 15 minute break”).
698
+ We use a rule-based system, based on regular expressions,
699
+ that filters the unnecessary information, keeping only the
700
+ transcriptions and the speaker names. The speaker labels are
701
+ created by transliterating their names and roles from Greek
702
+ to Greeklish using the “All Greek to Me!” tool [63]. Text is
703
+ lower-cased and normalized to remove multiple whitespaces.
704
+ The result is a text file containing the raw transcriptions, and
705
+ a mapping from speaker labels to their respective text parts.
706
+ 3) Aligment and Segmentation: The primary challenge of
707
+ exploiting the plenary sessions for ASR purposes is the length
708
+ of the plenary recordings, as their durations vary from 6
709
+ minutes to 16 hours in length. However, data samples used to
710
+ train ASR are generally less than 30 seconds long. Computa-
711
+ tional challenges have limited the length of training utterances
712
+ for HMM-GMM models [64], and continue to do so in the
713
+ contemporary neural network models. Therefore, we need to
714
+ segment the sessions into smaller pieces more suitable for ASR
715
+ training. A second challenge is posed by mismatches between
716
+ audio and transcripts. Parliamentary proceedings do not fully
717
+ capture everything that is said during the parliamentary ses-
718
+ sions, and do not account for speech disfluencies.
719
+ In order to obtain smaller, clean segments, that are suit-
720
+ able for ASR training we follow the segmentation procedure
721
+ proposed by [65]. Initially the raw recordings are segmented
722
+ into
723
+ 30 second segments and the transcriptions are split
724
+ into smaller segments of approximately 1000 words called
725
+ documents. Each segment is decoded using a seed acoustic
726
+ model trained on the Logotypografia corpus [66] and a 4-
727
+ gram biased LM trained on the corresponding transcription
728
+ of each recording. The best path transcript of each segment
729
+ is obtained and paired with the best matching document via
730
+ TF-IDF similarity. Finally each hypothesis is aligned with the
731
+ transcription using Smith-Waterman alignment [67] to select
732
+ the best matching sub-sequence of words. The above method
733
+ yields a list of text utterances, with their corresponding start
734
+ and end times in the source audio files. The procedure yields
735
+ 120 hours of useable segmented utterances out of the original
736
+ 303 hours of raw audio, or a ratio of 39.6%.
737
+ 4) Post-processing: After the segments are extracted, we
738
+ filter out extremely short segments (less than 2 words).
739
+ Moreover, the iterative alignment algorithm may replace some
740
+ intermediate words with a <spoken-noise> tag. When this
741
+ tag is inserted, we match the surrounding text with the raw
742
+ transcriptions and re-insert the missing words. Furthermore,
743
+ we match each segment to its corresponding speaker label.
744
+ Segments without a speaker label are discarded. Lastly, speak-
745
+ ers are associated to their gender based on name suffixes, using
746
+ a simple, Greek language-specific, rule: Speaker names which
747
+ end in a(α), h(η), w(ω) or is(ις) are classified as female, while
748
+ the rest as male. We format the segments, speaker and gender
749
+ mappings in the standard folder structure used by the Kaldi
750
+ speech recognition toolkit [36].
751
+ 5) Data Splitting: We provide an official train - devel-
752
+ opment - test split. The development set contains 3 plenary
753
+ sessions, one from 2018, one from 2019 and one from 2022,
754
+
755
+ Current Speaker
756
+ Parliament President7
757
+ resulting to 9 hours of segmented speech. Similarly, the test
758
+ set contains one session from each year, resulting to 11 hours
759
+ of segmented speech. The rest 99 hours of segmented speech
760
+ are assigned to the training set.
761
+ B. Including corpora from different domains
762
+ We merge HParl with two publicly available corpora to
763
+ create GREC-MD for multi-domain evaluation.
764
+ 1) Common Voice: Common Voice (CV) [68] is a crowd-
765
+ sourced, multi-lingual corpus of dictated speech, created by
766
+ Mozilla. The data collection is performed by use of a web
767
+ app or an iPhone app. Contributors are presented with a
768
+ prompt and are asked to read it. The prompts are taken from
769
+ public domain sources, i.e., books, wikipedia, user submitted
770
+ prompts and other public corpora. The maximum prompt
771
+ length is 15 words. A rating system is built into the plat-
772
+ form, where contributors can upvote or downvote submitted
773
+ <audio,transcript> pairs. A pair is considered valid, if
774
+ it receives two upvotes. Speaker independent train, develop-
775
+ ment and test splits are provided. The dataset is open to the
776
+ research community, released under a permisFsive Creative
777
+ Commons license (CC0). In this work, we use version 9.0
778
+ of CV, accessed on April 27, 2022. We keep only the valid
779
+ utterances, i.e., 16 hours of speech from 325 contributors
780
+ (19 − 49 years old, 67% male / 23% female).
781
+ 2) Logotypografia: Logotypografia [66] is one of the first
782
+ corpora for Large Vocabulary Continuous Speech Recognition
783
+ in Greek. The dataset contains 33, 136 newscast utterances, or
784
+ 72 hours of speech. The utterances were collected from 125
785
+ speakers (55 male, 70 female), who were staff of the popular
786
+ “Eleftherotypia” newspaper in Greece, under varied acoustic
787
+ conditions. Approximately one third of the utterances were
788
+ collected in a sound proof room, one third in a quiet room and
789
+ the last third in an office room. The average utterance duration
790
+ is 7.8 seconds. The transcriptions contain several speech and
791
+ non-speech events (e.g., <cough>), lower-cased Greek words
792
+ and stress marks. Numbers are expanded to full words. We
793
+ use the whole dataset, and perform light preprocessing in
794
+ the transcriptions, by discarding the annotated events and
795
+ punctuation.
796
+ We hence refer to each dataset by the abbreviations: HParl:
797
+ HP, CommonVoice: CV, Logotypografia: LG.
798
+ V. EXPERIMENTAL SETTINGS
799
+ For our experiments we use the following hyper-parameter
800
+ settings, unless explicitly stated otherwise. For model training,
801
+ we use AdamW optimizer [69] with learning rate 0.0003. We
802
+ apply warmup for the first 10% of the maximum training
803
+ steps, and a linear learning rate decay after that. Models
804
+ are finetuned for a maximum of 10000 steps. For speech
805
+ recognition training, we make use of the Connectionist Tem-
806
+ poral Classification (CTC) loss [70], optimized using the
807
+ available transcribed data in each scenario. Validation runs
808
+ every 500 steps on the development set, and early stopping
809
+ is employed on the development CTC loss with patience 5.
810
+ Batch size is set to 8 during finetuning for all scenarios,
811
+ except for M2DS2. In the case of M2DS2 we create mixed
812
+ batches of size 12, containing 4 transcribed source domain
813
+ samples and 8 unlabeled target domain samples and train
814
+ for 10, 000 CTC updates. For memory reasons we split the
815
+ mixed batches in mini-batches of 4 and interleave them during
816
+ model training. Gradients are accumulated over 3 interleaved
817
+ batches. For the self-supervised objective, we create masks
818
+ of maximum timestep length 10, with masking probability
819
+ 0.4. We weigh the contributions of the source and target
820
+ domain contrastive objectives, and bring them to the same
821
+ order of magnitude as the CTC loss, by setting α = 0.01 and
822
+ β = 0.02. The convolutional feature encoder is kept frozen
823
+ for all experiments. Our code is based on the huggingface 4
824
+ implementation of XLSR. For all experiments we resample
825
+ the audio files to 16 kHz and downsample to single channel
826
+ audio. We exclude utterances in the training set that are longer
827
+ than 12 seconds. All experiments are run on a single NVIDIA
828
+ RTX 3090 GPU, with mixed precision training.
829
+ For the Language model training, we create a large corpus
830
+ for the Greek language using a subset of the Greek part of CC-
831
+ Net [71] (approximately 11 billion tokens) and combine it with
832
+ 1.5 billion tokens from the Greek version of Wikipedia and the
833
+ Hellenic National Corpus (HNC) [72]. During preprocessing,
834
+ we remove all punctuation and accents, deduplicate lines and
835
+ convert all letters to lowercase. We will refer to this corpus as
836
+ the Generic Greek Corpus (GGC). We train a 4-gram language
837
+ model on GGC using KenLM [73] and prune bigrams, trigrams
838
+ and four-grams with counts less than 3, 5 and 7 respectively.
839
+ We incorporate the n-gram LMs at inference time using the
840
+ pyctcdecode framework5. We use language model rescoring
841
+ over a beam search decoder with 13 beams.
842
+ The evaluation metric is the Word Error Rate (WER) over
843
+ the target test set. For assessing the adaptation effectiveness we
844
+ also report the relative WER improvement over the unadapted
845
+ baseline in appropriate scenarios, which is defined in Eq. (5).
846
+ We refer to this metric as Relative Adaptation Improvement
847
+ (RAI) for the rest of this paper:
848
+ RAI = −WERadapted − WERunadapted
849
+ WERunadapted
850
+ × 100%
851
+ (5)
852
+ The minus sign is included, so that RAI takes negative
853
+ values when the adaptation fails, i.e., when WERunadapted <
854
+ WERadapted.
855
+ TABLE IV
856
+ ASR PERFORMANCE OF XLSR-53 OVER THE THREE CORPORA FOR FULLY
857
+ SUPERVISED IN-DOMAIN FINETUING (WER)
858
+ Dataset
859
+ LM
860
+ No LM
861
+ 4g GGC
862
+ HP
863
+ 26.21
864
+ 15.64
865
+ CV
866
+ 29.33
867
+ 9.52
868
+ LG
869
+ 31.94
870
+ 26.45
871
+ VI. SUPERVISED IN-DOMAIN TRAINING
872
+ In the first set of experiments, we explore the performance
873
+ of supervised finetuning of XLSR-53 for each domain. This
874
+ 4https://huggingface.co/docs/transformers/
875
+ 5https://github.com/kensho-technologies/pyctcdecode
876
+
877
+ 8
878
+ TABLE V
879
+ M2DS2 PERFORMANCE USING GREEDY DECODING FOR UDA BETWEEN HP, CV, AND LG. A → B INDICATES THAT A IS THE SOURCE DOMAIN AND B IS
880
+ THE TARGET DOMAIN. (G) INDICATES GREEDY DECODING. (LM) INDICATES BEAM SEARCH WITH LM RESCORING. WE REPORT THE WER ON THE
881
+ TARGET TEST SET, AS WELL AS THE RAI (%) OVER THE SO (UNADAPTED) BASELINE. WER: LOWER IS BETTER. RAI: HIGHER IS BETTER.
882
+ Method
883
+ SO (G)
884
+ CPT (G)
885
+ PSL (G)
886
+ M2DS2 (G)
887
+ SO (LM)
888
+ CPT (LM)
889
+ PSL (LM)
890
+ M2DS2 (LM)
891
+ Setting
892
+ WER
893
+ WER
894
+ RAI
895
+ WER
896
+ RAI
897
+ WER
898
+ RAI
899
+ WER
900
+ WER
901
+ RAI
902
+ WER
903
+ RAI
904
+ WER
905
+ RAI
906
+ HP → CV
907
+ 55.9
908
+ 59.68
909
+ −6.8
910
+ 55.3
911
+ 1.2
912
+ 52.95
913
+ 5.3
914
+ 25.26
915
+ 26.44
916
+ −4.7
917
+ 24.24
918
+ 4.0
919
+ 18.35
920
+ 27.4
921
+ HP → LG
922
+ 48.65
923
+ 52.63
924
+ −8.2
925
+ 57.68
926
+ −18.6
927
+ 58.99
928
+ −21.3
929
+ 30.34
930
+ 32.27
931
+ −6.4
932
+ 39.32
933
+ −29.6
934
+ 32.58
935
+ −7.4
936
+ LG → CV
937
+ 59.57
938
+ 66.43
939
+ −13.4
940
+ 81.90
941
+ −39.8
942
+ 51.31
943
+ 12.4
944
+ 25.96
945
+ 31.51
946
+ −21.4
947
+ 52.05
948
+ −100.5
949
+ 17.30
950
+ 33.4
951
+ LG → HP
952
+ 62.13
953
+ 67.51
954
+ −8.7
955
+ 71.46
956
+ −15.0
957
+ 60.09
958
+ 3.3
959
+ 31.48
960
+ 31.58
961
+ −0.3
962
+ 45.36
963
+ −44.1
964
+ 31.36
965
+ 0.4
966
+ CV → LG
967
+ 69.55
968
+ 71.12
969
+ −2.3
970
+ 71.34
971
+ −2.6
972
+ 63.40
973
+ 8.8
974
+ 50.80
975
+ 52.40
976
+ −3.2
977
+ 48.68
978
+ 4.2
979
+ 36.93
980
+ 27.3
981
+ CV → HP
982
+ 70.72
983
+ 73.83
984
+ −4.4
985
+ 78.05
986
+ −10.4
987
+ 68.70
988
+ 2.9
989
+ 52.09
990
+ 52.18
991
+ −0.2
992
+ 54.82
993
+ −5.2
994
+ 41.88
995
+ 19.6
996
+ will give an upper bound estimation for UDA performance.
997
+ We finetune XLSR-53 on CV, HP and LG (separately) and
998
+ perform in-domain evaluation on the respective test sets.
999
+ Results are summarized in Table IV. The first row indicates the
1000
+ performance of greedy decoding, while in the second row we
1001
+ report the performance of the beam search decoder, rescored
1002
+ using the scores of the 4-gram GGC language model. We
1003
+ observe that the greedy decoding performance is under 30
1004
+ WER for both HP and CV, while for LG we achieve ∼ 32
1005
+ WER. This makes sense, as LG is the most diverse dataset,
1006
+ with respect to the included acoustic conditions. Furthermore,
1007
+ we observe that the incorporation of a language model results
1008
+ in an impressive WER reduction on CV, followed by HP and
1009
+ then LG. While CV includes relatively simple phrases with
1010
+ common vocabulary, HP and LG contain more specialized
1011
+ terminology.
1012
+ VII. UNSUPERVISED DOMAIN ADAPTATION USING
1013
+ IN-DOMAIN AUDIO
1014
+ Here, we evaluate the effectiveness of M2DS2 for UDA.
1015
+ We compare with three baselines:
1016
+ 1) Source Only Training (SO): We perform supervised
1017
+ finetuning of XLSR-53 (CTC) using only the source-
1018
+ domain data, and run decoding on the target domain
1019
+ test set. No in-domain data are used for adaptation.
1020
+ 2) Continual Pre-Training (CPT): We perform a pre-
1021
+ training phase using the loss in Eq. (3) on the target
1022
+ domain train set, to create adapted versions of XLSR.
1023
+ Pre-training is run for 20000 steps with batch size
1024
+ 4. Only the audio is used, without transcriptions. The
1025
+ adapted checkpoints are then finetuned by use of CTC
1026
+ loss on the source domain transcribed data. Evaluation
1027
+ is performed on the target test set.
1028
+ 3) Pseudolabeling (PSL): We finetune XLSR-53 using the
1029
+ source domain data with CTC loss. Then we run infer-
1030
+ ence on the source model, to extract silver transcriptions
1031
+ for the target domain training set. We use the silver
1032
+ transcriptions for supervised finetuning on the target
1033
+ domain.
1034
+ In Table V we compare M2DS2 with the SO, CPT and
1035
+ PSL baselines for six adaptation scenarios, i.e., cross dataset
1036
+ evaluation between the three datasets in GREC-MD. The left
1037
+ half corresponds to greedy decoding, while for the right half
1038
+ we use the 4-gram LM trained on GGC. First, we observe
1039
+ the SO model performance. The SO models are the finetuned
1040
+ Fig. 3. Performance of M2DS2 (blue line) for the LG → CV setting, when
1041
+ reducing the amount of available target samples to 50%, 25%, and 10% of
1042
+ the original dataset (horizontal axis). SO performance is indicated with the
1043
+ orange line. Vertical axis: WER, Horizontal Axis: target audio percentage
1044
+ (100% → 0%)
1045
+ models from Table IV, evaluated in out-of-domain settings.
1046
+ We see that out-of-domain evaluation results in a large perfor-
1047
+ mance hit, e.g., while in the CV9 → CV9 in-domain setting
1048
+ we achieve 29.33 WER, in the CV9 → HP out-of-domain
1049
+ setting we get 69.55 WER. This confirms that for real-world
1050
+ ASR tasks, multi-domain evaluation is of essence. Second, we
1051
+ observe that in most adaptation scenarios both CPT and PSL
1052
+ fail to surpass the SO (unadapted) baseline. In the case of CPT,
1053
+ we hypothesize that is due to the relatively data constrained
1054
+ version of our setting. In the best-case scenario, we have 99
1055
+ hours of available target domain audio, which is not enough
1056
+ to perform a discrete CPT stage. Note that most of works in
1057
+ the literature use ∼ 1000 hours of target audio for CPT. In
1058
+ the case of PSL, the poor performance is due to the quality
1059
+ of the silver labels created by the seed model. While the
1060
+ performance would improve with more elaborate approaches
1061
+ (e.g., confidence filtering), in challenging adaptation scenarios
1062
+ PSL approaches are limited by the SO model’s performance.
1063
+ Lastly, we observe that M2DS2 is the only approach among
1064
+ our baselines that manages to achieve a positive RAI in most
1065
+ adaptation scenarios, by consistently outperforming the SO
1066
+ baseline by significant margins. This is exaggerated when
1067
+ we include a LM during inference. One exception in this
1068
+ pattern is the HP → LG scenario, where the SO baseline
1069
+ achieves the best performance. We attribute this to the fact that
1070
+ we performed minimal hyper-parameter tuning during model
1071
+ development.
1072
+
1073
+ 65
1074
+ 60
1075
+ WER
1076
+ 55
1077
+ 50
1078
+ 100%
1079
+ 90%
1080
+ 80%
1081
+ 70%
1082
+ 60%
1083
+ 50%
1084
+ 40%
1085
+ 30%
1086
+ 20%
1087
+ 10%
1088
+ 0%
1089
+ Percentage of In-Domain Audio Data9
1090
+ A. The sample efficiency of M2DS2
1091
+ One key observation in the literature, and in our experiments
1092
+ is that CPT requires a large amount of un-transcribed target
1093
+ domain audio. This raises the question, can we leverage self-
1094
+ supervision for domain adaptation in data constrained settings?
1095
+ In Fig. 3 we evaluate the performance of M2DS2, when
1096
+ we reduce the amount of target domain audio. Specifically
1097
+ we focus on the scenario of LG → CV. The full training
1098
+ corpus of CV contains 12 hours of audio. We train M2DS2
1099
+ with 50%, 25% and 10% of the available samples, or 6, 3
1100
+ and 1.2 hours of audio respectively, and plot the resulting
1101
+ WER on the target (CV) test set. In all cases, the full source
1102
+ (LG) training corpus is used. We observe that M2DS2 achieves
1103
+ lower WER than the SO baseline, even with only 3 hours of
1104
+ target domain audio. While CPT can suffer from catastrophic
1105
+ forgetting, as most multi-stage training approaches, M2DS2
1106
+ avoids this issue, being a single-stage approach with a mixed
1107
+ task-specific and self-supervised objective. This provides a
1108
+ promising avenue for adaptation, when collection of in-domain
1109
+ recordings is expensive, or infeasible.
1110
+ (a) Only target domain self-supervision
1111
+ (b) Target and source domain self-supervision
1112
+ Fig. 4. T-SNE scatter plots of code vectors extracted from M2DS2 without
1113
+ source domain self-supervision (top) and with source domain self-supervision
1114
+ (bottom) for LG (red) and CV (teal)
1115
+ B. The importance of Multi-Domain Self-Supervision
1116
+ In Section III-B we argue that it is essential to include both
1117
+ source and target domain data for the self-supervised objective
1118
+ of M2DS2. To illustrate the effect of this approach, we train
1119
+ two versions of M2DS2 for the LG → CV scenario. For the
1120
+ TABLE VI
1121
+ LANGUAGE ADAPTATION OF THE M2DS2 LG → CV MODEL, USING
1122
+ BIASED AND AUGMENTED LMS. WE USE THE VARIANT OF THE MODEL
1123
+ TRAINED WITH 3 HOURS OF IN-DOMAIN AUDIO. WE VARY THE AMOUNT
1124
+ OF IN-DOMAIN TEXT DATA FROM 752K TOKENS TO 38K TOKENS.
1125
+ Biased LM
1126
+ Augmented LM
1127
+ 100%
1128
+ 11.22
1129
+ 12.84
1130
+ 50%
1131
+ 15.13
1132
+ 15.05
1133
+ 25%
1134
+ 20.84
1135
+ 16.64
1136
+ 10%
1137
+ 27.75
1138
+ 18.47
1139
+ 5%
1140
+ 33.04
1141
+ 19.31
1142
+ Baseline (M2DS2 + Generic LM)
1143
+ 20.7
1144
+ Fig. 5. Language-only adaptation for LG → HP using the SO model finetuned
1145
+ on LG. In-domain text data range from 11M tokens (left) to 110K tokens
1146
+ (right). Blue/dashed: Baseline with generic LM. Purple/circles: Biased LM.
1147
+ Orange/diamonds: Augmented LM.
1148
+ first version we set α = 0.01, while for the second we set
1149
+ α = 0, removing the second term of Eq. (4). We extract the
1150
+ code vectors for the first 100 samples of both LG and CV, and
1151
+ flatten them across the time steps , resulting to 60000 × 768
1152
+ code vectors corresponding to individual timesteps. We plot
1153
+ these code vectors using T-SNE [74] in Fig. 4 for both models.
1154
+ We see that when we do not include the source domain self-
1155
+ supervision, the code vector space collapses in a few tight
1156
+ clusters, and most audio segments correspond to just a few
1157
+ code vectors. This is a visual clue that indicates the mode
1158
+ collapse problem. When we include the source domain term,
1159
+ we see that the that the code vector space has more structure,
1160
+ and coverage of the space is more complete, both for CV
1161
+ (target domain) and LG (source domain). Experimentally we
1162
+ train M2DS2 with α = 0 for all source / target domain pairs
1163
+ and we find that the mode collapse is destructive for target
1164
+ domain performance. During our experiments we got WER in
1165
+ the range 80−99, indicating failure to converge to acceptable
1166
+ solutions across all scenarios. The simple inclusion of both
1167
+ source and target domain self supervision stabilizes training,
1168
+ avoids mode collapse and leads to successful unsupervised
1169
+ adaptation between domains.
1170
+ VIII. UNSUPERVISED AND WEAKLY SUPERVISED
1171
+ LANGUAGE ADAPTATION
1172
+ When small amounts of in-domain textual data are avail-
1173
+ able, simple N-gram LM adaptation techniques can be very
1174
+ effective. In this brief set of experiments, we first explore
1175
+ the unsupervised language adaptation setting, where no in-
1176
+
1177
+ .
1178
+ :
1179
+
1180
+ .
1181
+
1182
+ .
1183
+ .
1184
+ .0
1185
+ C
1186
+ ·
1187
+ .
1188
+ .
1189
+ .
1190
+ .
1191
+ .
1192
+ .
1193
+ 008
1194
+ :
1195
+ .80
1196
+ ·
1197
+ .
1198
+ o43
1199
+ 41
1200
+ 39
1201
+ 37
1202
+ WER
1203
+ 35
1204
+ 33
1205
+ 31
1206
+ 29
1207
+ 100%
1208
+ 90%
1209
+ 80%
1210
+ 70%
1211
+ 60%
1212
+ 50%
1213
+ 40%
1214
+ 30%
1215
+ 20%
1216
+ 10%
1217
+ 0%
1218
+ Percentage of In-Domain Text Data10
1219
+ TABLE VII
1220
+ CLOSING THE GAP BETWEEN SO TRAINING AND FULLY SUPERVISED
1221
+ TRAINING FOR THE LG → CV ADAPTATION SCENARIO USING M2DS2,
1222
+ WITH VARYING AMOUNTS OF AVAILABLE UNPAIRED IN-DOMAIN AUDIO
1223
+ AND TEXT. (U): UNSUPERVISED ACOUSTIC OR LANGUAGE ADAPTATION.
1224
+ (W): WEAKLY SUPERVISED ADAPTATION.
1225
+ Method
1226
+ #Audio (h)
1227
+ #Tokens
1228
+ LM
1229
+ WER
1230
+ SO (U)
1231
+ -
1232
+ -
1233
+ N/A
1234
+ 59.57
1235
+ M2DS2 (U)
1236
+ 3
1237
+ -
1238
+ N/A
1239
+ 57.31
1240
+ M2DS2 (U)
1241
+ 12
1242
+ -
1243
+ N/A
1244
+ 51.31
1245
+ SO (U)
1246
+ -
1247
+ -
1248
+ Generic
1249
+ 25.96
1250
+ SO (U)
1251
+ -
1252
+ 38, 632
1253
+ Augmented
1254
+ 24.67
1255
+ SO (U)
1256
+ -
1257
+ 751, 953
1258
+ Augmented
1259
+ 20.46
1260
+ M2DS2 (U)
1261
+ 3
1262
+ -
1263
+ Generic
1264
+ 20.7
1265
+ M2DS2 (U)
1266
+ 12
1267
+ -
1268
+ Generic
1269
+ 17.3
1270
+ M2DS2 (W)
1271
+ 3
1272
+ 38, 632
1273
+ Augmented
1274
+ 19.31
1275
+ M2DS2 (W)
1276
+ 12
1277
+ 38, 632
1278
+ Augmented
1279
+ 16.29
1280
+ M2DS2 (W)
1281
+ 3
1282
+ 751, 953
1283
+ Augmented
1284
+ 12.84
1285
+ M2DS2 (W)
1286
+ 12
1287
+ 751, 953
1288
+ Augmented
1289
+ 10.61
1290
+ Supervised
1291
+ 12
1292
+ 751, 953
1293
+ Generic
1294
+ 9.52
1295
+ Supervised
1296
+ 12
1297
+ 751, 953
1298
+ Augmented
1299
+ 7.94
1300
+ domain audio is used, and then we relax the problem to
1301
+ the weakly supervised setting, where M2DS2 is combined
1302
+ with the adapted N-Gram LMs. These settings are described
1303
+ in Sections II-A2 and II-A3 respectively. We explore two
1304
+ approaches for LM adaptation: biased LMs, and in-domain
1305
+ data augmentation. To create biased LMs, we train a 4-gram
1306
+ LM on the available in-domain data. Then we replace the
1307
+ generic LM trained on GGC. For LM data augmentation we
1308
+ follow a perplexity filtering approach similar to [71]. We first
1309
+ train a biased LM using available target domain text, and
1310
+ then use it to calculate the perplexity of each line in the
1311
+ GGC corpus. We keep the 10% of the lines with the lowest
1312
+ perplexity. Then we train a 4-gram LM on the augmented “in-
1313
+ domain” corpus and use it for inference.
1314
+ Fig. 5 shows the performance of the SO LG → HP model
1315
+ with biased and augmented LMs, as we reduce the amount
1316
+ of available in-domain text data from 100% to 1% of the
1317
+ in-domain transcriptions (11B tokens to 110K tokens respec-
1318
+ tively). As a baseline we include the LG → HP SO model in
1319
+ combination with the generic LM trained on GGC. We observe
1320
+ that the use of biased LMs can lead to successful adaptation,
1321
+ when an adequate amount of in-domain text data is available.
1322
+ On the other hand the LM augmentation approach results to
1323
+ successful augmentation, even with very small amounts of in-
1324
+ domain text.
1325
+ In Table VI we see the results of LM adaptation, combined
1326
+ with the M2DS2 LG → CV model. To demonstrate the sample
1327
+ efficiency of the approach, we use the variant that was trained
1328
+ using only 25% of the target domain audio (3 hours). We
1329
+ compare with M2DS2 combined with the 4-gram GGC LM for
1330
+ inference. We draw similar conclusions, i.e., use of biased LMs
1331
+ performs well for sufficient text data. When we use augmented
1332
+ LMs we can leverage very small amounts of in-domain text.
1333
+ IX. DISCUSSION & CONCLUSIONS
1334
+ In this work, we have explored Unsupervised and Weakly
1335
+ Supervised Domain Adaptation of ASR systems in the con-
1336
+ text of an under-resourced language, i.e., Greek. We focus
1337
+ on domain adaptation through in-domain self-supervision for
1338
+ XLSR-53, a state-of-the-art multilingual ASR model. Specif-
1339
+ ically, we adopt a mixed task and self-supervised objective,
1340
+ inspired from NLP, and show that using only in-domain self-
1341
+ supervision can lead to mode collapse of the representa-
1342
+ tions created by the contrastive loss of XLSR-53. Therefore,
1343
+ we propose the use of mixed task and multi-domain self-
1344
+ supervision, M2DS2, where the contrastive loss leverages both
1345
+ the source and target domain audio data. For evaluation we
1346
+ create and release HParl, the largest to-date public corpus
1347
+ of transcribed Greek speech (120 hours), collected from the
1348
+ Greek Parliamentary Proceedings. HParl is combined with two
1349
+ other popular Greek speech corpora, i.e., Logotypografia and
1350
+ CommonVoice, for multi-domain evaluation.
1351
+ In our experiments, we find that while most UDA baselines
1352
+ fail in our low-resource setting, the proposed mixed task
1353
+ and multi-domain self-supervised finetuning strategy yields
1354
+ significant improvements for the majority of adaptation sce-
1355
+ narios. Furthermore, we focus our ablations on showcasing
1356
+ the sample efficiency of the proposed finetuning strategy,
1357
+ and demonstrating the necessity of including both source
1358
+ and target domain data for self-supervision. Finally, we show
1359
+ that M2DS2 can be combined with simple language model
1360
+ adaptation techniques in a relaxed weakly supervised setting,
1361
+ where we achieve significant performance improvements with
1362
+ a few hours of in-domain audio and a small, unpaired in-
1363
+ domain text corpus.
1364
+ More concretely, in Table VII we present a summary of
1365
+ the discussed unsupervised and weakly supervised adaptation
1366
+ combinations, for different amounts of available in-domain
1367
+ audio and text. Note that for the weakly supervised scenarios,
1368
+ the in-domain audio and text are unpaired. We see, that when
1369
+ no in-domain data are available, including an n-gram LM
1370
+ trained on large corpora is recommended. Furthermore, when
1371
+ in-domain audio is available, following a mixed multi-domain
1372
+ finetuning strategy using M2DS2 can yield significant WER
1373
+ reductions, even for a few hours of audio. When small amounts
1374
+ of in-domain text is available, using a corpus augmentation
1375
+ strategy, e.g., perplexity filtering, can produce adapted LMs
1376
+ and yield small improvements to the final WER. In the case
1377
+ of sufficient amounts of unpaired in-domain text and audio,
1378
+ independent adaptation of XLSR-53 using the audio data and
1379
+ the n-gram LM using the text data can yield comparable
1380
+ performance with a fully supervised finetuning pipeline.
1381
+ X. FUTURE WORK
1382
+ In the future we plan to explore the effectiveness of the
1383
+ proposed adaptation strategy for other languages, and different
1384
+ adaptation settings, e.g., accent or cross-lingual adaptation.
1385
+ Of special interest is the investigation of the effectiveness
1386
+ of our approach for endagered languages, e.g., Pomak. Fur-
1387
+ thermore, we plan to explore the combination of in-domain
1388
+ self-supervision, when combined with other popular UDA
1389
+ techniques, e.g., teacher student models, adversarial learning,
1390
+ and data augmentation approaches. On the language adaptation
1391
+ side, we plan to explore multi-resolution learning, which has
1392
+
1393
+ 11
1394
+ shown promise for ASR [75], and investigate more elaborate
1395
+ end-to-end weakly supervised adaptation methods. Finally, we
1396
+ plan to expand our study in a multimodal setting, where both
1397
+ audio and video are available, e.g., lip reading.
1398
+ REFERENCES
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1
+ MNRAS 000, 1–7 (2023)
2
+ Preprint 30 January 2023
3
+ Compiled using MNRAS LATEX style file v3.0
4
+ Formation of asymmetric arms in barred galaxies
5
+ P. Sánchez-Martín,1⋆ C. García-Gómez,2 J. J. Masdemont3 and M. Romero-Gómez4
6
+ 1 IUMA, Universidad de Zaragoza, Dept. de Matemática Aplicada, Pedro Cerbuna, 12, 50009 Zaragoza, Spain
7
+ 2D.E.I.M, Universitat Rovira i Virgili, Avd. Països Catalans, 26, 43007 Tarragona, Spain
8
+ 3IEEC & Universitat Politècnica de Catalunya, Dept. de Matemàtiques, Diagonal 647, E08028 Barcelona, Spain
9
+ 4Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, E08028 Barcelona, Spain
10
+ Accepted XXX. Received YYY; in original form ZZZ
11
+ ABSTRACT
12
+ We establish a dynamical mechanism to explain the origin of the asymmetry between the arms observed in some barred disk
13
+ galaxies, where one of the two arms emanating from the bar ends is very well defined, while the second one displays a ragged
14
+ structure, extending between its ridge and the bar. To this purpose, we study the invariant manifolds associated to the Lyapunov
15
+ periodic orbits around the unstable equilibrium points at the ends of the bar. Matter from the galaxy center is transported along
16
+ these manifolds to the periphery, forming this way the spiral arms that emanate from the bar ends. If the mass distribution in
17
+ the galaxy center is not homogeneous, because of an asymmetric bar with one side stronger than the other, or because of a
18
+ non-centered bulge, the dynamics about the two unstable Lagrange points at the ends of the bar will not be symmetric as well.
19
+ One of their invariant manifolds becomes more extended than the other, enclosing a smaller section and the escaping orbits on it
20
+ are fewer and dispersed in a wider region. The result is a weaker arm, and more ragged than the one at the other end of the bar.
21
+ Key words: galaxies: kinematics and dynamics – galaxies: structure – galaxies: spiral
22
+ 1 INTRODUCTION
23
+ A high fraction of disk galaxies appear to be barred (74% to 85%
24
+ in automatic classifications or 36% to 63% with alternative meth-
25
+ ods (Lee et al. 2019)). These authors also suggest that strongly
26
+ barred galaxies, classified as SBs, are preponderant in late-type
27
+ galaxies. Most of these strongly barred galaxies appear to be asym-
28
+ metric under a simple visual inspection. They usually present a
29
+ strong bar and two spiral arms emanating from the bar ends, and
30
+ frequently some inner rings as well. However, a detailed visual in-
31
+ spection of these galaxies, available in the STScI Digitized Sky Sur-
32
+ vey or in the Sloan Digital Sky Survey (SDSS) (Bundy et al. 2015),
33
+ reveals that some barred galaxies exhibit important asymmetries in
34
+ their spiral structure. While one of the two spiral arms is long and
35
+ strongly defined, the second arm shows a ragged structure, and the
36
+ matter is distributed irregularly between the ridge of the spiral arm
37
+ and the bar. A non-symmetric distribution of matter in the galaxy
38
+ disc can be a natural outcome, since disc galaxies may be formed
39
+ through a combination of secular evolution and violent events, in-
40
+ cluding smooth accretion, disc instabilities and minor and major
41
+ mergers (e.g. Tonini et al. 2016). In Fig. 1 we show some examples
42
+ of galaxies showing asymmetric discs.
43
+ In this paper we study the dynamic response of an asymmetric
44
+ mass distribution on the orbital structure of barred galaxies, mod-
45
+ elling the asymmetry of the central parts by a slightly off-centered
46
+ bulge which, viewing the galaxy as a whole, represents a simple but
47
+ realistic model of an asymmetric bar. In this scenario, the Lagrange
48
+ ⋆ E-mail: patricias@unizar.es
49
+ Figure 1. Images of some strongly barred galaxies showing the asymmetric
50
+ spiral arm patterns in the R-filter from the STScI Digitized Sky Survey. In the
51
+ top row we show the galaxies PGC 05849 (NGC 0613), PGC 12412 (NGC
52
+ 1300) and in the bottom one PGC 15941 (NGC 1672) and PGC 54849 (NGC
53
+ 5921).
54
+ invariant points located at the bar ends will present also an asym-
55
+ metric orbital structure, as initially shown in Colin & Athanassoula
56
+ (1989). The new step in this work is to additionally study the unsta-
57
+ ble periodic orbits around these Lagrange points and the associated
58
+ unstable manifolds which provide escape routes for orbits which
59
+ can transport material from the central regions to the outer parts of
60
+ the discs (Romero-Gómez et al. 2006; Sánchez-Martín et al. 2016).
61
+ © 2023 The Authors
62
+ arXiv:2301.11385v1 [astro-ph.GA] 26 Jan 2023
63
+
64
+ .PGC 05849(R)
65
+ PGC 12412 (R)
66
+ PGC 15941 (R)
67
+ PGC 54849 (R)2
68
+ P. Sánchez-Martín et al.
69
+ These unstable manifolds are the backbones of the spiral arms ema-
70
+ nating from the bar ends. In the case of asymmetric Lagrange points,
71
+ however, the unstable manifolds display also important differences.
72
+ The orbits following one of these manifolds are close together and
73
+ can explain the presence of a strong spiral arm. On the other hand,
74
+ the second manifold has a more open orbital structure, with its es-
75
+ caping orbits dispersed in a wider zone, causing the ragged struc-
76
+ ture of the second arm. Thus, the orbital structure of the unstable
77
+ Lagrange points is able to explain the asymmetric phenomenology
78
+ present in some of the strongly barred galaxies.
79
+ The paper is organized as follows: in Section 2 we quantify the
80
+ level of asymmetry in asymmetric barred galaxies using a two-
81
+ dimensional Fourier transform method. In Sections 3 and 4 we de-
82
+ scribe the asymmetric barred galaxy model used in this work and
83
+ show the orbital analysis and invariant manifolds, respectively. Dif-
84
+ ferent approaches are summarized and conclusions are given in Sec-
85
+ tion 5.
86
+ 2 ANALYSIS OF THE BAR ASYMMETRY
87
+ In this paper, we model the galactic asymmetric mass distribution
88
+ using a classical galactic model with three symmetric components
89
+ (disc, bar and bulge) but displacing slightly the bulge from the
90
+ galaxy centre. This results in a total mass distribution biased to-
91
+ wards one side of the bar. Many barred galaxies show some kind
92
+ of asymmetries in their inner mass distribution with one side of the
93
+ bar stronger than the other. An example is the galaxy NGC 1300,
94
+ studied by Patsis et al. (2010). They generated a numerical model
95
+ of the potential using K images of the galaxy. The resultant model
96
+ was clearly asymmetric, containing a bar with a stronger arm. This
97
+ model was used to study the orbits associated with this asymmetric
98
+ barred structure. Many barred galaxies show similar asymmetries.
99
+ Some of these galaxies are shown in Fig. 1. In order to quantify
100
+ these asymmetries, we can analyze in detail the mass distribution of
101
+ one of the galaxies showing this phenomenology, namely the galaxy
102
+ PGC 70419 (NGC 7479). For this purpose, we use a galaxy image
103
+ from the SDSS survey in the infrared z-filter. Assuming a constant
104
+ M/L ratio, the light distribution is a good tracer of the mass distribu-
105
+ tion in the disk. The image is previously cleaned from background
106
+ stars and then deprojected using the FFT method (García-Gómez et
107
+ al. 2004) giving a position angle (PA) of 38◦ and an inclination angle
108
+ of 45◦.
109
+ The image is then decomposed in its Fourier components using a
110
+ technique first introduced by Considère & Athanassoula (1982); Iye
111
+ et al. (1982), and further developed by García-Gómez et al. (2017).
112
+ For deprojected image of the galaxy I(u,θ), where u = ln(r), we cal-
113
+ culate the two-dimensional Fourier transform defined as
114
+ A(p,m) =
115
+ � umax
116
+ umin
117
+ � 2π
118
+ 0
119
+ I(u,θ)ei(pu+mθ) dθdu
120
+ (1)
121
+ Where m is the azimuthal frequency, associated with the multiplicity
122
+ of the structures, i.e., the number of arms while p is the radial fre-
123
+ quency, associated to the pitch angle of the structure i, through the
124
+ relation
125
+ p = −
126
+ m
127
+ tan(i)
128
+ (2)
129
+ In this way, the m = 1 spectrum contains the spiral components with
130
+ no symmetry, the m = 2 spectrum the components with a periodicity
131
+ of π radians or bisymmetric signals, and so on for the rest of the m
132
+ frequencies. Each of the azimuthal components m = 1,2,... can be
133
+ further decomposed in its radial components using a Gaussian fit to
134
+ the modulus and keeping the phase constant as follows:
135
+ | A(p,m) |=
136
+ Ng
137
+
138
+ j=1
139
+ C j exp−(p− pj)2
140
+ 2σ2
141
+ j
142
+ .
143
+ (3)
144
+ In this relation, pj represents the central frequency of the Gaussian,
145
+ σj its dispersion, and C j its amplitude. The number of Gaussians
146
+ used in each fit, Ng, will depend on the complexity of the spectrum.
147
+ In the upper panel of Fig. 2 we show the deprojected SDSS galaxy
148
+ image of NGC 7479 using the z-filter. Note that one of its arms is
149
+ very pronounced, while the opposite arm appears diffuse. On the left
150
+ of the middle and lower panels of this figure we show the modulus
151
+ of the Fourier spectrum of the m = 1 component which is associ-
152
+ ated to the asymmetries in the light distribution. The modulus of the
153
+ Fourier components are normalized to the modulus of the stronger
154
+ component, which in this case is the m = 2 containing the bisymmet-
155
+ ric signals of the strong bar and the spiral arms. The m = 1 spectrum
156
+ shown here contains the spiral components responsible for the asym-
157
+ metries in the mass distribution. The relative low values of the m = 1
158
+ in this scale shows that the mass asymmetry is a second order effect
159
+ for this galaxy. In red we superpose two of the Gaussian components
160
+ into which this signal is decomposed. These Gaussian components
161
+ can be transformed back to obtain the density distribution associated
162
+ to each particular spiral mode. The density distributions of these spi-
163
+ ral components are presented in the respective right panels in green,
164
+ superposed on the galaxy image. The isocontours come from the
165
+ normalized density corresponding to the Gaussian components and
166
+ show that the asymmetries are related to the southern end of the
167
+ bar and the strong arm emanating from its end. This shows that the
168
+ galaxy has an asymmetric mass distribution, biased to the southern
169
+ part of the galaxy.
170
+ 3 CHARACTERISTICS OF THE GALACTIC MODEL
171
+ The equations of motion of the classical galactic model (see e.g.
172
+ Pfenniger 1984; Skokos et al. 2002; Romero-Gómez et al. 2006;
173
+ Sánchez-Martín et al. 2016) describe the movement of a particle in
174
+ a gravitational potential φ . In the rotating frame, the equations of
175
+ motion are described by
176
+ ¨r = −2(Ωp × ˙r)−Ωp ×(Ωp ×r)−∇φ,
177
+ (4)
178
+ where r = (x, y, z) is the position of the particle, φ is the total grav-
179
+ itational potential of the system and Ωp is the angular velocity of
180
+ the bar around the z-axis, Ωp = (0,0,Ω). The origin of the reference
181
+ frame is located at the center of mass of the system and the frame is
182
+ aligned with the main axis of the bar.
183
+ The potential model φ used in this paper is a combination of three
184
+ analytical components: an axisymmetric Miyamoto-Nagai disc with
185
+ potential φd (Miyamoto & Nagai 1975), an ellipsoid Ferrers bar with
186
+ potential φb (Ferrers 1877) and a bulge structure represented by a
187
+ Plummer spheroid potential φbl (Plummer 1911). The total potential
188
+ is the addition of these three components, φ = φd +φb +φbl.
189
+ The disc potential is described by the equation
190
+ φd = −
191
+ GMd
192
+
193
+ R2 +(A+
194
+
195
+ B2 +z2)2
196
+ ,
197
+ (5)
198
+ were R2 = x2 + y2 is the cylindrical coordinate radius of the poten-
199
+ tial in the disc plane, and z is the vertical distance over the disk
200
+ component. The parameters G, Md, A and B denote the gravitational
201
+ MNRAS 000, 1–7 (2023)
202
+
203
+ Formation of asymmetric arms in barred galaxies
204
+ 3
205
+ Figure 2. Analysis of the m = 1 component of the SDSS galaxy image of
206
+ NGC 7479 in the z-filter. The upper panel shows the deprojected image of the
207
+ galaxy. On the left of the middle and lower panels we show the modulus of
208
+ the m = 1 component of the Fourier spectrum, and with red-dashed lines two
209
+ Gaussian components fitted to this modulus. In the respective right panels,
210
+ we show in green the density distribution associated to each of these single
211
+ components superimposed on the galaxy image.
212
+ constant, the disc mass and the shape of the disc, respectively. Tak-
213
+ ing A = 0 the potential becomes the Plummer potential. The bar is
214
+ modelled by an ellipsoid with density function
215
+ ρ =
216
+
217
+ ρ0(1−m2)nh,
218
+ m ≤ 1
219
+ 0,
220
+ m > 1
221
+ (6)
222
+ where m2 = x2/a2 + y2/b2 + z2/c2, a (semi-major axis), b (interme-
223
+ diate axis) and c (semi-minor axis) determine the shape of the bar,
224
+ nh is the homogeneity degree of the mass distribution (nh = 2 in our
225
+ work) and ρ0 is the density at the origin (ρ0 = 105
226
+ 32π
227
+ GMb
228
+ abc if nh = 2,
229
+ where Mb is the bar mass).
230
+ The unit of length considered is the kpc, the time unit is ut =
231
+ 2 × 106 yr, Ω is in [ut]−1, and the mass unit is uM = 2 × 1011M⊙,
232
+ where M⊙ denotes the mass of the Sun. G stands for the gravitational
233
+ constant.
234
+ In our model, we select a disc radius of A = 3 kpc, height B = 1 kpc
235
+ and mass giving a value of GMd = 0.52 kpc3/u2
236
+ t . The dimensions of
237
+ the bar are a = 6 kpc, b = 1.5 kpc, c = 0.4 kpc, and its mass is such
238
+ that GMb = 0.4 kpc3/u2
239
+ t . The Plummer bulge with a radius B = 1 kpc
240
+ and GMbl is set to have around 15% of the mass of the bar GMb, in
241
+ order that G(Md + Mb + Mbl) = 1. The bar pattern speed is fixed as
242
+ Ω = 0.0633 [ut]−1 (∼ 30.97 km/s/kpc).
243
+ In the rotating reference frame aligned with the main axis of the
244
+ bar, the equations of motion given in Eq.(4) are written as the fol-
245
+ lowing dynamical system:
246
+ ���������
247
+ ¨x = 2Ω ˙y+Ω2 x−φx
248
+ ¨y = −2Ω ˙x+Ω2 y−φy
249
+ ¨z = −φz .
250
+ (7)
251
+ The Jacobi first integral of Eq.(4) (which can be regarded as the
252
+ energy in the rotating frame) is given by
253
+ CJ(x,y,z, ˙x, ˙y, ˙z) = −(˙x2 + ˙y2 + ˙z2)+Ω2 (x2 +y2)−2φ,
254
+ (8)
255
+ and the effective potential is defined by φeff = φ− 1
256
+ 2 Ω2 (x2 +y2).
257
+ The goal of this paper is to analyze which is the effect of an asym-
258
+ metric distribution of mass in the central parts of the galaxy on the
259
+ external spiral structure. To introduce this asymmetry, we displace
260
+ the bulge potential along the x-axis (main axis of the bar) towards the
261
+ equilibrium point placed at the right end of the bar. The bulge center
262
+ is then located at (xd,0,0) using several values for the displacement
263
+ xd (in kpc), namely: (0,0,0), (0.5,0,0), (1,0,0) and (1.5,0,0). We
264
+ move the center of our potential model to the resulting center of
265
+ mass of the system. The plot of the equal density contours of the
266
+ resulting models are shown in Fig. 3. Note that the displacement of
267
+ the bulge along the main axis of the bar creates an asymmetry in the
268
+ central part of the model and around the libration points L1 and L2.
269
+ The rotation curves of the model, defined as V2
270
+ rot = r dφ
271
+ dr , where the
272
+ potential is φ = φd +φb +φbl with the selected values of the parame-
273
+ ters and for the above explained positions of the bulge displacement,
274
+ are shown in Fig. 4. The resulting rotation curve is reasonably flat in
275
+ the outer parts, and displays only minor differences in the position
276
+ of the maximum.
277
+ 4 DYNAMICS OF THE MODELS
278
+ The solutions of ∇φeff = 0 in rotating coordinates give five La-
279
+ grangian equilibrium points of the model (Li, i = 1,...,5). Points L1
280
+ and L2 are linearly unstable points and lie on the x-axis at the ends of
281
+ the bar. Point L3 is linearly stable and it is placed on the origin of co-
282
+ ordinates in the case of a symmetric model. Points L4 and L5 are also
283
+ linearly stable and located out of the x-axis. A detailed explanation
284
+ of the dynamics around these points can be found in Athanassoula
285
+ et al. (1983); Romero-Gómez et al. (2006); Sánchez-Martín et al.
286
+ (2016).
287
+ The regions where φeff > CJ are forbidden regions for a star of en-
288
+ ergy CJ. In the plane, these regions are delimited by the zero velocity
289
+ curves, which are defined by the level surfaces φeff = CJ intersected
290
+ with z = 0. In Fig. 5 we show the zero velocity curves corresponding
291
+ to an energy slightly above of that of the equilibrium point, CJ,Li +δ,
292
+ for the models with off-centered bulges in Fig. 4, i.e., setting the val-
293
+ ues xd = 0, 0.5, 1, 1.5. These zero velocity curves limit the inner and
294
+ outer regions in the galaxy. In the symmetric case (xd = 0), the en-
295
+ ergy of both equilibrium points L1 and L2 is the same, CJ,L1 = CJ,L2.
296
+ For the asymmetric cases, as xd grows CJ,L2 becomes smaller than
297
+ CJ,L1, which makes the zero velocity curves related to L1 more open
298
+ at the opposite point.
299
+ Particular attention is given in this work to the unstable points
300
+ L1 and L2. They are surrounded by planar and vertical families of
301
+ Lyapunov periodic orbits, which are unstable in the neighborhood of
302
+ the equilibrium point. The relevant family for the transport of matter
303
+ between the inner and outer regions of the galaxy is the planar family
304
+ (Romero-Gómez et al. 2009). Fig. 6 shows the (x, y) projection of
305
+ the planar family for the models with xd = 0 (solid black line) and
306
+ xd = 1.5 (dotted red line). The family around L2 is displayed at the
307
+ left panel and that around L1 at the right one. In the symmetric case
308
+ (xd = 0) both families coincide, in the asymmetric one (xd = 1.5) the
309
+ family around L2 is smaller than the one around L1.
310
+ These critical points are characterized by the superposition of a
311
+ saddle and two harmonic oscillations in the rotating frame. Conse-
312
+ quently, for a given Jacobi constant, stable and unstable invariant
313
+ MNRAS 000, 1–7 (2023)
314
+
315
+ PGC70419
316
+ (z)
317
+ NGC 7479
318
+ m=1
319
+ A=0.11
320
+ g=
321
+ 0.57
322
+ A
323
+ m=1
324
+ A=
325
+ 0.09
326
+ 1.86
327
+ =6
328
+ 0.86
329
+ A
330
+ -10
331
+ 0
332
+ 10
333
+ p4
334
+ P. Sánchez-Martín et al.
335
+ -5
336
+ 0
337
+ 5
338
+ x
339
+ -5
340
+ 0
341
+ 5
342
+ y
343
+ -5
344
+ 0
345
+ 5
346
+ x
347
+ -5
348
+ 0
349
+ 5
350
+ y
351
+ -5
352
+ 0
353
+ 5
354
+ x
355
+ -5
356
+ 0
357
+ 5
358
+ y
359
+ -5
360
+ 0
361
+ 5
362
+ x
363
+ -5
364
+ 0
365
+ 5
366
+ y
367
+ Figure 3. Isodensity curves of the potential φ = φd +φb +φbl. Equilibrium points of the system are marked with a cross. The Ferrers bar and the Plummer bulge
368
+ are outlined by dotted black curves. From left to right: Bulge centered at (0,0,0), (0.5,0,0), (1,0,0) and at (1.5,0,0).
369
+ 0
370
+ 2
371
+ 4
372
+ 6
373
+ 8
374
+ 10
375
+ R [kpc]
376
+ 0
377
+ 50
378
+ 100
379
+ 150
380
+ Vrot [km/s]
381
+ Figure 4. Rotation curve of the potential φ = φd + φb + φbl for the bulge
382
+ centered at (0,0,0) (in blue), (0.5,0,0) (in red), (1,0,0) (in magenta) and
383
+ (1.5,0,0) (in green).
384
+ manifolds emanate from the periodic Lyapunov orbit around each
385
+ point. The stable manifold is defined as the set of orbits that asymp-
386
+ totically tend to the periodic orbit forward in time, and the unstable
387
+ manifold consists of those orbits which depart asymptotically from
388
+ the periodic orbit. These latter manifold drive the escape orbits that
389
+ are responsible for the visible trajectories, in the form of arms and
390
+ rings. Fig. 7 shows the invariant manifolds associated to L1 and L2
391
+ for the set of models where xd = 0, 0.5, 1, 1.5. The effect of an asym-
392
+ metric mass distribution, modelled by the displacement of the bulge,
393
+ makes the exterior manifold that emanates from L2 to differ from the
394
+ invariant manifold associated with L1.
395
+ The transit orbits trapped inside the manifolds are in charge of
396
+ the transfer of matter, from the inner to the outer regions delimited
397
+ by the zero velocity curves (Gidea & Masdemont 2007; Romero-
398
+ Gómez et al. 2006; Sánchez-Martín et al. 2018). As the dynamics of
399
+ our system (4) takes place in a four dimensional phase space when
400
+ we consider orbits with z = ˙z = 0 (the plane z = 0 is invariant), the
401
+ intersection of the trajectories of the inner branch of the stable in-
402
+ variant manifold of a Lyapunov orbit with the hyperplane S defined
403
+ by the section x = 0 in phase space gives a closed curve in the (y, ˙y)
404
+ projection. Given a pair (y, ˙y) and an energy level, this lets us to de-
405
+ fine a state on S by selecting (0,y,0, ˙x, ˙y,0), where ˙x is determined
406
+ by the fixed energy level of the Lyapunov orbit and the orientation
407
+ of the crossing, taking into account Eq. (8),
408
+ ˙x =
409
+
410
+ −˙y2 +Ω2y2 −2φ(0,y,0)−(CJ,Li +δ),
411
+ (9)
412
+ with CJ,Li +δ the energy of the Lyapunov orbit around the point Li,
413
+ i = 1,2, slightly above of that of the equilibrium point. The forward
414
+ in time integration of initial conditions corresponding to (y, ˙y) points
415
+ inside the closed curve establishes the trajectories of the particles
416
+ confined inside the invariant manifold that transit from the inner to
417
+ the outer region.
418
+ This procedure enables us to quantify the amount of matter trans-
419
+ ferred inside each invariant manifold associated to any energy level
420
+ and, consequently, from each spiral arm of the galaxy. Fig. 8 repre-
421
+ sents the (y, ˙y) projection of the intersection with the hyperplane S
422
+ of the invariant manifolds arising from three orbits with different Ja-
423
+ cobi constants in the Lyapunov family around L2 (top), and the same
424
+ corresponding orbits for L1 (bottom). The three closed curves are
425
+ displayed with different colors, from red to yellow, according to the
426
+ increasing Jacobi constant. The initial conditions located inside each
427
+ of the closed curves are marked with crosses of the same color as the
428
+ curve. From left to right, the figure displays the (y, ˙y) projection for
429
+ the models with bulge center ranging from (0,0,0) to (1.5,0,0). The
430
+ main feature to notice in this figure is the narrowing and stretching
431
+ of the curves as the bulge moves away from the L2 equilibrium point.
432
+ This constriction marks the difference between initial conditions for
433
+ the escape orbits related with L2 and those related to L1. The initial
434
+ conditions emanating from near L2 are more extended along the y-
435
+ axis, resulting in a dispersion in space of the orbits enclosed by the
436
+ manifold. As we integrate forward in time these initial conditions,
437
+ we obtain fewer escape orbits, and dispersed in a wider region, in
438
+ comparison to those emanating near L1. So, the spiral arm defined
439
+ by these orbits becomes more spread out and less bright. The result
440
+ of these integrations is exhibited in Fig. 9 where it can be appreci-
441
+ ated how the orbits inside the invariant manifolds develop the arms.
442
+ The orbits corresponding to the L1 point are more concentrated as
443
+ the density of the bar increases in the region close to L1.
444
+ 5 DISCUSSION AND CONCLUSIONS
445
+ The goal of this work is to analyze the relation between the asym-
446
+ metric arms observed in certain barred galaxies and the mass distri-
447
+ bution in the central part of the galaxy. We propose and show that
448
+ there is a strong correlation between their asymmetries. The mod-
449
+ els used to study this fact consist of a superposition of a bar and an
450
+ off-centered bulge. The displacements of the bulge along the main
451
+ axis of the bar introduce an increasing asymmetry in the central part
452
+ of the model. When the center of mass of the galaxy is displaced
453
+ along the major axis of the bar, the zero velocity curves of the ef-
454
+ fective potential become asymmetric: the one at the opposite side
455
+ of the dispacement becomes more open (see Fig. 5). We show that
456
+ this asymmetry between the zero velocity curves in their opening
457
+ is carried to the orbits trapped by invariant manifolds around the
458
+ Lagrangian equilibrium points L1 and L2. This orbits become asym-
459
+ MNRAS 000, 1–7 (2023)
460
+
461
+ Formation of asymmetric arms in barred galaxies
462
+ 5
463
+ -5
464
+ 0
465
+ 5
466
+ x
467
+ -5
468
+ 0
469
+ 5
470
+ y
471
+ -5
472
+ 0
473
+ 5
474
+ x
475
+ -5
476
+ 0
477
+ 5
478
+ y
479
+ -5
480
+ 0
481
+ 5
482
+ x
483
+ -5
484
+ 0
485
+ 5
486
+ y
487
+ -5
488
+ 0
489
+ 5
490
+ x
491
+ -5
492
+ 0
493
+ 5
494
+ y
495
+ Figure 5. Zero velocity curves for a Jacobi constant slightly above to that of L1 (blue) and for one slightly above to that of L2 (red). Equilibrium points of
496
+ the system are marked with a cross. The Ferrers bar and the Plummer bulge are outlined by dotted black curves. From left to right: Bulge centered at (0,0,0),
497
+ (0.5,0,0), (1,0,0) and at (1.5,0,0).
498
+ -7
499
+ -6.5
500
+ -6
501
+ -5.5
502
+ x
503
+ -0.6
504
+ -0.4
505
+ -0.2
506
+ 0
507
+ 0.2
508
+ 0.4
509
+ y
510
+ 5.5
511
+ 6
512
+ 6.5
513
+ 7
514
+ x
515
+ -0.6
516
+ -0.4
517
+ -0.2
518
+ 0
519
+ 0.2
520
+ 0.4
521
+ y
522
+ Figure 6. Lyapunov family of periodic orbits around Li, i = 1, 2, for a range
523
+ of values of the Jacobi constant in (CJ,Li, CJ,Li +10−4). Solid black line: (x, y)
524
+ projection of the family around L2 (left) and L1 (right) for the symmetric
525
+ model with bulge centered at (0,0,0), CJ,Li = −0.2338, i = 1, 2. Dotted red
526
+ line: (x, y) projection of the family around L2 (left) and L1 (right) for the
527
+ asymmetric model with bulge centered at (1.5,0,0), CJ,L2 = −0.2358, CJ,L1 =
528
+ −0.2331.
529
+ metric as well, with the arm at the opposite side of the displacement
530
+ wider and more spread out. We quantify this effect by computing the
531
+ fraction of orbits trapped by the manifolds. The procedure consists
532
+ in intersecting the manifold with a hyperplane and, inside the result-
533
+ ing closed curve in the (y, ˙y) projection, obtaining the set of points
534
+ with the same energy of the manifold as initial conditions that char-
535
+ acterize the escaping orbits.
536
+ Indeed, the above closed curve turns out to be the most relevant
537
+ feature in order to predict the asymmetry of the arms. When the
538
+ bulge is displaced along the main axis of the bar, causing an asym-
539
+ metry in the density distribution of the model, the closed curves
540
+ around the equilibrium point at the end of the less dense side of
541
+ the bar, narrow. This leads to a smaller measure of states in phase
542
+ space that become initial conditions for escaping orbits. Moreover,
543
+ these closed curves are more stretched, making the distribution of
544
+ the points inside them more spread out in space. Both aspects are
545
+ responsible for the resulting ragged and dispersed spiral arms in the
546
+ less dense end of the bar, while the arm associated with the denser
547
+ end becomes brighter and well defined.
548
+ The dynamics of off-centered bars have been studied using ana-
549
+ lytical models (Colin & Athanassoula 1989), showing how the La-
550
+ grangian points vary as a function of the displacement with respect
551
+ to the center of mass of the galaxy. Łokas (2021) digs into the
552
+ barred galaxies in the IllustrisTNG simulations to check how com-
553
+ mon asymmetric and off-centered bars are and study its possible ori-
554
+ gin, concluding that asymmetric bars are persistent in time and this
555
+ asymmetric may be due to the interaction with a companion galaxy
556
+ or due to the disc itself being asymmetric. In either case, no further
557
+ development has been proposed to link the asymmetry in the bar
558
+ with the one-armed dominant spiral structure.
559
+ As shown in Fig. 1, this is a quite common phenomenon in barred
560
+ galaxies. In particular, the dynamics of NGC 1300 have been studied
561
+ in detail by (Patsis et al. 2010), including an orbital analysis. The iso-
562
+ contours of a smooth K-band image (see their Fig. 1) show a clear
563
+ asymmetric bar distribution leading to an asymmetry in the spiral
564
+ arms, which is reproduced by the different orbital models. Other ex-
565
+ amples of one armed dominated galaxies with an asymmetric bar
566
+ may be: NGC 4027 (Phookun et al. 1992), the density contours
567
+ show an asymmetric and off-centered bar leading to a mass distri-
568
+ bution dominated by an m = 1 mode, though a weak counter-part is
569
+ also clear; the Large Magellanic Cloud (de Vaucouleurs & Freeman
570
+ 1972), showing a rotational asymmetry, which is confirmed by more
571
+ recent studies (e.g. Jiménez-Arranz et al. 2022; Niederhofer et al.
572
+ 2022, and references therein).
573
+ To sum up, we show that asymmetric arms are a common feature
574
+ in barred galaxies and that there is a clear correlation between arm
575
+ asymmetry and the displacement of the center of mass caused by an
576
+ asymmetric central density distribution. Escaping orbits trapped in
577
+ the invariant manifolds of an asymmetric bar distribution are asym-
578
+ metric and the limiting case would be to have only one armed barred
579
+ spiral.
580
+ ACKNOWLEDGEMENTS
581
+ P.S.M. thanks the Spanish Ministry of Economy grants PID2020-
582
+ 117066GB-I00
583
+ and
584
+ PID2021-123968NB-I00.
585
+ J.J.M.
586
+ thanks
587
+ MINECO-FEDER for the grant PID2021-123968NB-I00 and
588
+ the Catalan government grant 2017SGR-1049. M.R.G. thanks
589
+ the Spanish Ministry of Science grant MICIU/FEDER RTI2018-
590
+ 095076-B-C21, and the Institute of Cosmos Sciences University
591
+ of Barcelona (ICCUB, Unidad de Excelencia "María de Maeztu")
592
+ grant CEX2019-000918-M.
593
+ Funding for the Sloan Digital Sky Survey IV has been provided
594
+ by the Alfred P. Sloan Foundation, the U.S. Department of En-
595
+ ergy Office of Science, and the Participating Institutions. SDSS
596
+ (www.sdss.org) acknowledges support and resources from the Cen-
597
+ ter for High-Performance Computing at the University of Utah.
598
+ DATA AVAILABILITY
599
+ The data underlying this article are available in the article.
600
+ MNRAS 000, 1–7 (2023)
601
+
602
+ 6
603
+ P. Sánchez-Martín et al.
604
+ -10
605
+ 0
606
+ 10
607
+ x
608
+ -10
609
+ -5
610
+ 0
611
+ 5
612
+ 10
613
+ y
614
+ -10
615
+ 0
616
+ 10
617
+ x
618
+ -10
619
+ -5
620
+ 0
621
+ 5
622
+ 10
623
+ y
624
+ -10
625
+ 0
626
+ 10
627
+ x
628
+ -10
629
+ -5
630
+ 0
631
+ 5
632
+ 10
633
+ y
634
+ -10
635
+ 0
636
+ 10
637
+ x
638
+ -10
639
+ -5
640
+ 0
641
+ 5
642
+ 10
643
+ y
644
+ Figure 7. Unstable invariant manifolds associated to the Lyapunov periodic orbits of L1 and L2. The position of the equilibrium points is marked with crosses.
645
+ The bar and the bulge are outlined by dotted black curves. The reference system is marked with a solid black line and the center of the bar with a dotted magenta
646
+ line. From left to right: Bulge centered at (0,0,0), (0.5,0,0), (1,0,0) and at (1.5,0,0).
647
+ 1.6
648
+ 1.8
649
+ 2
650
+ 2.2
651
+ 2.4
652
+ 2.6
653
+ 2.8
654
+ y
655
+ 0
656
+ 0.05
657
+ 0.1
658
+ 0.15
659
+ 0.2
660
+ 0.25
661
+ 1.6
662
+ 1.8
663
+ 2
664
+ 2.2
665
+ 2.4
666
+ 2.6
667
+ y
668
+ 0
669
+ 0.05
670
+ 0.1
671
+ 0.15
672
+ 0.2
673
+ 0.25
674
+ 1.2
675
+ 1.4
676
+ 1.6
677
+ 1.8
678
+ 2
679
+ 2.2
680
+ y
681
+ 0
682
+ 0.05
683
+ 0.1
684
+ 0.15
685
+ 0.2
686
+ 0.25
687
+ 0.6
688
+ 0.8
689
+ 1
690
+ 1.2
691
+ 1.4
692
+ 1.6
693
+ 1.8
694
+ y
695
+ 0.1
696
+ 0.15
697
+ 0.2
698
+ 0.25
699
+ 0.3
700
+ -2.6
701
+ -2.4
702
+ -2.2
703
+ -2
704
+ y
705
+ -0.06
706
+ -0.04
707
+ -0.02
708
+ 0
709
+ -2.8
710
+ -2.6
711
+ -2.4
712
+ -2.2
713
+ y
714
+ -0.06
715
+ -0.04
716
+ -0.02
717
+ 0
718
+ -3
719
+ -2.8
720
+ -2.6
721
+ -2.4
722
+ y
723
+ -0.06
724
+ -0.04
725
+ -0.02
726
+ 0
727
+ -3
728
+ -2.8
729
+ -2.6
730
+ -2.4
731
+ y
732
+ -0.06
733
+ -0.04
734
+ -0.02
735
+ 0
736
+ Figure 8. (y, ˙y) projection of the intersection of the plane S with the stable manifolds associated to three orbits with different Jacobi constant of the Lyapunov
737
+ family around L2 (top) and L1 (bottom). The intersection of each manifold is in a color, from red to yellow, according to the energy of the manifold. Initial
738
+ conditions distributed inside each curve are marked with a cross with the same color as the curve. From left to right: Bulge centered at (0,0,0), (0.5,0,0), (1,0,0)
739
+ and (1.5,0,0). Note that the axis limits are different but their scale and range length are constant in each row.
740
+ Figure 9. Orbits resulting from the integration of the initial conditions of Fig. 8. Equilibrium points marked in red. Bar and bulge outlined by dotted black
741
+ curves. The reference system is marked with a solid black line and the center of the bar with a dotted magenta line. From left to right: Bulge centered at (0,0,0),
742
+ (0.5,0,0), (1,0,0) and at (1.5,0,0).
743
+ REFERENCES
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+ X10
788
+ 5
789
+ 0
790
+ -5
791
+ -10
792
+ -10
793
+ 0
794
+ 10
795
+ X10
796
+ 5
797
+ 0
798
+ -5
799
+ -10
800
+ -10
801
+ 0
802
+ 10
803
+ XFormation of asymmetric arms in barred galaxies
804
+ 7
805
+ 2009, CNSNS, 14, 4123.
806
+ Sánchez-Martín P., Romero-Gómez M., Masdemont J. J., 2016, A&A, 588,
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+ A76.
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+ Sánchez-Martín P., Masdemont J. J., Romero-Gómez M., 2018, A&A, 618,
809
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+ Skokos Ch., Patsis P. A., Athanassoula E., 2002, MNRAS, 333, 847.
811
+ Tonini C., Mutch S. J., Croton D. J., Wyithe J. S. B., 2016, MNRAS, 459,
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+ 4109.
813
+ This paper has been typeset from a TEX/LATEX file prepared by the author.
814
+ MNRAS 000, 1–7 (2023)
815
+
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1
+ arXiv:2301.05060v1 [cs.CR] 12 Jan 2023
2
+ Evaluating the Fork-Awareness of Coverage-Guided Fuzzers
3
+ Marcello Maugeri1
4
+ a, Cristian Daniele2
5
+ b, Giampaolo Bella1
6
+ c, and Erik Poll2
7
+ d
8
+ 1Department of Maths and Computer Science, University of Catania, Catania, Italy
9
+ 2Department of Digital Security, Radboud University, Nijmegen, The Netherlands
10
+ marcello.maugeri@phd.unict.it, cristian.daniele@ru.nl, giampaolo.bella@unict.it, erikpoll@cs.ru.nl
11
+ Keywords:
12
+ Fuzzing, Fork, Security Testing, Software Security
13
+ Abstract:
14
+ Fuzz testing (or fuzzing) is an effective technique used to find security vulnerabilities. It consists of feeding a
15
+ software under test with malformed inputs, waiting for a weird system behaviour (often a crash of the system).
16
+ Over the years, different approaches have been developed, and among the most popular lies the coverage-based
17
+ one. It relies on the instrumentation of the system to generate inputs able to cover as much code as possible.
18
+ The success of this approach is also due to its usability as fuzzing techniques research approaches that do
19
+ not require (or only partial require) human interactions. Despite the efforts, devising a fully-automated fuzzer
20
+ still seems to be a challenging task. Target systems may be very complex; they may integrate cryptographic
21
+ primitives, compute and verify check-sums and employ forks to enhance the system security, achieve better
22
+ performances or manage different connections at the same time. This paper introduces the fork-awareness
23
+ property to express the fuzzer ability to manage systems using forks. This property is leveraged to evaluate 14
24
+ of the most widely coverage-guided fuzzers and highlight how current fuzzers are ineffective against systems
25
+ using forks.
26
+ 1
27
+ Introduction
28
+ In the last years, plenty of fuzzers have been devel-
29
+ oped to deal with sophisticated software and nowa-
30
+ days it is extremely common that network systems
31
+ employ forks to deal with different connections at the
32
+ same time. This leads to 1) the need to devise accurate
33
+ and ad-hoc fuzzers and 2) the need to evaluate these
34
+ fuzzer according to their ability to cope with such ad-
35
+ vanced systems.
36
+ Unfortunately,
37
+ as
38
+ pointed
39
+ out
40
+ in
41
+ [Hazimeh et al., 2020],
42
+ it is not easy to bench-
43
+ mark all of them since the fuzzers are very different
44
+ from each other. Metzman et al. faced this problem
45
+ by devising FuzzBench [Metzman et al., 2021], an
46
+ open-source service for the evaluations of state-
47
+ less fuzzers.
48
+ Later, Natella and Pham presented
49
+ ProFuzzBench
50
+ [Natella and Pham, 2021],
51
+ which
52
+ similarly to FuzzBench provides a service to evaluate
53
+ stateful fuzzers.
54
+ Although FuzzBench includes a sample of real
55
+ word programs and ProFuzzBench includes dif-
56
+ a
57
+ https://orcid.org/0000-0002-6585-5494
58
+ b
59
+ https://orcid.org/0000-0001-7435-4176
60
+ c
61
+ https://orcid.org/0000-0002-7615-8643
62
+ d
63
+ https://orcid.org/0000-0003-4635-187X
64
+ ferent network systems (i.e.
65
+ systems that often
66
+ employ forks to deal with multiple connections
67
+ [Tanenbaum, 2009]), they do not evaluate the ability
68
+ of the fuzzers to cope with programs that use forks.
69
+ Despite forks representing the only way to create
70
+ a new process [Tanenbaum, 2009], experimental re-
71
+ sults have shown that current fuzzers cannot deal with
72
+ forked processes.
73
+ The existing approach merely relies on code mod-
74
+ ifications to remove the forks. Unfortunately, this ap-
75
+ proach goes against the willingness to reduce manual
76
+ work and improve automation during a fuzzing cam-
77
+ paign. [Boehme et al., 2021].
78
+ In this work, we explore and classify the limita-
79
+ tions current fuzzers exhibit in front of forking pro-
80
+ grams.
81
+ In summary, this paper:
82
+ 1. devises a novel property capturing the ability of
83
+ fuzzers to deal with forks appropriately;
84
+ 2. evaluates 14 coverage-guided fuzzers based on
85
+ this property;
86
+ 3. proposes possible improvements to the current
87
+ state-of-the-art and future directions.
88
+ The paper is organised as follows. Section 2 de-
89
+ scribes the relevant background, Section 3 presents
90
+
91
+ our contributions to knowledge, Section 4 shows the
92
+ existing approaches that try to cope with the fork
93
+ problem and, eventually, Section 5 discuss the results
94
+ and propose possible future directions.
95
+ 2
96
+ Background
97
+ 2.1
98
+ Fuzz testing
99
+ Fuzzing is an automated testing technique pioneered
100
+ by Miller et al. [Miller et al., 1990] in 1990 to test
101
+ UNIX utilities. As outlined in Figure 1, coverage-
102
+ guided fuzzing is composed at least of seed selection,
103
+ input generation and system execution.
104
+ 1) Seeds selection. The user must provide some
105
+ input messages (seeds) representative of some usual
106
+ inputs for the system.
107
+ 2) Input generation. The core of every fuzzer is
108
+ the generation of slightly malformed input messages
109
+ to forward to the software under test. A fuzzer is as
110
+ efficient as the generated inputs are able to break the
111
+ system. According to the approach used to generate
112
+ the messages, the fuzzers may be classified into:
113
+ • dumb: generate random strings (as the first fuzzer
114
+ [Miller et al., 1995] did);
115
+ • dumb mutational: blindly mutate seed messages
116
+ provided by the user;
117
+ • grammar-based: leverage the grammar of the sys-
118
+ tem to craft the input messages;
119
+ • smart mutational (often called evolutionary): re-
120
+ quire a sample of inputs and leverage feedback
121
+ mechanisms to craft system-tailored messages.
122
+ An example of feedback mechanisms is the code
123
+ coverage feedback, explored in Section 2.2.
124
+ 3) System execution. Each execution of the fuzzer
125
+ involves three components:
126
+ • Bugs detector: it reports eventual bugs. The ma-
127
+ jority of the bugs detectors only report crashes,
128
+ however for many systems, also a weird deviation
129
+ from the happy flow of the protocol may represent
130
+ significant security issues;
131
+ • Hangs detector:
132
+ it detects program execution
133
+ hangs;
134
+ • Code coverage detector: as further explained in
135
+ Section 2.2, the code coverage represents one of
136
+ the feedbacks the fuzzer leverages to improve the
137
+ quality of the input messages.
138
+ 2.2
139
+ Coverage-Guided Fuzzing
140
+ Smart mutational fuzzers use feedback mecha-
141
+ nisms to steer the generation of the messages.
142
+ Different
143
+ types
144
+ of
145
+ feedback
146
+ mechanisms
147
+ exist
148
+ [Shahid et al., 2011], and often different terms are
149
+ used to express the same idea. To avoid further noise,
150
+ in this work we use the term code coverage to express
151
+ the lines of code that are reached by a specific mes-
152
+ sage.
153
+ Code coverage fuzzers need to recompile the code
154
+ with ad-hoc compilers (e.g. the AFL compiler) to in-
155
+ strument the code and obtain run-time information.
156
+ AFL [Zalewski, 2017], for example, instruments
157
+ the code to fill a bitmap that represents the lines of
158
+ the code covered by the inputs.
159
+ Later, it uses this bitmap to assign a higher score
160
+ to messages able to explore previously unseen lines of
161
+ code.
162
+ Start
163
+ Seeds selection I
164
+ Input
165
+ generation
166
+ System execution
167
+ Hangs
168
+ detector
169
+ Bugs
170
+ detector
171
+ Code
172
+ coverage
173
+ detector
174
+ Report O
175
+ End
176
+ Figure 1: Coverage-guided fuzzing process
177
+
178
+ 2.3
179
+ Inter-Process Communication
180
+ Operating systems provide system calls to perform
181
+ different tasks (e.g. writing and reading files, access-
182
+ ing hardware services, creating and executing new
183
+ processes). On UNIX systems, new processes are cre-
184
+ ated by using the fork system call [Tanenbaum, 2009].
185
+ In short, the first process, called parent process, gen-
186
+ erates a clone, called child process, that is an exact
187
+ copy of the parent process. After the fork, file de-
188
+ scriptors and registers are duplicated, thus a change
189
+ in one of the processes does not affect the other one.
190
+ Also, the parent and child process will follow sepa-
191
+ rate execution paths.
192
+ 3
193
+ Our contribution
194
+ This paper aims to understand how the state-of-the-
195
+ art coverage-guided fuzzers deal with software under
196
+ tests containing forks.
197
+ It was not obvious to come up with a way to com-
198
+ pare and contrast the various tools.
199
+ We devised a
200
+ novel property, the fork awareness, that must be sat-
201
+ isfied when a fuzzer deals with forks effectively and
202
+ efficiently. As we shall see below, fork awareness
203
+ rests upon three aspects representing the ability to
204
+ deal with child processes.
205
+ Also, we evaluate the novel property over the
206
+ most widely used fuzzers from two benchmark frame-
207
+ works, reaching a total of 14 evaluated tools, 11
208
+ drawn from FuzzBench and 3 from ProFuzzBench.
209
+ 3.1
210
+ Fork-awareness
211
+ Abstractly, fork awareness insists that every fuzzer
212
+ should address the child process as the parent one.
213
+ During the system execution, the system monitor
214
+ should detect bugs or hangs regardless of their loca-
215
+ tion and the coverage should be measured also in the
216
+ child process. This is formalised through Definition 1.
217
+ Definition 1. A coverage-guided fuzzer is fork-aware
218
+ if it can detect bugs and hangs and measure coverage
219
+ in the same way for both the child and the parent’s
220
+ branch.
221
+ The three aspects in this definition are called:
222
+ [C.1] Child bugs detection: any anomaly is reported
223
+ also if it occurs in child processes;
224
+ [C.2] Child hangs detection: any infinite hang is re-
225
+ ported also if it occurs in child processes;
226
+ [C.3] Child code coverage: code coverage is measured
227
+ also for child processes.
228
+ 3.2
229
+ Example challenges
230
+ We wrote three simple C programs to use as chal-
231
+ lenges for the fuzzers, namely to test whether the
232
+ fuzzers satisfy the aspects given above.
233
+ a) Bugs detection challenge:
234
+ 1
235
+ if(fork()==0){ //Child process
236
+ 2
237
+ raise(SIGSEGV); //Simulated crash
238
+ 3
239
+ } else { //Parent process
240
+ 4
241
+ wait(NULL); //Waiting child
242
+ 5
243
+ //termination
244
+ 6
245
+ }
246
+ The snippet sends a SIGSEGV signal to simulate
247
+ a bug in the child process. This signal is used to
248
+ report a segmentation fault, i.e. a memory access
249
+ violation, which is common in programs written
250
+ in low-level languages. The fuzzer must detect
251
+ this bug also after the parent’s termination.
252
+ b) Hangs detection challenge:
253
+ 1
254
+ if(fork()==0){ //Child process
255
+ 2
256
+ while(1){ ; } //Simulation of
257
+ 3
258
+ //blocking code
259
+ 4
260
+ }
261
+ The snippet simulates an infinite loop in the child
262
+ process. The fuzzers must report processes still
263
+ in execution after the loop and must kill child
264
+ processes at the end of the fuzzing campaign,
265
+ avoiding pending process executions.
266
+ c) Code coverage challenge:
267
+ 1
268
+ pid_t pid = fork();
269
+ 2
270
+ if(pid==0){ //Child process
271
+ 3
272
+ if(data %2 == 0){ do_something(); }
273
+ 4
274
+ else { do_something(); }
275
+ 5
276
+ if(data %3 == 0){ do_something(); }
277
+ 6
278
+ else { do_something(); }
279
+ 7
280
+ if(data %5 == 0){ do_something(); }
281
+ 8
282
+ else { do_something(); }
283
+ 9
284
+ if(data %7 == 0){ do_something(); }
285
+ 10
286
+ else { do_something(); }
287
+ 11
288
+ }
289
+ 12 else { //Parent process
290
+ 13
291
+ wait(NULL); //Waiting child
292
+ 14
293
+ //termination
294
+ 15
295
+ }
296
+ This snippet simulates a child with several branches.
297
+ A fuzzer must cover and consider every child’s
298
+ branches.
299
+ We run the 14 fuzzers over these challenges and
300
+ organised the results in Table 1. We noticed that none
301
+ of the fuzzers succeeded through all three challenges.
302
+
303
+ Fuzzer
304
+ Based on
305
+ Monitor technique
306
+ Bugs
307
+ Detection
308
+ (C1)
309
+ Hangs
310
+ Detection
311
+ (C2)
312
+ Code
313
+ coverage
314
+ (C3)
315
+ AFL [Zalewski, 2017]
316
+ -
317
+ POSIX signals
318
+ ×
319
+ ×
320
+
321
+ AFL++ [Fioraldi et al., 2020]
322
+ AFL
323
+ POSIX signals
324
+ ×
325
+ ×
326
+
327
+ AFLFast [Bohme et al., 2017]
328
+ AFL
329
+ POSIX signals
330
+ ×
331
+ ×
332
+
333
+ AFLSmart [Pham et al., 2021]
334
+ AFL
335
+ POSIX signals
336
+ ×
337
+ ×
338
+
339
+ Eclipser [Choi et al., 2019]
340
+ AFL
341
+ POSIX signals
342
+ ×
343
+ ×
344
+
345
+ FairFuzz [Lemieux and Sen, 2018]
346
+ AFL
347
+ POSIX signals
348
+ ×
349
+ ×
350
+
351
+ lafintel [Besler and Frederic, 2016]
352
+ AFL
353
+ POSIX signals
354
+ ×
355
+ ×
356
+
357
+ AFLnwe1
358
+ AFL
359
+ POSIX signals
360
+ ×
361
+ ×
362
+
363
+ AFLNet [Pham et al., 2020]
364
+ AFL
365
+ POSIX signals
366
+ ×
367
+ ×
368
+
369
+ MOpt-AFL [Lyu et al., 2019]
370
+ AFL
371
+ POSIX signals
372
+ ×
373
+ ×
374
+
375
+ StateAFL [Natella, 2022]
376
+ - AFL
377
+ - AFLNet
378
+ POSIX signals
379
+ ×
380
+ ×
381
+
382
+ LibFuzzer2
383
+ -
384
+ - UBSAN
385
+ - ASAN
386
+ - MSAN
387
+
388
+ ×
389
+
390
+ Entropic [Bohme et al., 2020]
391
+ LibFuzzer
392
+ - UBSAN
393
+ - ASAN
394
+ - MSAN
395
+
396
+ ×
397
+
398
+ Honggfuzz3
399
+ -
400
+ ptrace (Linux)
401
+
402
+ ×
403
+
404
+ Table 1: Coverage guided fuzzers evaluation
405
+ 3.3
406
+ Testbed
407
+ We decided to analyse only the coverage-guided
408
+ fuzzers present in FuzzBench [Metzman et al., 2021]
409
+ and ProFuzzBench [Natella and Pham, 2021] even
410
+ though the property applies to every coverage-guided
411
+ fuzzer. All fuzzers were executed on an Ubuntu 20.04
412
+ server machine and all our source codes are freely
413
+ available online4 so that our experiments are fully re-
414
+ producible.
415
+ 3.4
416
+ Fuzzers evaluation
417
+ We run all selected fuzzers against our three example
418
+ challenges. Table 1 summarises our findings.
419
+ All the fuzzers based on AFL use POSIX signals
420
+ and a bitmap respectively to report bugs and keep
421
+ track of the code coverage.
422
+ As shown in the Table 1, while the bitmaps are
423
+ able to keep track of the child’s code coverage, bugs
424
+ triggered in the child’s processes are not detected
425
+ since AFL catches signals from the main process
426
+ only, as pointed out in the documentation5. The only
427
+ fuzzers able to detect bugs in the child process are
428
+ LibFuzzer6, Entropic [Bohme et al., 2020] and Hong-
429
+ 4https://github.com/marcellomaugeri/forks-break-afl
430
+ 5https://github.com/google/AFL/blob/master/README.md
431
+ 6https://llvm.org/docs/LibFuzzer.html
432
+ fuzz7, as discussed in more detail below:
433
+ • LibFuzzer 8 and Entropic [Bohme et al., 2020]
434
+ employ a set of sanitizers9 to report bugs. These
435
+ mechanisms make the fuzzers able to find the
436
+ bug in Challenge 1 and measure the different
437
+ code paths in Challenge 3, thereby satisfying chal-
438
+ lenges C.1 and C.3, as seen above. Unfortunately,
439
+ challenge C.2 is not satisfied since the fuzzer can-
440
+ not detect hangs in the child process.
441
+ • Honggfuzz supports different software/hardware
442
+ feedback mechanisms and a low-level interface to
443
+ monitor targets.
444
+ When executed on Linux ma-
445
+ chines, Honggfuzz uses the ptrace system call to
446
+ manage processes.
447
+ This mechanism allows the
448
+ fuzzer to capture a wide range of signals.
449
+ As
450
+ shown in Table 1, the use of ptrace (along with
451
+ the SanitizerCoverage) allows the fuzzer to detect
452
+ bugs and to consider coverage also in the child
453
+ process. Unfortunately, neither this mechanism is
454
+ able to detect hangs in the child process.
455
+ In summary, while all selected fuzzers detect the
456
+ code coverage (C3), none detect hangs (C2) and only
457
+ a few detect bugs (C1) in the child process. The eval-
458
+ uation underlines that:
459
+ 7https://honggfuzz.dev/
460
+ 8https://llvm.org/docs/LibFuzzer.html
461
+ 9AddressSanitizer,
462
+ UndefinedBehaviorSanitizer
463
+ and
464
+ MemorySanitizer
465
+
466
+ • Loops detection challenge is the most difficult be-
467
+ cause fuzzers do not wait for all the child pro-
468
+ cesses but only for the main one;
469
+ • Code coverage challenge is the easiest because
470
+ the instrumentation allows measuring coverage
471
+ from the execution, regardless of the process in-
472
+ volved;
473
+ • Bug detection challenge depends on the technique
474
+ used to observe bugs, as well as the use of sanitis-
475
+ ers.
476
+ We interpret this general outcome as a clear call for
477
+ future research and developments.
478
+ 4
479
+ Existing solutions
480
+ Nowadays the only solutions to fuzz programs that
481
+ use forks are manually modifying the code or break-
482
+ ing the multi-process nature of the system (by em-
483
+ ploying tools like defork10) in order to get rid of the
484
+ forks.
485
+ Unfortunately, making modifications to the code,
486
+ as pointed out in the AFLNet documentation 11, to
487
+ remove all the forks is a challenging and error-prone
488
+ task and break the multi-process nature of the system
489
+ often leads to weird system behaviours. The only so-
490
+ lution, therefore, remains to modify the fuzzers.
491
+ 5
492
+ Conclusions
493
+ This paper analyses the fork awareness of the
494
+ coverage-guided fuzzers using three different aspects.
495
+ The analysis conducted on 14 well-known fuzzers
496
+ highlights that while is it clear how important is to
497
+ handle multi-process programs, the majority of the
498
+ fuzzers overlook the problem. 11 of 14 fuzzers are
499
+ not able to detect bugs in the child process. The intu-
500
+ ition behind these outcomes is related to the way these
501
+ fuzzers detect bugs. All the AFL-derived fuzzers use
502
+ signals (SIGSEGV, SIGABRT, etc) to detect bugs and
503
+ this mechanism misses bugs in child processes. We
504
+ noticed that dealing with forks is not the only problem
505
+ and other issues may be related to the IPC scheduling.
506
+ For example, the IPC may influence the success of
507
+ the fuzzing process since some bugs may be triggered
508
+ only after a specific process schedule and only after
509
+ access to a particular cell of memory. We believe this
510
+ paper represents a first step towards the devising of
511
+ fuzzers aware of the eventual multiprocess nature of
512
+ 10https://github.com/zardus/preeny/blob/master/src/defork.c
513
+ 11https://github.com/aflnet/aflnet
514
+ the software. The first step to achieve this goal might
515
+ be the implementation of a loop detector at an early
516
+ stage, e.g. by leveraging a dynamic library to keep
517
+ track of all process identifiers of forked processes. To
518
+ summarise, this work not only provides the first con-
519
+ crete way to evaluate the fuzzers according to their
520
+ fork awareness but sheds light for the first time on a
521
+ class of problems that have been ignored until now,
522
+ showing interesting future directions.
523
+ REFERENCES
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+
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+ ScienceandTechnologyPublications
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+ filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf,len=340
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+ page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='05060v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='CR] 12 Jan 2023 Evaluating the Fork-Awareness of Coverage-Guided Fuzzers Marcello Maugeri1 a, Cristian Daniele2 b, Giampaolo Bella1 c, and Erik Poll2 d 1Department of Maths and Computer Science, University of Catania, Catania, Italy 2Department of Digital Security, Radboud University, Nijmegen, The Netherlands marcello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='maugeri@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='unict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='it, cristian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='daniele@ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='nl, giampaolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='bella@unict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
11
+ page_content='it, erikpoll@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='nl Keywords: Fuzzing, Fork, Security Testing, Software Security Abstract: Fuzz testing (or fuzzing) is an effective technique used to find security vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
14
+ page_content=' It consists of feeding a software under test with malformed inputs, waiting for a weird system behaviour (often a crash of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
15
+ page_content=' Over the years, different approaches have been developed, and among the most popular lies the coverage-based one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
16
+ page_content=' It relies on the instrumentation of the system to generate inputs able to cover as much code as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
17
+ page_content=' The success of this approach is also due to its usability as fuzzing techniques research approaches that do not require (or only partial require) human interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
18
+ page_content=' Despite the efforts, devising a fully-automated fuzzer still seems to be a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
19
+ page_content=' Target systems may be very complex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
20
+ page_content=' they may integrate cryptographic primitives, compute and verify check-sums and employ forks to enhance the system security, achieve better performances or manage different connections at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
21
+ page_content=' This paper introduces the fork-awareness property to express the fuzzer ability to manage systems using forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
22
+ page_content=' This property is leveraged to evaluate 14 of the most widely coverage-guided fuzzers and highlight how current fuzzers are ineffective against systems using forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
23
+ page_content=' 1 Introduction In the last years, plenty of fuzzers have been devel- oped to deal with sophisticated software and nowa- days it is extremely common that network systems employ forks to deal with different connections at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
24
+ page_content=' This leads to 1) the need to devise accurate and ad-hoc fuzzers and 2) the need to evaluate these fuzzer according to their ability to cope with such ad- vanced systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
25
+ page_content=' Unfortunately, as pointed out in [Hazimeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
26
+ page_content=', 2020], it is not easy to bench- mark all of them since the fuzzers are very different from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
27
+ page_content=' Metzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
28
+ page_content=' faced this problem by devising FuzzBench [Metzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
29
+ page_content=', 2021], an open-source service for the evaluations of state- less fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
30
+ page_content=' Later, Natella and Pham presented ProFuzzBench [Natella and Pham, 2021], which similarly to FuzzBench provides a service to evaluate stateful fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
31
+ page_content=' Although FuzzBench includes a sample of real word programs and ProFuzzBench includes dif- a https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
32
+ page_content='org/0000-0002-6585-5494 b https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
33
+ page_content='org/0000-0001-7435-4176 c https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
34
+ page_content='org/0000-0002-7615-8643 d https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
35
+ page_content='org/0000-0003-4635-187X ferent network systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
36
+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
37
+ page_content=' systems that often employ forks to deal with multiple connections [Tanenbaum, 2009]), they do not evaluate the ability of the fuzzers to cope with programs that use forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
38
+ page_content=' Despite forks representing the only way to create a new process [Tanenbaum, 2009], experimental re- sults have shown that current fuzzers cannot deal with forked processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
39
+ page_content=' The existing approach merely relies on code mod- ifications to remove the forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
40
+ page_content=' Unfortunately, this ap- proach goes against the willingness to reduce manual work and improve automation during a fuzzing cam- paign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
41
+ page_content=' [Boehme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
42
+ page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
43
+ page_content=' In this work, we explore and classify the limita- tions current fuzzers exhibit in front of forking pro- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
44
+ page_content=' In summary, this paper: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
45
+ page_content=' devises a novel property capturing the ability of fuzzers to deal with forks appropriately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
46
+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
47
+ page_content=' evaluates 14 coverage-guided fuzzers based on this property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
48
+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
49
+ page_content=' proposes possible improvements to the current state-of-the-art and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
50
+ page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
51
+ page_content=' Section 2 de- scribes the relevant background, Section 3 presents our contributions to knowledge, Section 4 shows the existing approaches that try to cope with the fork problem and, eventually, Section 5 discuss the results and propose possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
52
+ page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
53
+ page_content='1 Fuzz testing Fuzzing is an automated testing technique pioneered by Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
54
+ page_content=' [Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
55
+ page_content=', 1990] in 1990 to test UNIX utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
56
+ page_content=' As outlined in Figure 1, coverage- guided fuzzing is composed at least of seed selection, input generation and system execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
57
+ page_content=' 1) Seeds selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
58
+ page_content=' The user must provide some input messages (seeds) representative of some usual inputs for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
59
+ page_content=' 2) Input generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
60
+ page_content=' The core of every fuzzer is the generation of slightly malformed input messages to forward to the software under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
61
+ page_content=' A fuzzer is as efficient as the generated inputs are able to break the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
62
+ page_content=' According to the approach used to generate the messages, the fuzzers may be classified into: dumb: generate random strings (as the first fuzzer [Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
63
+ page_content=', 1995] did);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
64
+ page_content=' dumb mutational: blindly mutate seed messages provided by the user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
65
+ page_content=' grammar-based: leverage the grammar of the sys- tem to craft the input messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
66
+ page_content=' smart mutational (often called evolutionary): re- quire a sample of inputs and leverage feedback mechanisms to craft system-tailored messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
67
+ page_content=' An example of feedback mechanisms is the code coverage feedback, explored in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
68
+ page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
69
+ page_content=' 3) System execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
70
+ page_content=' Each execution of the fuzzer involves three components: Bugs detector: it reports eventual bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
71
+ page_content=' The ma- jority of the bugs detectors only report crashes, however for many systems, also a weird deviation from the happy flow of the protocol may represent significant security issues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
72
+ page_content=' Hangs detector: it detects program execution hangs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
73
+ page_content=' Code coverage detector: as further explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
74
+ page_content='2, the code coverage represents one of the feedbacks the fuzzer leverages to improve the quality of the input messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
75
+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='2 Coverage-Guided Fuzzing Smart mutational fuzzers use feedback mecha- nisms to steer the generation of the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Different types of feedback mechanisms exist [Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2011], and often different terms are used to express the same idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' To avoid further noise, in this work we use the term code coverage to express the lines of code that are reached by a specific mes- sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Code coverage fuzzers need to recompile the code with ad-hoc compilers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' the AFL compiler) to in- strument the code and obtain run-time information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' AFL [Zalewski, 2017], for example, instruments the code to fill a bitmap that represents the lines of the code covered by the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Later, it uses this bitmap to assign a higher score to messages able to explore previously unseen lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Start Seeds selection I Input generation System execution Hangs detector Bugs detector Code coverage detector Report O End Figure 1: Coverage-guided fuzzing process 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='3 Inter-Process Communication Operating systems provide system calls to perform different tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' writing and reading files, access- ing hardware services, creating and executing new processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' On UNIX systems, new processes are cre- ated by using the fork system call [Tanenbaum, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' In short, the first process, called parent process, gen- erates a clone, called child process, that is an exact copy of the parent process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' After the fork, file de- scriptors and registers are duplicated, thus a change in one of the processes does not affect the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Also, the parent and child process will follow sepa- rate execution paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' 3 Our contribution This paper aims to understand how the state-of-the- art coverage-guided fuzzers deal with software under tests containing forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' It was not obvious to come up with a way to com- pare and contrast the various tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' We devised a novel property, the fork awareness, that must be sat- isfied when a fuzzer deals with forks effectively and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' As we shall see below, fork awareness rests upon three aspects representing the ability to deal with child processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Also, we evaluate the novel property over the most widely used fuzzers from two benchmark frame- works, reaching a total of 14 evaluated tools, 11 drawn from FuzzBench and 3 from ProFuzzBench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='1 Fork-awareness Abstractly, fork awareness insists that every fuzzer should address the child process as the parent one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' During the system execution, the system monitor should detect bugs or hangs regardless of their loca- tion and the coverage should be measured also in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' This is formalised through Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' A coverage-guided fuzzer is fork-aware if it can detect bugs and hangs and measure coverage in the same way for both the child and the parent’s branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' The three aspects in this definition are called: [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='1] Child bugs detection: any anomaly is reported also if it occurs in child processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='2] Child hangs detection: any infinite hang is re- ported also if it occurs in child processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='3] Child code coverage: code coverage is measured also for child processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='2 Example challenges We wrote three simple C programs to use as chal- lenges for the fuzzers, namely to test whether the fuzzers satisfy the aspects given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' a) Bugs detection challenge: 1 if(fork()==0){ //Child process 2 raise(SIGSEGV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' //Simulated crash 3 } else { //Parent process 4 wait(NULL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' //Waiting child 5 //termination 6 } The snippet sends a SIGSEGV signal to simulate a bug in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' This signal is used to report a segmentation fault, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' a memory access violation, which is common in programs written in low-level languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' The fuzzer must detect this bug also after the parent’s termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' b) Hangs detection challenge: 1 if(fork()==0){ //Child process 2 while(1){ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' } //Simulation of 3 //blocking code 4 } The snippet simulates an infinite loop in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' The fuzzers must report processes still in execution after the loop and must kill child processes at the end of the fuzzing campaign, avoiding pending process executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' c) Code coverage challenge: 1 pid_t pid = fork();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' 2 if(pid==0){ //Child process 3 if(data %2 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' } 4 else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' } 5 if(data %3 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' } 6 else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' } 7 if(data %5 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' } 8 else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' } 9 if(data %7 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' } 10 else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' } 11 } 12 else { //Parent process 13 wait(NULL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' //Waiting child 14 //termination 15 } This snippet simulates a child with several branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' A fuzzer must cover and consider every child’s branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' We run the 14 fuzzers over these challenges and organised the results in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' We noticed that none of the fuzzers succeeded through all three challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Fuzzer Based on Monitor technique Bugs Detection (C1) Hangs Detection (C2) Code coverage (C3) AFL [Zalewski, 2017] POSIX signals × × ✓ AFL++ [Fioraldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2020] AFL POSIX signals × × ✓ AFLFast [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2017] AFL POSIX signals × × ✓ AFLSmart [Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2021] AFL POSIX signals × × ✓ Eclipser [Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2019] AFL POSIX signals × × ✓ FairFuzz [Lemieux and Sen, 2018] AFL POSIX signals × × ✓ lafintel [Besler and Frederic, 2016] AFL POSIX signals × × ✓ AFLnwe1 AFL POSIX signals × × ✓ AFLNet [Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2020] AFL POSIX signals × × ✓ MOpt-AFL [Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2019] AFL POSIX signals × × ✓ StateAFL [Natella, 2022] AFL AFLNet POSIX signals × × ✓ LibFuzzer2 UBSAN ASAN MSAN ✓ × ✓ Entropic [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2020] LibFuzzer UBSAN ASAN MSAN ✓ × ✓ Honggfuzz3 ptrace (Linux) ✓ × ✓ Table 1: Coverage guided fuzzers evaluation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='3 Testbed We decided to analyse only the coverage-guided fuzzers present in FuzzBench [Metzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2021] and ProFuzzBench [Natella and Pham, 2021] even though the property applies to every coverage-guided fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' All fuzzers were executed on an Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='04 server machine and all our source codes are freely available online4 so that our experiments are fully re- producible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='4 Fuzzers evaluation We run all selected fuzzers against our three example challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Table 1 summarises our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' All the fuzzers based on AFL use POSIX signals and a bitmap respectively to report bugs and keep track of the code coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' As shown in the Table 1, while the bitmaps are able to keep track of the child’s code coverage, bugs triggered in the child’s processes are not detected since AFL catches signals from the main process only, as pointed out in the documentation5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' The only fuzzers able to detect bugs in the child process are LibFuzzer6, Entropic [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2020] and Hong- 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='com/marcellomaugeri/forks-break-afl 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='com/google/AFL/blob/master/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='md 6https://llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='org/docs/LibFuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='html fuzz7, as discussed in more detail below: LibFuzzer 8 and Entropic [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=', 2020] employ a set of sanitizers9 to report bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' These mechanisms make the fuzzers able to find the bug in Challenge 1 and measure the different code paths in Challenge 3, thereby satisfying chal- lenges C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='3, as seen above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Unfortunately, challenge C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='2 is not satisfied since the fuzzer can- not detect hangs in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Honggfuzz supports different software/hardware feedback mechanisms and a low-level interface to monitor targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' When executed on Linux ma- chines, Honggfuzz uses the ptrace system call to manage processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' This mechanism allows the fuzzer to capture a wide range of signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' As shown in Table 1, the use of ptrace (along with the SanitizerCoverage) allows the fuzzer to detect bugs and to consider coverage also in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Unfortunately, neither this mechanism is able to detect hangs in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' In summary, while all selected fuzzers detect the code coverage (C3), none detect hangs (C2) and only a few detect bugs (C1) in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' The eval- uation underlines that: 7https://honggfuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='dev/ 8https://llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='org/docs/LibFuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content='html 9AddressSanitizer, UndefinedBehaviorSanitizer and MemorySanitizer Loops detection challenge is the most difficult be- cause fuzzers do not wait for all the child pro- cesses but only for the main one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Code coverage challenge is the easiest because the instrumentation allows measuring coverage from the execution, regardless of the process in- volved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Bug detection challenge depends on the technique used to observe bugs, as well as the use of sanitis- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' We interpret this general outcome as a clear call for future research and developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' 4 Existing solutions Nowadays the only solutions to fuzz programs that use forks are manually modifying the code or break- ing the multi-process nature of the system (by em- ploying tools like defork10) in order to get rid of the forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Unfortunately, making modifications to the code, as pointed out in the AFLNet documentation 11, to remove all the forks is a challenging and error-prone task and break the multi-process nature of the system often leads to weird system behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' The only so- lution, therefore, remains to modify the fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' 5 Conclusions This paper analyses the fork awareness of the coverage-guided fuzzers using three different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' The analysis conducted on 14 well-known fuzzers highlights that while is it clear how important is to handle multi-process programs, the majority of the fuzzers overlook the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' 11 of 14 fuzzers are not able to detect bugs in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
185
+ page_content=' The intu- ition behind these outcomes is related to the way these fuzzers detect bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' All the AFL-derived fuzzers use signals (SIGSEGV, SIGABRT, etc) to detect bugs and this mechanism misses bugs in child processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' We noticed that dealing with forks is not the only problem and other issues may be related to the IPC scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' For example, the IPC may influence the success of the fuzzing process since some bugs may be triggered only after a specific process schedule and only after access to a particular cell of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' We believe this paper represents a first step towards the devising of fuzzers aware of the eventual multiprocess nature of 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
190
+ page_content='com/zardus/preeny/blob/master/src/defork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
191
+ page_content='c 11https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
192
+ page_content='com/aflnet/aflnet the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
193
+ page_content=' The first step to achieve this goal might be the implementation of a loop detector at an early stage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
194
+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
195
+ page_content=' by leveraging a dynamic library to keep track of all process identifiers of forked processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
196
+ page_content=' To summarise, this work not only provides the first con- crete way to evaluate the fuzzers according to their fork awareness but sheds light for the first time on a class of problems that have been ignored until now, showing interesting future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' Pearson Education, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' American fuzzy lop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+ page_content=' https://lcamtuf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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1
+ arXiv:2301.13792v1 [math.FA] 31 Jan 2023
2
+ Linear extension operators for Sobolev spaces on
3
+ uniform trees
4
+ Charles Fefferman1 and Bo’az Klartag2
5
+ Dedicated in friendship to David Jerison
6
+ Abstract
7
+ Let 1 < p < ∞ and suppose that we are given a function f defined on the leaves of a
8
+ weighted tree. We would like to extend f to a function F defined on the entire tree, so as
9
+ to minimize the weighted W 1,p-Sobolev norm of the extension. An easy situation is when
10
+ p = 2, where the harmonic extension operator provides such a function F. In this note
11
+ we record our analysis of the particular case of a uniform binary tree, which is a complete,
12
+ finite, binary tree with weights that depend only on the distance from the root. Neither the
13
+ averaging operator nor the harmonic extension operator work here in general. Nevertheless,
14
+ we prove the existence of a linear extension operator whose norm is bounded by a constant
15
+ depending solely on p. This operator is a variant of the standard harmonic extension operator,
16
+ and in fact it is harmonic extension with respect to a certain Markov kernel determined by p
17
+ and by the weights.
18
+ 1
19
+ Introduction
20
+ Consider a full binary tree of height N, whose set of vertices is denoted by
21
+ V =
22
+ N
23
+
24
+ k=0
25
+ {0, 1}k,
26
+ i.e., the vertices are strings of zeroes and ones of length at most N. For x ∈ {0, 1}k and ℓ ≤ k
27
+ we write πℓ(x) ∈ {0, 1}ℓ for the prefix of x of length ℓ. Thus for k ≥ 1, the parent of a vertex
28
+ x ∈ {0, 1}k ⊆ V is the vertex πk−1(x). The set {0, 1}0 is a singleton whose unique element
29
+ is denoted by ∅, the empty string, which is the root of the tree. The set of leaves of the tree is
30
+ 1Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544,
31
+ USA. Email: cf@math.princeton.edu. Supported by the Air Force Office of Scientific Research, grant number
32
+ FA9950-18-1-0069 and the National Science Foundation (NSF), grant number DMS-1700180.
33
+ 2Department of Mathematics,
34
+ Weizmann Institute of Science,
35
+ Rehovot 7610001,
36
+ Israel.
37
+ Email:
38
+ boaz.klartag@weizmann.ac.il. Supported by a grant from the Israel Science Foundation (ISF).
39
+ 1
40
+
41
+ {0, 1}N, and all other vertices in V are internal vertices. A vertex x ∈ {0, 1}k is said to have
42
+ depth
43
+ d(x) = k,
44
+ thus leaves have depth N and the root has depth 0. The collections of bijections from V to
45
+ V that preserve depth and parenthood relations form a group. This group is referred to as the
46
+ symmetry group of the tree. It has 22N−1 elements, and it is a 2-Sylow subgroup of the group of
47
+ all permutations of the leaves. The set
48
+ E =
49
+ N
50
+
51
+ k=1
52
+ {0, 1}k = V \ {∅}
53
+ is referred to as the set of edges of the tree, and the depth of an edge e ∈ {0, 1}k is d(e) = k.
54
+ That is, we think of e ∈ E as an undirected edge connecting the vertex whose string corresponds
55
+ to e to its unique parent. Each internal vertex other than the root is connected to three vertices,
56
+ which are its parent and its two children. Assume that we are given edge weights
57
+ W1, W1, . . . , WN > 0,
58
+ where we view Wk as the weight of all edges of depth k. For 1 < p < ∞, the associated
59
+ ˙W 1,p-seminorm is defined, for F : V → R, via
60
+ ∥F∥ ˙W 1,p(V ) =
61
+
62
+
63
+ N
64
+
65
+ k=1
66
+ Wk
67
+
68
+ x∈{0,1}k
69
+ |F(x) − F(πk−1(x))|p
70
+
71
+
72
+ 1/p
73
+ .
74
+ (1)
75
+ We write ∂V = {0, 1}N ⊆ V for the set of leaves of the tree. The trace of the ∥·∥ ˙W 1,p-seminorm
76
+ is defined, for f : ∂V → R, via
77
+ ∥f∥ ˙W 1,p(∂V ) = inf
78
+
79
+ ∥F∥ ˙W 1,p(V ) ; F|∂V = f
80
+
81
+ ,
82
+ (2)
83
+ i.e., the infimum of the ˙W 1,p-seminorm over all extensions of f from the leaves to the entire tree.
84
+ We write RV for the collection of all functions f : V → R, and similarly R∂V is the collection
85
+ of all functions f : ∂V → R. Our main result is the following:
86
+ Theorem 1.1. Let 1 < p < ∞ and let W1, . . . , WN > 0. Then there exists a linear operator
87
+ H : R∂V → RV with the following properties:
88
+ 1. It is a linear extension operator, i.e., (Hf)(x) = f(x) for any x ∈ ∂V and any function
89
+ f : ∂V → R.
90
+ 2. Its norm is bounded by a constant ¯Cp depending only on p, i.e., for any f : ∂V → R,
91
+ ∥Hf∥ ˙W 1,p(V ) ≤ ¯Cp∥f∥ ˙W 1,p(∂V ).
92
+ 2
93
+
94
+ In fact, we have the bound
95
+ ¯Cp ≤ 4p1/pq1/q ·
96
+
97
+ 1 + max{(p − 1)−1/p, (q − 1)−1/q}
98
+
99
+ ≤ C · max
100
+
101
+ 1
102
+ p − 1,
103
+ 1
104
+ q − 1
105
+
106
+ ,
107
+ (3)
108
+ where q = p/(p − 1) and where C > 0 is a universal constant.
109
+ The proof of Theorem 1.2 is constructive, and the extension operator H that we construct is
110
+ in fact a harmonic extension operator with respect to a certain random walk defined on the tree.
111
+ At each step the random walk jumps from a vertex to one of its neighbors, where of course the
112
+ neighbors of a vertex are its parent and its children. The Markov kernel corresponding to the
113
+ random walk is invariant under the symmetries of the tree, thus the probability to move from a
114
+ vertex to its neighbor depends only on the weights of the vertex and of its neighbor. The Markov
115
+ kernel of our random walk is determined by the following requirement: For any s = 1, . . . , N,
116
+ the probability that a random walk starting at some vertex of depth s will reach a leaf before
117
+ reaching a vertex of depth s − 1 equals
118
+ qs =
119
+ (2sWs)−1/(p−1)
120
+ �N
121
+ k=s(2kWk)−1/(p−1).
122
+ (4)
123
+ Thus the weights of our random walk typically depend on p ∈ (1, ∞), except for the case where
124
+ Ws is proportional to 2−s. This seems inevitable. Indeed, in some examples such as the 3rd
125
+ example in Section 2 below, the linear extension operator H : R∂V → RV that corresponds to
126
+ the parameter value p = p0, is not uniformly bounded for any p ∈ (1, ∞) \ {p0}. When p = 2,
127
+ our random walk coincides with the usual random walk corresponding to the given weights on
128
+ the edges of the binary tree, and thus in this case H is the standard harmonic extension operator
129
+ (hence ¯Cp = 1 for p = 2).
130
+ There is a certain range of weights where the averaging operator yields a uniformly bounded
131
+ linear extension operator, as proven by Bj¨orn ×2, Gill and Shanmugalingam [1]. The averaging
132
+ operator is the extension operator that assigns to each internal vertex v the average of the function
133
+ values on the leaves of the subtree whose root is v. This averaging operator seems natural also
134
+ from the point of view of Whitney’s extension theory, see the work by Shvartsman [5] on Sobolev
135
+ extension in W 1,p(Rn). However, there are examples of uniform binary trees where the averaging
136
+ operator does not provide a uniformly bounded operator, such as the case where Wk = 2−k for
137
+ all k.
138
+ What about non-uniform trees? Suppose that the edge weights are arbitrary positive numbers
139
+ (We)e∈E that are not necessarily determined by the depth of the edge. When p = 2, there is still
140
+ a harmonic extension operator of norm one from ˙W 1,p(∂V ) to ˙W 1,p(V ). However, for p ̸= 2 the
141
+ situation seems subtle. We conjecture that in the general case of non-uniform tree weights there
142
+ is no linear extension operator whose norm is bounded by a function of p alone. This conjecture
143
+ is closely related to questions about well-complemented subspaces of ℓn
144
+ p that are beyond the
145
+ scope of this note.
146
+ 3
147
+
148
+ In order to prove Theorem 1.1 we reformulate the problem in a way that brings us closer to
149
+ analysis in ℓp-spaces. For 1 < p < ∞, the associated Lp(E)-norm is defined, for f : E → R,
150
+ via
151
+ ∥f∥p = ∥f∥Lp(E) =
152
+
153
+
154
+ N
155
+
156
+ k=1
157
+ Wk
158
+
159
+ x∈{0,1}k
160
+ |f(x)|p
161
+
162
+
163
+ 1/p
164
+ .
165
+ (5)
166
+ We think of a function f : E → R as the gradient of a function ˜f : V → R, uniquely determined
167
+ up to an additive constant. Given f : E → R we thus define a function ˜f : V → R as follows:
168
+ For any x ∈ V other then the root,
169
+ ˜f(x) =
170
+ d(x)
171
+
172
+ i=1
173
+ f(πix),
174
+ (6)
175
+ while ˜f(x) = 0 if x = ∅ is the root. The only property of ˜f that matters is that for all x ∈ E,
176
+ f(x) = ˜f(x) − ˜f(πd(x)−1(x)).
177
+ We are interested in finding a linear operator T : Lp(E) → Lp(E), with a uniform bound on its
178
+ operator norm, that has the following properties:
179
+ 1. The operator T takes the form
180
+ Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)
181
+ (7)
182
+ for some linear operator ˜T : RV → RV . That is, ˜T takes functions on V to functions on
183
+ V , and T is induces from ˜T via formula (7).
184
+ 2. The operator ˜T is equivariant with respect to the tree symmetries and it satisfies ˜T(1) ≡ 1,
185
+ i.e., it maps the constant function 1 to itself.
186
+ 3. The function ˜Tg coincides with the function g on the leaves of the tree, i.e. ( ˜Tg)|∂V = g|∂V
187
+ for any g : V → R.
188
+ 4. The function ˜Tg is determined by the values of the function g on the leaves of the tree.
189
+ The operator norm of ˜T with respect to the ˙W 1,p-seminorm equals to the operator norm
190
+ of T with respect to the Lp(E)-norm. Defining H(f|∂V ) = ˜Tf, Theorem 1.1 may thus be
191
+ reformulated as follows:
192
+ Theorem 1.2. Let 1 < p < ∞ and let W1, . . . , WN > 0. Then there exists a linear operator
193
+ T : Lp(E) → Lp(E) with the above properties, whose operator norm is at most a certain
194
+ constant ¯Cp depending only on p. In fact, we have the bound (3) for the constant ¯Cp.
195
+ 4
196
+
197
+ The proof of Theorem 1.2 occupies the next three sections. In Section 2 we discuss invariant
198
+ random walks on a full binary tree and describe the corresponding harmonic extension operator.
199
+ In Section 3 we deal with the problem of bounding the norm of this operator, and use the sym-
200
+ metries of the problem in order to reduce it to a one-dimensional question. This one-dimensional
201
+ question is then answered in Section 4 using the Muckenhoupt criterion [4].
202
+ When analyzing the binary tree we use the following notation: We write dlca(x, y) for the
203
+ length of the maximal prefix shared by the strings x ∈ {0, 1}k and y ∈ {0, 1}ℓ, while lca(x, y)
204
+ is the maximal prefix itself. Thus for two vertices x, y ∈ V , their least common ancestor is
205
+ lca(x, y) ∈ V and its depth is dlca(x, y) ∈ {0, . . . , N}. Note that for any x ∈ E and ω ∈ ∂V ,
206
+ dlca(πd(x)−1x, ω) = min{d(x) − 1, dlca(x, ω)}.
207
+ (8)
208
+ Acknowledgements. We would like to thank Jacob Carruth, Arie Israel, Anna Skorobogatova
209
+ and Ignacio Uriarte-Tuero for helpful conversations. This research was conducted while BK
210
+ was visiting Princeton University’s Department of Mathematics; he is grateful for their gracious
211
+ hospitality.
212
+ 2
213
+ Invariant random walks
214
+ A Markov chain on V is a sequence of random variables R1, R2, . . . ∈ V such that the distribution
215
+ of Ri+1 conditioned on R1, . . . , Ri is the same as its distribution conditioned on Ri. A Markov
216
+ chain is time-homogeneous if for any x, y ∈ V , the probability that Ri+1 = x conditioned on the
217
+ event Ri = y does not depend on i. A random walk on V is a time-homogeneous Markov chain
218
+ R1, R2, . . . ∈ V such that Ri+1 is a neighbor of Ri with probability one.
219
+ We say that the random walk is invariant if the probability to jump from a vertex x to a vertex
220
+ y depends only on the depths d(x) and d(y). Our random walk will be invariant, and it will stop
221
+ when it reaches a leaf, i.e., we have the stopping time
222
+ τ = min{i ≥ 1 ; Ri is a leaf}.
223
+ For s ≥ 1 we define qs to be the probability of the following event: Assuming that R1 is a vertex
224
+ of depth s, the event is that Ri will remain at the subtree whose root is R1 for all 1 ≤ i ≤ τ.
225
+ Equivalently, define
226
+ Xi = d(Ri) ∈ {0, . . . , N}.
227
+ Then X1, X2, . . . ∈ {0, . . . , N} is a random walk, since |Xi+1 − Xi| = 1 for all i. Furthermore,
228
+ qs = P(∀1 ≤ i ≤ τ, Xi ≥ s | X1 = s).
229
+ (9)
230
+ Clearly
231
+ q0 = qN = 1.
232
+ (10)
233
+ For r, s ∈ {0, . . . , N} with r ≤ s we set
234
+ ps,r = P(min{Xi ; i ≤ τ} = r | X1 = s).
235
+ 5
236
+
237
+ That is, the number ps,r is the probability that r is the minimal node that the walker visits when
238
+ starting from node s, before reaching the terminal node N. Clearly �s
239
+ r=0 ps,r = 1.
240
+ Lemma 2.1. For 0 ≤ r ≤ s ≤ N − 1,
241
+ ps,r = qr ·
242
+ s
243
+
244
+ k=r+1
245
+ (1 − qk)
246
+ (11)
247
+ where an empty product equals one. Moreover, pN,r = δrN, where δrN is the Kronecker delta.
248
+ Proof. The expression on the right-hand side of (11) is the probability to ever reach s − 1 when
249
+ starting from X1 = s, and from s − 1 to ever reach s − 2, etc. until we finally reach r, yet from
250
+ r we require to never reach r − 1. Alternatively, when 0 ≤ r ≤ s, s ≥ 1 we have the recurrence
251
+ relation
252
+ ps,r = qsδs,r + (1 − qs) · ps−1,r.
253
+ (12)
254
+ This recurrence relation leads to another proof of (11).
255
+ Suppose that our random walk R1, R2, . . . ∈ V begins at a vertex R1 = x with d(x) = s.
256
+ Consider a leaf y ∈ ∂V with dlca(x, y) = r. What is the probability that our random walk will
257
+ reach the leaf y? We claim that this probability is
258
+ bs,r := P(Rτ = y) =
259
+ r
260
+
261
+ k=0
262
+ 2k−Nps,k.
263
+ (13)
264
+ Indeed, conditioning on the value of k = mini≤τ d(Ri), by symmetry we know that Rτ is dis-
265
+ tributed uniformly among the 2N−k leaf-descendants of the vertex πk(x). When k ≤ r, exactly
266
+ one of these leaf-descendants is the leaf y, since the vertex of minimal depth that (Ri) visits must
267
+ be the vertex πk(x), which is a prefix of y as k ≤ r = dlca(x, y). Hence the probability that
268
+ Rτ = y, conditioning on the value of k, equals to 1/2N−k when k ≤ r and it vanishes other-
269
+ wise. By using the definition of ps,k and the complete probability formula, we obtain (13). The
270
+ harmonic extension operator associated with our invariant random walk is given by
271
+ ˜Tg(x) =
272
+
273
+ ω∈{0,1}N
274
+ bd(x),dlca(x,ω) · g(ω)
275
+ (x ∈ V ).
276
+ (14)
277
+ The operator T is induced from ˜T via formula (7) above. Requirements 1,...,4 from Section 1 are
278
+ clearly satisfied.
279
+ We stipulate that the collection of descendants of a vertex x ∈ V , denoted by D(x) ⊆ V ,
280
+ includes the vertex x itself. Abbreviate a ∧ b = min{a, b} and a ∨ b = max{a, b}. The operator
281
+ T takes the form
282
+ Tf(x) =
283
+
284
+ y∈E
285
+ K(x, y)f(y),
286
+ (15)
287
+ where the kernel K is described next.
288
+ 6
289
+
290
+ Proposition 2.2. Let x, y ∈ E and denote s = d(x), t = d(y), r = dlca(x, y). Then the following
291
+ hold: If x ̸∈ D(y) and y ̸∈ D(x), then r ≤ s ∧ t − 1 and
292
+ K(x, y) = −qs · 2−t ·
293
+ r
294
+
295
+ k=0
296
+ 2kps−1,k ≤ 0.
297
+ (16)
298
+ Otherwise, i.e., if y ∈ D(x) or if x ∈ D(y) then r = s ∧ t and
299
+ K(x, y) = qs · 2−t ·
300
+ r−1
301
+
302
+ k=0
303
+ (2r − 2k)ps−1,k ≥ 0.
304
+ (17)
305
+ Proof. By (7) and (14) we have, for any x ∈ E,
306
+ Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)
307
+ =
308
+
309
+ ω∈{0,1}N
310
+ bd(x),dlca(x,ω) ˜f(ω) −
311
+
312
+ ω∈{0,1}N
313
+ bd(x)−1,dlca(πd(x)−1x,ω) ˜f(ω)
314
+ =
315
+
316
+ ω∈{0,1}N
317
+ ad(x),dlca(x,w) ˜f(ω)
318
+ (18)
319
+ where for any 0 ≤ r ≤ s, by (8) and (13),
320
+ as,r = bs,r − bs−1,min{s−1,r} =
321
+ r
322
+
323
+ k=0
324
+ 2k−Nps,k −
325
+ min{s−1,r}
326
+
327
+ k=0
328
+ 2k−Nps−1,k.
329
+ (19)
330
+ Hence, by (6) and (18),
331
+ Tf(x) =
332
+
333
+ ω∈{0,1}N
334
+ ad(x),dlca(x,ω)
335
+ N
336
+
337
+ i=1
338
+ f(πiω)
339
+ =
340
+
341
+ y∈E
342
+
343
+
344
+
345
+ ω∈{0,1}N;πd(y)ω=y
346
+ ad(x),dlca(x,ω)
347
+
348
+  f(y) =
349
+
350
+ y∈E
351
+ K(x, y)f(y),
352
+ where the kernel K of the operator T satisfies, for any x, y ∈ E,
353
+ K(x, y) =
354
+
355
+ ω∈{0,1}N ;πd(y)ω=y
356
+ ad(x),dlca(x,ω).
357
+ (20)
358
+ Fix x, y ∈ E with s = d(x), t = d(y) and r = dlca(x, y). Let us consider first the case where x
359
+ is not a descendant of y. This means that the prefix of y that is shared by x, is not the entire string
360
+ y. Hence for any ω ∈ {0, 1}N with πd(y)ω = y we have dlca(x, ω) = dlca(x, y) ≤ d(y) − 1.
361
+ Therefore, from (20) and (19),
362
+ K(x, y) = 2N−d(y)ad(x),dlca(x,y)
363
+ = 2N−t �
364
+ bs,r − bs−1,(s−1)∧r
365
+
366
+ =
367
+ r
368
+
369
+ k=0
370
+ 2k−tps,k −
371
+ (s−1)∧r
372
+
373
+ k=0
374
+ 2k−tps−1,k,
375
+ 7
376
+
377
+ as bs,r = �r
378
+ k=0 2k−Nps,k. Since ps,s = qs, we have
379
+ K(x, y) = δrs2s−tqs +
380
+ (s−1)∧r
381
+
382
+ k=0
383
+ 2k−t [ps,k − ps−1,k] = qs ·
384
+
385
+ δrs2s−t −
386
+ (s−1)∧r
387
+
388
+ k=0
389
+ 2k−tps−1,k
390
+
391
+  ,
392
+ (21)
393
+ where we used the relation (12), which implies that when k ≤ s − 1,
394
+ ps,k − ps−1,k = −qs · ps−1,k.
395
+ (22)
396
+ We may now prove the conclusion of the proposition in the case where x ̸∈ D(y) and y ̸∈ D(x).
397
+ Indeed, in this case r ≤ s ∧ t − 1 and formula (21) applies. Since δrs = 0 in this case, we deduce
398
+ formula (16) from (21).
399
+ The next case we consider is the case where r = s ≤ t − 1, or equivalently, where y ∈
400
+ D(x) \ {x}. Thus x ̸∈ D(y) and formula (21) applies. Recalling that �s−1
401
+ k=0 ps−1,k = 1 we obtain
402
+ from (21) that
403
+ K(x, y) = qs ·
404
+ s−1
405
+
406
+ k=0
407
+ (2s−t − 2k−t)ps−1,k = qs · 2−t ·
408
+ s−1
409
+
410
+ k=0
411
+ (2r − 2k)ps−1,k,
412
+ proving formula (17) in the case y ∈ D(x) \ {x}.
413
+ We move on to the case where x ∈ D(y) \ {y}, thus dlca(x, y) = t ≤ s − 1. In this case, by
414
+ applying (20), (19), (13) and then (22),
415
+ K(x, y) =
416
+
417
+ ω∈{0,1}N ;πd(y)ω=y
418
+ ad(x),dlca(x,ω) = 2N−d(x)ad(x),d(x) +
419
+ d(x)−1
420
+
421
+ k=d(y)
422
+ 2N−k−1ad(x),k
423
+ = 2N−sas,s +
424
+ s−1
425
+
426
+ k=t
427
+ 2N−k−1as,k = 2N−s(bs,s − bs−1,s−1) +
428
+ s−1
429
+
430
+ k=t
431
+ 2N−k−1(bs,k − bs−1,k)
432
+ = 2N−s
433
+
434
+ s
435
+
436
+ k=0
437
+ 2k−Nps,k −
438
+ s−1
439
+
440
+ k=0
441
+ 2k−Nps−1,k
442
+
443
+ +
444
+ s−1
445
+
446
+ k=t
447
+ 2N−k−1
448
+ k
449
+
450
+ ℓ=0
451
+ 2ℓ−N(ps,ℓ − ps−1,ℓ)
452
+ = qs − qs
453
+ s−1
454
+
455
+ k=0
456
+ 2k−sps−1,k − qs
457
+ s−1
458
+
459
+ ℓ=0
460
+ s−1
461
+
462
+ k=ℓ∨t
463
+ 2ℓ−k−1ps−1,ℓ
464
+ = qs
465
+
466
+ 1 −
467
+ s−1
468
+
469
+ k=0
470
+ 2k−sps−1,k −
471
+ s−1
472
+
473
+ ℓ=0
474
+ [2ℓ−ℓ∨t − 2ℓ−s]ps−1,ℓ
475
+
476
+ = qs
477
+
478
+ 1 −
479
+ s−1
480
+
481
+ k=0
482
+ 2k−k∨tps−1,k
483
+
484
+ = qs ·
485
+
486
+ 1 −
487
+ t−1
488
+
489
+ k=0
490
+ 2k−tps−1,k −
491
+ s−1
492
+
493
+ k=t
494
+ ps−1,k
495
+
496
+ = qs ·
497
+ t−1
498
+
499
+ k=0
500
+ (1 − 2k−t)ps−1,k = qs · 2−t ·
501
+ t−1
502
+
503
+ k=0
504
+ (2t − 2k)ps−1,k.
505
+ 8
506
+
507
+ Since r = t in this case, we have proved formula (17) in the case where y ∈ D(x) \ {x}. Finally,
508
+ the last case that remains is when x = y. In this case r = s = t and
509
+ K(x, y) =
510
+
511
+ ω∈{0,1}N;πd(y)ω=y
512
+ ad(x),dlca(x,ω) = 2N−d(x)ad(x),d(x) = 2N−sas,s = 2N−s(bs,s − bs−1,s−1)
513
+ = 2N−s
514
+
515
+ s
516
+
517
+ k=0
518
+ 2k−Nps,k −
519
+ s−1
520
+
521
+ k=0
522
+ 2k−Nps−1,k
523
+
524
+ = qs − qs
525
+ s−1
526
+
527
+ k=0
528
+ 2k−sps−1,k
529
+ = qs ·
530
+ s−1
531
+
532
+ k=0
533
+ (1 − 2k−s)ps−1,k,
534
+ completing the proof of formula (17).
535
+ Some examples.
536
+ 1. The simplest example is when qs = 1 for all s ≥ 1. In this case the operator ˜T is the
537
+ familiar averaging operator. That is, the extension operator ˜T is the operator that assigns
538
+ to each vertex the average of the values at the leaves of its subtree. In this case
539
+ ps,r = δs,r.
540
+ 2. Consider the case where the invariant random walk is such that Xi := d(Ri) is a symmetric
541
+ random walk on {0, . . . , N}, i.e., the probability to jump from i to i + 1 is exactly 1/2 for
542
+ i = 1, . . . , N − 1. Recall that qs is the probability to never leave the subtree when starting
543
+ at a vertex of depth s. We claim that in this example, for s = 1, . . . , N,
544
+ qs =
545
+ 1
546
+ N − s + 1.
547
+ (23)
548
+ Indeed, the function f(i) = i is harmonic on {0, 1, . . . , N} and hence f(Xi) is a mar-
549
+ tingale. Thus for any stopping time ˜τ we have f(X1) = Ef(X˜τ). We pick the stopping
550
+ time
551
+ ˜τ = min{ i; Xi ∈ {s − 1, N} }
552
+ and obtain (23) since
553
+ N · qs + (s − 1) · (1 − qs) = s.
554
+ Next we use Lemma 2.1 and find a formula for ps,r. Since formula (23) is valid for any
555
+ s ≥ 1, we conclude that for any r ≥ 0 and s ≥ r + 1,
556
+ s�
557
+ k=r+1
558
+ (1 − qk) =
559
+ s�
560
+ k=r+1
561
+ N − k
562
+ N − k + 1 = N − s
563
+ N − r.
564
+ (24)
565
+ 9
566
+
567
+ Formula (24) is actually valid for any 0 ≤ r ≤ s, since an empty product equals one.
568
+ Recall that q0 = 1. We thus conclude from Lemma 2.1 that for s = 0, . . . , N − 1,
569
+ ps,r = N − s
570
+ N − r ·
571
+
572
+ 1
573
+ r = 0
574
+ 1
575
+ N−r+1
576
+ 1 ≤ r ≤ s
577
+ while pN,r = δN,r.
578
+ 3. Let 0 < δ < 1, and consider the case where (Xi) is a random walk on {0, . . . , N} such
579
+ that the probability to jump from k to k + 1 equals 1/2 if k < N − 1, and it equals δ if
580
+ k = N − 1. A harmonic function here is
581
+ f(k) =
582
+
583
+ k
584
+ k ≤ N − 1
585
+ N − 2 + 1/δ
586
+ k = N
587
+ Therefore, for s = 1, . . . , N − 1,
588
+ (N − 2 + 1/δ) · qs + (s − 1)(1 − qs) = s.
589
+ Thus q0 = qN = 1 while for s = 1, . . . , N − 1,
590
+ qs =
591
+ 1
592
+ N − s + 1/δ − 1.
593
+ Hence for any s ≤ N − 1 and r ≤ s − 1,
594
+ s
595
+
596
+ k=r+1
597
+ (1 − qk) =
598
+ s
599
+
600
+ k=r+1
601
+ N − k + δ−1 − 2
602
+ N − k + δ−1 − 1 = N − s + δ−1 − 2
603
+ N − r + δ−1 − 2.
604
+ We conclude from Lemma 2.1 that for s = 0, . . . , N − 1 and 0 ≤ r ≤ s,
605
+ ps,r = qr ·
606
+ s�
607
+ k=r+1
608
+ (1 − qk) = N − s + δ−1 − 2
609
+ N − r + δ−1 − 2 ·
610
+
611
+ 1
612
+ r = 0
613
+ 1
614
+ N−r+1/δ−1
615
+ 1 ≤ r ≤ s
616
+ while pN,r = δN,r.
617
+ We conclude this section with the following:
618
+ Lemma 2.3. For any numbers q1, . . . , qN−1 ∈ (0, 1) there exists a random walk
619
+ X1, X2, . . . ∈ {0, . . . , N}
620
+ satisfying (9) with τ = min{i ≥ 1 ; Xi = N} for s = 1, . . . , N − 1.
621
+ Proof. Write xk for the probability that the random walk jumps from k to k + 1. Then for
622
+ s = 0, . . . , N − 1,
623
+ qs = xs(qs+1 + (1 − qs+1)xs),
624
+ (25)
625
+ where we set q0 = qN = 1. We claim that xs ∈ [0, 1] is determined by equation (25). Indeed, the
626
+ right-hand side of (25) is a continuous, increasing function of xs in the interval [0, 1], that maps
627
+ this interval to itself.
628
+ 10
629
+
630
+ 3
631
+ The ancestral and non-ancestral parts of the kernel
632
+ We need to bound the operator norm in Lp(E) of the operator T whose kernel is described in
633
+ Proposition 2.2. Let us consider first the non-ancestral part of the operator, given by the kernel
634
+ K0(x, y) = K(x, y) · 1{x̸∈D(y),y̸∈D(x)} = −1{r≤s∧t−1} · qs · 2−t ·
635
+ r
636
+
637
+ k=0
638
+ 2kps−1,k.
639
+ (26)
640
+ Here as usual s = d(x), t = d(y) and r = dlca(x, y). Write T0 : Lp(E) → Lp(E) for the
641
+ operator whose kernel is K0. A function f : E → R is invariant under the symmetries of the
642
+ tree, or invariant in short, if it takes the form
643
+ f(x) = F(d(x))
644
+ for some function F : {1, . . . , N} → R. The operator T0 is equivariant under the symmetries
645
+ of the tree. Therefore, if f(x) = F(d(x)) is an invariant function, then so is T0f. In fact, in the
646
+ case where f(x) = F(d(x)) we can write
647
+ T0f(x) =
648
+ N
649
+
650
+ t=1
651
+ L0(d(x), t)F(t)
652
+ (27)
653
+ for a certain kernel L0(s, t) defined for s, t = 1, . . . , N.
654
+ Lemma 3.1. For s, t = 1, . . . , N,
655
+ L0(s, t) = −qs ·
656
+ m−1
657
+
658
+ k=0
659
+ (1 − 2k−m)ps−1,k ≤ 0.
660
+ Proof. Let r ≤ min{t, s} − 1. A moment of reflection reveals that for x ∈ E with d(x) = s,
661
+ n(t; s, r) := #{y ∈ E ; d(y) = t, dlca(x, y) = r} = 2t−r−1.
662
+ By (26), (27) and the definition of T0, for any x ∈ E with d(x) = s,
663
+ L0(s, t) =
664
+
665
+ y∈E;d(y)=t
666
+ K0(x, y) = −
667
+ t∧s−1
668
+
669
+ r=0
670
+ n(t; s, r) · qs · 2−t ·
671
+ r
672
+
673
+ k=0
674
+ 2kps−1,k.
675
+ Denote m = s ∧ t. Then for s, t = 1, . . . , N,
676
+ L0(s, t) = −qs ·
677
+ t∧s−1
678
+
679
+ r=0
680
+ r
681
+
682
+ k=0
683
+ 2k−r−1ps−1,k = −qs ·
684
+ m−1
685
+
686
+ k=0
687
+ m−1
688
+
689
+ r=k
690
+ 2k−r−1ps−1,k
691
+ = −qs ·
692
+ m−1
693
+
694
+ k=0
695
+ (1 − 2k−m)ps−1,k.
696
+ 11
697
+
698
+ Write ΩN = {1, . . . , N} and for F : ΩN → R define
699
+ ∥F∥p = ∥F∥Lp(ΩN) =
700
+ � N
701
+
702
+ k=1
703
+ 2k · Wk · |F(k)|p
704
+ �1/p
705
+ .
706
+ (28)
707
+ Observe that ∥f∥Lp(E) = ∥F∥p if f(x) = F(d(x)). Let
708
+ S0F(s) =
709
+ N
710
+
711
+ t=1
712
+ L0(s, t)F(t)
713
+ so that by (27),
714
+ T0(F ◦ d) = (S0F) ◦ d.
715
+ (29)
716
+ For f, g : E → R we consider the scalar product
717
+ ⟨f, g⟩ =
718
+
719
+ x∈E
720
+ Wd(x)f(x)g(x)
721
+ while for F, G : ΩN → R we set
722
+ ⟨F, G⟩ := ⟨F ◦ d, G ◦ d⟩ =
723
+ N
724
+
725
+ k=1
726
+ 2kWkF(k)G(k).
727
+ The adjoint operators T ∗
728
+ 0 and S∗
729
+ 0 are defined with respect to these scalar products. The follow-
730
+ ing lemma is probably well-known to experts (see, e.g., Howard and Schep [2] for a related
731
+ argument), and its proof is provided for completeness.
732
+ Lemma 3.2. Let 1 < p < ∞. Then the norm of the operator T0 : Lp(E) → Lp(E) is attained at
733
+ an invariant, non-negative function f, and it equals to the norm of the operator S0 : Lp(Ωn) →
734
+ Lp(Ωn).
735
+ Proof. Denote momentarily T = −T0 and S = −S0. By Lemma 3.1 the kernel −L0 of the
736
+ operator S is non-negative, and by (26) the kernel of the operator T is non-negative as well.
737
+ By approximation, we may assume that these two kernels are strictly positive, while keeping
738
+ condition (29), thus
739
+ T (F ◦ d) = (SF) ◦ d.
740
+ (30)
741
+ We deduce that T ∗(F ◦ d) = (S∗F) ◦ d. By compactness,
742
+ sup
743
+ 0̸≡F ∈Lp(ΩN )
744
+ ∥SF∥p
745
+ ∥F∥p
746
+ is attained at some function F. Since the kernel of S is non-negative, we may assume that
747
+ the extremal function F is non-negative. By the Lagrange multipliers theorem, the function F
748
+ satisfies a certain eigenvalue equation, and in fact there exists λ ∈ R such that
749
+ S∗(SF)p−1 = λF p−1.
750
+ (31)
751
+ 12
752
+
753
+ Since the kernel of S is positive and F is non-negative and not identically zero, it follows from
754
+ (31) that F is actually positive. The norm of the operator S : Lp(Ωn) → Lp(Ωn) equals λ1/p > 0,
755
+ since
756
+ ∥SF∥p
757
+ p = ⟨(SF)p−1, SF⟩ = ⟨S∗(SF)p−1, F⟩ = λ⟨F p−1, F⟩ = λ∥F∥p
758
+ p.
759
+ Denoting f = F ◦ d, we find from (30) that f is a positive invariant function satisfying
760
+ T ∗(T f)p−1 = λf p−1.
761
+ Since the kernel of T is non-negative, we have the pointwise H¨older inequality
762
+ T (uv) ≤ T (up)1/p · T (vq)1/q,
763
+ valid for any non-negative functions u, v ∈ Lp(E), where q = p/(p − 1). The operator T has
764
+ a non-negative kernel, and hence its norm is attained at a non-negative function u ∈ Lp(E). By
765
+ the pointwise H¨older inequality,
766
+ (T u)p ≤ T (upf −p/q) · (T f)p/q = T (upf 1−p) · (T f)p−1.
767
+ Therefore,
768
+ ∥T u∥p
769
+ Lp(E) ≤ ⟨T (upf 1−p), (T f)p−1⟩ = ⟨upf 1−p, T ∗(T f)p−1⟩ = λ⟨upf 1−p, f p−1⟩ = λ∥u∥p
770
+ Lp(E).
771
+ Thus the norm of T : Lp(E) → Lp(E) is at most λ1/p, which is the norm of S : Lp(ΩN) →
772
+ Lp(ΩN). The two norms must therefore be equal, since the operator S is equivalent to the
773
+ restriction of T to the space of invariant functions.
774
+ We move on to the ancestral part of the operator, which according to Proposition 2.2 is given
775
+ by
776
+ K1(x, y) = K(x, y) − K0(x, y) = 1{r=s∧t} · qs · 2−t ·
777
+ r−1
778
+
779
+ k=0
780
+ (2r − 2k)ps−1,k ≥ 0.
781
+ (32)
782
+ Write T1 for the operator whose kernel is K1. As before, for an invariant function f(x) =
783
+ F(d(x)) we may write
784
+ T1f(x) =
785
+ N
786
+
787
+ t=1
788
+ L1(d(x), t)F(t)
789
+ (33)
790
+ for a certain kernel L1(s, t) defined for s, t = 1, . . . , N. We also write
791
+ S1F(s) =
792
+ N
793
+
794
+ t=1
795
+ L1(s, t)F(t).
796
+ It is possible to use formula (7) for the operator T and the definition (14) of the harmonic exten-
797
+ sion operator ˜T and deduce that
798
+ S0 + S1 ≡ 0,
799
+ (34)
800
+ essentially because the only invariant, harmonic function on the vertices of the tree is the constant
801
+ function. An alternative, more direct proof of (34) is provided in the following:
802
+ 13
803
+
804
+ Lemma 3.3. Let 1 < p < ∞. Then the norm of the operator T1 : Lp(E) → Lp(E) is equal to
805
+ the norm of S1 : Lp(Ωn) → Lp(Ωn). Additionally, for s, t = 1, . . . , N with m = s ∧ t we have
806
+ L1(s, t) = qs ·
807
+ m−1
808
+
809
+ k=0
810
+ (1 − 2k−m)ps−1,k = −L0(s, t).
811
+ Proof. The first assertion of the lemma follows from fact that the kernel of T1 is non-negative
812
+ and invariant under the symmetries of the tree, as in Lemma 3.2. For the second part, let s, t =
813
+ 1, . . . , N and denote m = s ∧ t. We claim that for x ∈ E with d(x) = s,
814
+ n(t; s, m) := #{y ∈ E ; d(y) = t, dlca(x, y) = m} = max{1, 2t−s}.
815
+ (35)
816
+ Indeed, assume first that t ≥ s. How many y’s are there with d(y) = t and dlca(x, y) = m?
817
+ Since m = s, the answer is 2t−s. Next, if s ≥ t, then the number of such y’s is one. This proves
818
+ (35). Therefore,
819
+ L1(s, t) =
820
+
821
+ y∈E;d(y)=t
822
+ K1(x, y) = n(t; s, m) · qs · 2−t ·
823
+ m−1
824
+
825
+ k=0
826
+ (2m − 2k)ps−1,k
827
+ = max{2−t, 2−s} · qs ·
828
+ m−1
829
+
830
+ k=0
831
+ (2m − 2k)ps−1,k.
832
+ = 2−m · qs ·
833
+ m−1
834
+
835
+ k=0
836
+ (2m − 2k)ps−1,k.
837
+ Corollary 3.4. We have
838
+ ∥T∥Lp(E)→Lp(E) ≤ 2∥S0∥Lp(ΩN)→Lp(ΩN).
839
+ Proof. This follows from the fact that T = T0 + T1 together with the facts that ∥T0∥ = ∥S0∥ and
840
+ ∥T1∥ = ∥S1∥ while S1 = −S0.
841
+ In view of Corollary 3.4, we are interested in bounds for the norm of the operator S = −S0 =
842
+ S1 : Lp(ΩN) → Lp(ΩN) whose non-negative kernel is
843
+ L(s, t) = qs ·
844
+ s∧t−1
845
+
846
+ k=0
847
+ (1 − 2k−s∧t)ps−1,k ≤ qs ·
848
+ s∧t−1
849
+
850
+ k=0
851
+ ps−1,k.
852
+ (36)
853
+ 14
854
+
855
+ From Lemma 2.1 we know that ps,r = qr · �s
856
+ k=r+1(1 − qk) for s ≤ N − 1. A little exercise in
857
+ probability shows that for any 1 ≤ m ≤ s ≤ N,
858
+ m−1
859
+
860
+ k=0
861
+ ps−1,k =
862
+ s−1
863
+
864
+ k=m
865
+ (1 − qk).
866
+ (37)
867
+ Alternatively, (37) holds true for m = 0 as q0 = 1, and it may be proven by induction on m since
868
+ ps−1,m +
869
+ s−1
870
+
871
+ k=m
872
+ (1 − qk) = qm ·
873
+ s−1
874
+
875
+ k=m+1
876
+ (1 − qk) +
877
+ s−1
878
+
879
+ k=m
880
+ (1 − qk) =
881
+ s�
882
+ k=m+1
883
+ (1 − qk).
884
+ From (36) and (37) we thus obtain
885
+ Corollary 3.5. For s, t = 1, . . . , N, with m = min{s, t},
886
+ 0 ≤ L(s, t) ≤ qs
887
+ s−1
888
+
889
+ k=m
890
+ (1 − qk),
891
+ where an empty product equals one.
892
+ Some examples (parallel to the ones discussed in Section 2).
893
+ 1. For the averaging operator, where qs = 1 and ps,r = δs,r, we have
894
+ L(s, t) = −1
895
+ 2 · 1{s≤t},
896
+ i.e., this is the matrix whose entries equal 0 below the diagonal and −1/2 on and above
897
+ the diagonal. This is a rather simple matrix, and it is bounded with respect to the weighted
898
+ Lp-norm for quite a few sequences of weights.
899
+ 2. For the symmetric random walk matrix, we have q0 = 1 while for 1 ≤ s ≤ N,
900
+ qs =
901
+ 1
902
+ N − s + 1.
903
+ Hence in view of Corollary 3.5, with m = min{s, t},
904
+ 0 ≤ L(s, t) ≤
905
+ 1
906
+ N − s + 1
907
+ s−1
908
+
909
+ k=m
910
+ N − k
911
+ N − k + 1 =
912
+ 1
913
+ N − m + 1.
914
+ 3. In the case where
915
+ qs =
916
+ 1
917
+ N − s + 1/δ − 1
918
+ for some 0 < δ < 1, we have
919
+ 0 ≤ L(s, t) ≤
920
+ 1
921
+ N − s + 1/δ − 1
922
+ s−1
923
+
924
+ k=m
925
+ N − k + 1/δ − 2
926
+ N − k + 1/δ − 1 =
927
+ 1
928
+ N − m + 1/δ − 1.
929
+ 15
930
+
931
+ All that remains is to bound the Lp(ΩN)-norm of the operator whose kernel is discussed
932
+ in Corollary 3.5. Recall from Lemma 2.3 that we have the freedom to choose the parameters
933
+ q1, . . . , qN−1 ∈ (0, 1) as we please.
934
+ How should we choose these parameters? Since L(N, t) ≤ �N−1
935
+ k=t (1−qk) and we are looking
936
+ for upper bounds for the norm, the qs should not be too tiny. On the other hand, L(s, t) ≤ qs for
937
+ s ≤ N − 1 and hence it is beneficial to choose qs rather small. We would therefore need some
938
+ balance for the qs, which is the subject of the next section.
939
+ 4
940
+ One-dimensional analysis
941
+ Let 1 < p < ∞. It will be slightly more convenient to denote
942
+ K(s, t) = L(N + 1 − s, N + 1 − t)
943
+ and
944
+ Qs = qN+1−s.
945
+ Recalling from (10) that qN = 1, we see that
946
+ Q1 = 1.
947
+ From Corollary 3.5 we know that for s, t = 1, . . . , N,
948
+ K(s, t) = L(N + 1 − s, N + 1 − t) ≤ qN+1−s
949
+ N−s
950
+
951
+ k=N+1−max{s,t}
952
+ (1 − qk) = Qs
953
+ max{s,t}
954
+
955
+ k=s+1
956
+ (1 − Qk).
957
+ From Corollary 3.5 we know that L ≥ 0. Consequently, for s, t = 1, . . . , N,
958
+ 0 ≤ K(s, t) ≤
959
+
960
+ Qs
961
+ t ≤ s
962
+ Qs · �t
963
+ k=s+1(1 − Qk)
964
+ t ≥ s + 1
965
+ (38)
966
+ Recall that we are given edge weights W1, . . . , WN > 0, and that the associated Lp(E)-norm is
967
+ given by (5). Denote
968
+ ws := 2N+1−s · WN+1−s > 0.
969
+ Consider the weighted ℓp-norm
970
+ ∥f∥p,w =
971
+ � N
972
+
973
+ k=1
974
+ wk|f(k)|p
975
+ �1/p
976
+ (39)
977
+ and the operator T whose kernel is K(s, t). We are allowed to choose the weights q1, . . . , qN−1 ∈
978
+ (0, 1) as we please, or equivalently, we have the freedom to determine Q2, . . . , QN ∈ (0, 1). We
979
+ must keep Q1 = 1. Based on considerations related to the Muckenhoupt criterion discussed
980
+ below, we set
981
+ Qs =
982
+ w−1/(p−1)
983
+ s
984
+ �s
985
+ k=1 w−1/(p−1)
986
+ k
987
+ .
988
+ (40)
989
+ It is clear that Q1 = 1 and that Qs ∈ (0, 1) for all s ≥ 2. Recall that q = p/(p − 1).
990
+ 16
991
+
992
+ Lemma 4.1. In order to prove Theorem 1.2, it suffices to show that the operator norm of T with
993
+ respect to the ∥ · ∥p,w-norm is bounded by a constant ˆCp > 0 depending only on p, where in fact
994
+ ˆCp ≤ 2p1/pq1/q ·
995
+
996
+ 1 + max{(p − 1)−1/p, (q − 1)−1/q}
997
+
998
+ .
999
+ (41)
1000
+ Proof. In view of Corollary 3.4, it suffices to bound the operator norm of −S0, whose kernel is
1001
+ L, with respect to the Lp(ΩN)-norm defined in (28). Under the transformation
1002
+ s �→ N + 1 − s
1003
+ the operator −S0 whose kernel is L transforms to the operator T whose kernel is K. The Lp(ΩN)-
1004
+ norm from (28) transforms to the ∥·∥p,w-norm defined in (39). Hence Theorem 1.2 would follow
1005
+ once we obtain the bound (41), where ¯Cp ≤ 2 ˆCp by Corollary 3.4.
1006
+ The remainder of this section is devoted to the proof of the following:
1007
+ Proposition 4.2. The operator norm of T with respect to the norm (39) is bounded by a number
1008
+ ˆCp depending only on p ∈ (1, ∞). In fact, we have the bound (41) for the constant ˆCp.
1009
+ Our main tool in the proof of Proposition 4.2 is the Muckenhoupt criterion [4], which is an
1010
+ indispensable tool for proving one-dimensional inequalities of Poincar´e-Sobolev type. For the
1011
+ reader’s convenience, we include here a statement and a proof of a straightforward modification
1012
+ of the Muckenhoupt criterion, with sums in place of integrals:
1013
+ Theorem 4.3. (Muckenhoupt) Let 1 < p < ∞ and write ΩN = {1, . . . , N}. Let U, V : ΩN →
1014
+ (0, ∞) and let A > 0 be such that for all r = 1, . . . , N,
1015
+ � N
1016
+
1017
+ k=r
1018
+ |U(k)|p
1019
+ �1/p
1020
+ ≤ A
1021
+
1022
+ r
1023
+
1024
+ k=1
1025
+ |V (k)|−q
1026
+ �−1/q
1027
+ .
1028
+ (42)
1029
+ Then for any function f : ΩN → R,
1030
+ � N
1031
+
1032
+ k=1
1033
+ �����U(k)
1034
+ k
1035
+
1036
+ ℓ=1
1037
+ f(ℓ)
1038
+ �����
1039
+ p�1/p
1040
+ ≤ CpA
1041
+ � N
1042
+
1043
+ k=1
1044
+ |V (k)f(k)|p
1045
+ �1/p
1046
+ ,
1047
+ (43)
1048
+ with Cp = p1/pq1/q.
1049
+ By continuity, the analog of Theorem 4.3 for p = 1, ∞ holds true with C1 = C∞ = 1. We
1050
+ remark that as in [4], this criterion is tight, in the sense that the infimum over all A > 0 satisfying
1051
+ (42) is equivalent to the best constant in inequality (43). For the proof of Theorem 4.3 we require
1052
+ the following:
1053
+ 17
1054
+
1055
+ Lemma 4.4. For α1, . . . , αN > 0 and r = 1, . . . , N,
1056
+ r
1057
+
1058
+ k=1
1059
+ αk
1060
+
1061
+ k
1062
+
1063
+ ℓ=1
1064
+ αℓ
1065
+ �−1/p
1066
+ ≤ q ·
1067
+
1068
+ r
1069
+
1070
+ k=1
1071
+ αk
1072
+ �1/q
1073
+ .
1074
+ (44)
1075
+ Proof. We will use the simple inequality
1076
+ (a + b)1/q − a1/q ≥ b ·
1077
+ min
1078
+ ξ∈(a,a+b)
1079
+ ξ1/q−1
1080
+ q
1081
+ = 1
1082
+ q(a + b)1/q−1 · b,
1083
+ (45)
1084
+ valid for any a, b ≥ 0 with a + b > 0. From (45), for k = 1, . . . , N,
1085
+ � k
1086
+
1087
+ ℓ=1
1088
+ αℓ
1089
+ �1/q
1090
+
1091
+ �k−1
1092
+
1093
+ ℓ=1
1094
+ αℓ
1095
+ �1/q
1096
+ ≥ 1
1097
+ q · αk ·
1098
+
1099
+ k
1100
+
1101
+ ℓ=1
1102
+ αℓ
1103
+ �1/q−1
1104
+ ,
1105
+ where an empty sum equals zero. By summing this for k = 1, . . . , r we obtain (44).
1106
+ Proof of Theorem 4.3 (Muckenhoupt). By the H¨older inequality, for any function f : ΩN → R
1107
+ and weights h : ΩN → (0, ∞),
1108
+ N
1109
+
1110
+ k=1
1111
+ �����U(k)
1112
+ k
1113
+
1114
+ ℓ=1
1115
+ f(ℓ)
1116
+ �����
1117
+ p
1118
+
1119
+ N
1120
+
1121
+ k=1
1122
+ Up(k) ·
1123
+ k
1124
+
1125
+ ℓ=1
1126
+ |f(ℓ)V (ℓ)h(ℓ)|p ·
1127
+
1128
+ k
1129
+
1130
+ j=1
1131
+ |V (j)h(j)|−q
1132
+ �p/q
1133
+ =
1134
+ N
1135
+
1136
+ ℓ=1
1137
+ |f(ℓ)V (ℓ)h(ℓ)|p
1138
+ N
1139
+
1140
+ k=ℓ
1141
+ Up(k)
1142
+ � k
1143
+
1144
+ j=1
1145
+ |V (j)h(j)|−q
1146
+ �p/q
1147
+ .
1148
+ (46)
1149
+ Set h(k) =
1150
+ ��k
1151
+ ℓ=1 V (ℓ)−q�1/(pq)
1152
+ . By applying Lemma 4.4 with αk = V (k)−q we obtain
1153
+ r
1154
+
1155
+ k=1
1156
+ |V (k)h(k)|−q =
1157
+ r
1158
+
1159
+ k=1
1160
+ αk
1161
+ � k
1162
+
1163
+ ℓ=1
1164
+ αℓ
1165
+ �−1/p
1166
+ ≤ q ·
1167
+
1168
+ r
1169
+
1170
+ k=1
1171
+ αk
1172
+ �1/q
1173
+ = q ·
1174
+
1175
+ r
1176
+
1177
+ k=1
1178
+ V (k)−q
1179
+ �1/q
1180
+ .
1181
+ Hence for any f : ΩN → R, the expression in (46) is at most
1182
+ qp/q ·
1183
+ N
1184
+
1185
+ ℓ=1
1186
+ |f(ℓ)V (ℓ)h(ℓ)|p
1187
+ N
1188
+
1189
+ k=ℓ
1190
+ Up(k)
1191
+
1192
+ k
1193
+
1194
+ j=1
1195
+ |V (j)|−q
1196
+ �p/q2
1197
+ .
1198
+ (47)
1199
+ By applying (42) and then Lemma 4.4 with αk = Up(N + 1 − k) and with p ∈ (1, ∞) playing
1200
+ the rˆole of q ∈ (1, ∞), we see that
1201
+ N
1202
+
1203
+ k=ℓ
1204
+ |U(k)|p
1205
+
1206
+ k
1207
+
1208
+ j=1
1209
+ |V (j)|−q
1210
+ �p/q2
1211
+ ≤ Ap/q
1212
+ N
1213
+
1214
+ k=ℓ
1215
+ |U(k)|p
1216
+ � N
1217
+
1218
+ j=k
1219
+ |U(j)|p
1220
+ �−1/q
1221
+ = Ap/q
1222
+ N+1−ℓ
1223
+
1224
+ k=1
1225
+ αk
1226
+ � k
1227
+
1228
+ j=1
1229
+ αj
1230
+ �−1/q
1231
+ ≤ p · Ap/q
1232
+ �N+1−ℓ
1233
+
1234
+ k=1
1235
+ αk
1236
+ �1/p
1237
+ = p · Ap/q
1238
+ � N
1239
+
1240
+ k=ℓ
1241
+ Up(k)
1242
+ �1/p
1243
+ .
1244
+ 18
1245
+
1246
+ Hence the expression in (47) is at most
1247
+ p · (qA)p/q ·
1248
+ N
1249
+
1250
+ ℓ=1
1251
+ |f(ℓ)V (ℓ)h(ℓ)|p ·
1252
+ � N
1253
+
1254
+ k=ℓ
1255
+ Up(k)
1256
+ �1/p
1257
+ .
1258
+ Applying (42) again we bound the last expression from above by
1259
+ p · (qA)p/q · A ·
1260
+ N
1261
+
1262
+ ℓ=1
1263
+ |f(ℓ)V (ℓ)h(ℓ)|p ·
1264
+
1265
+
1266
+
1267
+ k=1
1268
+ V −q(k)
1269
+ �−1/q
1270
+ = pqp/qAp
1271
+ N
1272
+
1273
+ ℓ=1
1274
+ |f(ℓ)V (ℓ)|p,
1275
+ completing the proof.
1276
+ Corollary 4.5. Let 1 < p < ∞ and ΩN = {1, . . . , N}. Let U, V : ΩN → (0, ∞) and let A > 0
1277
+ be such that for all r = 1, . . . , N,
1278
+
1279
+ r
1280
+
1281
+ k=1
1282
+ |U(k)|p
1283
+ �1/p
1284
+ ≤ A
1285
+ � N
1286
+
1287
+ k=r
1288
+ |V (k)|−q
1289
+ �−1/q
1290
+ .
1291
+ (48)
1292
+ Then for any f : ΩN → R,
1293
+ � N
1294
+
1295
+ k=1
1296
+ �����U(k)
1297
+ N
1298
+
1299
+ ℓ=k
1300
+ f(ℓ)
1301
+ �����
1302
+ p�1/p
1303
+ ≤ CpA
1304
+ � N
1305
+
1306
+ k=1
1307
+ |V (k)f(k)|p
1308
+ �1/p
1309
+ ,
1310
+ (49)
1311
+ with Cp = p1/pq1/q.
1312
+ Proof. Denote ˜U(k) = U(N + 1 − k) and ˜V (k) = V (N + 1 − k). Then from (48), for all
1313
+ r = 1, . . . , N,
1314
+ � N
1315
+
1316
+ k=r
1317
+ | ˜U(k)|p
1318
+ �1/p
1319
+ ≤ A
1320
+
1321
+ r
1322
+
1323
+ k=1
1324
+ | ˜V (k)|−q
1325
+ �−1/q
1326
+ .
1327
+ By Theorem 4.3, this implies that for any g : ΩN → R, denoting f(r) = g(N + 1 − r),
1328
+ � N
1329
+
1330
+ k=1
1331
+ �����
1332
+ ˜U(k)
1333
+ k
1334
+
1335
+ ℓ=1
1336
+ f(N + 1 − ℓ)
1337
+ �����
1338
+ p�1/p
1339
+ ≤ CpA
1340
+ � N
1341
+
1342
+ k=1
1343
+ | ˜V (k)f(N + 1 − k)|p
1344
+ �1/p
1345
+ ,
1346
+ or equivalently,
1347
+ � N
1348
+
1349
+ k=1
1350
+ �����
1351
+ ˜U(N + 1 − k)
1352
+ N
1353
+
1354
+ ℓ=k
1355
+ f(ℓ)
1356
+ �����
1357
+ p�1/p
1358
+ ≤ CpA
1359
+ � N
1360
+
1361
+ k=1
1362
+ | ˜V (N + 1 − k)f(k)|p
1363
+ �1/p
1364
+ .
1365
+ This implies (49).
1366
+ 19
1367
+
1368
+ Proposition 4.6. For f : Ωn → R and s = 1, . . . , N denote
1369
+ T0f(s) = Qs
1370
+ s
1371
+
1372
+ t=1
1373
+ f(t).
1374
+ Then the operator norm of T0 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by a
1375
+ number ˜Cp depending only on p ∈ (1, ∞). In fact, ˜Cp ≤ 21/pp1/pq1/q · max{1, (p − 1)−1/p}.
1376
+ The proof of Proposition 4.6 requires the following:
1377
+ Lemma 4.7. For any α1, . . . , αN > 0 and r = 1, . . . , N,
1378
+ N
1379
+
1380
+ k=r
1381
+ αk
1382
+ � k
1383
+
1384
+ ℓ=1
1385
+ αℓ
1386
+ �−p
1387
+
1388
+ 2
1389
+ min{1, p − 1} ·
1390
+
1391
+ r
1392
+
1393
+ k=1
1394
+ αk
1395
+ �−p+1
1396
+ .
1397
+ (50)
1398
+ Proof. We use the inequality
1399
+ (p − 1) · b(a + b)−p ≤ a−p+1 − (a + b)−p+1,
1400
+ which is valid for any a, b > 0. Then for k = 2, . . . , N,
1401
+ (p − 1) · αk
1402
+ � k
1403
+
1404
+ ℓ=1
1405
+ αℓ
1406
+ �−p
1407
+
1408
+ �k−1
1409
+
1410
+ ℓ=1
1411
+ αℓ
1412
+ �−p+1
1413
+
1414
+
1415
+ k
1416
+
1417
+ ℓ=1
1418
+ αℓ
1419
+ �−p+1
1420
+ .
1421
+ By summing this for k = r + 1, . . . , N we obtain
1422
+ (p − 1) ·
1423
+ N
1424
+
1425
+ k=r+1
1426
+ αk
1427
+ � k
1428
+
1429
+ ℓ=1
1430
+ αℓ
1431
+ �−p
1432
+
1433
+
1434
+ r
1435
+
1436
+ k=1
1437
+ αk
1438
+ �−p+1
1439
+
1440
+ � N
1441
+
1442
+ ℓ=1
1443
+ αℓ
1444
+ �−p+1
1445
+
1446
+
1447
+ r
1448
+
1449
+ k=1
1450
+ αk
1451
+ �−p+1
1452
+ ,
1453
+ where an empty sum equals zero. We conclude (50) by summing this with the trivial inequality
1454
+ min{1, p − 1} · αr
1455
+
1456
+ r
1457
+
1458
+ ℓ=1
1459
+ αℓ
1460
+ �−p
1461
+
1462
+
1463
+ r
1464
+
1465
+ k=1
1466
+ αk
1467
+ �−p+1
1468
+ .
1469
+ Proof of Proposition 4.6. Define
1470
+ U(k) = w1/p
1471
+ k
1472
+ · Qk
1473
+ and
1474
+ V (k) = w1/p
1475
+ k .
1476
+ Let us verify the condition of the Muckenhoupt criterion. We need to find A > 0 such that for
1477
+ all r = 1, . . . , N inequality (42) holds true, that is,
1478
+ N
1479
+
1480
+ k=r
1481
+ wkQp
1482
+ k ≤ Ap
1483
+
1484
+ r
1485
+
1486
+ k=1
1487
+ w−q/p
1488
+ k
1489
+ �−p/q
1490
+ .
1491
+ (51)
1492
+ 20
1493
+
1494
+ Recall that p/q = p − 1. From the definition (40) of Qk, we need
1495
+ N
1496
+
1497
+ k=r
1498
+ w−1/(p−1)
1499
+ k
1500
+
1501
+ k
1502
+
1503
+ ℓ=1
1504
+ w−1/(p−1)
1505
+
1506
+ �−p
1507
+ ≤ Ap
1508
+
1509
+ r
1510
+
1511
+ k=1
1512
+ w−1/(p−1)
1513
+ k
1514
+ �−(p−1)
1515
+ .
1516
+ Setting αk = w−1/(p−1)
1517
+ k
1518
+ and using Lemma 4.7, we see that (51) holds true with
1519
+ A = 21/p · max{1, (p − 1)−1/p}.
1520
+ From Theorem 4.3 we thus conclude that for any f : ΩN → R,
1521
+ � N
1522
+
1523
+ k=1
1524
+ wk
1525
+ �����Qk
1526
+ k
1527
+
1528
+ ℓ=1
1529
+ f(ℓ)
1530
+ �����
1531
+ p�1/p
1532
+ ≤ p1/pq1/qA ·
1533
+ � N
1534
+
1535
+ k=1
1536
+ wk|f(k)|p
1537
+ �1/p
1538
+ .
1539
+ This implies the required bound for the operator norm of f.
1540
+ Proposition 4.8. For f : Ωn → R and s = 1, . . . , N denote
1541
+ T1f(s) =
1542
+ N
1543
+
1544
+ t=s+1
1545
+
1546
+ Qs
1547
+ t�
1548
+ k=s+1
1549
+ (1 − Qk)
1550
+
1551
+ f(t).
1552
+ Then the operator norm of T1 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by
1553
+ 21/qp1/pq1/q · max{1, (q − 1)−1/q}.
1554
+ Proof. Denote αk = w−1/(p−1)
1555
+ k
1556
+ and recall from (40) that Qk = αk/ �k
1557
+ ℓ=1 αℓ. We claim that for
1558
+ any t ≥ s + 1,
1559
+ Qs
1560
+ t�
1561
+ k=s+1
1562
+ (1 − Qk) =
1563
+ αs
1564
+ �t
1565
+ k=1 αk
1566
+ .
1567
+ (52)
1568
+ Indeed, (52) holds true for t = s, since an empty product equals one, and for t ≥ s + 1 it is
1569
+ proven by an easy induction on t. Consequently,
1570
+ T1f(s) = αs
1571
+ N
1572
+
1573
+ t=s+1
1574
+ 1
1575
+ �t
1576
+ j=1 αj
1577
+ f(t).
1578
+ Since p, q ∈ (1, ∞), the elementary inequality of Lemma 4.7 is valid also when p is replaced by
1579
+ q. It implies that for r = 1, . . . , N,
1580
+
1581
+ r
1582
+
1583
+ k=1
1584
+ αk
1585
+ �−q+1
1586
+ ≥ min{1, q − 1}
1587
+ 2
1588
+ ·
1589
+ N
1590
+
1591
+ k=r
1592
+ αk
1593
+
1594
+ k
1595
+
1596
+ j=1
1597
+ αj
1598
+ �−q
1599
+ .
1600
+ (53)
1601
+ 21
1602
+
1603
+ Set A = 21/q min{1, q − 1}−1/q. Since αk = w−1/(p−1)
1604
+ k
1605
+ and q/p = q − 1 = 1/(p − 1), it follows
1606
+ from (53) that for r = 1, . . . , N,
1607
+
1608
+ r
1609
+
1610
+ k=1
1611
+ wkαp
1612
+ k
1613
+ �1/p
1614
+ ≤ A
1615
+
1616
+
1617
+ N
1618
+
1619
+ k=r
1620
+ w−q/p
1621
+ k
1622
+ � k
1623
+
1624
+ j=1
1625
+ αj
1626
+ �−q
1627
+
1628
+ −1/q
1629
+ .
1630
+ This is precisely the Muckenhoupt criterion from Corollary 4.5, with
1631
+ U(k) = w1/p
1632
+ k αk
1633
+ and
1634
+ V (k) = w1/p
1635
+ k
1636
+ k
1637
+
1638
+ j=1
1639
+ αj.
1640
+ Thus, by Corollary 4.5, for any g : ΩN → R,
1641
+ N
1642
+
1643
+ k=1
1644
+ wk
1645
+ �����αk
1646
+ N
1647
+
1648
+ ℓ=k
1649
+ g(ℓ)
1650
+ �����
1651
+ p
1652
+ ≤ (CpA)p
1653
+ N
1654
+
1655
+ k=1
1656
+ wk
1657
+ �������
1658
+
1659
+ k
1660
+
1661
+ j=1
1662
+ αj
1663
+
1664
+ g(k)
1665
+ �����
1666
+ p
1667
+ ,
1668
+ (54)
1669
+ with Cp = p1/pq1/q. By restricting attention to non-negative functions g in (54), we may alter
1670
+ (54) and replace �N
1671
+ ℓ=k by the shorter sum �N
1672
+ ℓ=k+1. Inequality (54) remains correct, for non-
1673
+ negative g, also after this modification. Denoting g(k) = f(k)/ �k
1674
+ j=1 αj, we conclude that for
1675
+ any non-negative function f : ΩN → R,
1676
+ N
1677
+
1678
+ k=1
1679
+ wk
1680
+ �����αk
1681
+ N
1682
+
1683
+ ℓ=k+1
1684
+ 1
1685
+ �ℓ
1686
+ j=1 αj
1687
+ f(ℓ)
1688
+ �����
1689
+ p
1690
+ ≤ (CpA)p
1691
+ N
1692
+
1693
+ k=1
1694
+ wk |f(k)|p .
1695
+ (55)
1696
+ Since the kernel of T1 is non-negative, its operator norm is attained at a non-negative function
1697
+ f : ΩN → R. Therefore (55) implies the required bound for the operator norm of T1.
1698
+ Proof of Proposition 4.2. The kernel of the operator T is given in (38). It is a non-negative
1699
+ kernel, and therefore the operator norm of T is at most the operator norm of the operator whose
1700
+ kernel is the expression on the right-hand side of (38). The latter operator equals
1701
+ T0 + T1
1702
+ with T0 from Proposition 4.6 and T1 from Proposition 4.8. From these two propositions it follows
1703
+ that the operator norm of T is at most
1704
+ 21/pp1/pq1/q· max{1, (p − 1)−1/p} + 21/qp1/pq1/q · max{1, (q − 1)−1/q}.
1705
+ Theorem 1.2 follows from Lemma 4.1 and Proposition 4.2.
1706
+ 22
1707
+
1708
+ Remarks.
1709
+ 1. In this paper we have left open several natural questions, including the existence of linear
1710
+ extension operators for ˙W 1,p(T) for weighted trees T in the extreme cases p = 1, p = ∞,
1711
+ as well as the analog of our result for the inhomogeneous Sobolev space W 1,p(T) in place
1712
+ of ˙W 1,p(T).
1713
+ 2. The problem of existence of linear Sobolev extension operators for weighted trees arose in
1714
+ connection with an extension problem for W 1,p(R2). More precisely, given E ⊆ R2, let
1715
+ ˙W 1,p(E) denote the space of restrictions to E of functions in ˙W 1,p(R2), endowed with the
1716
+ natural seminorm. Does there exist a linear extension operator from ˙W 1,p(E) to ˙W 1,p(R2)?
1717
+ The answer is affirmative for p ≥ 2; see A. Israel [3]. For 1 < p < 2, the answer
1718
+ is unknown. For a particular class of examples E, the problem reduces to the question
1719
+ answered by Theorem 1.1.
1720
+ References
1721
+ [1] Bj¨orn, A., Bj¨orn, J., Gill, J. T., Shanmugalingam, N., Geometric analysis on Cantor sets
1722
+ and trees. J. Reine Angew. Math., Vol. 725, (2017), 63–114.
1723
+ [2] Howard, R., Schep, A. R., Norms of positive operators on Lp-spaces. Proc. Amer. Math.
1724
+ Soc., Vol. 109, no. 1, (1990), 135–146.
1725
+ [3] Israel, A., A bounded linear extension operator for L2,p(R2). Ann. of Math. (2), Vol. 178,
1726
+ no. 1, (2013), 183–230.
1727
+ [4] Muckenhoupt, B., Hardy’s inequality with weights. Studia Math., Vol. 44, (1972), 31–38.
1728
+ [5] Shvartsman, P., Sobolev W 1
1729
+ p -spaces on closed subsets of Rn. Adv. Math., Vol. 220, no. 6,
1730
+ (2009), 1842–1922.
1731
+ 23
1732
+
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1
+ arXiv:2301.08639v1 [math.AC] 20 Jan 2023
2
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
3
+ ALESSANDRO LINZI ID
4
+ Abstract. The main aim of this article is to study and develop valuation theory for Krasner
5
+ hyperfields. In analogy with classical valuation theory for fields, we generalise the formalism of
6
+ valuation rings to describe equivalence of valuations on hyperfields. After proving basic results
7
+ and discussing several examples, we focus on the valued hyperfields that Krasner originally defined
8
+ in 1957. We find that these must have a particular additive structure which in turns implies the
9
+ existence of a valuation a’la Krasner. We note that given such a valued hyperfield (F, v), the
10
+ valuation induced by its additive structure does not have to be equivalent to v. We discuss the
11
+ cases in which it does.
12
+ 1. Introduction
13
+ In 1957, M. Krasner in [24] (the article is included in Krasner’s collected works [4, pages 413–
14
+ 490]) formulated for the first time an axiomatisation of structures that generalise fields by allowing
15
+ the additive operation to be multivalued (see [4, pages 407–490] for more details). He called these
16
+ hyperfields (in french hypercorps).
17
+ On the one hand, he found his inspiration in the work of Marty [35, 36, 37] which is considered
18
+ as the starting point of hypergroup theory. In fact, the additive part of a hyperfield is a hypergroup
19
+ (see also [48, 38] for a more detailed historical overview).
20
+ On the other hand, Krasner was motivated by his interest in valuation theory (in connection with
21
+ p-adic numbers) and he sensed the importance of some hyperfields which are canonically associated
22
+ to any valued field (cf. Example 4.11). Let us briefly recall some basic notions of classical valuation
23
+ theory.
24
+ Let K be a field and Γ a linearly ordered abelian group (written additively). A surjective map
25
+ v : K → Γ ∪ {∞}
26
+ is called a (Krull) valuation on K if it satisfies for all x, y ∈ K:
27
+ – v(x) = ∞ if and only if x = 0,
28
+ – v(xy) = v(x) + v(y),
29
+ – v(x + y) ≥ min{v(x), v(y)}.
30
+ Here, ∞ is a symbol such that γ + ∞ = ∞ + γ = ∞ > γ for all γ ∈ Γ. If a valuation v on a field
31
+ K is given, then (K, v) is called a valued field. One usually denotes Γ by vK and call it the value
32
+ group of (K, v). The value v(x) of x ∈ K will often be written as vx, if there is no risk of confusion.
33
+ 2020 Mathematics Subject Classification. Primary: 12J20, 20N20 Secondary: 13A18.
34
+ Key words and phrases. Hyperfield, multifield, hyperring, valuation, ordered abelian group, tropical hyperfield.
35
+ The author would like to spend a few words to thank H. Stojałowska, P. Touchard, Franz-Viktor and Katarzyna
36
+ Kuhlmann, I. Cristea as well as Ch. Massouros, who all, in one way or another, contributed to the realisation of the
37
+ final version of this manuscript.
38
+ 1
39
+
40
+ 2
41
+ LINZI, A.
42
+ If (K, v) is a valued field, then
43
+ Ov := {x ∈ K | vx ≥ 0}
44
+ is a subring of K, called the valuation ring of (K, v). It determines the valuation map v up to
45
+ equivalence, i.e., up to composition with an order preserving isomorphism of the value group. Any
46
+ valuation ring has a unique maximal ideal
47
+ Mv := {x ∈ K | vx > 0}
48
+ and the field Kv := Ov/Mv is called the residue field of (K, v). For further details, let us mention
49
+ [10, 44] as general references on classical valuation theory.
50
+ While it is quite natural to generalise the concept of valuation to hyperfields (see Definition 3.1),
51
+ Krasner noticed that the structures which attracted his attention come equipped with a map similar
52
+ to a valuation which satisfies two additional properties (see Definition 4.7). These properties would
53
+ be vacuous if postulated for valued fields. Thus, Krasner included them in his axiomatisation of
54
+ valued hyperfields (hypercorps valué).
55
+ Krasner’s motivation was not model theoretical. Nevertheless, it turns out that the structures he
56
+ studied play an important role in the model theory of valued fields; specifically for the problem of
57
+ quantifier elimination for henselian valued fields (of characteristic 0). In this setting, these objects
58
+ are known as RV-structures or leading-term structures and have been considered, independently of
59
+ Krasner, by Flenner in [11] (see also [3, 27, 33, 34] for more details). In addition, an interesting
60
+ application of hyperfields in the model theory of valued fields recently appeared in [32].
61
+ The interest for valued hyperfields may also be motivated by the following observations. Clas-
62
+ sically, real algebra, which studies real fields (i.e., linearly ordered fields, with an order compatible
63
+ with the operations) and was developed by E. Artin in his solution of Hilbert’s seventeenth problem,
64
+ is in relation with valuation theory. After the works of Marshall and Gładki [15, 16] on real hyper-
65
+ fields, it is to expect that a development of valuation theory for hyperfields would be beneficial for
66
+ this line of research. Significant developments have already been achieved with the generalisation
67
+ of classical results, such as the Baer–Krull Theorem, to the multivalued setting (see [26, 31]). The
68
+ reader interested in these aspects may look also at [12, 13, 14]. Furthermore, the unpublished work
69
+ of Viro [47], followed up by Jun and Jell et al. [20, 18], indicate applications in the realm of trop-
70
+ icalization maps and analytification, topics that are as well, classically, in relation with valuation
71
+ theory.
72
+ We believe that developing valuation theory for hyperfields will eventually lead to further appli-
73
+ cations back to the classical theory of valuations for fields.
74
+ The switch from singlevalued operations to multivalued ones may be not difficult to describe
75
+ and understand. Nevertheless, the consequences of this switch have to be handled very carefully.
76
+ For instance, valuation theorists are accostumed to the fact that v(x − y) always defines a metric
77
+ (in fact, an ultrametric, cf. Section 4.1) on a valued field (K, v), but this clearly ceases to be true
78
+ (in general) if the operation of addition (and hence of difference) is multivalued. Indeed, in the
79
+ latter case, the distance map may depend on the choice of some element of the set x − y and this
80
+ choice is not canonical, in general. This obstacle is overtaken using one of Krasner’s postulates
81
+ which forces all the elements of x − y, for x ̸= y, to have the same value. However, one may expect
82
+ the latter being a quite strong requirement. In fact, the restrictions due to Krasner’s axioms for
83
+ valued hyperfields are so strong that, e.g., the trivial valuation on a hyperfield F (Example 3.2)
84
+ satisfies them only if F is actually a field (Remark 4.10). For this and other reasons (which we
85
+ will discuss later in the paper), it makes sense to consider also valuations on hyperfields which do
86
+ not satisfy the two additional properties imposed by Krasner (cf. Example 4.17) and it turns out
87
+
88
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
89
+ 3
90
+ that this choice is more beneficial than harmful. As a consequence of these observations, the term
91
+ “valued hyperfield” will be used in this paper in a broader sense than the original one; the valued
92
+ hyperfields satisfying in addition Krasner’s axioms will be called “Krasner valued hyperfields”and
93
+ their valuations “Krasner valuations”.
94
+ The manuscript is organised as follows. In Section 2 we introduce the necessary concepts and
95
+ terminology from hypercompositional algebra, taking the opportunity to discuss several examples.
96
+ In Section 3, we show that the formalism of valuation rings to describe valuations up to equivalence
97
+ generalises without major modifications to the multivalued framework. We use this formalism to
98
+ state and prove one of our main results Theorem 4.22 on Krasner valued hyperfields in Section 4.
99
+ Our main theorem states that the existence of a Krasner valuation on a hyperfield F implies that
100
+ the additive structure of F satisfies certain additional axioms (cf. Proposition 4.20). Conversely, if
101
+ the additive structure of a hyperfield F satisfies those axioms, then F admits a Krasner valuation
102
+ (cf. Proposition 4.21). Let v be a Krasner valuation on a hyperfield F. Then the additive structure
103
+ of F induces a Krasner valuation w on F. We observe that, in general, w is not equivalent to
104
+ v (Example 4.25). After Section 5, where we put into our context the notion of coarsening of a
105
+ valuation, which turns out to be necessary for our discussion, we describe situations in which v and
106
+ w are equivalent in Section 6. The final Section 7 of this manuscript contains possible further lines
107
+ of investigation.
108
+ Besides presenting some new results, the article is also thought to serve as a unified reference for
109
+ basic hyperring and hyperfield theory and related terminology. On the one hand, these hypercom-
110
+ positional structures have recently attracted increasing interest from the mathematical community
111
+ (including top mathematicians such as A. Connes [7, 8]). On the other hand, we found the relevant
112
+ literature on hyperrings and hyperfields to be quite fragmented. After the time and effort spent
113
+ during the PhD studies and thanks to the fruitful connections with some of the mathematicians who
114
+ are part of the early history of hypercompositional algebra, we felt that we could give a contribution
115
+ from this point of view as well.
116
+ 2. Preliminaries
117
+ Let H be a non-empty set and P(H) its power set. A multivalued operation + on H is a function
118
+ which associates to every pair (x, y) ∈ H × H an element of P(H), denoted by x + y. If + is a
119
+ multivalued operation on H ̸= ∅, then for x ∈ H and A, B ⊆ H we set
120
+ A + B :=
121
+
122
+ a∈A,b∈B
123
+ a + b,
124
+ A + x := A + {x} and x + A := {x} + A. If A or B is empty, then so is A + B.
125
+ A hypergroup can be defined as a non-empty set H with a multivalued operation + which is
126
+ associative (see Definition 2.3 (CH1) below) and reproductive on H (i.e., x+H = H +x = H for all
127
+ x ∈ H). This notion was first considered by F. Marty in [35, 36, 37]. The theory of hypergroups,
128
+ with a detailed historical overview, is presented in [38], where an extensive bibliography is also
129
+ provided.
130
+ Definition 2.1. A hyperoperation + on H is a multivalued operation such that x + y ̸= ∅ for all
131
+ x, y ∈ H.
132
+ Lemma 2.2 (Theorem 12 in [38]). If (H, +) is a hypergroup, then + is a hyperoperation on H.
133
+ Proof. Aiming for a contradiction, suppose that x + y = ∅ for some x, y ∈ H. Then
134
+ H = x + H = x + (y + H) = (x + y) + H = ∅ + H = ∅,
135
+
136
+ 4
137
+ LINZI, A.
138
+ which is excluded.
139
+
140
+ The following special class of hypergroups (cf. Remark 2.4 below) will be of interest for us.
141
+ Definition 2.3. A canonical hypergroup is a triple (H, +, 0), where H ̸= ∅, + is a multivalued
142
+ operation on H and 0 is an element of H such that the following axioms hold:
143
+ (CH1) + is associative, i.e., (x + y) + z = x + (y + z) for all x, y, z ∈ H,
144
+ (CH2) x + y = y + x for all x, y ∈ H,
145
+ (CH3) for every x ∈ H there exists a unique x′ ∈ H such that 0 ∈ x + x′ (the element x′ will be
146
+ denoted by −x),
147
+ (CH4) z ∈ x + y implies y ∈ z − x := z + (−x) for all x, y, z ∈ H.
148
+ Lemma 2.4. Let (H, +, 0) be a canonical hypergroup. Then the multivalued operation + is repro-
149
+ ductive on H. In particular, (H, +) is a hypergroup and + is a hyperoperation.
150
+ Proof. Fix a ∈ H. For all x ∈ H + a there exists y ∈ H such that x ∈ y + a ⊆ H. Therefore,
151
+ H + a ⊆ H. For the other inclusion, observe that for all x ∈ H we have that
152
+ x ∈ x + 0 ⊆ x + (a − a) = (x − a) + a,
153
+ so there exists y ∈ x − a ⊆ H such that x ∈ y + a ⊆ H + a. We have proved that H = H + a
154
+ for an arbitrary a ∈ H. Now the conclusions of the lemma follow from (CH2), the definition of
155
+ hypergroups and Lemma 2.2.
156
+
157
+ Let (H, +, 0) be a canonical hypergroup. Then 0 is called the neutral element for + in H because
158
+ of the following observation.
159
+ Lemma 2.5 (Section III, (b) in [39]). Let (H, +, 0) be a canonical hypergroup. Then x + 0 = {x}
160
+ for all x ∈ H.
161
+ Proof. Take x ∈ H. If y ∈ x + 0, then 0 ∈ y − x follows by (CH4). Now y = x follows from the
162
+ uniqueness required in (CH3).
163
+
164
+ Remark 2.6. Note that an abelian group (G, ∗, 0) is not a priori a canonical hypergroup, because the
165
+ operation on G is not a multivalued operation, as it takes values in G and not in P(G). However,
166
+ it can be turned into a canonical hypergroup (G, +, 0) by setting x + y := {x ∗ y} for all x, y ∈ G.
167
+ Conversely, if a canonical hypergroup (H, +, 0) satisfies that x + y is a singleton for all x, y ∈ H,
168
+ then one can define on H a standard operation which makes it an abelian group with identity 0. In
169
+ these cases, we sometimes abusively say that (G, ∗, 0) is a canonical hypergroup or that (H, +, 0)
170
+ is a group.
171
+ 2.1. Hyperrings and hyperfields. Let us now define the structures of main interest in this paper.
172
+ Definition 2.7. A (commutative) hyperring is a tuple (R, +, ·, 0) which satisfies the following
173
+ axioms:
174
+ (HR1) (R, +, 0) is a canonical hypergroup,
175
+ (HR2) (R, ·) is a (commutative) semigroup and 0 is an absorbing element, i.e., 0 · x = x · 0 = 0, for
176
+ all x ∈ R,
177
+ (HR3) the operation · is distributive with respect to +. That is, for all x, y, z ∈ R,
178
+ x(y + z) = xy + xz,
179
+
180
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
181
+ 5
182
+ where, for x ∈ R and A ⊆ R, we have set
183
+ xA := {xa | a ∈ A}.
184
+ If the operation · has a neutral element 1 ̸= 0, then we say that (R, +, ·, 0, 1) is a hyperring with unity.
185
+ If (R, +, ·, 0, 1) is a hyperring with unity and (R \ {0}, ·, 1) is an abelian group, then (R, +, ·, 0, 1) is
186
+ called a hyperfield. If R is a hyperring with unity, then we denote by R× the multiplicative group
187
+ of the units of R, i.e.,
188
+ R× := {x ∈ R | ∃y ∈ R : xy = 1}.
189
+ In particular, if R is a hyperfield, then R× = R \ {0}.
190
+ Since we will only consider the commutative case, in the following sections we will call commu-
191
+ tative hyperrings simply hyperrings.
192
+ Remark 2.8. In the literature, the name “hyperring” can be found for structures (R, +, ·) where +
193
+ is an operation and · is a multivalued operation or where both + and · are multivalued operations.
194
+ Some authors refer to the structures (R, +, ·, 0) where only the addition is multivalued as Krasner
195
+ hyperrings (some more historical remarks on this are given in in [17, Section 4]).
196
+ Structures (R, +, ·, 0) where + is an operation and · is a multivalued operation (satisfying similar
197
+ axioms) were introduced in [45] and called multiplicative hyperrings (in italian iperanelli moltiplica-
198
+ tivi).
199
+ Structures (R, +, ·, 0) with both + and · multivalued operations (satisfying similar axioms) are
200
+ instead called general hyperrings (see e.g. [9, Section 2]).
201
+ In this paper, the name “hyperring” will be used for Krasner hyperrings exclusively, as indicated
202
+ in the above definition.
203
+ In the literature, one may also find the term multiring referring to a structure (R, +, ·, 0), where
204
+ + is a multivalued operation and · is an operation, satisfying (HR1), (HR2) and the following
205
+ weaker version of (HR3):
206
+ (MR) x · (y + z) ⊆ x · y + x · z, for all x, y, z ∈ M.
207
+ Thus, in this case, the different name reflects a difference in the axioms rather than in the structure.
208
+ Similarly, multifields are defined as hyperfields satisfying (MR) instead of (HR3). Multirings and
209
+ multifields have been considered for instance in [12], where, among other facts, it is observed that
210
+ all multifields are hyperfields, while there are several meaningful examples of multirings that are
211
+ not hyperrings.
212
+ In this paper, we preferred to use the name “hyperfield” solely.
213
+ We say that a hyperring (resp. hyperfield) R is a ring (resp. field) if the additive canonical
214
+ hypergroup of R is a group (cf. Remark 2.6). The next observation gives a necessary condition for
215
+ a hyperring with unity to be a ring. This fact is an immediate corollary of a result already noted
216
+ in [42, page 369] (see also Example 2.45 below). We wish to state it for later reference and we take
217
+ the opportunity to write a quick proof .
218
+ Lemma 2.9 ([42]). A hyperring with unity R is a ring if and only if 1 − 1 = {0}.
219
+ Proof. If R is a field, then 1 − 1 = {0} holds trivially. Conversely, by axiom (HR3) 1 − 1 = {0}
220
+ implies x − x = {0} for all x ∈ R. Take a, b ∈ R and x, y ∈ a + b. We have that
221
+ x − y ⊆ (a + b) − (a + b) = (a − a) + (b − b) = {0}
222
+ In particular, 0 ∈ x − y and x = y follows fro (CH3). We have proved that a + b is a singleton for
223
+ all a, b ∈ R.
224
+
225
+
226
+ 6
227
+ LINZI, A.
228
+ Example 2.10. The K-hyperfield K is the set {0, 1} with the hyperoperation + which has 0 as its
229
+ neutral element and satisfies 1 + 1 = {0, 1}. The multiplication is the obvious one.
230
+ Remark 2.11. It is customary among some authors to call the above hyperfield the Krasner hy-
231
+ perfield. We prefer to avoid that terminology which, in view of Remark 2.8 above, might lead to
232
+ unnecessary confusion.
233
+ Example 2.12. The sign hyperfield S is the set {−1, 0, 1} with the hyperoperation + which has
234
+ 0 as its neutral element and satisfies x + x = {x} for x ∈ {−1, 1} and 1 − 1 = {−1, 0, 1}. The
235
+ multiplication is the obvious one.
236
+ Example 2.13. The weak sign hyperfield W is the set {−1, 0, 1} with the hyperoperation + which
237
+ has 0 as its neutral element and satisfies x + x = {x, −x} for x ∈ {−1, 1} and 1 − 1 = {−1, 0, 1}.
238
+ The multiplication is the obvious one.
239
+ Other examples of finite hyperfields are described e.g. in [1]. Let us now consider some infinite
240
+ examples.
241
+ Example 2.14. Let (Γ, +, <, 0) be an ordered abelian group and ∞ be a symbol such that γ+∞ =
242
+ ∞+γ = ∞ > γ for all γ ∈ Γ. For x, y ∈ Γ∪{∞} such that x ≤ y (i.e., x < y or x = y), let us denote
243
+ by [x, y] the set consisting of all z ∈ Γ ∪ {∞} such that x ≤ z ≤ y. Consider the hyperoperation ⊞
244
+ defined on T (Γ) := Γ ∪ {∞} as x ⊞ ∞ = ∞ ⊞ x = {x} for all x ∈ T (Γ) and:
245
+ x ⊞ y :=
246
+
247
+ {min{x, y}}
248
+ if x ̸= y,
249
+ [x, ∞]
250
+ if x = y.
251
+ (x, y ∈ T (Γ))
252
+ It is not difficult to check that (T (Γ), ⊞, +, ∞, 0) is a hyperfield. The hyperfield T (R, +, 0, >),
253
+ where > denotes the standard order of the real numbers, is known as the tropical hyperfield (see
254
+ [47, Section 5.3]). Therefore, we call the hyperfields of the form T (Γ) generalised tropical hyperfields.
255
+ Example 2.15. Let (Γ, +, <, 0) be an ordered abelian group and ∞ be a symbol such that γ+∞ =
256
+ ∞ + γ = ∞ > γ for all γ ∈ Γ. For x, y ∈ Γ ∪ {∞} such that x < y, let us denote by (x, y] the set
257
+ consisting of all z ∈ Γ∪{∞} such that x < z ≤ y. Consider the hyperoperation ⊞′ defined on T (Γ)
258
+ as x + ∞ = ∞ + x = {x} for all x ∈ T (Γ) and:
259
+ x ⊞′ y :=
260
+
261
+ {min{x, y}}
262
+ if x ̸= y,
263
+ (x, ∞]
264
+ if x = y.
265
+ (x, y ∈ T (Γ))
266
+ It is not difficult to check that (T (Γ), ⊞′, +, ∞, 0) is a hyperfield. We will denote it by T ′(Γ) and
267
+ call it the strict generalised tropical hyperfield.
268
+ The above examples are (up to isomorphism) all instances of a general construction which yields
269
+ a hyperring from a ring and a subgroup of its multiplicative semigroup (see Proposition 2.17 below).
270
+ This construction was first described by Krasner in [25]. We briefly recall it in the following example.
271
+ Example 2.16. Let A be a commutative ring and T a subgroup of the commutative semigroup
272
+ (A, ·). Let AT denote the set of all multiplicative cosets xT for x ∈ A, in particular 0T = {0}.
273
+ Krasner showed that by setting
274
+ xT + yT := {(x + yt)T | t ∈ T }
275
+ and
276
+ xT · yT := xyT
277
+
278
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
279
+ 7
280
+ we have that (AT , +, ·, 0T ) is a hyperring where −(xT ) = (−x)T for all x ∈ A. In addition, if A
281
+ is a field, then this construction always yields a hyperfield. This is called the factor (or quotient)
282
+ hyperring (resp. hyperfield) of A modulo T .
283
+ For the next proposition we employ the notion of isomorphism of hyperfields (see Definition 2.33
284
+ below).
285
+ Proposition 2.17. The following assertions hold:
286
+ (i) For any field K with more than two elements, we have that K is isomorphic to KK×.
287
+ (ii) For any ordered field K with positive cone P, we have that S is isomorphic to KP .
288
+ (iii) If p > 3 is a prime such that p ≡ 3 mod 4, then W is isomorphic to (Fp)(F×
289
+ p )2.
290
+ (iv) Let Γ be an ordered abelian group. For any field k with more than two elements, let K :=
291
+ k((tΓ)) denote the Hahn series field and let v be its canonical t-adic valuation. Then KO×
292
+ v
293
+ is isomorphic to T (Γ).
294
+ (v) Let Γ be an ordered abelian group. Let K := F2((tΓ)) denote the Hahn series field and let
295
+ v be its canonical t-adic valuation. Then KO×
296
+ v is isomorphic to T ′(Γ).
297
+ Proof. Let K be a field with more than two elements. Then KK× contains precisely two elements,
298
+ the coset 0K× and the coset 1K×, the latter containing all non-zero elements of K. If x ∈ K×, then
299
+ −x ∈ K× as well and thus 0K× belongs to 1K× + 1K×. Since by assumption there are x, y ∈ K×
300
+ with x ̸= y, also (x − y)K× ̸= 0K× is an element of 1K× + 1K×. This shows (i).
301
+ Let K be an ordered field with positive cone P. Here, this means that P is a subset of K× such
302
+ that P + P, P · P ⊆ P and K× is the disjoint union of P and −P. It follows that KP contains
303
+ precisely three1 elements: 0P, 1P (containing all the elements of P) and −1P (containing all the
304
+ elements of −P). Since P + P ⊆ P, we have that 1P + 1P only contains 1P. By definition, if
305
+ x ∈ P, then −x ∈ −P. Thus, 1P − 1P contains 0P. It does also contain 1P and −1P since e.g.
306
+ 1 + 1 ∈ P (because 1 ∈ P) and (1 + 1) − 1 = 1 ∈ P while 1 − (1 + 1) = −1 ∈ −P. This shows (ii).
307
+ Let K be Fp (the finite field with p elements) for some prime number p > 3 such that p ≡ 3
308
+ mod 4. The prime number p is certainly odd and thus the cardinality of (K×)2 is p−1
309
+ 2 . If −1 is
310
+ a square in K, then K× contains an element of order 4 but this is excluded by the assumption
311
+ p ≡ 3 mod 4. We have that x ∈ K× is a square if and only if −x is not a square. It follows that
312
+ K(K×)2 contains precisely three elements: 0(K×)2, 1(K×)2 (containing all the non-zero squares)
313
+ and −1(K×)2 (containing all the non-squares). Moreover, 1(K×)2 − 1(K×)2 contains 0(K×)2.
314
+ This shows that −1(K×)2 is the hyperinverse of 1(K×)2 in K(K×)2. Since 1 + 1 ̸= 0 in K and
315
+ 1 = (1 + 1) − 1, we have that either 1(K×)2 or −1(K×)2 (depending on which among 1 + 1 and
316
+ −(1 + 1) is a square) belongs to 1(K×)2 − 1(K×)2. By the symmetry of 1(K×)2 − 1(K×)2 under
317
+ the application of − (i.e., multiplication by −1(K×)2), we obtain that both 1(K×)2 and −1(K×)2
318
+ must be elements of 1(K×)2 −1(K×)2. In a finite field, any non-square is the sum of two (non-zero)
319
+ squares. Thus, −1(K×)2 is an element of 1(K×)2 + 1(K×)2. In order to show that 1(K×)2 is an
320
+ element of 1(K×)2+1(K×)2 as well, we distinguish two cases. If 1+1 is a square in K, then 1(K×)2
321
+ is in 1(K×)2 + 1(K×)2 trivially. Otherwise, 1 + 1 is not a square. In this case, either 1 + 1 + 1 is a
322
+ square, in which case 1 = (1+1+1)−(1+1) shows that 1(K×)2 is an element of 1(K×)2 +1(K×)2,
323
+ or −(1 + 1 + 1) = −((1 + 1) + 1)) is a square, in which case −(1 + 1) = −(1 + 1 + 1) + 1 implies
324
+ that 1(K×)2 is an element of 1(K×)2 + 1(K×)2. This proves (iii).
325
+ 1In the literature, sometimes positive cones are defined as “non-negative cones”, i.e., containing 0. Notice that for
326
+ our statement to hold it is necessary to exclude 0 from positive cones.
327
+
328
+ 8
329
+ LINZI, A.
330
+ Let k be a field with more thatn 2 elements and consider the Hahn series field K = k((tΓ))
331
+ equipped with the t-adic valuation v. A non-zero element x of K× is represented as a formal series
332
+ x =
333
+
334
+ γ∈Γ
335
+ aγtγ
336
+ with well-ordered support, i.e., {γ ∈ Γ | aγ ̸= 0} is a well-ordered set with respect to the order of
337
+ Γ. The t-adic value of x under v is defined to be the minimum of the support of x (cf. [10, Exercise
338
+ 3.5.6]).
339
+ The value group of v is thus Γ and it follows from general valuation theory that K×/O×
340
+ v is
341
+ isomorphic to Γ as an ordered abelian group (cf. [10, Section 2.1] or Lemma 3.20 below).
342
+ For
343
+ x ∈ K×, we have that xO×
344
+ v contains all the elements of K with the same value of x under v. If
345
+ x, y ∈ K are such that vx ̸= vy (i.e., xO×
346
+ v ̸= yO×
347
+ v ), then
348
+ v(x + yt) = v(x + y) = min{vx, vy}
349
+ for any t ∈ O×
350
+ v (i.e., any t with vt = 0). This follows from [10, Section 1.3 (1.3.4)] (or Corollary 3.5
351
+ (iv) below). We now note that if x ∈ K has any positive value under v, then v(x − 1) = 0 and thus
352
+ vx = v(1 + (x − 1)) is the value of some element of K generating a coset in KO×
353
+ v which belongs to
354
+ 1O×
355
+ v +1O×
356
+ v . Moreover, by the assumption on the cardinality of k, there exists a ∈ k \{1}, so by the
357
+ definition of the t-adic valuation, we have that va = 0 and v(1 − a) = 0. Since v(1) = v(−1) = 0,
358
+ we conclude that the image of 1O×
359
+ v − aO×
360
+ v = 1O×
361
+ v + 1O×
362
+ v under v is [0, ∞]. By distributivity,
363
+ this suffices to show that T (Γ) ≃ KO×
364
+ v (see also Lemma 3.3 below). The proof of (iv) is now also
365
+ complete.
366
+ The proof for (iv) can easily be adapted to prove (v).
367
+
368
+ Remark 2.18. Let us mention at this point that not all hyperfields are factor hyperfields. This fact
369
+ was proved by Massouros in [39] who then further improved his results in [40]. According to [21,
370
+ Theorem 6] finite hyperfields such that 1 /∈ 1 + 1 and with the property that 0 does not belong to
371
+ any finite sum of 1 with itself are also not quotient hyperfields.
372
+ 2.2. Subhyperrings. To choose a good notion of subhyperring is not as straightforward as it might
373
+ seem. Two options are presented in the next definition.
374
+ Definition 2.19. Let (R, +, ·, 0) be a hyperring. A subset S of R is a relational subhyperring of R
375
+ if it is multiplicatively closed and with the induced multivalued operation:
376
+ x +S y := (x + y) ∩ S
377
+ (x, y ∈ S)
378
+ we have that (S, +S, ·, 0) is a hyperring as well.
379
+ A subset S of R is a (traditional) subhyperring of R if 0 ∈ S and for all x, y ∈ S we have that
380
+ x − y ⊆ S and xy ∈ S.
381
+ If R is a hyperring with unity 1, then we say that a relational subhyperring S of R is a relational
382
+ subhyperfield of R if 1 ∈ S and (S, +S, ·, 0, 1) is a hyperfield. In addition, a subhyperring S of R is
383
+ called a (traditional) subhyperfield if 1 ∈ S and (S \ {0}, ·, 1) is an abelian group.
384
+ Remark 2.20. The adjective “relational” has been chosen for the following reason. A multivalued
385
+ operation can be encoded in a first-order language via the ternary relation z ∈ x + y (as noticed
386
+ e.g. in [32]). If we consider a first-order language L with a constant symbol for 0, a binary function
387
+ symbol for · and a ternary relation symbol for +, then a hyperring naturally becomes a structure on
388
+ L, which is a model of all the axioms of Definitions 2.7 and 2.3 (written as sentences in L). Under
389
+ this interpretation, the relational subhyperrings of a hyperring R are precisely the submodels of R,
390
+
391
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
392
+ 9
393
+ i.e., the substructures of R (see [43, Section 2.3]) which satisfy the axioms. For more details on the
394
+ model theoretical point of view let us refer the reader to [33].
395
+ Our choices regarding terminology are also motivated by historical reasons. In fact, subhyper-
396
+ groups (and consequently subhyperfields) have been defined long time ago (see e.g. Definition 2 and
397
+ the subsequent remark in [22]2).
398
+ Remark 2.21. It is clear that a subhyperring S of a hyperring (R, +, ·, 0) is a relational subhyperring
399
+ of R and that +S = + in this case.
400
+ On the other hand, not all relational subhyperrings are
401
+ subhyperrings, as the following example shows.
402
+ Example 2.22. Consider an ordered abelian group (Γ, +, 0, <). Let R := T (Γ) be the hyperfield
403
+ obtained as in Example 2.14 and consider the subset S := {∞, 0} of R. It is straightforward to check
404
+ that S is a relational subhyperfield of R. However, S is not a subhyperfield of R since 0⊞0 = [0, ∞]
405
+ is not a subset of S.
406
+ Remark 2.23. Let Γ be an ordered abelian group. It is not difficult to see that the only subhyperfield
407
+ of T (Γ) is T (Γ) itself. On the other hand, if ∆ is a subgroup of Γ, then T (∆) is a relational
408
+ subhyperfield of T (Γ) (Example 2.22 above corresponds to the case ∆ = {0}).
409
+ 2.3. Homomorphisms. Next, let us consider a notion of homomorphism for hyperrings and hy-
410
+ perfields.
411
+ Definition 2.24. Let (R, +R, ·R, 0R) and (S, +S, ·S, 0S) be hyperrings. We say that a map σ :
412
+ R → S is a homomorphism of hyperrings if
413
+ (HH1) σ(0R) = 0S,
414
+ (HH2) σ(x ·R y) = σ(x) ·S σ(y), for all x, y ∈ R,
415
+ (HH3) σ(x +R y) ⊆ σ(x) +S σ(y), for all x, y ∈ R.
416
+ If (R, +R, ·R, 0R, 1R) and (S, +S, ·S, 0S, 1S) are hyperfields, then we say that a homomorphism of
417
+ hyperrings σ : R → S is a homomorphism of hyperfields when the following properties
418
+ (HH4) σ(1R) = 1S,
419
+ (HH5) σ(x−1) = σ(x)−1, for all x ∈ R \ {0},
420
+ hold as well.
421
+ Example 2.25. Let R be a hyperring. The map defined by σ(0) := 0 and σ(x) := 1 for x ∈ R\{0}
422
+ is a homomorphism of hyperrings R → K.
423
+ Example 2.26. Let (Γ, <, +, 0Γ) be an ordered abelian group. The map
424
+ ι : K → T (Γ)
425
+ 0K �→ ∞
426
+ 1K �→ 0Γ
427
+ is a homomorphism of hyperfields.
428
+ Example 2.27. Let K be a field and T a subgroup of K×. The function K → KT mapping x ∈ K
429
+ to [x]T ∈ KT is a homomorphism of hyperfields.
430
+ Remark 2.28. Let σ : R → S be a homomorphism of hyperrings. Then the kernel of σ
431
+ ker σ := {x ∈ R | σ(x) = 0S}
432
+ 2This article is included in Krasner’s collected works [4, pages 280–406]
433
+
434
+ 10
435
+ LINZI, A.
436
+ is a subhyperring of R. Indeed, for x, y ∈ ker σ, we have that
437
+ σ(x − y) ⊆ σ(x) − σ(y) = 0S − 0S = {0S}.
438
+ Thus, σ(z) = 0S for all z ∈ x − y, i.e., x − y ⊆ ker σ.
439
+ Remark 2.29. Let (R, +, ·, 0, 1) be a hyperring and S ⊆ R. Consider the inclusion map ι : S ֒→ R.
440
+ It is straightforward to verify that if S is a relational subhyperring of R, then ι is a homomoprhism
441
+ of hyperrings.
442
+ On the other hand, if (R, +R, ·R, 0R) and (S, +S, ·S, 0S) are hyperrings with S ⊆ R and the
443
+ inclusion ι : S ֒→ R is a homomorphism of hyperrings, then one cannot conclude that S is a
444
+ relational subhyperring of R, as the following example shows.
445
+ Example 2.30. The identity S → W is a homomorphism of hyperrings. However, S is not a
446
+ subhyperring of W as the hypersum of 1 with itself in S is {1}, while the hypersum of 1 with itself
447
+ in W intersected with S is {−1, 1}.
448
+ The above example motivates the following definition.
449
+ Definition 2.31. Let (R, +R, ·R, 0R) and (S, +S, ·S, 0S) be hyperrings (resp. hyperfields with uni-
450
+ ties 1R and 1S). An injective homomormphism of hyperrings (resp. hyperfields) σ : R → S is called
451
+ an embedding of hyperrings (resp. hyperfields) if
452
+ (EM1) σ(x +R y) = (σ(x) +S σ(y)) ∩ Im σ, for all x, y ∈ R.
453
+ We leave the straightforward proof of following lemma to the reader.
454
+ Lemma 2.32. If S is a relational subhyperring of a hyperring R, then the inclusion map S ֒→ R is
455
+ an embedding of hyperrings. Conversely, If R and S are hyperrings with S ⊆ R and the inclusion
456
+ map S ֒→ R is an embedding of hyperrings, then S is a relational subhyperring of R.
457
+ Definition 2.33. A homomorphism of hyperrings (resp. hyperfields) σ : R → S is called an
458
+ isomorphism of hyperrings (resp. hyperfields) if it is a bijective map and σ−1 : S → R is also a
459
+ homomorphism of hyperrings (resp. hyperfields). We say that two hyperrings (resp. hyperfields) R
460
+ and S are isomorphic and we write R ≃ S if there exists an isomorphism σ : R → S.
461
+ Lemma 2.34. Let R and S be hyperrings (resp. hyperfields) and denote by +R and +S their
462
+ hyperoperations, respectively. A map σ : R → S is an isomorphism of hyperrings (resp. hyperfields)
463
+ if and only if σ is a surjective embedding of hyperrings (resp. hyperfields).
464
+ Proof. If σ is an isomorphism, then it is bijective by definition, in particular Im σ = S. Since σ is
465
+ a homomorphism, we have that σ(x +R y) ⊆ σ(x) +S σ(y) holds. On the other hand, since σ−1
466
+ satisfies (HH3) as well, we obtain that
467
+ σ−1(σ(x) +S σ(y)) ⊆ x +R y.
468
+ Applying σ to both sides then yields σ(x +R y) ⊇ σ(x) +S σ(y).
469
+ Assume now that σ is a surjective embedding and let us show that σ−1 is a homomorphism.
470
+ Clearly, (HH1) and (HH2) hold for σ−1.
471
+ Since by (EM1) and the surjectivity we have that
472
+ σ(x +R y) = σ(x) +S σ(y), property (HH3) follows applying σ−1 to both sides and using again
473
+ the surjectivity of σ.
474
+
475
+ Remark 2.35. Note that a bijective homomorphism is not necessarily an isomorphism as it can be
476
+ deduced from Example 2.30.
477
+
478
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
479
+ 11
480
+ Example 2.36. Let (Γ, <, +, 0Γ) be an ordered abelian group. Consider the relational subhyper-
481
+ field S := {∞, 0} of T (Γ) as in Example 2.22. Then (S, +S, ·, ∞, 0Γ) is isomorphic to K.
482
+ As usual we identify isomorphic structures: Examples 2.22 and 2.36 can be expressed by saying
483
+ that K is a subhyperfield of T (Γ) for any ordered abelian group Γ.
484
+ 2.4. Hyperideals. We now briefly discuss how the classical ring theory notion of ideal generalises
485
+ in the multivalued setting.
486
+ Definition 2.37 (Definition 2.1 in [5] and Definition 2.9 in [31]). Let R be a hyperring. A subhy-
487
+ perring I of R is a hyperideal of R if it satisfies
488
+ (HID1) xy ∈ I, for all x ∈ R and y ∈ I.
489
+ Lemma 2.38. Let R be a hyperring with no zero divisors, i.e., xy = 0R implies x = 0R or y = 0R.
490
+ A relational subhyperring I of R satisfying (HID1) is a hyperideal of R.
491
+ Proof. Pick x, y ∈ I and let z ∈ x − y. It suffices to prove that z ∈ I. If x = 0R or y = 0R,
492
+ then there is nothing to show. Otherwise, by distributivity in R we have that zy ∈ xy − y2 and by
493
+ (HID1) we obtain that zy ∈ I, since y ∈ I. Therefore, zy ∈ (xy − y2) ∩ I. Now, distributivity in I
494
+ (with respect to the induced multivalued operation +I) yields zy ∈
495
+
496
+ (x − y) ∩ I
497
+
498
+ y, so there exists
499
+ z′ ∈ (x − y) ∩ I such that zy = z′y. Hence, 0R ∈ zy − z′y = (z − z′)y. Since R has no zero divisors,
500
+ this implies that 0R ∈ z − z′ and hence z = z′ ∈ I.
501
+
502
+ Remark 2.39. We wish to justify the choice of requiring hyperideals to be traditional subhyperrings.
503
+ In [19, Section 3.1], Jun provides a generalisation of the classical quotient construction of a ring
504
+ modulo an ideal in the multivalued setting. One property that is certainly desirable to preserve is for
505
+ the canonical epimorphism from a hyperring R to the quotient hyperring R/I modulo a hyperideal
506
+ I to be a homomorphism of hyperrings R → R/I having I as its kernel. We have no examples
507
+ of hyperrings with a relational subhyperring satisfying (HID1), but at the same time we did not
508
+ succeed in proving Lemma 2.38 without the assumption on zero divisors. If such an example would
509
+ exist, then that relational subhyperring could not be the kernel of a homomorphism of hyperrings
510
+ by Remark 2.28 and so we should not call it a hyperideal.
511
+ Remark 2.40. Let σ : R → S be a homomorphism of hyperrings. Then it is easy to show that ker σ
512
+ is a hyperideal of R. Conversely, for all hyperideals I of a hyperring R the map σ : R → K defined
513
+ as
514
+ σ(x) :=
515
+
516
+ 0
517
+ if x ∈ I,
518
+ 1
519
+ otherwise.
520
+ is a homomorphism of hyperrings such that ker σ = I.
521
+ The following statements can be shown to hold as in the classical theory of rings.
522
+ Lemma 2.41 (Lemma 2.12 in [31]). If a hyperideal I of a hyperring R contains a unit, then I = R.
523
+ Corollary 2.42 (Corollary 2.13 in [31]). The only hyperideals of a hyperfield F are {0} and F.
524
+ Definition 2.43 (Definition 2.14 (ii) in [31]). Let I be a hyperideal of a hyperring R. Then I is
525
+ called maximal if I ⊊ R and for all hyperideals J of R we have that I ⊊ J implies J = R.
526
+ We recall without proof the following result.
527
+ Proposition 2.44 (Proposition 2.16 (ii) in [31]). Assume that R is a hyperring with unity. Let I
528
+ be a hyperideal R. Then I is maximal if and only if R/I is a hyperfield.
529
+
530
+ 12
531
+ LINZI, A.
532
+ Example 2.45 ([42]). Let R be a hyperring with unity. An element s of R is called a scalar if
533
+ x + s is a singleton for all x ∈ R. As in the proof of Lemma 2.9, it is not difficult to show that
534
+ s ∈ R is a scalar if and only if s − s = {0}. Interestingly, the set of all scalar elements of R forms a
535
+ hyperideal of R.
536
+ Assume now that R is a hyperring with unity. If 1 is a scalar, then by Lemma 2.41 the hyperideal
537
+ of scalars is R and thus R is a ring (as all of its elements are scalars). Moreover, we deduce from
538
+ Corollary 2.42 that if F is a hyperfield, but not a field, then the only scalar of F is 0, i.e., x − x is
539
+ not a singleton for all x ∈ F ×.
540
+ 3. Valued hyperfields
541
+ Some of the results presented in this section can be found in [31].
542
+ Some proofs have been
543
+ improved, others we will repeat for the convenience of the reader.
544
+ 3.1. Valuations. The next definition is a straightforward generalisation of the definition of valua-
545
+ tion for fields.
546
+ Definition 3.1 (Definition 4.1 in [31]). Take a hyperfield F and an ordered abelian group Γ (written
547
+ additively). A surjective map v : F → Γ ∪ {∞} is called a valuation on F if it has the following
548
+ properties:
549
+ (V1) vx = ∞ ⇐⇒ x = 0, for all x ∈ F,
550
+ (V2) v(xy) = vx + vy, for all x, y ∈ F,
551
+ (V3) z ∈ x + y =⇒ vz ≥ min{vx, vy}, for all x, y, z ∈ F.
552
+ If v is a valuation on a hyperfield F we call (F, v) a valued hyperfield and we denote by vF the
553
+ ordered abelian group v(F ×), i.e., the value group of (F, v).
554
+ Example 3.2. Let F be a hyperfield. The trivial valuation vx = 0 for all x ∈ F × is always a
555
+ valuation on F with value group {0}.
556
+ The next result provides further examples of valued hyperfields.
557
+ Lemma 3.3 (Corollary 4.2 in [31]). Let (K, v) be a valued field and take a subgroup T of K×.
558
+ Denote by O×
559
+ v the group of units of the valuation ring of K, i.e., O×
560
+ v := {x ∈ K | vx = 0}. If
561
+ T ⊆ O×
562
+ v , then
563
+ vT : KT → vK ∪ {∞}
564
+ xT �→ vx
565
+ is a valuation on KT.
566
+ Proof. The assumption T ⊆ O×
567
+ v guarantees that the map vT is well-defined. Moreover, surjectivity,
568
+ (V1) and (V2) follow immediately from the corresponding properties of v and the definition of the
569
+ factor hyperfield KT . It remains to verify (V3). Take x, y ∈ K and t ∈ T . We obtain that
570
+ v(x + yt) ≥ min{vx, v(yt)} = min{vx, vy + vt} = min{vx, vy}.
571
+ From this it easily follows that (V3) holds for (KT , vT ).
572
+
573
+ The next lemma gives an alternative definition of valuation on (hyper)fields (cf. [2, Example
574
+ 1.8]).
575
+ Lemma 3.4. Let F be a hyperfield and Γ an ordered abelian group. Then (F, v) is a valued hyperfield
576
+ with vF = Γ if and only if the map v : F → T (Γ) is a surjective homomorphism of hyperfields.
577
+
578
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
579
+ 13
580
+ Proof. Assume that (F, v) is a valued hyperfield. Then v : F → T (vF) is a surjective map, (HH1)
581
+ follows from (V1), (HH2) follows from (V2), (HH3) follows from (V3). Property (HH4) follows
582
+ because vx = vx + v(1) for all x ∈ F and (HH5) follows since, for x ∈ F ×, we have that
583
+ 0 = v(1) = v(xx−1) = vx + v(x−1).
584
+ Thus, v is a homomorphism of hyperfields.
585
+ Conversely, let F be a hyperfield and v : F → T (Γ) a surjective homomoprhism of hyperfields.
586
+ Property (V2) for v follows from (HH2) and (V3) follows from (HH3) together with the definition
587
+ of the hyperoperation of T (Γ). Property (V1) follows from Corollary 2.42 since v(1) = 0 ̸= ∞ and
588
+ thus ker v = {x ∈ F | vx = ∞} is a proper hyperideal of F. We proved that v is a valuation on F
589
+ with vF = Γ.
590
+
591
+ On the basis of the previous lemma, the following result is almost immediate to verify. Thus,
592
+ we omit its proof.
593
+ Corollary 3.5 (Lemma 4.5 in [31]). Let v : F → Γ ∪ {∞} be a valuation on a hyperfield F. Then:
594
+ (i) v(1) = v(−1) = 0,
595
+ (ii) v(−x) = vx for all x ∈ F,
596
+ (iii) vx−1 = −vx for all x ∈ F ×,
597
+ (iv) if vx ̸= vy, then vz = min{vx, vy}, for every x, y ∈ F and z ∈ x + y.
598
+ Lemma 3.6. Let (KT , w) be a valued hyperfield which is a factor hyperfield. Then there exists a
599
+ valuation v on K such that T ⊆ O×
600
+ v and w = vT .
601
+ Proof. Let Γ be the value group of (KT , w). Since K → KT , x �→ [x]T and w : KT → T (Γ) are
602
+ surjective homomorphisms of hyperfields, their composition v : K → T (Γ) is a valuation on K
603
+ satisfying the conditions of the statement.
604
+
605
+ Example 3.7. By Proposition 2.17 (iv) we have that T (Γ) is isomorphic to all factor hyperfields of
606
+ the form k((tΓ))O×
607
+ vt for some field k. By the above lemmas, the isomorphism σ : k((tΓ))O×
608
+ vt → T (Γ)
609
+ is a valuation on k((tΓ))O×
610
+ vt and the identity map id = σ ◦ σ−1 : T (Γ) → T (Γ) is a valuation on
611
+ T (Γ).
612
+ 3.2. Valuation hyperrings. The next definition is inspired by classical valuation theory.
613
+ Definition 3.8 (Definition 4.6 in [31]). Let F be a hyperfield. A relational subhyperring O of F
614
+ is called a valuation hyperring in F if for all x ∈ F × we have that either x ∈ O or x−1 ∈ O.
615
+ Observe that, by definition, it follows that 1 ∈ O for any valuation hyperring O in F. Let us
616
+ now prove some more basic properties of valuation hyperrings.
617
+ Lemma 3.9 (Proposition 4.7 in [31]). A valuation hyperring O in a hyperfield F is a subhyperring
618
+ of F.
619
+ Proof. It suffices to show that a − b ⊆ O for all a, b ∈ O. Take a, b ∈ O and x ∈ a − b. If x ∈ O,
620
+ then there is nothing to show (note that this case also includes x = 0). Otherwise, we have that
621
+ x−1 ∈ O and thus ax−1, bx−1 ∈ O. Since x ∈ a − b we obtain from (CH4) that a ∈ x + b, so, using
622
+ axiom (HR3),
623
+ ax−1 ∈ (x + b)x−1 = 1 + bx−1.
624
+
625
+ 14
626
+ LINZI, A.
627
+ We have obtained that ax−1 ∈ (1 + bx−1) ∩ O = 1 +O bx−1. By (CH4) and (HR3) applied to the
628
+ hyperring (O, +O, ·, 0), it follows that
629
+ xx−1 = 1 ∈ ax−1 +O (−bx−1) = (a +O (−b))x−1.
630
+ Therefore, x ∈ a +O (−b) ⊆ O. This shows that a − b ⊆ O.
631
+
632
+ Lemma 3.10 (Lemma 4.8 in [31]). Let O be a valuation hyperring in a hyperfield F.
633
+ Then
634
+ M := O \ O× is the unique maximal hyperideal of O.
635
+ Proof. Take a ∈ M and c ∈ O. If ca is invertible in O, then there exists x ∈ O such that x(ca) = 1.
636
+ Hence (xc)a = 1 and a−1 = xc ∈ O contradicting a ∈ M. This proves that ca ∈ M and shows that
637
+ M satisfies (HID1).
638
+ Take a, b ∈ M. We may assume that ab−1 ∈ O (otherwise ba−1 ∈ O and we can interchange the
639
+ roles of a and b). Since O is a subhyperring of F (cf. Lemma 3.9), we obtain that 1 − ab−1 ⊆ O
640
+ and therefore, using what we have just proved, we conclude that
641
+ b − a = b(1 − ab−1) ⊆ M.
642
+ We have shown that M is a hyperideal of O.
643
+ Since, by the definition of M, we have that O\M = O×, by Lemma 2.41, every proper hyperideal
644
+ of O must be contained in M, showing that M is the unique maximal hyperideal of O.
645
+
646
+ 3.3. Residue hyperfield. Any valuation on a hyperfield F induces a valuation hyperring in F.
647
+ Proposition 3.11 (Proposition 4.11 in [31]). Let v : F → Γ ∪ {∞} be a valuation on a hyperfield
648
+ F. Then
649
+ Ov := {x ∈ F | vx ≥ 0}
650
+ is a valuation hyperring in F and
651
+ Mv := {x ∈ F | vx > 0}
652
+ is its unique maximal hyperideal.
653
+ Proof. We first prove that Ov is a subhyperring of F. Take a, b ∈ Ov. By (V3), for all c ∈ a − b we
654
+ have vc ≥ min{va, v(−b)} = min{va, vb} ≥ 0, so a − b ⊆ Ov. Further, we have ab ∈ Ov by (V2).
655
+ By Corollary 3.5 (iii) we conclude that if x /∈ Ov, then x−1 ∈ Ov so Ov is a valuation hyperring in
656
+ F.
657
+ Next we show that Mv is the unique maximal hyperideal of Ov. Observe that, by virtue of
658
+ Corollary 3.5 (iii),
659
+
660
+ v = {x ∈ Ov | vx = 0}.
661
+ Hence, Mv = Ov \ O×
662
+ v and then Mv is the unique maximal hyperideal of Ov by Lemma 3.10.
663
+
664
+ Remark 3.12. Let (F, v) be a valued hyperfield. Note that Ov can be described in terms of the
665
+ hyperoperation ⊞ of T (vF) as the preimage in F of v(1) ⊞ v(1) under v:
666
+ Ov = v−1(v(1) ⊞ v(1)).
667
+ It follows from Proposition 2.44 that, for a valuation hyperring O with maximal hyperideal M,
668
+ the quotient hyperring
669
+ O/M = {x + M | x ∈ O},
670
+ with the multivalued operation ⊕ defined as
671
+ (x + M) ⊕ (y + M) := {z + M | z ∈ x + y},
672
+
673
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
674
+ 15
675
+ is a hyperfield (see [19, Section 3] and Remark 2.39 above).
676
+ Definition 3.13. If (F, v) is a valued hyperfield, then we call the hyperfield Ov/Mv the residue
677
+ hyperfield of (F, v) and we denote it by Fv. For an element x ∈ Ov, we denote by xv its natural
678
+ image in Fv.
679
+ Proposition 3.14. Let (K, v) be a valued field and T ⊆ O×
680
+ v a subgroup of K×. Then KT vT ≃
681
+ (Kv)(T v), where T v := {tv | t ∈ T }.
682
+ Proof. By definition vT [x]T = vx for all x ∈ K, thus OvT = (Ov)T and MvT = (Mv)T . It follows
683
+ that
684
+ KT vT = OvT /MvT = (Ov)T /(Mv)T .
685
+ Let us define
686
+ σ : (Ov)T /(Mv)T → (Kv)(T v)
687
+ [x]T + (Mv)T �→ [xv]T v
688
+ and show that σ is an isomorphism of hyperfields. By [19, Lemma 3.3] we have that [x]T +(Mv)T =
689
+ [y]T + (Mv)T if and only if v(x − yt) > 0 for some t ∈ T . This implies that
690
+ 0 = (x − yt)v = xv − yv · tv ∈ [xv]T v − [yv]T v ,
691
+ where we have used the assumption T ⊆ O×
692
+ v . This shows that σ is well-defined and injective. The
693
+ surjectivity of σ is clear. Moreover, σ is easily seen to be a homomorphism of the corresponding
694
+ multiplicative groups. Finally, we observe that
695
+ σ
696
+
697
+ [x]T + (Mv)T ⊕ ([y]T + (Mv)T )
698
+
699
+ = {σ
700
+
701
+ [z]T + (Mv)T
702
+
703
+ | [z]T ∈ [x]T + [y]T }
704
+ = {[zv]T v | z = x + yt for some t ∈ T }
705
+ = {[xv + yv · tv]T v | t ∈ T }
706
+ = [xv]T v + [yv]T v
707
+ hence σ is an isomorphism of hyperfields by Lemma 2.34.
708
+
709
+ Example 3.15. By Proposition 2.17, T (Γ) ≃ KO×
710
+ v , where K = k((tΓ)) for some field k with more
711
+ than two elements and v is its canonical t-adic valuation. Since Kv = k and O×
712
+ v v = k×, by the
713
+ above proposition and Proposition 2.17 (i) we have that the residue field of T (Γ) with respect to
714
+ its valuation given by the identity map is isomorphic to kk× ≃ K which, moreover, is a relational
715
+ subhyperfield of T (Γ) (cf. Examples 2.22 and 2.36).
716
+ For a valued field (K, v) we denote by 1 + Mv the set of 1-units. That is, those x ∈ K such that
717
+ v(x − 1) > 0 or equivalently, xv = 1v.
718
+ Proposition 3.16. Let (K, v) be a valued field and T ⊆ O×
719
+ v a subgroup of K×. Then the map
720
+ ι : KT vT → KT
721
+ [xv]T v �→ [x]T
722
+ is an embedding of hyperfields if and only if 1 + Mv ⊆ T .
723
+ Proof. If 1 + Mv ⊆ T and x, y ∈ O×
724
+ v , then [xv]T v = [yv]T v if and only if xv = yv · tv = (yt)v for
725
+ some t ∈ T if and only if v(1 − ytx−1) = v(x − yt) > 0 if and only if ytx−1 ∈ 1 + Mv ⊆ T . It
726
+ follows that [y]T [x]−1
727
+ T
728
+ = [ytx−1]T = [1]T and ι is well-defined. On the other hand if [x]T = [y]T for
729
+
730
+ 16
731
+ LINZI, A.
732
+ some x, y ∈ O×
733
+ v , then for some t ∈ T we have that x = yt and thus xv = (yt)v = yv · tv. It follows
734
+ that ι is injective. Moreover, we have that
735
+ ι
736
+
737
+ [xv]T v · [yv]−1
738
+ T v
739
+
740
+ = ι
741
+
742
+ [xv · y−1v]T v
743
+
744
+ = [xy−1]T = [x]T [y]−1
745
+ T
746
+ and hence ι satisfies (HH2) and (HH5). It clearly satisfies (HH1) and (HH4). It remains to show
747
+ that (EM1) holds. First observe that Im ι consists of classes [x]T where x = 0 or x ∈ R ⊆ O×
748
+ v ,
749
+ where R is a set of some selected representatives for Kv. Take again x, y ∈ O×
750
+ v . We have that
751
+ ι ([xv]T v + [yv]T v) = {[x + yt]T | x, y ∈ R, t ∈ T ∩ R} = ([x]T + [y]T ) ∩ Im ι.
752
+ This completes the proof of one implication.
753
+ If a ∈ 1 + Mv is not in T , then av = 1v and thus [av]T v = [1v]T v holds. On the other hand, we
754
+ have that
755
+ ι[av] = [a]T ̸= [1]T = ι[1v]T
756
+ and thus ι is not well-defined and in particular not an embedding of hyperfields.
757
+
758
+ 3.4. Equivalence of valuations. In analogy with classical valuation theory, our aim is now to
759
+ show that valuation hyperrings can be used to describe valuations up to composition with an order
760
+ preserving isomorphism of the value group.
761
+ Proposition 3.17 (Proposition 4.12 in [31]). Let F be a hyperfield and O a valuation hyperring
762
+ in F. Consider the multiplicative group Γ := F ×/O× and define a relation ≤ on Γ as follows:
763
+ aO× ≤ bO× ⇐⇒ ba−1 ∈ O.
764
+ Then (Γ, ·, ≤) is an ordered abelian group and the canonical projection
765
+ π : F → Γ ∪ {∞},
766
+ extended so that π(0F ) = ∞, is a valuation on F. Furthermore, Oπ = O.
767
+ Proof. First we show that ≤ is an ordering for (Γ, ·). Since aa−1 = 1F ∈ O, reflexivity is clear. If
768
+ ab−1, ba−1 ∈ O, then ab−1 ∈ O× so aO× = bO×. Hence ≤ is antisymmetric. If ab−1, bc−1 ∈ O,
769
+ then ac−1 = ab−1bc−1 ∈ O, showing that ≤ is transitive. Take now a, b ∈ F × such that aO× ≤ bO×
770
+ and c ∈ F ×. We have that bc(ac)−1 = bcc−1a−1 = ba−1 ∈ O, whence acO× ≤ bcO×. This shows
771
+ that ≤ is compatible with the operation of Γ. Finally, ≤ is a total order since O is a valuation
772
+ hyperring, so that ab−1 ∈ O or ba−1 ∈ O for all a, b ∈ F ×.
773
+ We now show that π is a valuation on F. Clearly, π is a surjective map, onto the ordered abelian
774
+ group Γ with ∞ and (V1) holds. Since π is a homomorphism of groups we obtain (V2). It remains
775
+ to show that (V3) holds for π. Take x, y ∈ F. If one of them is 0F , then (V3) is straightforward.
776
+ We may then assume that x, y ∈ F × and that xO× ≤ yO×. Take z ∈ x + y. We wish to show that
777
+ zx−1 ∈ O. By assumption we have that yx−1 ∈ O, thus
778
+ zx−1 ∈ (x + y)x−1 = 1 + yx−1 ⊆ O,
779
+ where we used Lemma 3.9.
780
+ Finally, we observe that, by definition
781
+ Oπ = {x ∈ F | πx ≥ 1} = {x ∈ F | xO× ≥ 1O×} = {x ∈ F | x ∈ O} = O.
782
+
783
+ Definition 3.18. Let (Γi, <i) be (partially) ordered sets for i = 1, 2. A map σ : Γ1 → Γ2 is said
784
+ to be order preserving if γ1 ≤1 γ2 implies that σ(γ1) ≤2 σ(γ2) for all γ1, γ2 ∈ Γ1.
785
+
786
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
787
+ 17
788
+ Definition 3.19. For i = 1, 2 let vi : F → Γi ∪ {∞} be valuations on a hyperfield F. We say
789
+ that v1 and v2 are equivalent if there exists an isomorphism of groups σ : Γ1 → Γ2 which is order
790
+ preserving and such that v2 = σ ◦ v1.
791
+ Lemma 3.20. Let v : F → Γ ∪ {∞} be a valuation on a hyperfield F. Then Γ ≃ F ×/O×
792
+ v with an
793
+ isomorphism of groups which is order preserving.
794
+ Proof. We consider F ×/O×
795
+ v as an ordered abelian group with the ordering defined in Proposition
796
+ 3.17. Using the surjectivity of v, we define a map
797
+ σ : Γ → F ×/O×
798
+ v
799
+ by σ(va) = aO×
800
+ v for all a ∈ F ×. This is well-defined since if va = vb, then va − vb = v(ab−1) = 0
801
+ so that ab−1 ∈ O×
802
+ v and then aO×
803
+ v = bO×
804
+ v . Using property (V2) of valuations, we obtain that σ is
805
+ a homomorphism of groups. Further, if va ≤ vb, then ba−1 ∈ Ov which means that σ(va) ≤ σ(vb).
806
+ Thus, σ is order preserving. It is clear that σ is surjective. It therefore remains to show that σ is
807
+ injective. To this end, assume that aO×
808
+ v = bO×
809
+ v for some a, b ∈ F ×. Then there exists c ∈ O×
810
+ v such
811
+ that a = bc. Since vc = 0, by (V2) we obtain that va = vb. This completes the proof.
812
+
813
+ Remark 3.21. Observe that by construction of σ in the above proof, we have that σ ◦ v = π where
814
+ π is the canonical epimorphism F × → F ×/O×
815
+ v .
816
+ Corollary 3.22. For i = 1, 2 let vi : F → Γi ∪ {∞} be valuations on a hyperfield F. Then v1 and
817
+ v2 are equivalent if and only if Ov1 = Ov2.
818
+ Proof. By the previous lemma we obtain for i = 1, 2 that Γi ≃ F ×/O×
819
+ vi as ordered abelian groups
820
+ with isomorphisms σi such that σi ◦ vi = πi where πi : F × → F ×/O×
821
+ vi is the canonical projection
822
+ for i = 1, 2. Thus, if Ov1 = Ov2, then π1 = π2 and σ := σ−1
823
+ 2
824
+ ◦ σ1 is an isomorphism of ordered
825
+ abelian groups Γ1 → Γ2. Further we have that
826
+ σ ◦ v1 = σ−1
827
+ 2
828
+ ◦ (σ1 ◦ v1) = σ−1
829
+ 2
830
+ ◦ π2 = v2.
831
+ Hence, v1 and v2 are equivalent.
832
+ On the other hand, if v1 and v2 are equivalent, then we obtain that F ×/O×
833
+ v1 ≃ F ×/O×
834
+ v2 as ordered
835
+ abelian groups. In particular, for a ∈ F × we have that 1O×
836
+ v1 ≤ aO×
837
+ v1 if and only if 1O×
838
+ v2 ≤ aO×
839
+ v2.
840
+ Using the definition of the ordering in F ×/O×
841
+ vi we see that this means that a ∈ Ov1 if and only if
842
+ a ∈ Ov2. Since 0 ∈ Ovi for i = 1, 2, we conclude that Ov1 = Ov2 as claimed.
843
+
844
+ In the following sections we will always consider valuations on hyperfields up to equivalence.
845
+ 4. Krasner valued hyperfields
846
+ In this section, we focus our attention on those valued hyperfields which satisfy the original more
847
+ restrictive axioms of Krasner and were called by him hypercorps valué. These valued hyperfields
848
+ still attract the attention of mathematicians. For instance, they have recently been considered by
849
+ Tolliver and Lee (see [46, 32]) who called them simply “valued hyperfields”.
850
+ 4.1. Ultrametric spaces. We begin by presenting some basic theory of ultrametric spaces, a no-
851
+ tion as well studied by Krasner (cf. [23]). The axioms for an ultrametric distance can be formulated
852
+ using just the linear order of non-negative real numbers where 0 is a bottom element. Since nothing
853
+ but the order is used from the structure of real numbers, we will use the term ultrametric space in a
854
+ broader sense allowing ultrametric distances to take values in any linearly ordered set. In addition,
855
+ since the value group of a valuation has a top element, our definition of ultrametric space below
856
+
857
+ 18
858
+ LINZI, A.
859
+ is a modification which better fits into our context (the same approach can be found e.g. in [28]).
860
+ In the case of real-valued distances, this modification corresponds to a replacement of the linearly
861
+ ordered set (R≥0, <) with (R ∪ {∞}, >).
862
+ Definition 4.1. An ultrametric distance (or simply an ultrametric) on a set X is a function
863
+ d : X × X → Γ ∪ {∞}, where (Γ, <) is a linearly ordered set and ∞ satisfies γ < ∞ for all γ ∈ Γ,
864
+ such that for all x, y, z ∈ X
865
+ (U1) d(x, y) = ∞ if and only if x = y,
866
+ (U2) d(x, y) = d(y, x),
867
+ (U3) d(x, z) ≥ min{d(x, y), d(y, z)}.
868
+ We call (X, d) an ultrametric space whenever d is an ultrametric on X. We call the set dX :=
869
+ {d(x, y) | x, y ∈ X, x ̸= y} ⊆ Γ the value set of d.
870
+ Example 4.2. Let (K, v) be a valued field. Then the function
871
+ K × K → Γ ∪ {∞}
872
+ (x, y) �→ v(x − y)
873
+ is an ultrametric on K.
874
+ Definition 4.3. Let (X, d) be an ultrametric space. A subset B ⊆ X is called a ball if for all
875
+ y, z ∈ B and all x ∈ X we have that the following implication
876
+ d(x, y) ≥ d(y, z)
877
+ =⇒
878
+ x ∈ B
879
+ holds for all x, y, z ∈ X.
880
+ Definition 4.4. A subset ρ of a linearly ordered set (Γ, <) is called an initial segment (resp. final
881
+ segment) if for all δ ∈ ρ and all γ ∈ Γ if γ < δ (resp. γ > δ), then γ ∈ ρ.
882
+ Lemma 4.5. For every element x of an ultrametric space (X, d) and every final segment ρ of
883
+ dX ∪ {∞}, we have that
884
+ Bρ(x) := {y ∈ X | d(x, y) ∈ ρ}.
885
+ is a ball in X. Conversely, if B is a ball in X and ρ is the smallest (with respect to inclusion) final
886
+ segment of dX ∪ {∞} containing d(y, z) for all y, z ∈ B, then for every x ∈ B,
887
+ B = Bρ(x).
888
+ In particular, Bρ(x) = Bρ(y) for every y ∈ Bρ(x).
889
+ Proof. Assume that y, z ∈ Bρ(x), that is, d(x, y) ∈ ρ and d(x, z) ∈ ρ.
890
+ If t ∈ X is such that
891
+ d(y, t) ≥ d(y, z), then the inequalities
892
+ d(x, t) ≥ min{d(x, y), d(y, t)} ≥ min{d(x, y), d(y, z)} ≥ min{d(x, y), d(y, x), d(x, z)} ∈ ρ
893
+ follow from (U2) and (U3). Since ρ is a final segment of dX ∪ {∞}, we conclude that d(x, t) ∈ ρ.
894
+ Hence, t ∈ Bρ(x)and Bρ(x) is a ball.
895
+ For the converse, assume that B is a ball and let ρ be as in the assertion. Further, let x be any
896
+ element in B. If y ∈ B, then d(x, y) ∈ ρ and thus y ∈ Bρ(x). On the other hand, if y ∈ Bρ(x),
897
+ then d(x, y) ∈ ρ. So by definition of ρ, there is some z ∈ B such that d(x, z) ≤ d(x, y). Since B is
898
+ a ball, it follows that y ∈ B. We have proved that B = Bρ(x).
899
+
900
+ Corollary 4.6. Let (X, d) be an ultrametric space. Every two balls with non-empty intersection
901
+ are comparable by inclusion.
902
+
903
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
904
+ 19
905
+ Proof. Take two balls B and B′ and suppose that z ∈ B ∩ B′. By Lemma 4.5 there are final
906
+ segments ρ, ς of dX ∪ {∞} such that B = Bρ(z) and B′ = Bς(z). Since ρ and ς are final segments,
907
+ we must have ρ ⊆ ς or ς ⊆ ρ. Hence, B ⊆ B′ or B′ ⊆ B.
908
+
909
+ 4.2. Krasner valuations. Let (Γ, <, +, 0) be an ordered abelian group, ρ be an initial segment of
910
+ Γ and γ an element of Γ. We will sometimes write γ > ρ to indicate that γ /∈ ρ.
911
+ The following class of valued hyperfields is of special interest.
912
+ Definition 4.7 (Section 3 of [24], Definition 1.4 in [46] and Definition 2.4 in [32]). We call a valued
913
+ hyperfield (F, v) a Krasner valued hyperfield if
914
+ (KVH1) For all x, y ∈ F, v(x + y) is a singleton unless 0 ∈ x + y.
915
+ (KVH2) There exists an initial segment ρv of vF such that 0 ∈ ρv and for all x, y, z, t ∈ F we have
916
+ that z ∈ x + y implies that t ∈ x + y if and only if vs > ρv + min{vx, vy} for all s ∈ z − t.
917
+ The initial segment ρv is called the norm of v. We will also say that v is a Krasner valuation on F
918
+ when (F, v) is a Krasner valued hyperfield.
919
+ Example 4.8. Let (K, v) be a valued field and consider K as a hyperfield. Then v is a Krasner
920
+ valuation on K with norm vK.
921
+ Proposition 4.9. Let (F, v) be a Krasner valued hyperfield. For all x, y ∈ F such that x ̸= y, by
922
+ axiom (KVH1), v(x−y) contains a unique element γx,y ∈ vF. Define a map dv : F ×F → Γ∪{∞}
923
+ as
924
+ dv(x, y) :=
925
+
926
+ γx,y
927
+ if x ̸= y,
928
+
929
+ otherwise.
930
+ Then dv is an ultrametric on F. Moreover, for all x, y ∈ F and any z ∈ x + y, in the ultrametric
931
+ space (F, dv) we have that
932
+ x + y = Bρv+min{vx,vy}(z).
933
+ Proof. Axiom (U1) follows from axiom (V1), axiom (U2) follows from Corollary 3.5 (ii) and axiom
934
+ (U3) is a direct consequence of axiom (V3). The last statement is just a reformulation of axiom
935
+ (KVH2).
936
+
937
+ We call the ultrametric dv on a Krasner valued hyperfield (F, v) the ultrametric induced by v on
938
+ F.
939
+ Remark 4.10. Let F be a hyperfield and let v be the trivial valuation on F. If v is a Krasner
940
+ valuation of norm ρv, then by (KVH2) for all x, y ∈ F × we have that x ∈ 1 − 1 if and only if
941
+ 0 = vx > ρv. Since 0 ∈ ρv, it follows that x − x = {0} for all x ∈ F and then F is a field by Lemma
942
+ 2.9.
943
+ Conversely, if K is a field, then the map v(0) := ∞ and vx := 0 for all x ∈ K× is a Krasner
944
+ valuation on K with value group vK = {0} and norm vK.
945
+ Krasner’s main motivation probably came from the following example.
946
+ Example 4.11. For a valued field (K, v) and an initial segment ρ ⊆ vK containing 0, let us
947
+ consider the (multiplicative) group of 1-units of level ρ:
948
+ 1 + Mρ
949
+ v = {x ∈ K | v(x − 1) > ρ} ⊆ O×
950
+ v .
951
+ Then vρ := v1+Mρ
952
+ v is a Krasner valuation on Kρ := K1+Mρ
953
+ v and the norm of vρ is ρ.
954
+
955
+ 20
956
+ LINZI, A.
957
+ Actually, any Krasner valued hyperfield which is a factor hyperfield is of this form, as we show
958
+ below.
959
+ Proposition 4.12. Let F be a factor hyperfield admitting a Krasner valuation w. Then F = Kρ
960
+ and w is vρ for some valued field (K, v) and some initial segment ρ of vK = wF containing 0.
961
+ Proof. By Lemma 3.6 there is a valued field (K, v) and a subgroup T ⊆ O×
962
+ w of K× such that
963
+ vT = w. Suppose that t ∈ T is not a 1-unit in K. Then vt = 0 and v(t − 1) = 0 must hold.
964
+ now, on the one hand, since w = vT is a Krasner valuation, for all [x]T ∈ [1]T − [1]T we have that
965
+ vx = vT [x]T > 0 by (KVH2). On the other hand, since t ∈ T we have that [t − 1]T ∈ [1]T − [1]T .
966
+ This contradiction proves that T ⊆ 1 + Mv must hold and the result follows.
967
+
968
+ Remark 4.13. We do not know if all Krasner valued hyperfields are factor hyperfields. Some results
969
+ connected to this problem are provided in [34].
970
+ Example 4.14. Consider the Hahn series field K := F2((tΓ)) for some non-trivial ordered abelian
971
+ group Γ and let v denote its canonical t-adic valuation. In this case, since Kv ≃ F2 we have that
972
+
973
+ v = 1 + Mv. We conclude that
974
+ T ′(Γ) ≃ Kρ,
975
+ where ρ = {γ ∈ Γ | γ ≤ 0}. We have that the identity map vρ is the identity map on T (Γ) = T ′(Γ)
976
+ and it is a Krasner valuation on T ′(Γ) with norm ρ. Note that, the same map is not a Krasner
977
+ valuation on T (Γ) as 0 ∈ [0, ∞] = 0 ⊞ 0 violates (KVH2).
978
+ We now study the residue hyperfield of a Krasner valued hyperfield.
979
+ Proposition 4.15. The residue hyperfield Fv of a Krasner valued hyperfield (F, v) is a field.
980
+ Proof. Take xv ∈ 1v − 1v. This means that x ∈ 1 − 1 and by axiom (KVH2), we deduce that
981
+ vx > ρv. Since 0 ∈ ρv, it follows that xv = 0v and therefore Fv is a field by Lemma 2.9.
982
+
983
+ Example 4.16. Let (K, v) be a valued field. It follows from Proposition 3.14 that, for any initial
984
+ segment ρ of vK containing 0, the residue hyperfield of Kρ is a field isomorphic to Kv.
985
+ Let us now consider a valued hyperfield which is not a Krasner valued hyperfield.
986
+ Example 4.17. Consider the rational function field K := Q(X) over the rationals, and its mul-
987
+ tiplicative subgroup T := Q×. Under the canonical X-adic valuation v, we have that T ⊆ O×
988
+ v .
989
+ Hence, Lemma 3.3 yields a valued hyperfield (KT , vT ). By Proposition 3.14 and Proposition 2.17
990
+ (i), we have that
991
+ KT vT ≃ QQ× ≃ K.
992
+ Since K is not a field, the Proposition 4.15 implies that (KT , vT ) is not a Krasner valued hyperfield.
993
+ 4.3. Mittas and superiorly canonical hypergroups. J. Mittas was a student of Krasner. He
994
+ introduced superiorly canonical hypergroups (defined below) and published (sometimes without
995
+ proofs) some results which connect them to Krasner valuations (see e.g. [41]). Some of the contents
996
+ of this subsection were inspired by his work.
997
+ Definition 4.18 ([41] and page 81 of [38]). A canonical hypergroup H is called superiorly canonical
998
+ if
999
+ (SCH1) For all x, y ∈ H, if x ∈ x + y, then x + y = {x}.
1000
+ (SCH2) For all x, y, z, t ∈ H, if (x + y) ∩ (z + t) ̸= ∅, then x + y ⊆ z + t or z + t ⊆ x + y.
1001
+ (SCH3) For all x, y ∈ H such that x ̸= y we have that z, t ∈ x − y implies z − z = t − t.
1002
+
1003
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
1004
+ 21
1005
+ (SCH4) For all x, y, z ∈ H, if x ∈ z − z and y /∈ z − z, then x − x ⊆ y − y.
1006
+ Example 4.19. Any abelian group is a superiorly canonical hypergroup.
1007
+ Other examples of superiorly canonical hypergroups are provided by the following result.
1008
+ Proposition 4.20. Let (F, v) be a Krasner valued hyperfield. Then the additive canonical hyper-
1009
+ group of F is superiorly canonical.
1010
+ Proof. We verify the axioms one by one. Assume that x ∈ x+ y for some x, y ∈ F. By reversibility,
1011
+ y ∈ x − x, so the inequalities vy > ρv + vx ≥ vx follow from axiom (KVH2) and 0 ∈ ρv. Therefore,
1012
+ if z ∈ x + y, then vx = vz by Corollary 3.5 (iv) and since by reversibility y ∈ z − x and dv(y, 0) =
1013
+ vy > ρv + min{vx, vz}, axiom (KVH2) implies that 0 ∈ z − x and so x = z must hold. This shows
1014
+ (SCH1).
1015
+ Axiom (SCH2) follows from Proposition 4.9 and Corollary 4.6.
1016
+ For (SCH3), take x, y ∈ F and assume that x ̸= y, i.e., 0 /∈ x − y. Take z, t ∈ x − y. We
1017
+ claim that vz = vt. Suppose that vz < vt, then by Corollary 3.5 (iv) we have that va = vz for all
1018
+ a ∈ z − t, thus
1019
+ dv(z, 0) = vz = dv(z, t) > ρv + min{vx, vy}.
1020
+ Axiom (KVH2) now implies that 0 ∈ x − y, a contradiction. Therefore, vz ≥ vt. Symmetrically,
1021
+ vt ≥ vz and so vz = vt as claimed.
1022
+ Now, by (KVH2) we have that a ∈ z − z if and only if
1023
+ va > ρv + vz and a ∈ t − t if and only if va > ρv + vt. Since vz = vt we conclude that z − z = t − t.
1024
+ This shows that (SCH3) holds.
1025
+ For (SCH4), take x ∈ z − z and y /∈ z − z. By axiom (KVH2) it follows that vx > ρv + vz and
1026
+ vy ∈ ρv +vz. In particular, vx > vy. Now, again by axiom (KVH2), if a ∈ x−x, then va > ρv +vx
1027
+ which implies va > ρv + vy and so a ∈ y − y. Hence, x − x ⊆ y − y. This completes the proof.
1028
+
1029
+ The theory of superiorly canonical hypergroups and the theory of Krasner valued hyperfields are
1030
+ even more deeply related.
1031
+ Proposition 4.21. Let F be a hyperfield with a superiorly canonical additive hypergroup. Then
1032
+ O := {x ∈ F | x − x ⊆ 1 − 1}
1033
+ is a valuation hyperring in F. Moreover, if v is the valuation such that Ov = O, then (F, v) is a
1034
+ Krasner valued hyperfield.
1035
+ Proof. If F is a field, then O = F and thus v is the trivial valuation on F. Since F is a field, (F, v)
1036
+ is a Krasner valued hyperfield. We can then assume that F is not a field.
1037
+ If x − x ⊆ 1 − 1 and y − y ⊆ 1 − 1, then
1038
+ xy − xy = (x − x)y ⊆ (1 − 1)y = y − y ⊆ 1 − 1.
1039
+ Hence, O is multiplicatively closed.
1040
+ If x − x ⊈ 1 − 1, then 1 − 1 ⊊ x − x by axiom (SCH2), since 0 ∈ (x − x) ∩ (1 − 1). Multiplying
1041
+ by x−1 we obtain that x−1 − x−1 ⊊ 1 − 1. Therefore, x−1 ∈ O.
1042
+ Assume that x − x ⊆ 1 − 1 and y − y ⊆ 1 − 1. We claim that if z ∈ x + y, then z − z ⊆ 1 − 1.
1043
+ Pick a ∈ z − z. Since z ∈ x + y we have that
1044
+ a ∈ z − z ⊆ (x − x) + (y − y) ⊆ (1 − 1) + (1 − 1),
1045
+ so there exists x′ ∈ 1 − 1 and y′ ∈ 1 − 1 such that a ∈ x′ + y′. Hence, 1 ∈ x′ + 1 by reversibility, and
1046
+ a ∈ x′ + y′ ⊆ (x′ + 1) − 1 = 1 − 1,
1047
+
1048
+ 22
1049
+ LINZI, A.
1050
+ where we have used axiom (SCH1). We proved that O is a valuation hyperring in F.
1051
+ Clearly, O× = {x ∈ F | x−x = 1−1}. Set vF := F ×/O× and let v : F × → vF be the canonical
1052
+ epimorphism. We have to show that (F, v) is a Krasner valued hyperfield. It is a valued hyperfield
1053
+ by Proposition 3.17. Let us now verify the validity of axioms (KVH1) and (KVH2).
1054
+ Take x, y ∈ F × and assume that 0 /∈ x + y. Pick z, t ∈ x + y and suppose that vz < vt. This
1055
+ means that tz−1 ∈ O \ O×, so
1056
+ tz−1 − tz−1 ⊊ 1 − 1.
1057
+ But then t − t ⊊ z − z which contradicts axiom (SCH3). We have proved that (KVH1) holds for
1058
+ (F, v).
1059
+ We now have to verify (KVH2). Our first claim is that v(1 − 1) is a final segment of vF which
1060
+ does not contain 0. Pick a ∈ 1 − 1 and γ > va. Let b ∈ F × be such that vb = γ. Since vb > va, we
1061
+ have that b − b ⊊ a − a. Now, if b /∈ 1 − 1, then axiom (SCH4) implies a − a ⊆ b − b. Therefore,
1062
+ b ∈ 1 − 1 must hold and thus γ = vb ∈ v(1 − 1) and v(1 − 1) is a final segment.
1063
+ For x ∈ F ×, if x ∈ x − x, then x − x = {x} by axiom (SCH1). However, since 0 ∈ x − x, this
1064
+ cannot be. An element x ∈ F × has value 0 if and only if x − x = 1 − 1. Hence it cannot belong to
1065
+ 1 − 1 as x /∈ x − x for all x ∈ F ×. This shows that v(1 − 1) does not contain 0.
1066
+ We let ρv be the complement in vF of v(1 − 1). Then ρv is an initial segment of vF which
1067
+ contains 0.
1068
+ Observe that for all x ∈ F we have that a ∈ x − x = (1 − 1)x if and only if there exists y ∈ 1 − 1
1069
+ such that a = yx. This implies that va = vy + vx > ρv + vx. Conversely, if va > ρv + vx, then
1070
+ v(ax−1) ∈ v(1 − 1), so there exists b ∈ 1 − 1 such that vb = v(ax−1). Therefore,
1071
+ b ∈ 1 − 1 = bxa−1 − bxa−1 = b(xa−1 − xa−1),
1072
+ so 1 ∈ xa−1 − xa−1 implying that a ∈ x − x.
1073
+ Take now x, y ∈ F such that x ̸= y and vx ≤ vy. Fix z ∈ x − y. We have to show that t ∈ x − y
1074
+ if and only if dv(z, t) > ρv + vx.
1075
+ Assume first that t ∈ x − y. If t = z there is nothing to show. Otherwise, it suffices to show that
1076
+ z − t ⊆ x − x by what we have already shown above. Take a ∈ z − t. Since z ∈ x − y, we have that
1077
+ a ∈ z − t ⊆ x − (y + t).
1078
+ Hence, there exists b ∈ y+t such that a ∈ x−b. We obtain that b ∈ (x−a)∩(y+t), so y+t ⊆ x−a
1079
+ or x − a ⊆ y + t by axiom (SCH2). In the first case, since x ∈ y + t we have that x ∈ x − a and
1080
+ a ∈ x − x follows. It remains to deal with the case x − a ⊊ y + t. In this case, since t ∈ x − y and
1081
+ y − y ⊆ x − x, we obtain that
1082
+ x − a ⊊ y + t ⊆ y + x − y ⊆ x + (x − x).
1083
+ Take c ∈ x − a. There exists x′ ∈ x − x such that c ∈ x + x′. This implies that
1084
+ a ∈ x − c ⊆ x − (x + x′) = x − x
1085
+ where we have used axiom (CH4) and axiom (SCH1) to conclude that x + x′ = {x}.
1086
+ Now, assume that dv(z, t) > ρv + vx. We have to prove that t ∈ x − y. If z = t, then there is
1087
+ nothing to show. Hence, we can assume that z ̸= t and so z − t ⊊ x − x must hold. We have that
1088
+ t ∈ t + 0 ⊆ t + z − z ⊊ x − x + z.
1089
+ Hence, there is b ∈ z − x such that t ∈ x + b. We conclude that b ∈ (z − x) ∩ (t − x). By axiom
1090
+ (SCH2) we then obtain that z − x ⊆ t − x or t − x ⊆ z − x. In the first case, −y ∈ z − x ⊆ t − x
1091
+
1092
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
1093
+ 23
1094
+ and t ∈ x − y follows. It remains to deal with the case t − x ⊊ z − x. In this case, since z ∈ x − y,
1095
+ we have that
1096
+ t − x ⊊ z − x ⊆ x − y − x = x − x − y.
1097
+ Take a ∈ t − x. There exists x′ ∈ x − x such that a ∈ x′ − y. Now, by reversibility and since
1098
+ a ∈ t − x, we have that
1099
+ −y ∈ a − x′ ⊆ t − (x + x′) = t − x,
1100
+ where for the last equality we have used axiom (SCH1). Now, t ∈ x−y follows by axiom (CH4).
1101
+
1102
+ Directly from Proposition 4.20 and Proposition 4.21 above, we deduce the following characteri-
1103
+ zation theorem for the hyperfields which admit a Krasner valuation.
1104
+ Theorem 4.22. Let F be a hyperfield.
1105
+ Then F admits a Krasner valuation if and only if the
1106
+ additive hypergroup of F is superiorly canonical.
1107
+ Example 4.23. The additive hypergroup of the hyperfield that we have considered in Example
1108
+ 4.17 above is not superiorly canonical. Indeed,
1109
+ 1T + 1T = {0T, 1T }
1110
+ and thus (SCH1) fails. It follows from Theorem 4.22 that this hyperfield does not admit Krasner
1111
+ valuations at all.
1112
+ Example 4.24. Consider a generalised tropical hyperfield T (Γ), where Γ is some non-trivial ordered
1113
+ abelian group (see Example 2.14). By Proposition 2.17 (iv) and Lemma 3.3 we conclude that T (Γ)
1114
+ is a valued hyperfield (see also the discussion after Proposition 5.9). Nevertheless, by Theorem 4.22,
1115
+ T (Γ) does not admit Krasner valuations as its additive hypergroup does not satisfy (SCH1).
1116
+ Let us now analyse another example.
1117
+ Example 4.25. Take K = Q(X) with the valuation v := vp ◦ vX where vX denotes the X-adic
1118
+ valuation on Q(X) and vp denotes the p-adic valuation on Q, for some prime number p. More
1119
+ explicitly, for a polynomial f(X) = �n
1120
+ i=0 aiXi in Q[X], we have
1121
+ vf(X) =
1122
+
1123
+ vXf(X), vp(avXf(X))
1124
+
1125
+ ∈ Z × Z.
1126
+ The order relation in vK is the lexicographic order of Z × Z, that is, (n1, n2) < (m1, m2) if and
1127
+ only if n1 < m1 ∨ (n1 = m1 ∧ n2 < m2).
1128
+ Consider the Krasner valued hyperfield F := Kρ where ρ is the smallest initial segment of Z × Z
1129
+ which contains {0} × Z (cf. Example 4.11). Let us denote its Krasner valuation vρ by w.
1130
+ Note that, if (m1, m2) ∈ ρ and m1 > 0, then, since ρ is an initial segment, we have that
1131
+ {0} × Z ⊆ {(k1, k2) ∈ Z × Z | (k1, k2) ≤ (m1 − 1, m2)} ⊊ ρ,
1132
+ in contradiction with the minimality of ρ. Therefore, m1 ≤ 0 for all (m1, m2) ∈ ρ. Conversely, if
1133
+ m1 ≤ 0, then (m1, m2) ≤ (0, m2) for all m2 ∈ Z and therefore (m1, m2) ∈ ρ since ρ is an initial
1134
+ segment containing {0} × Z. It follows that
1135
+ (1)
1136
+ ρ = {(m1, m2) ∈ Z × Z | m1 ≤ 0} = {(m1, m2 + n) ∈ Z × Z | m1 ≤ 0} = ρ + (0, n)
1137
+ for any (0, n) ∈ {0} × Z.
1138
+ By Proposition 4.20, the additive hypergroup of F is superiorly canonical. Therefore, by Propo-
1139
+ sition 4.21, we have the valuation hyperring
1140
+ Ou = {x ∈ F | x − x ⊆ 1 − 1}
1141
+
1142
+ 24
1143
+ LINZI, A.
1144
+ for some Krasner valuation u on F. Let us now show that w and u are not equivalent valuations
1145
+ on F.
1146
+ In fact, we will show that Ow ⊊ Ou. Pick x ∈ Ow, i.e., wx ≥ (0, 0). By axiom (KVH2) we have
1147
+ that y ∈ x − x if and only if wy > ρ + wx ≥ ρ. Another application of axiom (KVH2) shows that
1148
+ x − x ⊆ 1 − 1 so that x ∈ Ou. Now, consider e.g. the element x of F corresponding to the rational
1149
+ number p−1 in K. Since vp(p−1) = −1 we have that wx = (0, −1) < (0, 0) so that x /∈ Ow. On
1150
+ the other hand, since wx = (0, −1) ∈ {0} × Z, we have that ρ + wx = ρ by (1) and thus using
1151
+ (KVH2), we obtain that y ∈ x − x if and only if wy > ρ if and only if y ∈ 1 − 1. This implies that
1152
+ x − x ⊆ 1 − 1 and thus x ∈ Ou.
1153
+ The above example constitutes the starting point for the discussion that we will have in Section
1154
+ 6.
1155
+ 5. More on ordered abelian groups
1156
+ In this section, we characterise generalised tropical hyperfields and discuss the concept of coars-
1157
+ ening of a valuation in the multivalued setting. We derive some conclusions on the relation between
1158
+ ordered abelian groups and generalised tropical hyperfields.
1159
+ 5.1. Characterisation of generalised tropical hyperfields. To characterise generalised tropi-
1160
+ cal hyperfields, we will apply another remarkable characterisation result. A hyperfield is stringent
1161
+ if x + y is a singleton whenever 0 /∈ x + y. Bowler and Su in [6] characterised stringent hyperfields
1162
+ and we will now briefly recall their result.
1163
+ In [6, Section 4] a natural construction of a hyperfield arising from a short exact sequence
1164
+ (2)
1165
+ {1}
1166
+ F ×
1167
+
1168
+ Γ
1169
+ {0}
1170
+ ϕ
1171
+ ψ
1172
+ where Γ is an ordered abelian group and F is any hyperfield is described.
1173
+ The thus obtained
1174
+ hyperfield has H× as multiplicative group and is called the Γ-layering of F along the short exact
1175
+ sequence (2). After that the following theorem is proved.
1176
+ Theorem 5.1 (Theorem 4.10 in [6]). A hyperfield H is stringent if and only if it is the Γ-layering
1177
+ of F along the short exact sequence (2) with F being (isomorphic to) either K, S or a field.
1178
+ From the details of the construction (which we omit for brevity) it is not difficult to verify that
1179
+ the extension of the map ϕ sending 0F to 0H is in any case an embedding of hyperfields.
1180
+ We say that a hyperfield has characteristic 2 if 0 ∈ 1 + 1 and that it has C-characteristic 1 if
1181
+ 1 ∈ 1 + 1. For example, S satisfies the latter but not the former while K satisfies both. For more
1182
+ information on characteristic and C-characteristic of hyperfields the reader can see [21].
1183
+ We are now ready to state and prove our characterisation theorem.
1184
+ Theorem 5.2 (Characterisation of generalised tropical hyperfields). Generalised tropical hyper-
1185
+ fields are precisely stringent hyperfields of characteristic 2 and C-characteristic 1.
1186
+ Proof. The reader can easily verify from the relevant definitions that a generalised tropical hyperfield
1187
+ T (Γ) is a stringent hyperfield of characteristic 2 and C-characteristic 1.
1188
+ For the converse, let H be a stringent hyperfield of characteristic 2 and C-characteristic 1. By
1189
+ Theorem 5.1 and our observations on the extension of the map ϕ above, we have that either K, S or
1190
+ a field embed into H. Since H has characteristic 2 we have that 1 = −1, thus S cannot embed into
1191
+ H. On the other hand, any field cannot embed into H neither. Indeed, our assumption 0, 1 ∈ 1 + 1
1192
+
1193
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
1194
+ 25
1195
+ implies that {0, 1} = (1 + 1) ∩ {0, 1} ⊆ (1 + 1) ∩ F = 1 +F 1 is not a singleton. It follows that H is
1196
+ the Γ-layering of K along a short exact sequence
1197
+ {1}
1198
+
1199
+
1200
+ Γ
1201
+ {0}
1202
+ ϕ
1203
+ ψ
1204
+ for some ordered abelian group Γ. Now, since K× = {1}, we obtain that the multiplicative group
1205
+ of H is isomorphic to Γ and the details of the construction given in [6, Section 4] show that the
1206
+ hyperoperation of H is the one of T (Γ). We conclude that H is a generalised tropical hyperfield.
1207
+
1208
+ An analogous reasoning yields to the following (but maybe less interesting) characterisation of
1209
+ strict generalised tropical hyperfields.
1210
+ Theorem 5.3. Generalised tropical hyperfields are precisely the Γ-layerings of F along the short
1211
+ exact sequence (2), with F = F2.
1212
+ 5.2. Coarsenings.
1213
+ Definition 5.4. Let (Γ, <, +, 0) be an ordered abelian group. A subgroup ∆ of Γ is convex if for
1214
+ all γ ∈ Γ we have that if there exist δ1, δ2 ∈ ∆ such that δ1 < γ < δ2, then γ ∈ ∆.
1215
+ It is not difficult to see that the intersection of a family of convex subgroups of an ordered
1216
+ abelian group is again a convex subgroup and that the collection of all convex subgroups of an
1217
+ ordered abelian group is linearly ordered by inclusion. Let us recall another basic fact which makes
1218
+ convex subgroups important.
1219
+ Fact 5.5. Let (Γ, <, +, 0) be an ordered abelian group and ∆ a convex subgroup of Γ. Then (Γ/∆, ≺
1220
+ , +, 0) is an ordered abelian group, where
1221
+ x + ∆ ≺ y + ∆ ⇐⇒ x < y and y − x /∈ ∆.
1222
+ In particular, the canonical epimorphism Γ → Γ/∆ is order preserving.
1223
+ The following notion will play a fundamental role later.
1224
+ Definition 5.6. Let (Γ, <, +, 0) be an ordered abelian group and ρ an initial segment of Γ. We
1225
+ call the set
1226
+ ig(ρ) := {γ ∈ Γ | ρ + γ = ρ}
1227
+ the invariance group of ρ. If ig(ρ) = {0}, then we say that ρ has trivial invariance group.
1228
+ Remark 5.7. In [29, 30] F.-V. Kuhlmann proves several results about invariance groups associated
1229
+ to initial segments in ordered abelian groups. In particular, it follows from [30, Lemma 2.3] that
1230
+ the invariance group of an initial segment ρ of an ordered abelian group Γ such that 0 ∈ ρ is a
1231
+ convex subgroup of Γ contained in ρ.
1232
+ Definition 5.8. Let v, w be two valuations on a hyperfield F. We say that w is a coarsening of v
1233
+ if Ov ⊆ Ow.
1234
+ In classical valuation theory for fields, it is well-known that to any convex subgroup of the value
1235
+ group one can associate a coarsening. We note that this result generalises to valued hyperfields,
1236
+ actually with a much more conceptual proof.
1237
+ Proposition 5.9. Let (F, v) be a valued hyperfield and let ∆ be a convex subgroup of vF. Then
1238
+ v∆ : F → (vF/∆) ∪ {∞}
1239
+ x �→ vx + ∆
1240
+ is a valuation on F which is a coarsening of v.
1241
+
1242
+ 26
1243
+ LINZI, A.
1244
+ Proof. By Lemma 3.4, v : F → T (vF) is a surjective homomorphism of hyperfields. In addition,
1245
+ the canonical epimorphism vF → vF/∆ extends to a surjective map T (vF) → T (vF/∆). From the
1246
+ fact that the latter is order preserving, it easily follows that it is a homomorphism of hyperfields.
1247
+ We conclude that the composition v∆ : F → T (vF/∆) of v with the latter map is a surjective
1248
+ homomorphism of hyperfields. This proofs the first part of the statement. Now, since
1249
+ Ov∆ = {x ∈ F | v∆x ⪰ 0vF/∆}
1250
+ = {x ∈ F | vx + ∆ ⪰ ∆}
1251
+ = {x ∈ F | vx ≥ 0 or vx ∈ ∆}
1252
+ ⊇ {x ∈ F | vx ≥ 0} = Ov,
1253
+ we conclude that v∆ is a coarsening of v.
1254
+
1255
+ In the above proof, we have seen that if ∆ is a convex subgroup of an ordered abelian group Γ,
1256
+ then the natural map
1257
+ π∆ : T (Γ) → T (Γ/∆)
1258
+ is a surjective homomorphism of hyperfields, which, by Lemma 3.4, is a valuation on T (Γ) with
1259
+ value group Γ/∆. The valuation hyperring of this valuation is
1260
+ O∆ = {γ ∈ Γ | γ + ∆ ⪰ 0 + ∆} = {γ ∈ Γ | γ ∈ ∆ or γ > ∆}
1261
+ and its unique maximal hypeideal is
1262
+ M∆ = {γ ∈ Γ | γ > ∆}
1263
+ It follows that the residue hyperfield T (Γ)π∆ = O∆/M∆ of T (Γ) with respect to the valuation π∆
1264
+ is isomorphic to T (∆). An isomorphism is given by the map
1265
+ T (∆) → T (Γ)π∆
1266
+ ∞ �→ ∞
1267
+ δ �→ δπ∆
1268
+ In the next result we show that the valuations of T (Γ) are (up to equivalence) all of this form.
1269
+ Proposition 5.10. Let Γ be an ordered abelian group and assume that v : T (Γ) → T (Γ′) is a
1270
+ valuation. Then there exists a convex subgroup ∆ of Γ such that Γ/∆ ≃ Γ′ via an order preserving
1271
+ isomorphism.
1272
+ Proof. Let 0′ ∈ Γ′ denote the neutral element of the group Γ′, ⊞ the hyperoperation of T (Γ) and ⊞′
1273
+ the hyperoperation of T (Γ′). Set ∆ := v−1(0′). Since v is a homomorphism of groups Γ → Γ′, we
1274
+ have that ∆ is a subgroup of Γ. Moreover, since v is surjective, we have that Γ/∆ ≃ Γ′ as groups
1275
+ by the first homomorphism theorem for groups. Assume that δ1, δ2 ∈ ∆ and that γ ∈ Γ is such
1276
+ that δ1 ≤ γ ≤ δ2. Hence, δ2 ∈ γ ⊞ γ and so 0′ = v(δ2) ∈ v(γ) ⊞′ v(γ) since v is a homomorphism
1277
+ of hyperfields. On the other hand, γ ∈ δ1 ⊞ δ1 so that v(γ) ∈ 0′ ⊞′ 0′ = [0′, ∞]. If v(γ) > 0′, then
1278
+ 0′ /∈ [v(γ), ∞] = v(γ) ⊞′ v(γ), a contradiction. We conclude that v(γ) = 0′ must hold and thus
1279
+ γ ∈ ∆. We have shown that ∆ is a convex subgroup of Γ. It remains to show that the isomorphism
1280
+ of groups σ : γ + ∆ �→ v(γ) is order preserving. Assume that γ1 + ∆ ≺ γ2 + ∆. This means that
1281
+ γ1 < γ2 and γ2 −γ1 /∈ ∆ hold in Γ. Thus, {γ1} = γ1 ⊞γ2 and then v(γ1) ∈ v(γ1)⊞′ v(γ2). Since σ is
1282
+ bijective and γ1 ̸= γ2, we have that v(γ1) ̸= v(γ2), so v(γ1) ∈ v(γ1) ⊞′ v(γ2) =
1283
+
1284
+ min{v(γ1), v(γ2)}
1285
+
1286
+ .
1287
+ This shows that v(γ1) < v(γ2) must hold in Γ′ and thus σ is order preserving and the proof is
1288
+ complete.
1289
+
1290
+
1291
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
1292
+ 27
1293
+ The next result now follows from Corollary 3.22.
1294
+ Corollary 5.11. Let Γ be an ordered abelian group. There is a bijective correspondence between
1295
+ convex subgroups of Γ and valuation hyperrings of T (Γ).
1296
+ 6. Krasner valuations induced by the additive structure
1297
+ In this section we apply some of the results obtained in the previous section to Krasner valued
1298
+ hyperfields. We begin by observing that coarsenings of Krasner valuations might not be Krasner
1299
+ valuations.
1300
+ Example 6.1. Let Γ be a non-trivial ordered abelian group with a non-trivial convex subgroup ∆
1301
+ and consider the identity map as a Krasner valuation v : T ′(Γ) → T (Γ). As a map, the coarsening
1302
+ v∆ : T ′(Γ) → T (Γ/∆)
1303
+ is just the canonical epimorphism Γ → Γ/∆ extended to send ∞ to ∞. Suppose that v∆ is a
1304
+ Krasner valuation of norm ρ for some initial segment ρ of Γ/∆ containing 0Γ/∆. Then by (KVH2)
1305
+ for all γ ∈ Γ we would have that
1306
+ γ > 0 ⇐⇒ γ ∈ 0 ⊞′ 0 ⇐⇒ γ + ∆ > ρ + (0 + ∆).
1307
+ On the other hand, since 0Γ/∆ ∈ ρ , any positive γ ∈ ∆ would violate this equivalence.
1308
+ Nevertheless, we have the following result.
1309
+ Theorem 6.2. Let (F, v) be a Krasner valued hyperfield and let w be the Krasner valuation on F
1310
+ determined by
1311
+ Ow = {x ∈ F | x − x ⊆ 1 − 1}.
1312
+ Then w is equivalent to the coarsening of v corresponding to ig(ρv). In particular, the coarsening
1313
+ of a Krasner valuation v corresponding to ig(ρv) is a Krasner valuation.
1314
+ Proof. We show that these two valuations have the same valuation hyperring. Indeed, for all x ∈ F
1315
+ we have that vx + ig(ρv) ⪰ 0vF/ ig(ρv) if and only if vx ∈ ig(ρv) or vx > ig(ρv). In both cases by
1316
+ (KVH2) applied to v we have that
1317
+ y ∈ x − x ⇐⇒ vy > ρv + vx ⊇ ρv =⇒ y ∈ 1 − 1,
1318
+ i.e., y ∈ Ow. Conversely, vx + ig(ρv) ≺ 0vF/ ig(ρv) if and only if vx /∈ ig(ρv) and vx < 0. Therefore,
1319
+ ρv + vx ⊊ ρv and there exists y ∈ F such that vy ∈ ρv and vy /∈ ρv + vx. By (KVH2) applied to v,
1320
+ the former implies y /∈ 1 − 1 while the latter is equivalent to y ∈ x − x. We conclude that x /∈ Ow.
1321
+ We have proved that vig(ρv) and w have the same valuation hyperring in F. The theorem follows
1322
+ from Corollary 3.22.
1323
+
1324
+ Corollary 6.3. Let F be a hyperfield admitting two Krasner valuations v1 and v2 of norm ρ1 and
1325
+ ρ2, respectively. Then the coarsening of v1 corresponding to ig(ρ1) is equivalent to the coarsening
1326
+ of v2 corresponding to ig(ρ2).
1327
+ Corollary 6.4. Let (F, v) be a Krasner valued hyperfield. If ρv has trivial invariance group, then
1328
+ Ov = {x ∈ F | x − x ⊆ 1 − 1}.
1329
+ Corollary 6.5. Let F be a hyperfield with a superiorly canonical additive hypergroup. Then the
1330
+ norm of the Krasner valuation v determined on F by the valuation hyperring
1331
+ Ov = {x ∈ F | x − x ⊆ 1 − 1}
1332
+ has trivial invariance group.
1333
+
1334
+ 28
1335
+ LINZI, A.
1336
+ Corollary 6.6. Let F be a hyperfield with a superiorly canonical additive hypergroup. There is a
1337
+ unique (up to equivalence) Krasner valuation v on F such that ig(ρv) = {0}.
1338
+ Remark 6.7. In the model theory of valued fields, the RV-structure (mentioned in the introduction)
1339
+ of level γ of a valued field (K, v), where γ is a non-negative element of vK, is essentially the Krasner
1340
+ valued hyperfield3 Kγ := Kργ, where ργ := {δ ∈ vK | δ ≤ γ}. Since ig(ργ) = {0}, it follows, as a
1341
+ consequence of Corollary 6.4, that the valuation hyperring of the Krasner valuation on Kγ induced
1342
+ by v is always definable in the language of hyperfields, e.g., using the ternary relation z ∈ x + y to
1343
+ encode the multivalued operation +.
1344
+ 7. Further research
1345
+ There are at least three interesting points that have been touched but not developed further in
1346
+ this manuscript. Below we briefly describe them.
1347
+ (1) In the multivalued setting, (F, v), vF and Fv can all be described as (valued) hyperfields
1348
+ and they are not distinct structures. This applies in particular when F is a field.
1349
+ (2) In the literature there are not many examples of infinite hyperfields that are not factor
1350
+ hyperfields. Essentially, only the work of Massouros [39, 40] provides such examples. In
1351
+ all those examples M we have by definition x + x = M \ {x} for all x ∈ M ×. If v is a
1352
+ valuation on M and x ̸= 1, then x, x−1 ∈ 1 + 1, so by (V3) and Corollary 3.5 (iii) we have
1353
+ that vx = 0 must hold for all x ∈ M ×. It follows that M only admits the trivial valuation.
1354
+ Is it true that any hyperfield admitting a non-trivial valuation is a factor hyperfield?
1355
+ (3) In view of Theorem 5.2, a non-trivial valuation on a hyperfield F is a surjective homomor-
1356
+ phism onto a hyperfield H satisfying the following three conditions:
1357
+ • H has characteristic 2;
1358
+ • H has C-characteristic 1;
1359
+ • H is stringent.
1360
+ From this point of view, the image of a valuation is a quite special hyperfield. For example,
1361
+ one could think that responsible for the fact that finite (hyper)fields only admit the trivial
1362
+ valuation is the third property of H, since there are many examples of finite hyperfields
1363
+ satisfying 0, 1 ∈ 1+1. We speculate that investigating weakenings of the stringency property
1364
+ for H may lead to a notion of generalised valuation which have the potential to be applied
1365
+ also in the classical singlevalued setting e.g. to use (generalised) valuation-theoretic methods
1366
+ over finite fields.
1367
+ References
1368
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+ 5(6):6552–6579, 2020.
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+ [2] M. Baker and O. Lorscheid. Descartes’ rule of signs, Newton polygons, and polynomials over hyperfields. J.
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+ Algebra, 569:416–441, 2021.
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+ [3] S. A. Basarab. Relative elimination of quantifiers for Henselian valued fields. Ann. Pure Appl. Logic, 53(1):51–74,
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+ [7] A. Connes and C. Consani. From monoids to hyperstructures: in search of an absolute arithmetic. In Casimir
1379
+ force, Casimir operators and the Riemann hypothesis, pages 147–198. Walter de Gruyter, Berlin, 2010.
1380
+ 3This hyperfield is called the valued γ-hyperfield in [32]
1381
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1382
+ NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
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1399
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1400
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1401
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1402
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1404
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1405
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1406
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1419
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+ In Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches
1423
+ Mathématiques, pages 129–206. Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris, 1957.
1424
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+ Israel J. Math., 85(1-3):277–306, 1994.
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1438
+ (Uniwersytet Szczeciński), https://bip.usz.edu.pl/doktorat-habilitacja/16282/alessandro-linzi, 2022.
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+ arXiv:2211.05082, 2022.
1441
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+ Stockholm, pages 45–49, 1934.
1443
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1444
+ 201:636–638, 1935.
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1446
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1448
+ , 2021.
1449
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1453
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1454
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1455
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1456
+ [43] A. Prestel and C. N. Delzell. Mathematical logic and model theory. Universitext. Springer, London, 2011. A brief
1457
+ introduction, Expanded translation of the 1986 German original.
1458
+ [44] A. Prestel and P. Roquette. Formally p-adic fields, volume 1050 of Lecture Notes in Mathematics. Springer-
1459
+ Verlag, Berlin, 1984.
1460
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1461
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+ 2016.
1463
+ [47] O. Viro. Hyperfields for tropical geometry I. hyperfields and dequantization. arXiv:1006.3034, 2010.
1464
+ [48] M. Vuković. Remembering Professor Marc Krasner. Sarajevo J. Math., 12(25)(2, suppl.):283–298, 2016.
1465
+ Center for Information Technologies and Applied Mathematics, University of Nova Gorica, Slove-
1466
+ nia.
1467
+ Email address: alessandro.linzi@ung.si
1468
+
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1
+ Fault-tolerant error correction for a universal non-Abelian topological quantum
2
+ computer at finite temperature
3
+ Alexis Schotte,1, ∗ Lander Burgelman,2 and Guanyu Zhu1, 3, †
4
+ 1IBM Quantum, IBM Almaden Research Center, San Jose, CA 95120, USA
5
+ 2Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
6
+ 3IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA
7
+ We study fault-tolerant error correction in a quantum memory constructed as a two-dimensional model of Fi-
8
+ bonacci anyons on a torus, in the presence of thermal noise represented by pair-creation processes and measure-
9
+ ment errors. The correction procedure is based on the cellular automaton decoders originating in the works of
10
+ G´acs [1] and Harrington [2]. Through numerical simulations, we observe that this code behaves fault-tolerantly
11
+ and that threshold behavior is likely present. Hence, we provide strong evidence for the existence of a fault-
12
+ tolerant universal non-Abelian topological quantum computer.
13
+ I.
14
+ INTRODUCTION
15
+ Anyons are emergent quasi-particles that exist in
16
+ two-dimensional condensed matter systems and whose
17
+ exchange statistics generalize that of Bosons and
18
+ Fermions. These particles have spurred much interest
19
+ due to their potential applications for quantum compu-
20
+ tation. In particular, it was found that with certain types
21
+ of non-Abelian anyons, a universal quantum compu-
22
+ tation can be performed by braiding and fusing these
23
+ particles [3–5]. An intriguing benefit of this paradigm
24
+ is that, due to their topological nature, computations
25
+ are intrinsically robust to perturbations at zero tem-
26
+ perature. At non-zero temperature, however, thermal
27
+ anyonic excitations can corrupt the computation by
28
+ performing non-trivial braids with the computational
29
+ anyons. Since systems exhibiting anyonic excitations
30
+ have a spectral gap ∆, this source of errors can be sup-
31
+ pressed to some extent at temperatures T ≪ ∆/kB
32
+ as the density of thermal anyons scales as e−∆/kBT .
33
+ Alas, this passive protection does not suffice, because
34
+ the presence of thermal anyons is unavoidable at non-
35
+ zero temperatures when scaling up the size of com-
36
+ putations. Therefore, proficient active error correction
37
+ schemes for non-Abelian models are paramount for the
38
+ realization of topological quantum computers.
39
+ Besides their envisaged use for topological quantum
40
+ computation, topologically ordered systems (i.e., those
41
+ that support anyonic excitations on top of their ground
42
+ space) are also of much interest for quantum error cor-
43
+ rection. In particular, one of the characteristics of such
44
+ systems is a robust ground space degeneracy, which al-
45
+ ∗ alexis.schotte@posteo.net
46
+ † guanyu.zhu@ibm.com
47
+ lows one to use their ground space as the code space of
48
+ an error correcting code. This realization led to the dis-
49
+ covery of topological quantum error correcting codes,
50
+ which encode logical quantum states in topologically
51
+ ordered states of a system of qudits (typically arranged
52
+ on a two-dimensional lattice). Since their discovery
53
+ in the 90s, most research has focused exclusively on
54
+ Abelian topological codes such as the surface code and
55
+ the color code [5–14], which admit an elegant charac-
56
+ terization in terms of the stabilizer formalism [15]. Due
57
+ to their geometrical locality and high error thresholds,
58
+ these codes are considered to be promising candidates
59
+ for protecting quantum information from noise in error-
60
+ corrected quantum computers. One of the drawbacks
61
+ of Abelian topological codes, however, is that they do
62
+ not allow one to execute a universal set of logical gates
63
+ in a protected fashion in two dimensions. Hence, they
64
+ must be supplemented with additional protocols such
65
+ as magic state distillation [16] or code switching to
66
+ higher-dimensional codes [17, 18] , which introduce a
67
+ large space-time overhead [19]. Alternatively, there ex-
68
+ ist non-Abelian topological codes which do not suffer
69
+ from this inherent limitation, and are able to perform
70
+ a universal gate set natively within their code space in
71
+ two dimensions [3]. The trade-off is that such codes go
72
+ beyond the stabilizer formalism and are therefore very
73
+ hard to simulate classically.
74
+ While active error correction in Abelian anyon mod-
75
+ els and Abelian topological codes has been studied
76
+ extensively, quantum error correction based on non-
77
+ Abelian anyon models has not enjoyed the same fo-
78
+ cus. Nevertheless, important progress has been made
79
+ over the last decade, including both analytical proofs
80
+ and numerical demonstrations of threshold behavior
81
+ for various non-Abelian topological error correcting
82
+ codes [20–24]. Moreover, syndrome extraction circuits
83
+ for such non-Abelian string-net codes have been devel-
84
+ arXiv:2301.00054v1 [quant-ph] 30 Dec 2022
85
+
86
+ 2
87
+ oped in recent years [24, 25]. In addition, state prepa-
88
+ ration for non-Abelian codes based on the Kitaev quan-
89
+ tum double models via measurements has also been
90
+ proposed recently for the experimental implementa-
91
+ tion on qubit lattices [26, 27], although further devel-
92
+ opment is still needed in the context of fault-tolerant
93
+ state preparation. Notably, previous studies in this field
94
+ already include codes based on the Fibonacci anyon
95
+ model, which is universal for quantum computation
96
+ [23, 24]. In particular, a quantum memory of qubits
97
+ supporting doubled Fibonacci anyonic excitations was
98
+ found to have a threshold that lies remarkably close to
99
+ that of the surface code under similar assumptions [24].
100
+ These results, however, all assume perfect syndrome
101
+ measurements, which are topological charge measure-
102
+ ments in this context. As we aim to model more re-
103
+ alistic scenarios, we must take faulty measurements
104
+ into consideration. Again, much is known in the case
105
+ of Abelian topological codes [2, 7, 28–31]. For their
106
+ non-Abelian counterparts, one key result stands out: in
107
+ Ref. [32] a proof was formulated that topological codes
108
+ based on non-cyclic anyon models admit a error cor-
109
+ rection thresholds with faulty topological charge mea-
110
+ surements. While this result is remarkable, non-cyclic
111
+ anyon models are not universal for quantum computa-
112
+ tion, and it remains an open question whether similar
113
+ claims can be made for universal models.
114
+ In this work, we take a step towards demonstrat-
115
+ ing that fault-tolerance is indeed possible for universal
116
+ non-Abelian topological codes. To this end, we define
117
+ a quantum memory constructed as a two-dimensional
118
+ model of Fibonacci anyons on a torus. We study ac-
119
+ tive continuous quantum error correction on this model
120
+ in the presence of thermal noise represented by pair-
121
+ creation processes, and with faulty syndrome measure-
122
+ ments. The correction procedure is based on the cel-
123
+ lular automaton decoders originating in the works of
124
+ G´acs [1] and Harrington [2], and further studied in the
125
+ context of non-Abelian models in Ref. [32]. Through
126
+ numerical simulations, we study how the average mem-
127
+ ory lifetime changes with the error rate. The results in-
128
+ dicate that this code is indeed fault-tolerant, which is
129
+ strong evidence for the existence of fault-tolerant uni-
130
+ versal non-Abelian codes.
131
+ The structure of this work is as follows. In Sec. II
132
+ we introduce the topological Fibonacci code. We then
133
+ describe the details of the noise model in Sec. III and
134
+ introduce the cellular automaton decoder in Sec. IV.
135
+ We proceed by giving an outline of the numerical sim-
136
+ ulations performed in this work in Sec. V. Finally, we
137
+ present the corresponding numerical results in Sec. VI
138
+ and conclude with a discussion in Sec. VII.
139
+ II.
140
+ THE FIBONACCI CODE
141
+ We consider a two-dimensional model comprised of
142
+ hexagonal tiles laid out on the surface of a torus. The
143
+ resulting geometry can be represented as an L × L
144
+ hexagonal lattice with periodic boundary conditions in
145
+ both directions (Fig. 2). Each of these hexagonal tiles
146
+ can contain an excitation known as a Fibonacci anyon.
147
+ Anyons are point-like quasi-particle excitations
148
+ which can be characterized algebraically in terms of a
149
+ unitary modular tensor category (UMTC). A thorough
150
+ description of anyon models using UMTCs goes be-
151
+ yond the scope of this work, however, some details are
152
+ given in Sec. A. For now, it is sufficient to state that
153
+ an anyon model specifies a set of anyon labels, also
154
+ referred to as particle types, which can fuse according
155
+ to a specific set of fusion rules. The Fibonacci anyon
156
+ model considered in this work contains two labels, 1
157
+ and τ, which obey the fusion rules
158
+ 1×1 = 1 ,
159
+ 1×τ = τ×1 = τ ,
160
+ τ×τ = 1+τ . (1)
161
+ In general, one can associate a vector space to a
162
+ given set of anyons, where the basis vectors are la-
163
+ beled by the different ways in which the anyons can
164
+ fuse. This fusion space has a topological degeneracy,
165
+ and can therefore be used to robustly encode quan-
166
+ tum information. In particular, for the Fibonacci anyon
167
+ model the anyonic vacuum on a two-dimensional torus
168
+ has a twofold degeneracy [33]. Starting from our two-
169
+ dimensional model, we can therefore define an error
170
+ correcting code whose code space is identified with the
171
+ anyonic vacuum on the torus and which encodes a sin-
172
+ gle logical qubit. A basis for this code space can be
173
+ defined using Wilson line operators along the homo-
174
+ logically non-trivial cycles x and y shown in Fig. 1:
175
+ W a
176
+ x |1⟩x = Sa1
177
+ S11 |1⟩x ,
178
+ W a
179
+ x |τ⟩x = Saτ
180
+ S1τ |τ⟩x ,
181
+ a ∈ {1, τ} ,
182
+ (2)
183
+ We note that a different basis, {|0⟩y , |1⟩y}, can be
184
+ defined analogously by swapping the x and y labels
185
+ above, where the two bases are related through the
186
+ modular S matrix Sab =
187
+ y⟨a|b⟩x. For the Fibonacci
188
+ anyon model its numerical values are
189
+ S =
190
+ 1
191
+
192
+ 1 + φ2
193
+
194
+ 1
195
+ φ
196
+ φ −1
197
+
198
+ ,
199
+ (3)
200
+ where φ = 1 +
201
+
202
+ 5
203
+ 2
204
+ .
205
+ The action of the mapping class group on the any-
206
+ onic vacuum then corresponds to unitary operations on
207
+
208
+ 3
209
+ x
210
+ y
211
+ Figure 1: The two homologically non-trivial cycles on
212
+ a torus.
213
+ the code space. For the Fibonacci category, any logical
214
+ unitary operator can be realized in this way, up to arbi-
215
+ trary precision [3]. Therefore, the quantum error cor-
216
+ recting code defined above natively supports universal
217
+ quantum computation.
218
+ Errors in this code appear as spurious anyonic ex-
219
+ citations, which can corrupt the encoded informa-
220
+ tion if their world lines between creation and re-
221
+ annihilation are topologically non-trivial, i.e., form a
222
+ non-contractible cycle 1. The objective of error cor-
223
+ rection is then to systematically remove these spurious
224
+ excitations, without corrupting the quantum memory in
225
+ the process. This correction is performed in an active
226
+ and continuous manner, and can be broken down into a
227
+ series of discrete steps. At each step, a suitable recov-
228
+ ery operation is performed based on a measured list of
229
+ positions and types of the excitations, called the error
230
+ syndrome.
231
+ We conclude this section by noting that the numer-
232
+ ical simulation of the error-correction process requires
233
+ the introduction of some additional manipulations on
234
+ fusion states of multiple Fibonacci anyons. As these
235
+ are technical details that do not contribute to the in-
236
+ tuition of the procedure, we defer their definition to
237
+ Sec. A.
238
+ III.
239
+ NOISE MODEL AND CORRECTABILITY
240
+ Having defined our model and code space, we now
241
+ turn to the description of the noise model used in our
242
+ 1 Note that a pair of Fibonacci anyons can also fuse to a single non-
243
+ trivial anyon when one member of the pair has been transported
244
+ along a non-trivial cycle. Since the resulting state is no longer in
245
+ the code space, this does not constitute a logical operation on the
246
+ encoded information. However, one can show that the encoded
247
+ information is irrevocably lost in case of such an event [21].
248
+ simulations.
249
+ We model continuous active error cor-
250
+ rection in our Fibonacci code as a sequence of time
251
+ steps, where each time step itself consists of three parts:
252
+ the application of pair-creation noise, faulty syndrome
253
+ measurement, and error correction respectively.
254
+ At each time step, first, for each edge of the hexago-
255
+ nal lattice graph a pair of anyons is created across this
256
+ edge with a probability p. Immediately after each pair
257
+ creation event, the resulting charge in the two affected
258
+ tiles is sampled, effectively collapsing all superposi-
259
+ tions of anyonic charge within each tile to either 1 or
260
+ τ. After the pair creation noise has been applied, faulty
261
+ syndrome extraction is simulated by generating a list of
262
+ the anyon charge in all tiles, and flipping each outcome
263
+ individually with a probability q. In addition to the
264
+ charges that are correctly detected, the resulting faulty
265
+ syndrome can contain both “ghost defects” (indicating
266
+ a non-trivial charge when none is truly present) and
267
+ “missing defects” (failing to report a true non-trivial
268
+ charge). Finally, this faulty syndrome is passed to a de-
269
+ coder, introduced in the following section, which then
270
+ performs a set of local operations based on the current
271
+ (and past) syndrome information in an attempt to move
272
+ the system back towards the initial state.
273
+ After each time step, the current state of the sys-
274
+ tem is copied and it is checked whether it is still cor-
275
+ rectable. This is done by passing the copy to a clus-
276
+ tering decoder [22–24] and simulating a decoding pro-
277
+ cedure with perfect syndrome measurements starting
278
+ from this given initial state. If this perfect decoding
279
+ is successful, the memory is considered intact and the
280
+ simulation is continued. If perfect decoding is unsuc-
281
+ cessful, the memory is considered corrupted and the
282
+ simulation is aborted. The memory lifetime is then de-
283
+ fined as the number of time steps after which a perfect
284
+ clustering decoder can no longer successfully restore
285
+ the initial state.
286
+ For a given pair of tiles which share an edge, the
287
+ process of pair creation across this edge corresponds to
288
+ the matrix elements
289
+ ⟨a′, b′; c′| Upc |a, b; c⟩ = δc,c′F aa1
290
+ ττa′F a′τa
291
+ b c b′ ,
292
+ (4)
293
+ where we have used the F-symbols of the Fibonacci
294
+ category, given in (A5). Here, |a, b; c⟩ represents the
295
+ state where the affected tiles have anyon charges a
296
+ and b, respectively, with total charge c. This then de-
297
+ fines the probability distribution according to which
298
+ outcomes a′ and b′ are sampled.
299
+ Since our noise
300
+ model does not allow any superposition in the any-
301
+ onic charge of individual tiles, it should be considered
302
+ semi-classical rather than fully quantum-mechanical.
303
+
304
+ 4
305
+ (a)
306
+ (b)
307
+ Figure 2: (a) Pair-creation events creating anyonic
308
+ excitations in neighboring tiles. The dotted ellipse
309
+ represents a collapse to the total charge of the anyons
310
+ it contains. A ghost defect is shown in blue, a missing
311
+ defect is highlighted in orange with a cross. (b) The
312
+ outcome of this noise process represented on the
313
+ decoding graph. Note that the missing defect
314
+ (highlighted in orange) will (by definition) not be
315
+ visible in the syndrome.
316
+ Note, however, that this does not render our model
317
+ completely classical. Indeed, superpositions in the total
318
+ charge c of the affected tiles are an inherent part of the
319
+ state evolution that cannot be captured faithfully by any
320
+ classical probabilistic process. Furthermore, while the
321
+ extreme decoherence assumption for the anyon charge
322
+ in individual tiles greatly simplifies the numerical sim-
323
+ ulation outlined in this work, it was argued in Ref. [22]
324
+ that this decoherence is unlikely to have any tangible
325
+ influence on the observed memory lifetimes, as the es-
326
+ sential topological nature of the noise processes is still
327
+ captured correctly.
328
+ We note that this type of noise can be understood as
329
+ originating from the connection to a thermal bath with
330
+ inverse temperature β = 1/(kBT) determined by the
331
+ error rate p through the relation
332
+ p
333
+ 1 − p = e−β∆ .
334
+ (5)
335
+ Here, ∆ represents the energy required to create a pair
336
+ of anyonic excitations and place them in neighboring
337
+ tiles.
338
+ To conclude this section, we emphasize that in the
339
+ case of non-Abelian error correction, even decoding
340
+ with perfect syndrome measurements is still an inher-
341
+ ently stochastic procedure due to the indeterminacy of
342
+ anyonic charge measurements. This means that perfect
343
+ decoding can sometimes either be successful or unsuc-
344
+ cessful even starting from the same initial state. Our
345
+ definition of the memory lifetime therefore simply cor-
346
+ responds to a statistical estimate of the actual memory
347
+ lifetime. Furthermore, it is not known which decoder
348
+ is optimal for the Fibonacci code. Hence, the choice
349
+ for the clustering decoder to verify the correctability of
350
+ states is, in a way, an arbitrary one. This choice, how-
351
+ ever, is motivated by the recent discovery that the clus-
352
+ tering decoder yields high thresholds for a related er-
353
+ ror correcting code exhibiting doubled Fibonacci any-
354
+ onic excitations, and performed significantly better in
355
+ this context than decoders based on a perfect matching
356
+ strategy [24]. In any case, one should keep in mind that
357
+ the memory lifetime as defined above, does not repre-
358
+ sent the true memory lifetime. Instead the sub-optimal
359
+ verification process entails that it merely provides us
360
+ with a lower bound on the true value.
361
+ IV.
362
+ HARRINGTON’S CELLULAR AUTOMATON
363
+ DECODER
364
+ The model described above is paired with a decoder
365
+ which is a straight-forward adaptation of the cellu-
366
+ lar automaton decoder introduced in Ref. [2]. Previ-
367
+ ously, this decoder has also been used for a similar
368
+ phenomenological model of Ising anyons in Ref. [32],
369
+ where the existence of an error correction threshold
370
+ was proven analytically. At each time step during the
371
+ error correction simulation, based on the reported mea-
372
+ surement outcomes in the faulty syndrome, the decod-
373
+ ing algorithm will apply local transition rules to fuse
374
+ neighboring anyons or to move anyons to neighboring
375
+ tiles.
376
+ Intuitively these transition rules work as follows.
377
+ The lattice is divided into square colonies of size Q×Q.
378
+ At each time step, the transition rules will attempt to
379
+ fuse neighboring non-trivial anyons, as observed in
380
+ the faulty syndrome.
381
+ If a non-trivial anyon has no
382
+ neighbors, the transition rules will move it to the cen-
383
+ ter of its colony.
384
+ At larger timescales, higher-level
385
+ transition rules are applied on a renormalized lattice
386
+ where anyons located at colony centers will be fused
387
+ with anyons at neighboring colony centers, or moved
388
+ toward the center of their respective super-colonies,
389
+ which consist of Q × Q colonies. This renormalization
390
+ scheme is then continued at higher levels until eventu-
391
+ ally the Qn × Qn super-colony covers the entire lattice
392
+ for some integer n. To ensure the latter is possible, we
393
+ will always assume that the linear lattice size satisfies
394
+ L = Qn for some integer n. An example of these pro-
395
+ cesses is shown in Fig. 4.
396
+ To describe the action of the decoding algorithm
397
+ more precisely, we will define its action at different
398
+ renormalization levels k. The level-0 transition rules
399
+
400
+ 5
401
+ are those already discussed above and are applied at
402
+ every time step based on the reported faulty syndrome
403
+ obtained from the most recent round of faulty measure-
404
+ ments. The transition rules are applied to one location
405
+ at a time and take into consideration only the anyon
406
+ content of that site and of its eight neighbors. A de-
407
+ tailed definition of these rules is given in App. C. When
408
+ an anyon is moved from a site l to a neighboring site
409
+ l′, the (true) anyon content of site l is fused with that
410
+ of site l′ and the resulting charge is placed on site l′
411
+ while the charge of site l is restored to the vacuum.
412
+ This happens irrespective of whether or not the syn-
413
+ drome for both sites was correct. Hence, when the de-
414
+ coder attempts to move a ghost defect (a trivial charge
415
+ misidentified as a non-trivial one) to a neighboring site,
416
+ this process does not create additional excitations. This
417
+ does not, however, mean that mistaking a trivial charge
418
+ for a non-trivial one has no negative consequences. In-
419
+ deed, these wrong syndromes may cause the decoder
420
+ to stretch out existing errors.
421
+ The level-1 transition rules are not applied in every
422
+ time step, but only when t is a multiple of a param-
423
+ eter called the work period, which we will denote by
424
+ U. We require that U = b2 for some positive integer
425
+ b. One should think of U as the time scale at which a
426
+ coarse-graining is performed. Level-1 transition rules
427
+ act at on a coarse-grained lattice where the sites corre-
428
+ spond to the centers of the level-0 colonies, and these
429
+ are grouped into level-1 colonies of size Q2 × Q2.
430
+ Hence, the actions determined by the level-1 transition
431
+ rules involve pairs of level-0 colony centers separated
432
+ by a distance Q. An example of such a move is pro-
433
+ vided in Fig. 4(d). The transition rules themselves are
434
+ nearly identical to the level-0 rules, but are based on
435
+ two sets of level-1 syndromes s1,c and s1,n (defined
436
+ below) rather than one. For a site l (which is a level-
437
+ 0 colony center), the transition rules use s1,c(l) as the
438
+ anyon content of site, while the anyon content of its
439
+ neighbors (that is, the neighboring level-0 colony cen-
440
+ ters) is taken to be s1,n(l′).
441
+ The definitions of the level-1 syndromes s1,c and
442
+ s1,n require a pair of variables fc, fn ∈ [0, 1].
443
+ In-
444
+ tuitively these variables serve as detection thresholds
445
+ for the level-1 syndromes by determining the fraction
446
+ of measurements that must return a non-trivial out-
447
+ come at a site before it qualifies as a non-trivial level-1
448
+ syndrome. The proper definition, however, is slightly
449
+ more complicated and uses a coarse-grained counting
450
+ method. Below, we give the precise definition of s1,c,
451
+ the definition of s1,n is entirely analogous (using fn
452
+ instead of fc). We start by dividing the work period
453
+ U = b2, into b intervals of b time steps each. For
454
+ each of these intervals, we say a non-trivial syndrome
455
+ is present at a colony center l if a non-trivial charge
456
+ was reported there for at least fcb of the b time steps in
457
+ the interval. When at least fcb of the b intervals have
458
+ a non-trivial syndrome, s1,c(l) is set to one. A visual
459
+ example of this coarse-grained counting procedure is
460
+ shown in Fig. 3.
461
+ ≥ fc · b
462
+ < fc · b
463
+ ≥ fc · b
464
+ Figure 3: A visual example of the coarse-grained
465
+ counting procedure to determine the level-1 syndrome
466
+ for a level-0 colony center. The row of dots represent
467
+ U time steps, divided in b intervals of size b. The time
468
+ steps during which a non-trivial measurement
469
+ outcome was reported are indicated by the colored
470
+ dots. The crosses in the second row indicate in which
471
+ intervals the fraction of non-trivial measurement
472
+ outcomes is equal to or higher than fc.
473
+ The motivation for using two types of syndromes for
474
+ k > 0 is as follows. Suppose that an error spans across
475
+ two neighboring colonies, which we will label ρ and ρ′.
476
+ The level-0 transition rules will transport all resulting
477
+ anyons to the respective colony centers, where they can
478
+ now be acted upon by level-1 transition rules at the end
479
+ of the work period. Imagine that a non-trivial anyon
480
+ is now present at both colony centers. When consid-
481
+ ering the level-1 transition rules acting on ρ, there are
482
+ four possible scenarios for the syndromes s1,c(ρ) and
483
+ s1,n(ρ′). In case s1,c(ρ) = 0, the transition rules act
484
+ trivially on ρ. If both s1,c(ρ) = 1 and s1,n(ρ′) = 1
485
+ then the transition rules will be applied correctly and
486
+ the anyons will be fused. However, if s1,c(ρ) = 1 but
487
+ s1,n(ρ′) = 0, the transition rules may move the anyon
488
+ in ρ away from ρ′, thereby increasing the weight of
489
+ the error. Hence, it is desirable to set fc > fn to de-
490
+ crease the odds that when a level-k syndrome reports
491
+ an non-trivial anyon at a colony center, the level-k syn-
492
+ drome for its neighbors are false negatives. We must
493
+ be careful not to set fc too high or fn too low, how-
494
+ ever. If we choose fc to high, low-weight errors could
495
+ cause s1,c to never report any non-trivial charges, de-
496
+ laying any necessary corrections. Similarly, setting fn
497
+ too low will result in low-weight error triggering many
498
+ false positives for s1,n, which can cause the decoder to
499
+ make wrong decisions.
500
+
501
+ 6
502
+ Level-k
503
+ transition
504
+ rules
505
+ are
506
+ applied
507
+ when
508
+ t
509
+ mod U k = 0. They operate on a renormalized lattice
510
+ that uses the centers of level-(k − 1) colonies as sites,
511
+ and groups these into level-k colonies of size Qk ×Qk.
512
+ The level-k syndromes sk,c and sk,n are determined by
513
+ the coarse-grained counting method described above,
514
+ using bk intervals of bk time steps each. For linear sys-
515
+ tem size L, k ranges from 0 to kmax = logQ(L).
516
+ It is important to note that non-Abelian anyons do
517
+ not allow for instantaneous moves. Indeed, while one
518
+ can construct a unitary string operator for Abelian
519
+ anyons, no such operator can be constructed for the
520
+ non-Abelian case. This discrepancy can be traced back
521
+ to the fact that fusion outcomes are non-deterministic
522
+ for non-Abelian anyons, implying it is not possible to
523
+ move an non-Abelian anyon by annihilating it with one
524
+ member of a particle-antiparticle pair (as is done in
525
+ e.g., the surface code).
526
+ Therefore, the actions determined by level-k transi-
527
+ tion rules, for k > 0 cannot be applied withing a sin-
528
+ gle time step. Instead, they will be broken up into a
529
+ sequence of moves involving only pairs of neighbor-
530
+ ing sites which will be applied in Qk consecutive time
531
+ steps. We further limit the model by requiring that the
532
+ number recovery operations affecting a single tile in the
533
+ lattice (or site in the decoding graph), is no greater than
534
+ one in each time step. This allows all recovery opera-
535
+ tions applied in one time step to be performed in par-
536
+ allel. Hence, we must define a hierarchy determining
537
+ which actions (moves or fusions between neighboring
538
+ tiles) get prioritized based on the renormalization level
539
+ from which they originated. In our case, it was opted to
540
+ always prioritize correction processes from the highest
541
+ renormalization level 2
542
+ It was argued in [32] that the prohibition of instanta-
543
+ neous corrections would likely not influence the thresh-
544
+ old behavior other than slightly lowering the memory
545
+ lifetimes relative to a hypothetical case where this re-
546
+ striction is dropped. We explicitly verify this claim for
547
+ our Fibonacci model below in Sec. VI.
548
+ V.
549
+ OUTLINE OF THE SIMULATION
550
+ The goal of this work is to numerically determine
551
+ a fault-tolerant error threshold for the error correcting
552
+ 2 Note that if one were to prioritize the level-0 corrections, higher-
553
+ level correction could never be completed, as they would be un-
554
+ done immediately after their first action is applied.
555
+ (a)
556
+ (b)
557
+ (c)
558
+ (d)
559
+ Figure 4: Illustration of the transition rules on the
560
+ decoding graph. The gray disks represent non-trivial
561
+ syndromes, and the blue arrows represent the actions
562
+ suggested by the decoder. The blue dotted lines
563
+ represent the 3 × 3 colonies. (a-c) show a sequence of
564
+ level-0 transition rules and possible outcomes of those
565
+ actions. In (d) non-trivial anyons have been
566
+ transported to two neighboring colony centers, the
567
+ blue arrow represent a level-1 transition which could
568
+ be applied at the end of the work period.
569
+ code defined in Sec. II with pair-creation noise and
570
+ measurement noise as outlined in Sec. III, and with the
571
+ cellular automaton decoder introduced in Sec. IV. This
572
+ is achieved by performing Monte-Carlo simulations to
573
+ determine the average memory lifetime for a range of
574
+ system sizes and error rates. These results then allow
575
+ one to estimate the value of the error threshold.
576
+ A single Monte-Carlo sample (with some fixed val-
577
+ ues for the noise strength p and the measurement er-
578
+ ror rate q) is obtained as follows. First, the state of
579
+ the system is initialized as a ground state (i.e.: con-
580
+ taining no anyons). Then a sequence of time steps is
581
+ performed consisting of the application of pair-creation
582
+ noise with rate p, a round of faulty syndrome measure-
583
+ ments with error probability q, and finally a sequence
584
+ of recovery operations. At the end of each time step,
585
+ it is verified whether or not the state is considered cor-
586
+ rectable, according to the criteria specified in Sec. III.
587
+
588
+ 7
589
+ (a)
590
+ (b)
591
+ (c)
592
+ Figure 5: (a) Level-0 colonies of size Q × Q. (b)
593
+ Level-1 colonies defined as Q × Q level-0 colonies.
594
+ (c) Renormalized lattice used for the level-1 transition
595
+ rules.
596
+ This sequence of time steps is continued until one of
597
+ the following three outcomes occurs: (1) The largest
598
+ connected group of anyons grows too large, rendering
599
+ its classical simulation intractable 3; (2) A noise pro-
600
+ cess or recovery operation induces a logical error by
601
+ fusing a pair of anyons along a path that forms a non-
602
+ contractible loop when combined with their fusion tree;
603
+ (3) The verification procedure at the end of a time step
604
+ fails. The memory lifetime is then set as the number of
605
+ time steps that were completed. The course of a single
606
+ Monte-Carlo sample in the simulation is summarized
607
+ as pseudo-code in Alg. 1.
608
+ 3 Note that such cases are likely to correspond configurations in
609
+ which the initial state cannot be recovered.
610
+ Algorithm 1 Numerical simulation
611
+ initialize state
612
+ t = 0
613
+ while correctable with clustering decoder & no logical
614
+ errors made do
615
+ t ← t + 1
616
+ apply pair-creation noise
617
+ perform faulty measurements
618
+ for k = 0 : kmax do
619
+ if t mod U k = 0 then
620
+ update level-k syndromes
621
+ apply level-k transition rules
622
+ end if
623
+ end for
624
+ end while
625
+ memory lifetime = t
626
+ VI.
627
+ NUMERICAL RESULTS
628
+ The Monte Carlo simulation described above were
629
+ performed for various system sizes with p = q. The
630
+ following parameters were used:
631
+ Q = 3 ,
632
+ b = 7 ,
633
+ Fc = 0.7 ,
634
+ Fn = 0.2 .
635
+ The resulting average memory lifetimes for L = 3,
636
+ L = 9 and L = 27 are shown below in Fig. 6(a).
637
+ These results clearly indicate that the code pre-
638
+ sented in this work is indeed fault-tolerant. Further-
639
+ more, while the current data is not sufficient to demon-
640
+ strate a clear-cut fault-tolerant threshold, it still ex-
641
+ hibits threshold behavior and is remarkably similar to
642
+ the results previously obtained for the toric code [2]
643
+ and the Ising topological code [32]. We estimate that
644
+ the fault-tolerant threshold for the Fibonacci topologi-
645
+ cal with pair-creation noise and measurement noise lies
646
+ between p = 10−4 and p = 5 · 10−4, which corre-
647
+ sponds to an inverse temperature between β = 9.2/∆
648
+ and β = 7.6/∆. This is comparable to the threshold
649
+ found for the Ising topological code [32], and only one
650
+ order of magnitude below that for the surface code un-
651
+ der similar circumstances [2]. For physical error rates
652
+ near p = q = 10−4, corresponding to a temperature
653
+ 1/β one order of magnitude below the spectral gap, a
654
+ code of linear size L = 27 yields logical error rates of
655
+ the order 10−8.
656
+
657
+ 8
658
+ (a) Average memory lifetime in function of the error strength,
659
+ with p = q, for various system sizes. Each data point
660
+ represents the average over 1000 Monte Carlo samples. The
661
+ blue line shows the coherence time of a single physical qubit.
662
+ The average memory lifetimes for p ≤ 10−3 were fitted to a
663
+ function of the form f(p) ∼ p−a. The results for L = 3,
664
+ L = 9 and L = 27 are shown as the green, yellow and red
665
+ lines respectively.
666
+ (b) Average lifetime in function of the error strength with
667
+ p = q for various system sizes and (unphysical)
668
+ instantaneous corrections. Each data point represents the
669
+ average over 1000 Monte Carlo samples. The blue line shows
670
+ the coherence time of a single physical qubit.
671
+ Figure 6
672
+ A second round of simulations was performed to de-
673
+ termine average memory lifetimes with the assumption
674
+ that all corrections happen instantaneously. While this
675
+ is akin to the Abelian topological codes, where dis-
676
+ tant anyons can be fused using unitary string-operators,
677
+ this scenario is unphysical for non-Abelian anyons as
678
+ they do not admit unitary string-operators.
679
+ Never-
680
+ theless, it is worth studying to which extend the re-
681
+ sults in Fig. 6(a) are influenced by the restriction to
682
+ non-instantaneous recovery operations.
683
+ In Ref. [32]
684
+ it was conjectured that allowing instantaneous correc-
685
+ tions does not significantly change the qualitative be-
686
+ havior of the average memory lifetimes as a function
687
+ of the error rate, but mostly just increases the memory
688
+ lifetimes. Our results, shown in Fig. 6(b), confirm this
689
+ hypothesis.
690
+ VII.
691
+ DISCUSSION AND OUTLOOK
692
+ The results presented in this work demonstrate that
693
+ fault-tolerant error correction is possible for non-
694
+ Abelian topological quantum error correcting codes
695
+ supporting a universal logical gate set within their code
696
+ space.
697
+ For a code consisting of Fibonacci anyons
698
+ in hexagonal tiles on a two-dimensional torus, sub-
699
+ jected to pair creation noise and measurement noise,
700
+ we demonstrated that the cellular automaton decoder
701
+ detailed in this work is fault-tolerant. In particular, for
702
+ physical error rates p ≤ 10−3, it was found that the
703
+ logical memory lifetime surpasses the physical coher-
704
+ ence time for all system sizes. When interpreting the
705
+ pair-creation noise as resulting from a non-zero tem-
706
+ perature, this pseudo-threshold corresponds to an in-
707
+ verse temperature β = 6.9/∆, where ∆ is the energy
708
+ required to create a pair of Fibonacci anyons. Further-
709
+ more, our results suggest that this code admits a fault-
710
+ tolerant quantum error correction threshold around p =
711
+ 10−4, or β = 9.2/∆, which is similar to the fault-
712
+ tolerant threshold found for the Ising topological code
713
+ [32].
714
+ Several future research directions present them-
715
+ selves. First, more research on a possible fault-tolerant
716
+ threshold is necessary. Wile the numeric results pre-
717
+ sented in this work provide a strong indication that a
718
+ fault-tolerant error correction threshold exists, they do
719
+ not conclusively prove its existence, nor do they pro-
720
+ vide a precise estimate of its value. Hence, an impor-
721
+ tant open problem is the formulation of a mathematical
722
+ proof of its existence. Such proofs were previously for-
723
+ mulated for the toric code [2] and for non-cyclic non-
724
+ Abelian anyon models such as in the Ising topological
725
+
726
+ 9
727
+ code [32]. Due to the cyclic nature of Fibonacci anyons
728
+ (or any universal anyon model), however, the existing
729
+ proofs are not sufficient.
730
+ Second, it would be interesting to study different
731
+ decoders in an identical setting.
732
+ This includes both
733
+ different cellular-automaton decoders such as those in
734
+ Refs. [31, 34], as well as new decoders tailored to the
735
+ Fibonacci topological code.
736
+ Third, while this work demonstrates that the Fi-
737
+ bonacci topological code can be operated fault-
738
+ tolerantly as a quantum memory, results regarding
739
+ its use for fault-tolerant quantum computing are still
740
+ lacking.
741
+ We envisage that fault-tolerant topological
742
+ quantum computing at non-zero temperatures could be
743
+ achieved by combining the code and decoding proce-
744
+ dure presented in this work with the scheme for per-
745
+ forming Dehn twists presented in Refs. [35–37]. Alter-
746
+ natively, one can also perform transversal logical gates
747
+ in a folded Fibonacci code [38].
748
+ Finally, it would be of great interest to expand the
749
+ current results to microscopic models for non-Abelian
750
+ topological quantum error correction, such as the Fi-
751
+ bonacci Turaev-Viro code [24].
752
+ ACKNOWLEDGMENTS
753
+ The authors would like to thank Guillaume Dauphi-
754
+ nais and Jim Harrington for enlightening discussions
755
+ on the cellular automaton decoder. The computational
756
+ resources (Stevin Supercomputer Infrastructure) and
757
+ services used in this work were provided by the Flem-
758
+ ish Supercomputer Center (VSC), funded by Ghent
759
+ University, the Research Foundation Flanders (FWO),
760
+ and the Flemish Government. AS was supported by a
761
+ fellowship of the Belgian American Educational Foun-
762
+ dation. LB was supported by a PhD fellowship from
763
+ the FWO. GZ was supported by the U.S. Department
764
+ of Energy, Office of Science, National Quantum In-
765
+ formation Science Research Centers, Co-design Center
766
+ for Quantum Advantage (C2QA) under contract num-
767
+ ber DE-SC0012704.
768
+ Appendix A: Fibonacci anyons
769
+ Below, we give with a brief overview of the topolog-
770
+ ical aspects of our model. A thorough exposition of the
771
+ theory of anyon models [39–42] is beyond the scope
772
+ of this work, and we restrict to a basic description of
773
+ the aspects of the Fibonacci model that are required for
774
+ the specific purpose of our simulations. We refer to
775
+ Ref. [33] for an in-depth discussion of anyonic fusion
776
+ states on the torus.
777
+ The Fibonacci anyon model has two particle types,
778
+ 1 and τ, that satisfy the fusion rules
779
+ 1 × 1 = 1 ,
780
+ 1 × τ = τ × 1 = τ ,
781
+ τ × τ = 1 + τ .
782
+ (A1)
783
+ It is a non-Abelian anyon model, as the fusion of two
784
+ τ-anyons can yield two distinct outcomes. In this case,
785
+ the fusion space V c
786
+ ab associated to the fusion of anyons
787
+ a and b to c is one-dimensional, and is spanned by the
788
+ state vector |a, b; c⟩ which we will represent graphi-
789
+ cally as
790
+ |a, b; c⟩ →
791
+ b
792
+ a
793
+ c
794
+ .
795
+ (A2)
796
+ When fusing several anyons, their total charge is a col-
797
+ lective property of the anyons that does not depend on
798
+ the specific order in which they are fused. Mathemati-
799
+ cally, this is expressed through the associativity of the
800
+ fusion rules,
801
+ (a × b) × c = a × (b × c) .
802
+ (A3)
803
+ If we consider the case of three anyons a, b and c that
804
+ fuse to a total charge d then this fusion may be car-
805
+ ried out in two distinct ways, implying the existence
806
+ of two equivalent decompositions of the associated fu-
807
+ sion space V d
808
+ abc in terms of fusion states (A2). These
809
+ equivalent decompositions are related through a uni-
810
+ tary transformation called an F-move, which is repre-
811
+ sented graphically as
812
+ b
813
+ c
814
+ e
815
+ d
816
+ a
817
+ =
818
+
819
+ f
820
+ F abe
821
+ cdf
822
+ f
823
+ d
824
+ b
825
+ c
826
+ a
827
+ .
828
+ (A4)
829
+ The coefficients F abe
830
+ cdf in this expression are called F-
831
+ symbols of the anyon model. For the Fibonacci model
832
+ they are given by
833
+ F ττ1
834
+ ττ1 = 1
835
+ φ ,
836
+ F τττ
837
+ ττ1 = F ττ1
838
+ τττ =
839
+ 1
840
+ √φ ,
841
+ F τττ
842
+ τττ = − 1
843
+ φ ,
844
+ (A5)
845
+ where all other F-symbols consistent with the fusion
846
+ rules (A1) are equal to 1, and 0 otherwise.
847
+ In addition to this recoupling one can also consider
848
+ the exchange or braiding of pairs of anyons, which pre-
849
+ serves their total charge.
850
+ At the level of the fusion
851
+
852
+ 10
853
+ space, such an exchange corresponds to a basis trans-
854
+ formation to a basis associated to a different linear or-
855
+ dering of the anyons4. Within V c
856
+ ab there are two such
857
+ possible basis transformations,
858
+ b
859
+ a
860
+ c
861
+ = Rab
862
+ c
863
+ c
864
+ a
865
+ b
866
+ =
867
+
868
+ Rba
869
+ c
870
+ �∗
871
+ c
872
+ b
873
+ a
874
+ ,
875
+ (A6)
876
+ which we will refer to as a clockwise and a counter-
877
+ clockwise swap respectively. For the Fibonacci model
878
+ the R-symbols appearing in these expressions are given
879
+ by
880
+ Rττ
881
+ 1
882
+ = e
883
+ 4πi
884
+ 5 ,
885
+ Rττ
886
+ τ
887
+ = e− 3πi
888
+ 5 ,
889
+ (A7)
890
+ where all other Rab
891
+ c
892
+ allowed by the fusion rules are
893
+ equal to 1, and 0 otherwise.
894
+ As we are dealing with a system that allows for an
895
+ extensive amount of anyonic excitations, we will be in-
896
+ terested in fusion states of many anyons, a1, a2, ..., an,
897
+ with some total charge c.
898
+ This gives rise to an ex-
899
+ ponentially large topological Hilbert space V c
900
+ a1a2···an
901
+ spanned basis states of the form
902
+ a1
903
+ a2
904
+ c
905
+ b1
906
+ a3
907
+ an−1
908
+ an
909
+ b2
910
+ bn−3
911
+ bn−2
912
+ .
913
+ (A8)
914
+ When dealing with anyonic fusion states on surfaces of
915
+ higher genus, one must take into account additional de-
916
+ grees of freedom in these states that are related to the
917
+ anyonic charge that runs along non-contractible cycles.
918
+ On a torus, this results in two distinct descriptions of
919
+ fusion states, which are related through a basis change.
920
+ These are known as the inside and outside bases, and
921
+ are depicted in Fig. 7. A detailed discussion can be
922
+ found in Ref. [33]. We will simply refer to these ad-
923
+ ditional degrees of freedom as the handle labels of the
924
+ state.
925
+ 4 As opposed to the more conventional view of braiding as an active
926
+ transformation that maps between different fusion spaces, we have
927
+ opted for the equivalent framework of braiding as a passive basis
928
+ transformation, as the latter is more appropriate in view of our
929
+ specific model and numerical simulations.
930
+ (a)
931
+ (b)
932
+ Figure 7: Two possible basis choices for anyons on a
933
+ torus: a) the inside basis, b) the outside basis.
934
+ For our current purpose, we won’t need the full de-
935
+ scription of said handle labels. It is sufficient for us to
936
+ pick one basis, and subsequently set the total charge of
937
+ all anyons to the vacuum. We are left with only a single
938
+ label (1 or τ) representing the anyonic charge flowing
939
+ along the non-contractible cycle associated with our
940
+ basis choice. This is precisely the origin of the twofold
941
+ degeneracy of the anyonic vacuum on the torus, which
942
+ we have taken to be our code space.
943
+ In the this work we will always start from the code
944
+ state corresponding to a trivial handle label.
945
+ As a
946
+ change in handle label at any point during the error cor-
947
+ rection procedure must be the consequence of a topo-
948
+ logically non-trivial process which constitutes a logi-
949
+ cal error, any simulation is aborted at the occurrence
950
+ of such an event, meaning that all handle labels can be
951
+ safely ignored in the remainder of our discussion.
952
+ A basic functionality required for the simulation of
953
+ the error correction procedure is the ability to correctly
954
+ sample a measurement of the total charge of a given set
955
+ of anyons. Starting from a given fusion state, this can
956
+ be achieved by first transforming to a basis in which the
957
+ relevant anyons are fused sequentially. For a system
958
+ of many anyons these reordering basis transformations
959
+ are obtained by combining Eqs. (A4) and (A6), giving
960
+
961
+ 11
962
+ rise to the clockwise swap
963
+ aj
964
+ b3
965
+ b2
966
+ b1
967
+ aj+1
968
+ =
969
+
970
+ b′
971
+ 2
972
+ Bb1ajb2
973
+ aj+1b3b′
974
+ 2
975
+ aj
976
+ aj+1
977
+ b3
978
+ b2
979
+ b1
980
+ ,
981
+ (A9)
982
+ and the counterclockwise swap
983
+ aj
984
+ b3
985
+ b2
986
+ b1
987
+ aj+1
988
+ =
989
+
990
+ b′
991
+ 2
992
+
993
+ Bb1ajb2
994
+ aj+1b3b′
995
+ 2
996
+ �∗
997
+ aj
998
+ aj+1
999
+ b3
1000
+ b2
1001
+ b1
1002
+ ,
1003
+ (A10)
1004
+ where
1005
+ Bb1ajb2
1006
+ aj+1b3b′
1007
+ 2 =
1008
+
1009
+ c
1010
+ F aj+1ajc
1011
+ b3b1b′
1012
+ 2
1013
+ Rajaj+1
1014
+ c
1015
+ F b1ajb2
1016
+ aj+1b3c .
1017
+ (A11)
1018
+ By performing a certain set of these basis transforma-
1019
+ tions, the fusion order can always be made consistent
1020
+ with the group of anyons of which we want to mea-
1021
+ sure the total charge. Subsequently, the fusion state is
1022
+ recoupled such that the relevant group of anyons is con-
1023
+ nected to the rest of the state by a single edge c. For a
1024
+ charge measurement of a pair of anyons this recoupling
1025
+ takes the form
1026
+ aj
1027
+ b3
1028
+ b2
1029
+ b1
1030
+ aj+1
1031
+ =
1032
+
1033
+ c
1034
+ F b1ajb2
1035
+ aj+1b3c
1036
+ aj
1037
+ b3
1038
+ b1
1039
+ aj+1
1040
+ c
1041
+ ,
1042
+ (A12)
1043
+ and charge measurements of larger groups of anyons
1044
+ simply require consecutive applications of the recou-
1045
+ pling identity (A4). Finally, the charge outcome is ob-
1046
+ tained sampling the total charge c from the probability
1047
+ distribution corresponding to the resulting superposi-
1048
+ tion of fusion states.
1049
+ Appendix B: Classical simulatbility
1050
+ It is well known that Fibonacci anyons are universal
1051
+ for quantum computation [3]. One might therefore be
1052
+ tempted to conclude that the classical simulation of the
1053
+ topological Fibonacci code described in Sec. II with
1054
+ pair-creation noise is unlikely to succeed. However,
1055
+ as was noted in Ref. [23], the simulation of noise and
1056
+ error correction processes does not require the simula-
1057
+ tion of general anyon dynamics. In particular, individ-
1058
+ ual noise processes create distinct connected groups of
1059
+ anyons with vacuum total charge (or extend such exist-
1060
+ ing groups). These groups correspond to anyons that
1061
+ have interacted at some point during their lifetime, and
1062
+ must thus only be merged whenever a noise or error
1063
+ correction process involves two members from discon-
1064
+ nected groups. Since each connected group has a triv-
1065
+ ial total charge, braiding between disconnected groups
1066
+ is trivial. Hence, the total fusion space factorizes into a
1067
+ tensor product of fusion spaces of individual connected
1068
+ groups, and we are only required to simulate anyon dy-
1069
+ namics within each of these groups separately. This
1070
+ factorization of the fusion space is illustrated in Fig. 8.
1071
+ The creation and subsequent merging of discon-
1072
+ nected groups of anyons by noise and recovery pro-
1073
+ cesses can be thought of as a kind of percolation pro-
1074
+ cess.
1075
+ Hence, below the percolation threshold, one
1076
+ expects that the size of the largest connected group
1077
+ scales as O(log(L)) (with variance O(1)), where L
1078
+ is the linear system size [43]. As this is a probabilis-
1079
+ tic statement, there will be instances where the largest
1080
+ connected group has a size larger than O(log(L), but
1081
+ the probability of such events is suppressed exponen-
1082
+ tially with the system size L. This logarithmic scal-
1083
+ ing of the largest cluster size s = O(log(L)) coun-
1084
+ ters the exponential scaling of the dimension of the fu-
1085
+ sion space d = O(exp(s)) for individual connected
1086
+ groups. Therefore, the fusion spaces of individual con-
1087
+ nect groups will have dimension dim = O(poly(L)),
1088
+ meaning that the dynamics within connected groups
1089
+ can be simulated efficiently.
1090
+ Exploiting the tensor product structure within the to-
1091
+ tal fusion space, requires the use of basis for the any-
1092
+ onic fusion space which reflects this structure and can
1093
+ be updated dynamically to keep track of noise and re-
1094
+ covery processes. This is best achieved by using the
1095
+ framework of curve diagrams, which were introduced
1096
+ in Ref. [23] (also see Ref. [44] for a more rigorous
1097
+ treatment using the language of modular functors), and
1098
+ also discussed extensively in Ref. [24]. Since these are
1099
+ merely a technical tool for keeping track of the most
1100
+ appropriate basis during the numerical simulations, we
1101
+ will refrain from discussing them here. Interested read-
1102
+ ers are referred to the aforementioned references for
1103
+ details.
1104
+
1105
+ 12
1106
+ V ab
1107
+ 1 ⊗ V cde
1108
+ 1
1109
+ ⊗ V fg
1110
+ 1
1111
+ V ab
1112
+ 1 ⊗ V cde
1113
+ 1
1114
+ ⊗ V fg
1115
+ 1
1116
+ (a)
1117
+ a
1118
+ b
1119
+ c
1120
+ d
1121
+ e
1122
+ f
1123
+ g
1124
+ x
1125
+ y
1126
+ V abcdefg
1127
+ 1
1128
+ = ⊕x,y
1129
+
1130
+ V ab
1131
+ x ⊗ V x cde
1132
+ y
1133
+ ⊗ V y fg
1134
+ 1
1135
+
1136
+ (b)
1137
+ V ab
1138
+ 1 ⊗ V cde
1139
+ 1
1140
+ ⊗ V fg
1141
+ 1
1142
+ a
1143
+ b
1144
+ c
1145
+ d
1146
+ e
1147
+ f
1148
+ g
1149
+ (c)
1150
+ Figure 8: (a) A set of noise processes. (b) A generic
1151
+ state in the full fusion space of all anyons created by
1152
+ the noise processes involves superpositions in the
1153
+ labels x and y. (c) The factorized Hilbert space which
1154
+ is sufficient to represent the state.
1155
+ Appendix C: Transition rules for Q = 3
1156
+ Below, we give a full definition of the transition rules
1157
+ for Q = 3. It is possible to define more general tran-
1158
+ sition rules that apply for any colony size, examples of
1159
+ such rules can be found in Refs. [2] and [32]. However,
1160
+ since the numerical simulations performed in this work
1161
+ were performed to Q = 3 we have taken the freedom
1162
+ to tailor the transition rules to this case specifically.
1163
+ • North-West
1164
+ if sk,c(ρ) = 0, do nothing;
1165
+ else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
1166
+ else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
1167
+ else if sk,n(ρ + (0, 1)) ̸= 0, move east;
1168
+ else if sk,n(ρ + (1, 0)) ̸= 0, move south;
1169
+ else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;
1170
+ else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
1171
+ else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;
1172
+ else move south;
1173
+ • North
1174
+ if sk,c(ρ) = 0, do nothing;
1175
+ else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
1176
+ else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
1177
+ else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;
1178
+ else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;
1179
+ else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
1180
+ else move south;
1181
+ • North-East
1182
+ if sk,c(ρ) = 0, do nothing;
1183
+ else if sk,n(ρ + (0, 1)) ̸= 0, move east;
1184
+ else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;
1185
+ else if sk,n(ρ + (1, 1)) ̸= 0, move south;
1186
+ else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
1187
+ else if sk,n(ρ + (0, −1)) ̸= 0, move west;
1188
+ else if sk,n(ρ + (1, 0)) ̸= 0, move south;
1189
+ else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
1190
+ else move south-west;
1191
+ • East
1192
+ if sk,c(ρ) = 0, do nothing;
1193
+ else if sk,n(ρ + (0, 1)) ̸= 0, move east;
1194
+ else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;
1195
+ else if sk,n(ρ + (1, 1)) ̸= 0, move south;
1196
+ else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
1197
+ else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;
1198
+ else move west;
1199
+ • South-East
1200
+ if sk,c(ρ) = 0, do nothing;
1201
+ else if sk,n(ρ + (0, 1)) ̸= 0, move east;
1202
+ else if sk,n(ρ + (1, 0)) ̸= 0, move south;
1203
+ else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;
1204
+ else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;
1205
+ else if sk,n(ρ + (1, 1)) ̸= 0, move south;
1206
+ else if sk,n(ρ + (0, −1)) ̸= 0, move west;
1207
+ else move north;
1208
+
1209
+ 13
1210
+ • South
1211
+ if sk,c(ρ) = 0, do nothing;
1212
+ else if sk,n(ρ + (1, 0)) ̸= 0, move south;
1213
+ else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;
1214
+ else if sk,n(ρ + (1, 1)) ̸= 0, move east;
1215
+ else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
1216
+ else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;
1217
+ else move north;
1218
+ • South-West
1219
+ if sk,c(ρ) = 0, do nothing;
1220
+ else if sk,n(ρ + (1, 0)) ̸= 0, move south;
1221
+ else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;
1222
+ else if sk,n(ρ + (1, 1)) ̸= 0, move east;
1223
+ else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
1224
+ else if sk,n(ρ + (−1, 0)) ̸= 0, move north;
1225
+ else if sk,n(ρ + (0, 1)) ̸= 0, move east;
1226
+ else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
1227
+ else move north-east;
1228
+ • West
1229
+ if sk,c(ρ) = 0, do nothing;
1230
+ else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
1231
+ else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
1232
+ else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;
1233
+ else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;
1234
+ else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
1235
+ else move east;
1236
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@@ -0,0 +1,2058 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:2301.03101v1 [cs.IT] 8 Jan 2023
2
+ 1
3
+ Massive MIMO and NOMA
4
+ Bits-per-Antenna Efficiency under Power
5
+ Allocation Policies
6
+ Thiago A. Bruza Alves, Taufik Abrão
7
+ State University of Londrina (UEL), Department of Electrical Engineering, Londrina-PR, Brazil.
8
+ Abstract—A comparative resource allocation analysis in terms
9
+ of received bits-per-antenna spectral efficiency (SE) and energy
10
+ efficiency (EE) in downlink (DL) single-cell massive multiple-input
11
+ multiple-output (mMIMO) and non-orthogonal multiple access
12
+ (NOMA) systems considering a BS equipped with many (M)
13
+ antennas, while K devices operate with a single-antenna, and the
14
+ loading of devices ρ = K
15
+ M ranging in 0 < ρ ≤ 2 is carried out under
16
+ three different Power Allocations (PA) strategies: the inverse
17
+ of the channel power allocation (PICPA), a modified water-
18
+ filling (∆-WF) allocation method, and the equal power allocation
19
+ (EPA) reference method. Since the two devices per cluster are
20
+ overlapped in the power domain in the NOMA system, the channel
21
+ matrix requires transformation to perform the zero-forcing (ZF)
22
+ precoding adopted in mMIMO. Hence, NOMA operating under
23
+ many antennas can favor a group of devices with higher array gain,
24
+ overcoming the mMIMO and operating conveniently in the higher
25
+ loading range 0.6 < ρ < 2.0. In such a scenario, a more realistic
26
+ and helpful metric consists in evaluating the area under SE and
27
+ EE curves, by measuring the bit-per-antenna and bit-per-antenna-
28
+ per-watt efficiency, respectively. Our numerical results confirm a
29
+ superiority of NOMA w.r.t. mMIMO of an order of 3x for the
30
+ SE-area and 2x for the EE-area metric.
31
+ Index Terms—Non-Orthogonal Multiple Access (NOMA); mas-
32
+ sive Multiple-Input Multiple-Output (mMIMO); Energy Efficiency
33
+ (EE); Spectral Efficiency (SE).
34
+ I. Introduction
35
+ The beyond Fifth Generation (5G) of wireless communica-
36
+ tion systems must allow ultra-dense connections with vastly
37
+ heterogeneous requirements. The challenges in networks per-
38
+ sist, including the Spectral Efficiency (SE) and the Energy
39
+ Efficiency (EE) joint improvement, the increase in the SE-
40
+ EE trade-off, and Quality of Service (QoS), always aiming
41
+ to meet the growing number of devices connected to the
42
+ network. Among the proposals to solve these challenges, the
43
+ massive Multiple-Input Multiple-Output (mMIMO) system is
44
+ the primary proposed system that allows the increase of the
45
+ link capacity, exploring the propagation of multiple paths with
46
+ the use of a large number of antennas at the Base Station
47
+ (BS) [1], [2]. Another relevant enabling technology is the
48
+ Non-Orthogonal Multiple Access (NOMA), which explores the
49
+ power domain as an alternative way in terms of multiple ac-
50
+ cess technology, helping to mitigate the spectrum exhaustion
51
+ problem and serving more than one device per resource block
52
+ [3].
53
+ Although in many works mMIMO is classified as an or-
54
+ thogonal technique, allocating the signal from devices in the
55
+ same resource block, possible by spatial diversity, allows us to
56
+ This work was partly supported by The National Council for Scien-
57
+ tific and Technological Development (CNPq) of Brazil under Grants
58
+ 310681/2019-7, partly by the CAPES- Brazil - Finance Code 001, and
59
+ the Londrina State University - Paraná State Government (UEL).
60
+ T. A. Bruza Alves and Taufik Abrão are with the State University of
61
+ Londrina (UEL), Department of Electrical Engineering, Londrina-PR, E-
62
+ mails: thiagobruza@hotmail.com, taufik@uel.br
63
+ classify it as a non-orthogonal technique too [4]. There is a
64
+ vast literature demonstrating the superior performance of the
65
+ Spectral Efficiency of NOMA when compared to Orthogonal
66
+ Multiple Access (OMA) techniques [5]. Previous aims to
67
+ improve the communication system performance by combining
68
+ MIMO (with a small number of antennas M) and NOMA have
69
+ been discussed in [6]–[9].
70
+ Studies comparing NOMA and mMIMO in a single cell are
71
+ proposed in [4], [10], [11]. The acquisition of Channel State
72
+ Information (CSI) through pilot acquisition to NOMA system
73
+ is proposed in [10]. In [11], the application of NOMA in the
74
+ mMIMO scheme is proposed, and better results are achieved
75
+ in the proposed comparative. Moreover, in [4] is analyzed the
76
+ performance of NOMA and mMIMO in line of sight and non-
77
+ line of sight.
78
+ The canonical mMIMO refers to the systems with BSs
79
+ formed by a large number of antennas M when compared
80
+ to the number of actives devices, K, succinctly M ≫ K is
81
+ considered a mMIMO setup. The typical NOMA improves the
82
+ SE by superposing the signals of the selected devices to form
83
+ a cluster in the power domain, multiplexing it over the same
84
+ signal and served by the same beamforming. Nonetheless,
85
+ the success of NOMA depends on the Successive Interference
86
+ Cancellation (SIC).
87
+ Power-domain NOMA can be a candidate technology in
88
+ dense networks [12]. To improve performance and minimize
89
+ the impact to assume the perfect SIC [13], devices are divided
90
+ into two groups. After grouping in pairs and forming a cluster,
91
+ each pair forms a cluster with a high difference between
92
+ channel conditions. The device with a higher channel condition
93
+ can decode the signal sent to the device with the lower channel
94
+ condition. The interference can thus be eliminated by SIC. The
95
+ use of NOMA in BS equipped with a large number of antennas
96
+ was investigated in terms of SE [4], [10] we propose in a similar
97
+ configuration system increasing the loading up to 2 times the
98
+ number of antennas in BS and analyze the SE, EE, and SE-EE
99
+ trade-off.
100
+ The EE metric is a popular figure of merit employed to
101
+ analyze the balance between power consumption and data
102
+ rate. The EE is the ratio between the effectively transmitted
103
+ data rate and the total power expended during the transmis-
104
+ sion process, including instantaneous and static components.
105
+ With the EE metric, it is possible to evaluate the efficiency
106
+ with which a system uses the limited energy resource to
107
+ communicate data and optimize this ratio. Can show the
108
+ tendency of energy consumption in the case of seeking justice
109
+ among devices.
110
+ The Zero Forcing (ZF) is simple and popular alternative
111
+ interference suppression beamforming under perfect CSI con-
112
+ dition and achieving a satisfactory condition in real situations
113
+ when imperfect CSI, in this work, we adopt perfect CSI, for
114
+
115
+ 2
116
+ that the pilots are needed. The adoption of NOMA system
117
+ with a large number of antennas requires a defined equivalent
118
+ channel to be deployed for interference mitigation; and ac-
119
+ cording to the NOMA principle, makes the equivalent channel
120
+ matrix smaller than the original one due to the exploration of
121
+ power domain in NOMA.
122
+ Various transmission topologies already deal with the EE
123
+ problem in mMIMO, finding the optimal number of antennas,
124
+ number of devices in a cell, and the maximal EE [2], [14]. The
125
+ EE analysis in the NOMA system is carried out in [15], and its
126
+ superiority is demonstrated when compared with conventional
127
+ orthogonal multiple access (OMA) systems. Recent researches
128
+ seek to improve the NOMA performance, e.g. in [16], the
129
+ minimum pairing distance is defined and compared to the
130
+ OMA, while in [17] it’s presented a comparison between OMA
131
+ and cell-free system equipped with mMIMO-NOMA. An EE
132
+ analysis in Terahertz (THz)-NOMA-Multiple-Input Multiple-
133
+ Output (MIMO) was proposed in [18]. Still, the number of
134
+ active devices is much smaller than the number of antennas
135
+ in the BS, and [19] is a survey about Power Domain NOMA
136
+ and makes clear the vacuum of EE analysis and comparison
137
+ between NOMA with many antennas and mMIMO.
138
+ Recent works propose the deployment of NOMA combined
139
+ with other techniques a more effective transmission scheme;
140
+ e.g., in [20] NOMA and mMIMO are jointly considered in a
141
+ two-tier network for accommodating colossal traffic. Further-
142
+ more, in [21], authors apply NOMA in Distributed Antenna
143
+ Systems (DAS), aiming to achieve better performance when
144
+ compared to the conventional NOMA or DAS technique alone.
145
+ While [22] shows an in-depth survey of the state-of-the-art
146
+ of power-domain NOMA variants; moreover, several open
147
+ issues and research challenges of NOMA-based applications
148
+ are systematized. The NOMA system presents drawbacks,
149
+ such as hardware (including SIC) complexity, channel feed-
150
+ back, receiver design, and careful power and pilot allocation
151
+ strategies [12], [19], [23].
152
+ This work focus on revealing the advantages of applying
153
+ the mMIMO scheme versus NOMA scheme with a massive
154
+ number of BS antennas, and varying the loading of devices,
155
+ i.e., the ratio of the number of mobile devices to the number
156
+ of BS antennas, ρ =
157
+ K
158
+ M , while we change the PA strategy.
159
+ Besides, we adopt a realistic model for the system’s power
160
+ consumption as in [2] but adapted to our needs, aiming at
161
+ providing a suitable analysis of the system resource allocation.
162
+ Contributions: the contributions of this work are fourfold.
163
+ a) an extensive and comparative analysis on the spectral
164
+ efficiency (SE) performance of mMIMO system against NOMA
165
+ system, varying the system loading under specific (three dif-
166
+ ferent) power allocation methods and making use of the area
167
+ under the SE (Ssyst) curve of the system as an effective, useful
168
+ and fair metric of performance and efficiency; b) we develop
169
+ an energy efficiency (EE) analysis using a detailed model of
170
+ energy consumption, with fixed and variable terms related to
171
+ circuitry power consumption with number of antennas and
172
+ devices, respectively, providing an extensive and comparative
173
+ analysis on both the NOMA and mMIMO systems under
174
+ realistic operation scenarios and making use of the area under
175
+ the EE (Esyst) curve of system; c) an analysis on the SE-EE
176
+ trade-off is developed considering a wide range of loading of
177
+ devices, verifying the fairness between devices; d) finally, under
178
+ mild conditions, we provide evidences for the NOMA’s ability
179
+ to serve a greater number of devices than mMIMO system.
180
+ The remainder of the paper is organized as follows. Section
181
+ II describes the system models for NOMA and mMIMO
182
+ adopted in this work. In Section III we present the proposed
183
+ EE-SE formulation for NOMA and massive MIMO systems.
184
+ Numerical results are analyzed in IV. Section V concludes the
185
+ paper.
186
+ Notation. In this work, boldface lower case and upper case
187
+ characters denote vectors and matrices, respectively. The
188
+ operator (x)+ = max(0, x). The operators [·]T, E[·] and
189
+ |·| denote transpose, expectation and cardinality, respectively.
190
+ A random vector x ∼ CN {0, Im} is circularly symmetric
191
+ Gaussian distributed with mean 0 and covariance matrix Im.
192
+ Im is m × m identity matrix.
193
+ II. System Models
194
+ Let us consider a multi-user single-cell Downlink (DL)
195
+ transmission operating in a Time Division Duplex (TDD) with
196
+ K single-antenna actives devices, communicating with one BS,
197
+ which is equipped with M transmit antennas in Non-Line-Of-
198
+ Sight (NLOS). The set K is formed by K devices, these devices
199
+ are randomly distributed in a radius disk dmax, the disk area
200
+ is formed by two sub-disk with the same number of devices
201
+ in each sub-area; both subsets are identified as KH and KL.
202
+ In the first subset, KH represents devices’ indexes having the
203
+ higher channel coefficient and sort in descending order, while
204
+ the other subset KL are formed by the devices with lower
205
+ channel coefficient and sort in ascending order; the indexes
206
+ k ∈ KH and k ∈ KL such that:
207
+ K = KH ∪ KL,
208
+ where
209
+ KH = {1, ..., K/2} and KL = {K/2 + 1, ..., K}.
210
+ (1)
211
+ The channel vector modeling of device k can be described liked
212
+ as:
213
+ hk =
214
+
215
+ βkh′
216
+ k,
217
+ k = 1, ..., K,
218
+ (2)
219
+ where βk is the large-scale fading coefficient and satisfy
220
+ βj > βi,
221
+ ∀j ∈ KH, ∀i ∈ KL.
222
+ (3)
223
+ Herein, the pathloss model in [dB] is defined as:
224
+ βk = β0 + 10 · ξ · log10(dk),
225
+ (4)
226
+ where dk is the distance of user k to BS, ξ is the pathloss
227
+ coefficient, and β0 is the attenuation at the distance of
228
+ reference.
229
+ In each coherence interval, h′
230
+ k in (2) for device k is an
231
+ independent random small-scale fading realization from an in-
232
+ dependent Rayleigh fading distribution, h′
233
+ k ∼ CN(0, IM), k =
234
+ 1, ..., K . The transmitted signal xk ∈ CM is the beamformed
235
+ data symbol of device k:
236
+ xk = gk
237
+ √pksk,
238
+ (5)
239
+ where gk is a normalized beamforming vector, pk normalized
240
+ transmission power and sk ∼ CN(0, 1) is the data symbol of
241
+ device k, and period Ts. The signal received at the k-th device:
242
+ yk =
243
+ K
244
+
245
+ k′=1
246
+ hT
247
+ k xk′ + nk,
248
+ =
249
+
250
+ βkh′T
251
+ k
252
+ K
253
+
254
+ k′=1
255
+ gk′√pk′sk′ + nk,
256
+ (6)
257
+ =
258
+
259
+ βkpkh′T
260
+ k gksk +
261
+
262
+ βkh′T
263
+ k
264
+ K
265
+
266
+ k′̸=k
267
+ k′=1
268
+ gk′√pk′sk′ + nk,
269
+ where nk ∼ CN(0, 1) is the additive noise. Notice that this
270
+ modeling applies to both NOMA and mMIMO systems, but
271
+ beamforming is selected differently, and this topic will be
272
+ addressed in the next sections.
273
+
274
+ 3
275
+ A. Prior Actions
276
+ Because the BS needs to know a priori crucial information
277
+ related to the channel and devices distributed in the cell,
278
+ including device location, rate demanded, and channel coeffi-
279
+ cient, such required a priori information may differ depending
280
+ on the multiple access scheme considered [12].
281
+ The option for the mMIMO and NOMA systems was carried
282
+ out with the guarantee that the needs of the devices would
283
+ be met, and this initial step was carried out successfully.
284
+ Subsection III-E briefly discusses the preliminary information
285
+ required to proceed with different Power Allocation (PA)
286
+ procedures in both mMIMO and NOMA systems.
287
+ B. Pilot Overhead for Channel State Information
288
+ Fig. 1 compares the pilot-data transmission structure along
289
+ the one-channel coherence interval for both mMIMO and
290
+ NOMA systems considered. Notice that Ts is the time required
291
+ to transmit a data symbol (Data). Moreover, the channel
292
+ coherence time interval T is assumed to be a multiple of the
293
+ Data symbol period, T = ι · Ts. The power allocated to each
294
+ pilot in the training step is enough. In contrast, the number of
295
+ pilots, and the dedicated portion of coherence interval for data
296
+ transmission are assumed to be the same in both systems.
297
+ !"
298
+ !"
299
+ #"
300
+ $%&'(
301
+ #)()
302
+ #"
303
+ #"
304
+ *+,+-
305
+ .-+/
306
+ -0123'41510612,0(157)&
307
+ (%*1
308
+ Figure 1: Coherence time interval structure: the training and
309
+ data transmission structure for mMIMO and NOMA schemes
310
+ under TDD NLOS setup.
311
+ Notice that in NOMA transmission, the pilot transmission
312
+ step is split into two portions, half for the UL transmission
313
+ pilots and receive the DL pilot confirmation; this happens
314
+ because to perform SIC, the cell-center devices need to learn
315
+ the effective channels that are established by the beamforming.
316
+ Additionally, the beamforming vectors are based on cell-center
317
+ devices, producing limited rates achieved in cell-edge devices.
318
+ On the other hand, in the mMIMO scheme, a significant
319
+ advantage is that there is no need for DL pilots since the
320
+ effective channels created by the beamforming are highly
321
+ predictable, i.e. nearly deterministic gain and phase due to
322
+ channel hardening effect [4].
323
+ Assumption 1: In the NOMA system, the power allocated to
324
+ each downlink pilot is sufficient to reach the destination device.
325
+ C. Beamforming for NOMA and mMIMO systems
326
+ At the mMIMO system, each device is served by a sin-
327
+ gle beamforming vector. The ZF technique is a popular
328
+ interference-suppressing beamforming scheme in the mMIMO
329
+ system since it eliminates all inter-user interference using
330
+ individual beamforming for each device, while the favorable
331
+ propagation facilitates such interference suppressing in massive
332
+ MIMO configurations. Besides, to perform ZF precoding in
333
+ NOMA system, it is essential to understand the NOMA user-
334
+ pairing concept.
335
+ User-pairing: Inherent to the NOMA system, user clustering
336
+ can be performed in several ways after the user-sorting and
337
+ the user classification in center-users and edge-users subsets.
338
+ Because we know that the SE of NOMA is directly proportional
339
+ to the difference between the pathloss of the users, a natural
340
+ choice consists in pairing users with as higher as possible
341
+ pathloss differences [6]:
342
+ ∆βk = βk − βK+1−k,
343
+ (7)
344
+ forming the cluster k for k = 1, ..., K/2. With the pair formed,
345
+ carefully beamforming vectors selection is required. Hence, in
346
+ NOMA we assume that the beamforming vector for paired
347
+ users is the same, i.e., gk = gK+1−k for all k = 1, ..., K/2.
348
+ Assumption 2: In user-pairing procedure, we assume that
349
+ the paired users are aligned with the BS so that the same
350
+ beamforming can serve all paired users simultaneously. Hence,
351
+ by admitting that each pair of devices is spatially aligned with
352
+ the BS, and using localizing tools described, for instance, in
353
+ [23], [24], one should assume further a priori user-pairing step
354
+ in NOMA systems.
355
+ Assumption 3: In NOMA system, beamforming serves more
356
+ than one aligned device simultaneously; specifically, in this
357
+ paper, two aligned devices per cluster are admitted according
358
+ to the user-pairing step, while eliminating the inter-cluster
359
+ interference (favorable propagation) under adopted perfect
360
+ CSI conditions.
361
+ In this work we adopt the linear ZF precoding as defined by
362
+ the vector:
363
+ gk = h′k(h′T
364
+ k h′k)−1,
365
+ (8)
366
+ and satisfying h′T
367
+ i gk
368
+ =
369
+ 0, ∀ i
370
+ ̸=
371
+ k, i.e., the favorable
372
+ propagation effect between users belonging to distinct clusters.
373
+ III. SE-EE in NOMA and mMIMO systems
374
+ We discuss the SE and EE configurations in the NOMA and
375
+ mMIMO systems. The operation of the NOMA system requires
376
+ pairing devices so that the channel coefficients of the devices
377
+ in the same cluster must be appropriately different, enabling
378
+ power domain usage. As already mentioned, the interference
379
+ cancellation process via beamforming presents problems that
380
+ we will demonstrate below.
381
+ A. Data Rates in NOMA with ZF
382
+ Devices are divided into two sets like described in Eq. (1),
383
+ and these groups are represented by Eq. (9) by their large-
384
+ scale fading coefficient and are grouped into pairs forming a
385
+ cluster as Fig. 2, the cluster k is formed by one device in cell
386
+ center set KH and one device in cell edge set KL. Hence, the
387
+ devices are grouped into two subsets:
388
+ KH ={β1 > β2 > ... > βk/2},
389
+ (center devices set)
390
+ (9)
391
+ KL ={βK < βk−1 < ... < βk/2+1}. (edge devices set)
392
+ The user-pairing adopted in Eq. (9) is the same as proposed
393
+ in [6], creating the largest possible difference in channel
394
+ coefficients for devices not yet paired.
395
+ Assumption 4: In this paper, we assume perfect SIC, and only
396
+ one perfect SIC stage per cluster is performed, since just 2
397
+ devices per cluster are admitted.
398
+
399
+ 4
400
+ !"
401
+ Figure 2: System Model indicating the Paring formation in
402
+ NOMA system. Both mMIMO and NOMA systems deploy the
403
+ same massive number of antennas at base-station, M.
404
+ The instantaneous Signal to Interference plus Noise Ratio
405
+ (SINR) of devices in cluster k is defined as:
406
+ SINRk =
407
+ βkpk|h′T
408
+ k gk|2
409
+ βk
410
+ �K
411
+ k′̸=k pk′|h′T
412
+ k gk′|2 + 1
413
+ .
414
+ (10)
415
+ In each cluster, the cell-edge devices treat the interference as
416
+ noise and decode their data symbols, whereas the cell-center
417
+ device can decode the data symbols of the cell-edge device
418
+ and perform SIC, hence effectively removing the interference
419
+ due to the cell-edge device under Assumption 3.
420
+ To perform SIC, the cell-center device needs to be able to
421
+ decode data signal intended for the cell-edge device, i.e., the
422
+ ergodic SINR of the cell-edge device, sinrK+1−k, at device
423
+ k, defined as sinrk,K+1−k, must be greater than or equal
424
+ to the ergodic SINR of the k-th cell-center device. Hence,
425
+ given the uplink (mMIMO and NOMA) and downlink (NOMA)
426
+ pilot overhead and assuming perfect CSI in all receivers, and
427
+ admitting Assumption 4, the following condition must be
428
+ satisfied [4], [10]:
429
+ E[SINRk,K+1−k] ≥ E[SINRk],
430
+ (11)
431
+ where
432
+ SINRk,K+1−k =
433
+ βkpK+1−k|h′T
434
+ k gk|2
435
+ βk
436
+ �K
437
+ k′̸=K+1−k pk′|h′T
438
+ k gk′|2 + 1
439
+ .
440
+ (12)
441
+ Herein, the condition in (11) must be satisfied by selecting the
442
+ transmit powers appropriately.
443
+ The achievable ergodic rate of devices in cluster k, i.e.
444
+ device k in KH subset and device K+1−k in KL subset, under
445
+ Assumptions 1–4, is given by the ergodic rate contribution of
446
+ user-center device:
447
+ Rnoma
448
+ k
449
+ = τE [log2 (1 + SINRk)] ,
450
+ ∀k ∈ KH
451
+ (13)
452
+ in [bits/s/Hz], and for the user-edge device:
453
+ Rnoma
454
+ K+1−k = τE [log2 (1 + SINRK+1−k)] ,
455
+ ∀k ∈ KL (14)
456
+ where τ = (1− K·Ts
457
+ T ), is the portion of each channel coherence
458
+ interval (T) that is used for data transmission.
459
+ Assuming perfect channel state information, ZF precoding
460
+ for inter-clusters interference elimination, and using random
461
+ matrix theory results [25], the k-th cluster NOMA achievable
462
+ rate is obtained plugging eq. (10), (13) and (14):
463
+ Rnoma
464
+ cl-k
465
+ = τE
466
+
467
+ log2
468
+
469
+ 1 + ¯
470
+ Mβkpk
471
+ ��
472
+ +
473
+ (15)
474
+ τE
475
+
476
+ log2
477
+
478
+ 1 + βK+1−kpK+1−k
479
+ βK+1−kpk + 1
480
+
481
+ ,
482
+
483
+ ∀k ∈ K and ¯
484
+ M = M + 1 − K/2. Hence, the NOMA system
485
+ can operate until K < 2M − 1. A detailed derivation of the
486
+ expressions on this section can be found in [4] and [10].
487
+ B. Data Rates in mMIMO with ZF
488
+ In the mMIMO system with ZF precoding the ergodic
489
+ achievable rate for device k is given by:
490
+ Rm-mimo
491
+ k
492
+ = τE [log2 (1 + SINRk)] ,
493
+ [bits/s/Hz]
494
+ (16)
495
+ where SINRk is defined in (10). Hence, the above mMIMO
496
+ achievable rate equation becomes:
497
+ Rm-mimo
498
+ k
499
+ = τE [log2 (1 + (M − K)pkβk)] ,
500
+ [bits/s/Hz],
501
+ (17)
502
+ where (M − K) is obtained using random matrix theory,
503
+ representing the coherent array gain of the received signal [25].
504
+ Under linear precoding and combiners, the mMIMO system
505
+ operates consistently when K < M. Finally, the average
506
+ system sum-rate (avg-sum-rate) is defined simply by:
507
+ Rm-mimo =
508
+ K
509
+
510
+ k=1
511
+ Rm-mimo
512
+ k
513
+ and
514
+ Rnoma =
515
+ KH
516
+
517
+ k=1
518
+ Rnoma
519
+ cl-k .
520
+ The mMIMO system equations have been thoroughly investi-
521
+ gated in literature and can be found in [25] and [26].
522
+ C. Energy Efficiency
523
+ EE metric is the ratio of the number of effective bits of
524
+ information received over the total energy consumed by the
525
+ overall system to transmit and receive/decode such informa-
526
+ tion. The system data rate can determine the number of
527
+ effective information bits received at the destination. Power
528
+ consumption required for processing the signal at the trans-
529
+ mitter and receiver side is often neglected; in this sense, it
530
+ is calculated just as proportional to the radiated transmitted
531
+ power. The growth in the number of antennas in the BS and
532
+ the increased number of devices in 5G systems can lead to
533
+ unattainable EE goals. In general, the average EE can be
534
+ expressed as:
535
+ EE =
536
+ �K
537
+ k=1 Rk
538
+ Ptot
539
+ ,
540
+ [bits/Joule/Hz],
541
+ (18)
542
+ where Ptot is the total power consumption across the com-
543
+ munication system. It should consider transmission power
544
+ consumption, such as RF power amplifier inefficiency, base-
545
+ band signal processing, and cooling, among others. Therefore,
546
+ a more realistic and detailed energy consumption model is
547
+ required.
548
+ Based on [2], the adopted power consumption model in our
549
+ work considers two power terms: a) fixed-term; b) terms scaled
550
+ with the number of antennas M and the number of devices
551
+ K. The scaled terms occur because of the transceiver chains,
552
+ coding/decoding, channel estimation, and precoding. Let the
553
+ computational efficiency be L operations per joule in BS. We
554
+ describe it as follows:
555
+ RF Power: Prf is the power consumed to transmit the signal
556
+ to active devices achieved the SINR target and 0 < ̟ ≤ 1 is
557
+ the efficiency of the power amplifier.
558
+
559
+ β1dcenl
560
+ 50mβ2K/2
561
+ β
562
+ K
563
+ +1
564
+ 2KOK-1
565
+ umac5
566
+ Fixed consumption: Pfixed is the power consumed at the
567
+ BS which is independent of the number of transmit antennas
568
+ and devices in the cell, is formed of term P0 including the
569
+ power consumption of backhaul infrastructure, control signal-
570
+ ing, baseband processor, and term Psyn a single oscillator used
571
+ in all BS.
572
+ Pfixed = P0 + Psyn
573
+ Dependence only on K: PK is formed by the consumption
574
+ to coding and modulation of information symbols to devices,
575
+ represented by Pcod, the consumption to BS decoded the K
576
+ sequences of information symbols, defined by Pdec, and the
577
+ received power, represented by PRX, still composes this term,
578
+ multiplied by K as well. In addition, a portion of the ZF
579
+ precoding cost [27] depends only on K3.
580
+ PK = K(Pcod + Pdec + PRX) + K3
581
+ 2
582
+ 3LT
583
+ Dependence only on M. The term PM represents the
584
+ transmitted power (PTX), hence
585
+ PM = MPTX
586
+ Dependence on K and M: the term PKM is the cost of the
587
+ ZF precoding (due to LU-based matrix inversion) [27], which
588
+ depends on the number of devices, the number of antennas,
589
+ and the vector information symbol.
590
+ PKM = MK 3 + T
591
+ T L
592
+ + MK2 2
593
+ T L
594
+ Adding the portions, we obtain the overall power consump-
595
+ tion of the system:
596
+ Ptot = Prf
597
+ ̟ + Pfixed + PK + PM + PKM
598
+ [W].
599
+ (19)
600
+ D. Power Allocation Strategies
601
+ In the sequel, we present three well-known and frequently
602
+ applied strategies for power allocation. Still, due to the in-
603
+ herent characteristics of NOMA, we propose modifications on
604
+ the classical water-filling (WF) algorithm to enable application
605
+ in the NOMA system. Such modifications, namely ∆-WF,
606
+ ensure that none of the paired devices are dropped-out without
607
+ undoing the pairing of devices. To guarantee a certain level of
608
+ power disparities in each paired device, the power allocation ∆-
609
+ WF procedure in the NOMA system has two steps: a) first, we
610
+ allocated power for the clusters; b) we allocate power between
611
+ paired devices. Thereby, we could analyze the behavior of the
612
+ systems and compare their results.
613
+ Notice that both mMIMO and NOMA systems deploy the
614
+ same massive number of antennas at base-station, M. Hence,
615
+ due to the channel hardening effect [1], [25] inherent to
616
+ massive MIMO configuration, the small-scale fading vanishes
617
+ across the M antennas equipped with a linear ZF precoding
618
+ with vector as eq. (8). Hence, one can consider just the
619
+ pathloss coefficients βk as the main parameter in the power
620
+ allocation policies of systems based on a massive number of
621
+ antennas.
622
+ 1) Equal Power Allocation (EPA): Equal Power Allocation
623
+ (EPA) power allocation is deployed as a simple, naive strategy,
624
+ where all devices are served with the same power. In mMIMO,
625
+ all devices are served with the same transmission power
626
+ regardless of their distance from the BS. In NOMA, power
627
+ allocation has two steps. In the first step, the power is allocated
628
+ equally between the pairs, and then we allocate each device’s
629
+ power equally. The EPA strategy applied to mMIMO can be
630
+ defined by:
631
+ pk = Prf
632
+ K
633
+ [W],
634
+ ∀ k ∈ K.
635
+ (20)
636
+ In the case of EPA procedure applied to NOMA, it is composed
637
+ of two steps: in the first step, power reference to each cluster
638
+ can be defined simply as:
639
+ pcl
640
+ ref = 2 · Prf
641
+ K
642
+ [W],
643
+ ∀ k ∈ KH.
644
+ (21)
645
+ In the second step, the power allocation among the devices in
646
+ the same cluster is defined as:
647
+ pcl-k
648
+ k
649
+ = pcl-k
650
+ K+1−k = pcl
651
+ ref
652
+ 2 .
653
+ (22)
654
+ 2) Proportional
655
+ Channel
656
+ Inversion
657
+ Power
658
+ Allocation
659
+ (PICPA): is another power allocation technique adopted in
660
+ this study. Unlike the EPA technique, which applies the same
661
+ power to all devices, PICPA applies more power to devices
662
+ with the worst channel conditions, favoring fairness across
663
+ the devices. Such power allocation penalizes the average sum
664
+ rate in favor of fairness among all users.
665
+ The Proportional to the Inverse of the Channel Power Al-
666
+ location (PICPA) strategy applied to mMIMO can be defined
667
+ as:
668
+ pk = Prf
669
+ β−1
670
+ k
671
+ �K
672
+ k=1 β−1
673
+ k
674
+ [W],
675
+ ∀k ∈ K,
676
+ (23)
677
+ while the PICPA procedure applied to NOMA follows two
678
+ steps; in the first step, the power is allocated equally among
679
+ the K/2 clusters:
680
+ pcl
681
+ ref = 2 · Prf
682
+ K
683
+ [W],
684
+ ∀k ∈ KH,
685
+ (24)
686
+ after that, the power of each device within the k-th cluster is
687
+ defined by allocating more power to the device with smaller
688
+ large-scale fading βk:
689
+ pcl-k
690
+ k
691
+ = pcl
692
+ ref
693
+ βK+1−k
694
+ βk − βK+1−k
695
+ ,
696
+ and
697
+ pcl-k
698
+ K+1−k = pcl
699
+ ref − pcl-k
700
+ k
701
+ ,
702
+ (25)
703
+ where pcl-k
704
+ k
705
+ is the power allocated to the device k in the cl-k
706
+ cluster, and pcl-k
707
+ K+1−k is the power allocated to the device (K +
708
+ 1 − k), also belonging to the k-th cluster.
709
+ 3) Classical Water-Filling (WF) Algorithm: The application
710
+ of the Water-Filling (WF) algorithm in mMIMO system results
711
+ in an optimal (maximum) system sum-rate solution. However,
712
+ some devices are dropped out of the service due to the dete-
713
+ riorated channel condition. The WF power allocation strategy
714
+ for mMIMO is described as:
715
+ µ = 1
716
+ |K|
717
+
718
+ Prf +
719
+ |K|
720
+
721
+ k=1,k∈K
722
+ 1
723
+ βk
724
+
725
+  ,
726
+ (26)
727
+ Prf =
728
+ |K|
729
+
730
+ k=1,k∈K
731
+ pk ,
732
+ where
733
+ pk =
734
+
735
+ µ − 1
736
+ βk
737
+ �+
738
+ , ∀k ∈ K
739
+ and
740
+ p = [p1, p2, ..., p|K|],
741
+ with the operator (z)+ = max(0, z). Notice that the con-
742
+ strained value for the total power available is set to Prf [W].
743
+ The Algorithm 1 describes the classical WF procedure.
744
+ On the other hand, the direct application of WF algorithm
745
+ in the NOMA system implies harming the pair formation, i.e.,
746
+ devices present in the KL set are effectively dropped-out of
747
+ the service, undoing the pair. We propose a modification in
748
+ classical WF like the following to allow some comparison.
749
+
750
+ 6
751
+ Algorithm 1: Classical Water Filling (WF) for mMIMO
752
+ Input: K,Prf, K = |K|
753
+ 1 NP ̸= ⊘;
754
+ 2 while (NP ̸= ⊘) do
755
+ 3
756
+ solve Eq. (26) → p;
757
+ 4
758
+ NP ← identify null positions in p;
759
+ 5
760
+ K/{k}NP ← exclude from p devices labeled as NP
761
+ 6 end
762
+ Output: p = [p1, p2, . . . , p|K|]
763
+ 4) ∆-WF for NOMA: The application of classical WF in
764
+ NOMA in the same way as it is applied to mMIMO causes
765
+ some formed pairings to be broken, since after WF algorithm
766
+ application, some devices are dropped-out from the system,
767
+ making the NOMA power difference (∆) in the devices of the
768
+ same cluster vanish. Hence, we suggest modifying the classical
769
+ WF procedure to be applied to NOMA accordingly. The steps
770
+ of the ∆-WF algorithm are described as follows.
771
+ In NOMA, the power allocation has two steps, in the first
772
+ step, the allocation is between clusters. Hence, to prevent the
773
+ formed pairs from being broken, we propose the application of
774
+ WF based on the large-scale fading differences of the paired
775
+ devices, as defined in eq. (7): ∆βk = (βk − βK+1−k) inside
776
+ each KL and KH subsets, eq. (9). In the second step of
777
+ the procedure, the power is allocated between the devices
778
+ inside the group, assuming perfect successive interference
779
+ cancellation (SIC); for this to be possible, the condition in Eq.
780
+ (11) must be satisfied. The new water-level in the modified
781
+ ���-WF power allocation strategy for NOMA is defined by
782
+ µ = 2
783
+ KH
784
+
785
+ Prf +
786
+ |KH|
787
+
788
+ k=1,k∈KH
789
+ 1
790
+ ∆βk
791
+
792
+  ,
793
+ (27)
794
+ Prf =
795
+ |KH|
796
+
797
+ k=1,k∈KH
798
+ pcl-k,
799
+ ∀k ∈ KH
800
+ where
801
+ pcl-k =
802
+
803
+ µ −
804
+ 1
805
+ ∆βk
806
+ �+
807
+ ,
808
+ and
809
+ pcl-k = [p1, p2, ..., p|KH|],
810
+ In the second step, the power allocation to both devices in the
811
+ k-th cluster is defined as:
812
+ pcl-k
813
+ K+1−k = pcl-k
814
+ k
815
+ = pcl-k
816
+ 2
817
+ (28)
818
+ Algorithm 2 summarize the proposed ∆-WF power allocation
819
+ procedure, aiming to improve the SE of NOMA systems.
820
+ Algorithm 2: ∆-WF (modified) for NOMA systems
821
+ Input: KH, KL, Prf
822
+ 1 NP ̸= ⊘;
823
+ 2 while (NP ̸= ⊘) do
824
+ 3
825
+ solve Eq. (27) → pcl-k;
826
+ 4
827
+ NP ← identify null position in pcl-k;
828
+ 5
829
+ KH/{k}NP ← exclude from pcl-k devices labeled as
830
+ NP
831
+ 6 end
832
+ Output: pcl-k = [p1, p2, . . . , p|KH|]
833
+ Complexity analysis: In a comparative analysis of complexity,
834
+ the ∆-WF algorithm for power allocation in NOMA system
835
+ (Algorithm 2) performs two simple additional operations com-
836
+ pared to the classical WF procedure (Algorithm 1): a) in eq.
837
+ (27) the subtraction in (βk − βK+1−k); and b) the division
838
+ by two in (28). Besides, NOMA vs. mMIMO systems require
839
+ different a priori information to proceed accordingly with the
840
+ PA procedure.
841
+ E. Prior Information for Power Allocation Step
842
+ For implementing the PA policies, prior information is re-
843
+ quired at the BS, as defined in Table I. Some of this necessary
844
+ information can be obtained through the dedicated pilot trans-
845
+ mission step, at the cost of some overhead, as described in
846
+ Section II-B. Moreover, a preliminary step is known, in which
847
+ the spatial localization and path loss estimation of all devices
848
+ must be realized. With such a priori information availability,
849
+ the user-sorting and user-pairing steps can be performed.
850
+ Table I: Prior information required to PA procedure
851
+ PA
852
+ βk
853
+ h′
854
+ k
855
+ K
856
+ KH
857
+ KL
858
+ pk, eq.
859
+ EPA
860
+ mMIMO
861
+
862
+
863
+ ▲∗
864
+
865
+
866
+ (20)
867
+ NOMA
868
+
869
+
870
+
871
+
872
+
873
+ (21), (22)
874
+ PICPA
875
+ mMIMO
876
+ ▲∗
877
+
878
+
879
+ ▲∗
880
+ ▲∗
881
+ (23)
882
+ NOMA
883
+
884
+
885
+
886
+ ▲∗
887
+
888
+ (24), (25)
889
+ WF
890
+ mMIMO
891
+ ▲∗
892
+
893
+
894
+ ▲∗
895
+ ▲∗
896
+ (26), Alg. 1
897
+ ∆-WF
898
+ NOMA
899
+
900
+
901
+
902
+ ▲∗
903
+
904
+ (27, 28), Alg. 2
905
+ ▲ information needed a prior
906
+ ∗ obtained via Pilot Overhead
907
+ − Information not needed
908
+ IV. Numerical Results
909
+ The numerical evaluations for the proposed analyses of
910
+ NOMA and mMIMO systems are presented in this section.
911
+ The simulation system and channel parameter values de-
912
+ ployed along this section are depicted in Table II. The BS
913
+ is located at the center of cell and equipped with massive
914
+ M BS transmit antennas in typical non-line-of-sight (NLOS)
915
+ channel propagation scenario. At the same time, the devices
916
+ are randomly distributed in the cell area and split into two
917
+ subsets, KL and KH, as illustrated in Fig. 2. In all simulations,
918
+ we consider a block fading model where the time-frequency
919
+ resources are divided into coherence time intervals (T), in
920
+ which the channels remain constant and frequency flat, and
921
+ it is measured in multiple of symbol transmit period (Ts).
922
+ The system and channel scenarios have been simulated using
923
+ Matlab 2019 software running under one Intel HD Graphics
924
+ 6000 GPU, Intel(R) Dual-Core(TM) I5 CPU @ 1.6 GHz, and
925
+ 8 GB RAM.
926
+ Table II:
927
+ Simulation Parameters
928
+ Parameter
929
+ Value
930
+ BS antennas
931
+ M = 64, 128 and 256
932
+ Max. # Devices in the cell
933
+ K = ζ · M (NOMA)
934
+ K = M (mMIMO)
935
+ Cell loading
936
+ ρ = K/M
937
+ Total RF power available
938
+ Prf = 1W
939
+ Pairs NOMA / Clusters
940
+ N = K/ζ = K/2
941
+ NOMA devices per cluster
942
+ ζ = 2
943
+ # antennas per device
944
+ 1
945
+ Cell edge length
946
+ dmax = 350 m
947
+ Strong device position
948
+ [dmin; d1] ∈ [50; 100] m
949
+ Weak device position
950
+ [d2; dmax] ∈ [150; 350] m
951
+ Array gain MIMO device
952
+ M − K
953
+ Array gain NOMA kH
954
+ M + 1 − K/2
955
+ Data symbol period
956
+ Ts
957
+ Coherence time interval
958
+ T = 512 · Ts,
959
+ ι = 512
960
+ Channel
961
+ Pathloss exponent
962
+ ξ = 3.78
963
+ Attenuation at a d0 reference
964
+ β0 = 130 [dB]
965
+ # Monte-Carlo realizations
966
+ 1000
967
+
968
+ 7
969
+ A. Spectral Efficiency Comparison
970
+ The mMIMO and NOMA SE performance analysis is carried
971
+ out in this subsection, by increasing the number of devices
972
+ two by two until the loading limit ρ = 2. The results consider
973
+ M = 64, 128 and 256 BS antennas. In Fig. 3. (a) the results
974
+ of SE are achieved when the RF power available is allocated
975
+ following the EPA strategy, where each device receives the
976
+ same PA values. The mMIMO system overcame NOMA in all
977
+ situations when the loading ρ < 0.6. However, the NOMA
978
+ system achieves a higher SE than the mMIMO for each M
979
+ scenario when the loading of devices increases, ρ > 0.6. The
980
+ maximum avg-SE is 373 [bits/s/Hz], being attained with ZF-
981
+ NOMA M = 256 antennas and ρ ≈ 0.76. Besides, one can
982
+ infer that the mMIMO does not work with a loading higher
983
+ than 1, due to the array gain reaching 0 at full loading of
984
+ devices, while NOMA works suitably until the loading attains
985
+ M · ζ, where ζ is number of devices per cluster.
986
+ Fig. 3.(b) depicts the results of SE achieved in the mMIMO
987
+ and NOMA systems when the PICPA method is applied to
988
+ allocate the available RF power per device along the BS an-
989
+ tennas. The maximum avg-SE in mMIMO system overcomes
990
+ the NOMA counterpart until the loading ρ exceeds ≈ 0.62 for
991
+ the three BS antenna configurations, M = 64, 128, and 256.
992
+ This PA technique provides more power to devices with the
993
+ worst channel condition, making the SE result reach maximum
994
+ values below the EPA.
995
+ Fig. 3.(c) depicts the conventional WF algorithm applied
996
+ to mMIMO. Under such a power allocation approach, we
997
+ highlight that forming pairs is unfeasible in the NOMA sys-
998
+ tem. Indeed, the WF algorithm can maximize the system SE
999
+ since it allocates more power to devices with better channel
1000
+ conditions. In contrast, devices under bad channel conditions
1001
+ (below the water level) are dropped out of the service.
1002
+ The classical WF algorithm has been adapted to the NOMA
1003
+ system dropped-out always a pair of devices. Such adaptation
1004
+ reveals substantial improvements of avg-sum-rate when M
1005
+ is low compared to classical WF PA in mMIMO. The ∆-
1006
+ WF power allocation procedure preserves the pairs clustering
1007
+ formation in the NOMA system, allocating more power to
1008
+ the cluster with a higher difference between coefficients of
1009
+ large-scale fading. Fig. 3.(c) shows that the maximum avg-SE
1010
+ ≈ 361 [bits/s/Hz], which is achieved under ρ = 1 (K ≈ 256
1011
+ devices) when the modified WF is deployed in NOMA system.
1012
+ Moreover, when the number of BS antennas is lower (M = 64
1013
+ or 128), the NOMA achieved a peak higher than mMIMO,
1014
+ e.g., for M = 64 antennas, the peak of SE mMIMO occurs
1015
+ at loading ρ ≈ 0.7, while the NOMA SE peaks at ρ ≈ 1.2.
1016
+ However, as the number of BS antennas grows, the NOMA
1017
+ SE advantage decreases.
1018
+ Number of active devices after PA procedure. Fig. 4 shows
1019
+ the number of actives device after applying PA methods: in
1020
+ the EPA and PICPA algorithms, all devices remain activated.
1021
+ However, in classical WF mMIMO system when the number
1022
+ of device increases beyond ρ ≈ 0.25, half of the devices are
1023
+ dropped-out; while in ∆-WF NOMA PA, the percentage of
1024
+ active devices is always higher, e.g. higher than 70% for ρ ≈
1025
+ 1.1 and M = 256 antennas, the worst case.
1026
+ Fig. 5 summarizes avg-sum-rate surfaces in terms of SE
1027
+ ×ρ × M results achieved by NOMA with EPA, mMIMO
1028
+ with WF, and NOMA with modified ∆-WF. In the initial
1029
+ loading part, ρ < 0.65, the classical WF PA in ZF-mMIMO
1030
+ achieves better results until the loading (pink surface). When
1031
+ the number of antennas is low as M = 64 and ρ is in
1032
+ between 0.7 and 1.8, the EPA PA applied to NOMA (green
1033
+ 0
1034
+ 0.2
1035
+ 0.4
1036
+ 0.6
1037
+ 0.8
1038
+ 1
1039
+ 1.2
1040
+ 1.4
1041
+ 1.6
1042
+ 1.8
1043
+ 2
1044
+ Loading = K/M
1045
+ 0
1046
+ 50
1047
+ 100
1048
+ 150
1049
+ 200
1050
+ 250
1051
+ 300
1052
+ 350
1053
+ 400
1054
+ Average Sum Rate (bits/s/Hz)
1055
+ ZF-mMIMO
1056
+ ZF-NOMA
1057
+ M = 256
1058
+ M = 128
1059
+ M = 64
1060
+ (a) EPA
1061
+ 0
1062
+ 0.2
1063
+ 0.4
1064
+ 0.6
1065
+ 0.8
1066
+ 1
1067
+ 1.2
1068
+ 1.4
1069
+ 1.6
1070
+ 1.8
1071
+ 2
1072
+ Loading = K/M
1073
+ 0
1074
+ 30
1075
+ 60
1076
+ 90
1077
+ 120
1078
+ 150
1079
+ 180
1080
+ Average Sum Rate (bits/s/Hz)
1081
+ ZF-mMIMO
1082
+ ZF-NOMA
1083
+ M = 256
1084
+ M = 128
1085
+ M = 64
1086
+ (b) PICPA
1087
+ 0
1088
+ 0.2
1089
+ 0.4
1090
+ 0.6
1091
+ 0.8
1092
+ 1
1093
+ 1.2
1094
+ 1.4
1095
+ 1.6
1096
+ 1.8
1097
+ 2
1098
+ Loading = K/M
1099
+ 0
1100
+ 50
1101
+ 100
1102
+ 150
1103
+ 200
1104
+ 250
1105
+ 300
1106
+ 350
1107
+ 400
1108
+ 450
1109
+ Average Sum Rate (bits/s/Hz)
1110
+ ZF-mMIMO-WF
1111
+ ZF-NOMA-
1112
+ -WF
1113
+ M = 256
1114
+ M = 128
1115
+ M = 64
1116
+ (c) Classical WF in ZF-mMIMO and ∆-WF in ZF-NOMA
1117
+ Figure 3: The average sum-rate with the loading of devices 0 <
1118
+ ρ ≤ 2, considering four power allocation methods: EPA, PICPA,
1119
+ WF, and ∆-WF The average is obtained over 1000 random devices
1120
+ locations.
1121
+
1122
+ 8
1123
+ 0
1124
+ 0.1
1125
+ 0.2
1126
+ 0.3
1127
+ 0.4
1128
+ 0.5
1129
+ 0.6
1130
+ 0.7
1131
+ 0.8
1132
+ 0.9
1133
+ 1
1134
+ 1.1
1135
+ 1.2
1136
+ 1.3
1137
+ 1.4
1138
+ 1.5
1139
+ 1.6
1140
+ 1.7
1141
+ 1.8
1142
+ 1.9
1143
+ 2
1144
+ Loading = K/M
1145
+ 0
1146
+ 0.1
1147
+ 0.2
1148
+ 0.3
1149
+ 0.4
1150
+ 0.5
1151
+ 0.6
1152
+ 0.7
1153
+ 0.8
1154
+ 0.9
1155
+ 1
1156
+ Actives Users
1157
+ ZF-mMIMO-WF - M = 64
1158
+ ZF-NOMA-
1159
+ -WF - M = 64
1160
+ ZF-mMIMO-WF - M = 128
1161
+ ZF-NOMA-
1162
+ -WF - M = 128
1163
+ ZF-mMIMO-WF - M = 256
1164
+ ZF-NOMA-
1165
+ -WF - M = 256
1166
+ ALL-EPA
1167
+ ALL-PICPA
1168
+ Figure 4: The average of active devices after PA procedure
1169
+ versus loading of devices in the range 0 < ρ ≤ 2. The average
1170
+ is obtained over 1000 random devices locations.
1171
+ Figure 5: Average sum rate with loading ρ and M.
1172
+ surface) achieved superior results. Moreover, when M = 128
1173
+ and 0.8 < ρ < 1.6, the ZF-NOMA-EPA achieve superior
1174
+ results (green surface). For a higher number of antennas in
1175
+ BS, i.e., M = 256 only in a short loading of devices range,
1176
+ 0.86 < ρ < 0.97, the ZF-NOMA-EPA achieves superior SE
1177
+ results. Finally, when ρ > 0.97, the modified ∆-WF achieves
1178
+ competitive results (blue surface).
1179
+ B. Jain’s Fairness Index
1180
+ Another critical analysis developed was to analyze the
1181
+ fairness between the devices, i.e., to know the difference in
1182
+ the transmission rate achieved by active devices in the cell. For
1183
+ this measure, we use the Jain’s Fairness index like described
1184
+ in [28] and can be defined as:
1185
+ Fsyst
1186
+ m
1187
+ =
1188
+ ��M
1189
+ k=1 Rk
1190
+ �2
1191
+ M �M
1192
+ k1 R2
1193
+ k
1194
+ .
1195
+ (29)
1196
+ The Fig. 6. depicts the fairness curves attainable by NOMA
1197
+ and mMIMO systems with EPA, PICPA, WF, and ∆-WF PA
1198
+ procedures when the loading of devices grows until ρ = 2. Fig.
1199
+ 6.(a) shows the Jain’s Fairness Index when EPA policy is used,
1200
+ the mMIMO system performs better F results than NOMA for
1201
+ ρ < 1, on the other hand, NOMA can attain Fnoma
1202
+ m
1203
+ ≈ 0.5 in
1204
+ almost every loading of devices, independent of M.
1205
+ Fig. 6.(b) reveals the Jain’s Fairness Index when the PICPA
1206
+ method is applied, despite the SE result being lower in this PA
1207
+ method, the mMIMO obtains the best fairness result, keeping
1208
+ the Fmmimo
1209
+ m
1210
+ consistently above 0.85, still the NOMA under
1211
+ 0.5.
1212
+ The Jain’s Fairness Index when WF and δ-WF are depicted
1213
+ in Fig.6(c), It is intrinsic to these algorithms to allocate more
1214
+ power to devices with better channel conditions, which causes
1215
+ fairness between devices to be impaired. In this method, it is
1216
+ possible to observe a significant influence of the number of
1217
+ antennas M in the BS and the fairness result.
1218
+ C. Energy Efficiency Comparison
1219
+ Energy efficiency (EE) is another important figure of merit
1220
+ used to compare systems’ performance. In this section, a power
1221
+ consumption model based on fixed circuitry power part and
1222
+ that one varying according to the number of antennas M and
1223
+ the number of devices K has been adopted, following eq. (19).
1224
+ Table III [2] presents the adopted parameter values for the
1225
+ EE analysis and comparison discussed in this subsection. Fig.
1226
+ 7 depicts the performance of EE with EPA, PICPA, WF, and
1227
+ ∆-WF PA procedures, considering the exact three quantities
1228
+ of antennas. Fig. 7.(a) shows the EE performance with EPA. In
1229
+ this method, all devices receive the same power. The avg-EE
1230
+ mMIMO overcomes the NOMA around 13% to 20%. It was
1231
+ possible to observe that adding antennas in the BS increases
1232
+ the power consumption, harming the EE result. Again, under
1233
+ loading of devices 0.6 < ρ ≤ 2.0, the NOMA overcomes the
1234
+ mMIMO system.
1235
+ Table III: Adopted Parameters values for EE analysis [2]
1236
+ Parameter
1237
+ Value
1238
+ Backhaul Infrastruture
1239
+ P0 = 2 W
1240
+ Single oscillator
1241
+ Psyn = 2 W
1242
+ Coding and modulation
1243
+ PCOD = 4 W per device
1244
+ Decoding and demodulation
1245
+ PDEC = 0.5 W per device
1246
+ Receive power
1247
+ PRX = 0.3 W per device
1248
+ Transmitted power
1249
+ PT X = 1 W per antenna
1250
+ Efficiency of Power Amplifier
1251
+ ̟ = 0.3
1252
+ Operations/Joule
1253
+ L = 109 oper. per joule
1254
+ Fig. 7.(b) reveals the EE performance with PICPA. In this
1255
+ method, more power is allocated to devices with poor channel
1256
+ coefficients, resulting in reduced poor EE performance for both
1257
+ NOMA and M-MIMO systems, attaining a maximum of 0.25
1258
+ and 0.16 [bits/W] for mMIMO and NOMA, respectively. The
1259
+ maximum EE attained by mMIMO is generally around 50%
1260
+ higher than NOMA. However, for loading of devices ρ ≥ 0.65,
1261
+ NOMA overcomes mMIMO EE performance.
1262
+ Fig. 7.(c) depicts the EE performance in the mMIMO with
1263
+ classical WF and in the NOMA with ∆-WF algorithm. It
1264
+ is possible to confirm the superiority of energy efficiency of
1265
+ mMIMO within the range where it operates consistently, i.e.,
1266
+ 0 < ρ < 1. Notice that the maximum EE achieved by mMIMO
1267
+ is about 70 % higher than NOMA for different BS antennas.
1268
+ Finally, NOMA becomes more energy efficient than mMIMO
1269
+ only when the loading of devices is high, ρ > 0.95.
1270
+ In all analyzed system scenarios, the mMIMO equipped with
1271
+ classical WF PA procedure achieves higher maximum EE. The
1272
+ mMIMO attains better EE results than NOMA for ρ < 1. On
1273
+ the other hand, NOMA can serve a more significant number
1274
+ of devices (twice) than mMIMO.
1275
+ Fig. 8 summarizes the best EE results in a surface plotting
1276
+ for the mMIMO with WF overcoming NOMA across the entire
1277
+
1278
+ IOMA-Z-VWF
1279
+ ZF-NOMA-EPA450
1280
+ n Rate (bits/s/Hz)
1281
+ 400
1282
+ 350
1283
+ 300
1284
+ 200256128
1285
+ M
1286
+ 64Average Sur
1287
+ 150
1288
+ 100
1289
+ 50
1290
+ 0
1291
+ 2
1292
+ 1.8
1293
+ 1.6
1294
+ 1.4
1295
+ 1.2
1296
+ p
1297
+ 0.8
1298
+ 0.6
1299
+ 0.4
1300
+ 0.29
1301
+ 0
1302
+ 0.2
1303
+ 0.4
1304
+ 0.6
1305
+ 0.8
1306
+ 1
1307
+ 1.2
1308
+ 1.4
1309
+ 1.6
1310
+ 1.8
1311
+ 2
1312
+ Loading = K/M
1313
+ 0
1314
+ 0.1
1315
+ 0.2
1316
+ 0.3
1317
+ 0.4
1318
+ 0.5
1319
+ 0.6
1320
+ 0.7
1321
+ 0.8
1322
+ 0.9
1323
+ 1
1324
+ Jain Fairness Index
1325
+ ZF-mMIMO - M = 64
1326
+ ZF-NOMA - M = 64
1327
+ ZF-mMIMO - M = 128
1328
+ ZF-NOMA - M = 128
1329
+ ZF-mMIMO - M = 256
1330
+ ZF-NOMA - M = 256
1331
+ (a) EPA
1332
+ 0
1333
+ 0.2
1334
+ 0.4
1335
+ 0.6
1336
+ 0.8
1337
+ 1
1338
+ 1.2
1339
+ 1.4
1340
+ 1.6
1341
+ 1.8
1342
+ 2
1343
+ Loading = K/M
1344
+ 0
1345
+ 0.1
1346
+ 0.2
1347
+ 0.3
1348
+ 0.4
1349
+ 0.5
1350
+ 0.6
1351
+ 0.7
1352
+ 0.8
1353
+ 0.9
1354
+ 1
1355
+ Jain Fairness Index
1356
+ ZF-mMIMO - M = 64
1357
+ ZF-NOMA - M = 64
1358
+ ZF-mMIMO - M = 128
1359
+ ZF-NOMA - M = 128
1360
+ ZF-mMIMO - M = 256
1361
+ ZF-NOMA - M = 256
1362
+ (b) PICPA
1363
+ 0
1364
+ 0.2
1365
+ 0.4
1366
+ 0.6
1367
+ 0.8
1368
+ 1
1369
+ 1.2
1370
+ 1.4
1371
+ 1.6
1372
+ 1.8
1373
+ 2
1374
+ Loading = K/M
1375
+ 0
1376
+ 0.1
1377
+ 0.2
1378
+ 0.3
1379
+ 0.4
1380
+ 0.5
1381
+ 0.6
1382
+ 0.7
1383
+ 0.8
1384
+ 0.9
1385
+ 1
1386
+ Jain Fairness Index
1387
+ ZF-mMIMO - M = 64
1388
+ ZF-NOMA - M = 64
1389
+ ZF-mMIMO - M = 128
1390
+ ZF-NOMA - M = 128
1391
+ ZF-mMIMO - M = 256
1392
+ ZF-NOMA - M = 256
1393
+ (c) WF and ∆-WF
1394
+ Figure 6: The fairness of NOMA and mMIMO system under three
1395
+ power allocation procedures: (a) EPA; b) PICPA; (c) WF and ∆-
1396
+ WF. The average is obtained over 1000 random device locations.
1397
+ 0
1398
+ 0.2
1399
+ 0.4
1400
+ 0.6
1401
+ 0.8
1402
+ 1
1403
+ 1.2
1404
+ 1.4
1405
+ 1.6
1406
+ 1.8
1407
+ 2
1408
+ Loading = K/M
1409
+ 0.1
1410
+ 0.2
1411
+ 0.3
1412
+ 0.4
1413
+ 0.5
1414
+ 0.6
1415
+ EE (bits/Joule/Hz)
1416
+ ZF-mMIMO - M=64
1417
+ ZF-NOMA - M=64
1418
+ ZF-mMIMO - M=128
1419
+ ZF-NOMA - M=128
1420
+ ZF-mMIMO - M=256
1421
+ ZF-NOMA - M=256
1422
+ (a) EPA
1423
+ 0
1424
+ 0.2
1425
+ 0.4
1426
+ 0.6
1427
+ 0.8
1428
+ 1
1429
+ 1.2
1430
+ 1.4
1431
+ 1.6
1432
+ 1.8
1433
+ 2
1434
+ Loading = K/M
1435
+ 0.03
1436
+ 0.06
1437
+ 0.09
1438
+ 0.12
1439
+ 0.15
1440
+ 0.18
1441
+ 0.21
1442
+ 0.24
1443
+ 0.27
1444
+ EE (bits/Joule/Hz)
1445
+ ZF-mMIMO - M=64
1446
+ ZF-NOMA - M=64
1447
+ ZF-mMIMO - M=128
1448
+ ZF-NOMA - M=128
1449
+ ZF-mMIMO - M=256
1450
+ ZF-NOMA - M=256
1451
+ (b) PICPA
1452
+ 0
1453
+ 0.2
1454
+ 0.4
1455
+ 0.6
1456
+ 0.8
1457
+ 1
1458
+ 1.2
1459
+ 1.4
1460
+ 1.6
1461
+ 1.8
1462
+ 2
1463
+ Loading = K/M
1464
+ 0.1
1465
+ 0.2
1466
+ 0.3
1467
+ 0.4
1468
+ 0.5
1469
+ 0.6
1470
+ 0.7
1471
+ 0.8
1472
+ EE (bits/Joule/Hz)
1473
+ ZF-mMIMO-WF - M=64
1474
+ ZF-NOMA-
1475
+ -WF - M=64
1476
+ ZF-mMIMO-WF - M=128
1477
+ ZF-NOMA-
1478
+ -WF - M=128
1479
+ ZF-mMIMO-WF - M=256
1480
+ ZF-NOMA-
1481
+ -WF - M=256
1482
+ (c) WF and ∆-WF
1483
+ Figure 7: EE for NOMA vs. mMIMO under three power allocation
1484
+ procedures: (a) EPA; b) PICPA; (c) WF and ∆-WF. Average EE
1485
+ obtained over 1000 random devices locations.
1486
+
1487
+ 10
1488
+ loading range where it operates consistently. For device loading
1489
+ ρ > 1, the NOMA operates under lower EE until the loading
1490
+ ρ = 2. Moreover, considering the smallest number of antennas
1491
+ in the BS, the NOMA with EPA overcame the NOMA with
1492
+ modified ∆-WF; despite that, as the number of antennas in
1493
+ the BS increases, the NOMA with ∆-WF achieves marginal
1494
+ superior EE results.
1495
+ Figure 8: EE with loading ρ and M.
1496
+ D. Area Under Curves SE and EE
1497
+ For a fair comparison, one can consider a wide range
1498
+ of average SE and EE along the loading of devices, and
1499
+ normalized per antenna, which can be attainable by NOMA
1500
+ and mMIMO systems. Hence, let us consider the corresponding
1501
+ areas under the SE and EE curves in Fig. 3 and Fig. 7, such
1502
+ that:
1503
+ Ssyst
1504
+ M
1505
+ = 1
1506
+ M ·
1507
+ � ρ=2
1508
+ 0
1509
+ SE(ρ) dρ
1510
+ �bits/antenna
1511
+ s · Hz
1512
+
1513
+ and
1514
+ Esyst
1515
+ M
1516
+ = 1
1517
+ M ·
1518
+ � ρ=2
1519
+ 0
1520
+ EE(ρ) dρ
1521
+ �bits/antenna
1522
+ Joule · Hz
1523
+
1524
+ ,
1525
+ respectively, where SE(ρ) is the average overall system sum-
1526
+ rate, and EE(ρ) is the average overall system energy efficiency
1527
+ achieved under specific loading of devices ρ. Hence, comparing
1528
+ the values of corresponding areas under the SE and EE curves
1529
+ of Fig. 3 and Fig. 7, we obtained Fig. 9.
1530
+ From the SE perspective, and considering EPA policy, the
1531
+ higher area-under-SE-curve ratio gain is achieved when the
1532
+ number of BS antennas is M = 64:
1533
+ Snoma
1534
+ M=64 ≈ 2.7 · SmMIMO
1535
+ M=64 .
1536
+ Notice that when the number of antennas M grows, the ratio
1537
+ above decreases. In the same way, considering WF policy, the
1538
+ gain trend remains. In contrast, considering the PICPA policy,
1539
+ the ratio practically remains the same.
1540
+ Furthermore, considering now the EE perspective, under
1541
+ EPA policy, the higher ratio is achieved when the BS is
1542
+ equipped with M = 256 antennas:
1543
+ Enoma
1544
+ M=256 ≈ 1.8 · EmMIMO
1545
+ M=256 .
1546
+ As one can conclude, in almost all scenarios, NOMA is more
1547
+ spectrally and energetically efficient than mMIMO over an
1548
+ extensive range of loading of devices 0 < ρ ≤ 2, roughly,
1549
+ in average, 80% in terms of energy efficiency, and 170% more
1550
+ efficient in terms of spectral efficiency.
1551
+ EPA-64
1552
+ EPA-128
1553
+ EPA-256
1554
+ WF-64
1555
+ WF-128
1556
+ WF-256
1557
+ PICPA-64
1558
+ PICPA-128
1559
+ PICPA-256
1560
+ 0
1561
+ 0.5
1562
+ 1
1563
+ 1.5
1564
+ 2
1565
+ 2.5
1566
+ 3
1567
+ 3.5
1568
+ 4
1569
+ 0
1570
+ 0.5
1571
+ 1
1572
+ 1.5
1573
+ 2
1574
+ 2.5
1575
+ 3
1576
+ 3.5
1577
+ 4
1578
+ ZF-NOMA
1579
+ mMIMO
1580
+ EPA
1581
+ WF
1582
+ PICPA
1583
+ (a) S - Bar chart of Area Under SE curves.
1584
+ EPA-64
1585
+ EPA-128
1586
+ EPA-256
1587
+ WF-64
1588
+ WF-128
1589
+ WF-256
1590
+ PICPA-64
1591
+ PICPA-128
1592
+ PICPA-256
1593
+ 0
1594
+ 0.002
1595
+ 0.004
1596
+ 0.006
1597
+ 0.008
1598
+ 0.01
1599
+ 0.012
1600
+ 0
1601
+ 0.2
1602
+ 0.4
1603
+ 0.6
1604
+ 0.8
1605
+ 1
1606
+ 1.2
1607
+ 1.4
1608
+ 1.6
1609
+ 1.8
1610
+ 2
1611
+ ZF-NOMA
1612
+ mMIMO
1613
+ EPA
1614
+ WF
1615
+ PICPA
1616
+ (b) E - Bar chart of Area Under EE curves.
1617
+ Figure 9: The Area Under curve of NOMA and mMIMO
1618
+ system under three power allocation procedures: (a) SE curves;
1619
+ b) EE curves. The average is obtained over 1000 random
1620
+ devices locations.
1621
+ E. Resource Efficiency (SE-EE Trade-off)
1622
+ The NOMA and mMIMO are analyzed in terms of SE
1623
+ and EE trade-off, namely resource efficiency (RE), considering
1624
+ loading of devices increasing up to 2. From Fig. 10, one
1625
+ can find graphically the best loading of devices range that
1626
+ maximizes the SE-EE trade-off for each BS antenna configu-
1627
+ ration M. The left y-axis depicts avg-SE, and the right y-axis
1628
+ shows the avg-EE. Table IV summarizes the optimal loading
1629
+ of devices that maximizes the SE-EE trade-off and shows the
1630
+ percentage of active users after Power Allocations and Jain’s
1631
+ Fairness Index. Fig. 10.(a) reveals the results when M = 64,
1632
+ NOMA with EPA achieved SE-EE trade-off with the highest
1633
+ loading of devices and the highest SE in trade-off; on the
1634
+ other hand, mMIMO with classical WF achieved the highest
1635
+ SE-EE trade-off; however, the percentage of actives devices is
1636
+ around 0.5. Fig. 10.(b) depicts results when M = 128, NOMA
1637
+ with EPA achieved SE-EE trade-off with the highest loading
1638
+ of devices, in contrast, mMIMO with classical WF with lower
1639
+ loading of devices achieved higher values of SE and EE in
1640
+ trade-off with half of the active devices. Fig 10.(c) showed
1641
+ the results when M=256, NOMA with ∆-WF achieved SE-EE
1642
+ trade-off with the highest loading of devices, and one more
1643
+ time mMIMO with WF achieved higher SE-EE trade-off with
1644
+ 47% of active devices. It is possible to demonstrate that the
1645
+
1646
+ ZE.I0.8
1647
+ 0.7256128
1648
+ M
1649
+ 64
1650
+ 0.6
1651
+ 0.4
1652
+ 0.2
1653
+ 0EE(bits/Joule
1654
+ 0.4
1655
+ 0.3
1656
+ 0.2
1657
+ 0.1 ~
1658
+ 0
1659
+ 2
1660
+ 1.8
1661
+ 1.6
1662
+ 1.4
1663
+ 1.2
1664
+ 1
1665
+ 0.8
1666
+ p11
1667
+ increase of antennas in the BS improves the SE result, on the
1668
+ other hand, it worsens the EE result.
1669
+ 0
1670
+ 0.2
1671
+ 0.4
1672
+ 0.6
1673
+ 0.8
1674
+ 1
1675
+ 1.2
1676
+ 1.4
1677
+ 1.6
1678
+ 1.8
1679
+ 2
1680
+ 0
1681
+ 134
1682
+ 0
1683
+ 0.781
1684
+ ZF-mMIMO-WF
1685
+ 0
1686
+ 0.2
1687
+ 0.4
1688
+ 0.6
1689
+ 0.8
1690
+ 1
1691
+ 1.2
1692
+ 1.4
1693
+ 1.6
1694
+ 1.8
1695
+ 2
1696
+ Loading = K/M
1697
+ 0
1698
+ 158
1699
+ 0
1700
+ 0.475
1701
+ ZF-NOMA-
1702
+ -WF
1703
+ 0
1704
+ 0.2
1705
+ 0.4
1706
+ 0.6
1707
+ 0.8
1708
+ 1
1709
+ 1.2
1710
+ 1.4
1711
+ 1.6
1712
+ 1.8
1713
+ 2
1714
+ 0
1715
+ 163
1716
+ Average Sum rate (bits/s/Hz)
1717
+ 0
1718
+ 0.476
1719
+ EE (bits/Joule/Hz)
1720
+ ZF-NOMA-EPA
1721
+ (a) M = 64 BS antennas
1722
+ 0
1723
+ 0.2
1724
+ 0.4
1725
+ 0.6
1726
+ 0.8
1727
+ 1
1728
+ 1.2
1729
+ 1.4
1730
+ 1.6
1731
+ 1.8
1732
+ 2
1733
+ 0
1734
+ 249
1735
+ 0
1736
+ 0.767
1737
+ ZF-mMIMO-WF
1738
+ 0
1739
+ 0.2
1740
+ 0.4
1741
+ 0.6
1742
+ 0.8
1743
+ 1
1744
+ 1.2
1745
+ 1.4
1746
+ 1.6
1747
+ 1.8
1748
+ 2
1749
+ Loading = K/M
1750
+ 0
1751
+ 251
1752
+ 0
1753
+ 0.449
1754
+ ZF-NOMA-
1755
+ -WF
1756
+ 0
1757
+ 0.2
1758
+ 0.4
1759
+ 0.6
1760
+ 0.8
1761
+ 1
1762
+ 1.2
1763
+ 1.4
1764
+ 1.6
1765
+ 1.8
1766
+ 2
1767
+ 0
1768
+ 267
1769
+ Average Sum rate (bits/s/Hz)
1770
+ 0
1771
+ 0.453
1772
+ EE (bits/Joule/Hz)
1773
+ ZF-NOMA-EPA
1774
+ (b) M = 128 BS antennas
1775
+ 0
1776
+ 0.2
1777
+ 0.4
1778
+ 0.6
1779
+ 0.8
1780
+ 1
1781
+ 1.2
1782
+ 1.4
1783
+ 1.6
1784
+ 1.8
1785
+ 2
1786
+ 0
1787
+ 412
1788
+ 0
1789
+ 0.706
1790
+ ZF-mMIMO-WF
1791
+ 0
1792
+ 0.2
1793
+ 0.4
1794
+ 0.6
1795
+ 0.8
1796
+ 1
1797
+ 1.2
1798
+ 1.4
1799
+ 1.6
1800
+ 1.8
1801
+ 2
1802
+ Loading = K/M
1803
+ 0
1804
+ 361
1805
+ 0
1806
+ 0.405
1807
+ ZF-NOMA-
1808
+ -WF
1809
+ 0
1810
+ 0.2
1811
+ 0.4
1812
+ 0.6
1813
+ 0.8
1814
+ 1
1815
+ 1.2
1816
+ 1.4
1817
+ 1.6
1818
+ 1.8
1819
+ 2
1820
+ 0
1821
+ 373
1822
+ Average Sum rate (bits/s/Hz)
1823
+ 0
1824
+ 0.412
1825
+ EE (bits/Joule/Hz)
1826
+ ZF-NOMA-EPA
1827
+ (c) M = 256 BS antennas
1828
+ Figure 10: SE-EE trade-off points when M = 64, 128 and 256.
1829
+ V. Conclusion and Future Works
1830
+ This work proposes a comparative SE and EE analysis in
1831
+ DL single-cell between mMIMO and NOMA with BS equipped
1832
+ with three antenna configurations. Under the SE perspective,
1833
+ mMIMO with the classical WF algorithm achieved better low-
1834
+ and medium-loading results. On the other hand, when the
1835
+ Table IV: SE-EE Trade-off
1836
+ M = 64
1837
+ ρ
1838
+ SE
1839
+ EE
1840
+ Actives Users
1841
+ F
1842
+ mMIMO-WF
1843
+ 0.652
1844
+ 131.15
1845
+ 0.762
1846
+ .50
1847
+ .485
1848
+ NOMA-EPA
1849
+ 0.875
1850
+ 147.45
1851
+ 0.428
1852
+ 1.0
1853
+ .485
1854
+ NOMA-∆-WF
1855
+ 0.844
1856
+ 143.48
1857
+ 0.441
1858
+ 1.0
1859
+ .475
1860
+ M = 128
1861
+ ρ
1862
+ SE
1863
+ EE
1864
+ Actives Users
1865
+ F
1866
+ mMIMO-WF
1867
+ 0.625
1868
+ 243.21
1869
+ 0.745
1870
+ .50
1871
+ .475
1872
+ NOMA-EPA
1873
+ 0.734
1874
+ 241.54
1875
+ 0.411
1876
+ 1.0
1877
+ .49
1878
+ NOMA-∆-WF
1879
+ 0.703
1880
+ 226.32
1881
+ 0.405
1882
+ .98
1883
+ .45
1884
+ M =256
1885
+ ρ
1886
+ SE
1887
+ EE
1888
+ Actives Users
1889
+ F
1890
+ mMIMO-WF
1891
+ 0.578
1892
+ 404.84
1893
+ 0.691
1894
+ .47
1895
+ .43
1896
+ NOMA-EPA
1897
+ 0.523
1898
+ 344.70
1899
+ 0.380
1900
+ 1.0
1901
+ .495
1902
+ NOMA-∆-WF
1903
+ 0.594
1904
+ 333.31
1905
+ 0.375
1906
+ .85
1907
+ .398
1908
+ system loading is higher as ρ > 0.6 the NOMA achieves better
1909
+ results in the range 0.6 < ρ ≤ 2.
1910
+ The analyzed PA methods applied to the NOMA system,
1911
+ (EPA, PICPA, and ∆-WF) result in different SE performance.
1912
+ Indeed, when the channel hardening condition is fully attained,
1913
+ and the amount of BS antennas increases(M = 128 and 256),
1914
+ the best SE results are attained with the proposed ∆-WF
1915
+ algorithm, but, as expected, the fairness index is harmed.
1916
+ Under the EE perspective, the mMIMO achieved better
1917
+ results when employing the three EPA, PICPA, and WF PA
1918
+ methods under K < M. However, the NOMA can operate
1919
+ under higher system loading, i.e., K < 2M − 1.
1920
+ In terms of area-under-SE-curve and EE-curve metrics, S
1921
+ and E, respectively, the NOMA system attained better results,
1922
+ due to its ability to serve a larger number of users than
1923
+ mMIMO. Such numerical results confirm NOMA’s ability to
1924
+ operate with high loading of devices. On the other hand,
1925
+ achieving high fairness with NOMA is impossible.
1926
+ From the perspective of SE-EE trade-off, mMIMO achieved
1927
+ the best results, because of the superiority in EE; always
1928
+ achieved in loading of devices ρ = 0.6 in all M setups.
1929
+ NOMA systems present exciting features and have been
1930
+ intensively investigated as a promising technique in devising
1931
+ future wireless generations. As future works, hybrid NOMA
1932
+ systems and alternative techniques such as rate-splitting mul-
1933
+ tiple access (RSMA) can improve the overall EE of massive
1934
+ MIMO systems.
1935
+ Acknowledgement
1936
+ This work was partly supported by The National Council for
1937
+ Scientific and Technological Development (CNPq) of Brazil
1938
+ under Grants 310681/2019-7, partly by the CAPES- Brazil -
1939
+ Finance Code 001, and the Londrina State University - Paraná
1940
+ State Government (UEL).
1941
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1
+ Certification Design for a Competitive Market
2
+ Andreas Haupt
3
+ Nicole Immorlica
4
+ Brendan Lucier∗
5
+ February 1, 2023
6
+ Abstract
7
+ Motivated by applications such as voluntary carbon markets and edu-
8
+ cational testing, we consider a market for goods with varying but hidden
9
+ levels of quality in the presence of a third-party certifier. The certifier
10
+ can provide informative signals about the quality of products, and can
11
+ charge for this service.
12
+ Sellers choose both the quality of the product
13
+ they produce and a certification. Prices are then determined in a compet-
14
+ itive market. Under a single-crossing condition, we show that the levels of
15
+ certification chosen by producers are uniquely determined at equilibrium.
16
+ We then show how to reduce a revenue-maximizing certifier’s problem to
17
+ a monopolistic pricing problem with non-linear valuations, and design an
18
+ FPTAS for computing the optimal slate of certificates and their prices. In
19
+ general, both the welfare-optimal and revenue-optimal slate of certificates
20
+ can be arbitrarily large.
21
+ 1
22
+ Introduction
23
+ Many markets sell goods that have different features but appear identical to
24
+ consumers.
25
+ Examples include the market for carbon credits, the market for
26
+ contract workers such as electricians or plumbers, or even some markets for
27
+ physical goods like milk and eggs. In each of these markets, the goods have
28
+ hidden, hard-to-verify properties that distinguish them. Nature-based carbon
29
+ credits, for example, are sourced from a variety of forests with different pro-
30
+ tection and longevity properties.
31
+ Contract workers have differing skill levels
32
+ and amount of knowledge. And physical goods are produced in a variety of
33
+ circumstances, some more ethical and sustainable than others.
34
+ In such settings, sellers seek to distinguish their products through costly
35
+ certification. In the example of carbon credits, large certifiers issue certification
36
+ standards for carbon emissions.
37
+ Contract workers can enroll in certification
38
+ programs or take exams to document their skill level. And farms can reach out
39
+ to third-party organizations that certify the conditions, such as free-range or
40
+ vegetarian-fed, under which their food is produced.
41
+ ∗We thank Robert Maddox, Yanika Meyer-Oldenburg and a seminar audience at Harvard.
42
+ 1
43
+ arXiv:2301.13449v1 [cs.GT] 31 Jan 2023
44
+
45
+ In each of these cases, a downstream market allocates goods based on cer-
46
+ tification.
47
+ Carbon exchanges allow for the trade of carbon certificates, and
48
+ platforms for local services allow the trade of electrical or plumbing services.
49
+ These markets have price formation that is independent of, and not controllable
50
+ by, the certifier. Therefore, both welfare- and revenue-maximizing certifiers will
51
+ need to reason about how the certification options they provide affect trade,
52
+ and how this in turn influences the demand for certification.
53
+ What impact does the presence of certification have on the downstream
54
+ market? We answer this question in the context of a competitive market with
55
+ production. In our model there is a mass of heterogeneous producers each creat-
56
+ ing a single unit of a vertically-differentiated product. Producers can select the
57
+ quality level of their products. The corresponding production cost is determined
58
+ by the producer’s type and is increasing in quality. A mass of unit-demand con-
59
+ sumers purchases these goods. Consumers also have varying types which deter-
60
+ mine the strength of their preference for quality. Types are totally ordered and
61
+ satisfy a single-crossing condition: higher-type producers are relatively more
62
+ efficient at producing higher-quality goods, which higher-type consumers have
63
+ relatively stronger preference for.
64
+ Quality cannot be verified directly; instead, a third-party certifier offers a
65
+ menu of certifications, each with its own requirements and certification price.
66
+ Producers can purchase certifications of their products’ qualities from this menu.
67
+ If their product surpasses the quality level of the certificate, it will be marketed
68
+ as such. The certified goods, together with indistinguishable uncertified ones,
69
+ are then sold in a competitive market.
70
+ As in the Economics literature on certification, we assume that market par-
71
+ ticipants are able form correct beliefs about the quality implications of a certifi-
72
+ cate Milgrom (1981); Grossman (1981); Lizzeri (1999); DeMarzo et al. (2019). A
73
+ big strand of empirical work considers the question of whether certifications are
74
+ correctly interpreted in various markets (Wimmer and Chezum (2003) for race
75
+ horses, Tadelis and Zettelmeyer (2015) on used-car auctions, Ramanarayanan
76
+ and Snyder (2012) for dialysis screening centers, Luca (2016) for restaurant
77
+ reviews, Elfenbein et al. (2015) for seller ratings on Ebay, Conte and Kotchen
78
+ (2010) for voluntary carbon credits). In our model, certificates are based on ver-
79
+ ifiable certification requirements, and at perfect Bayesian equilibrium all market
80
+ participants hold rational beliefs about the distribution of quality implied by
81
+ any given certificate. We note that such rational beliefs need not be aligned
82
+ with official certificate descriptions. For example, in voluntary carbon markets,
83
+ several empirical contributions point out that carbon certificate descriptions
84
+ might not correspond to counterfactual mitigation outcomes, compare West
85
+ et al. (2020, 2023); Guizar-Couti˜no et al. (2022); Clarke and Barratt ([n. d.]);
86
+ Greenfield ([n. d.]b,n).
87
+ The presence of certificates clearly impacts the market equilibrium because
88
+ it influences consumers’ beliefs and hence willingness-to-pay. Our first result
89
+ shows that, for any menu of certificates offered by the certifier, the equilibrium
90
+ choice of certificates and the allocation outcome of the downstream market
91
+ equilibrium is unique. Furthermore better producers (that is, those who can
92
+ 2
93
+
94
+ produce higher quality at lower cost) always purchase higher (more restrictive)
95
+ levels of certification.
96
+ This implies that the producer-consumer matching in
97
+ equilibrium is assortative. Better producers sell to consumers with higher value
98
+ for quality, and at higher quality levels.
99
+ This analysis immediately implies that the welfare-optimal menu offers every
100
+ certificate at cost, and suggests a dynamic program for finding the welfare-
101
+ optimal menu subject to a cardinality restriction on the number of certification
102
+ levels that can be offered. But in many markets, certification is provided by
103
+ third-party firms with profit-maximizing incentives.
104
+ Our main result shows
105
+ how to approximately construct the revenue-optimal menu in polynomial time.
106
+ Specifically, we provide an FPTAS: we show how to compute a menu whose
107
+ revenue for the certifier is an additive ϵ less than the optimal revenue in time
108
+ polynomial in 1/ϵ. Our algorithm can optionally take as input a cardinality
109
+ constraint k on the menu size (i.e., a maximum number of certification levels),
110
+ in which case the revenue benchmark is the optimal slate of at most k certificates
111
+ and their prices.
112
+ Our proof contains technical insights that may be of independent interest.
113
+ Namely, we reduce the certifier’s problem to one of a seller who wishes to sell
114
+ a divisible good, facing buyers whose valuations are non-linear and potentially
115
+ non-monotone in quantity. The seller corresponds to the certifier, and the buy-
116
+ ers correspond to producer-consumer pairs. This projection of producers and
117
+ consumers into a single economic agent is enabled by the fact that the matching
118
+ in equilibrium is unique and assortative. The buyer valuations in this reduced
119
+ problem are concave but not necessarily monotone as a function of quantity,
120
+ meaning that for different buyer types the valuation may reach its maximum at
121
+ different quantities. The valuations do satisfy a single-crossing property, which
122
+ implies that types are totally ordered and higher-type buyers have pointwise
123
+ higher valuations and have weakly higher preferred quantities.
124
+ Under these
125
+ conditions, we show that the revenue-optimal menu may be non-linear but will
126
+ exhibit prices that are monotone in quantity. Monotonicity of prices enables
127
+ the use of dynamic programming to construct an approximately optimal menu,
128
+ though some care must be taken when discretizing the space of quantities and
129
+ prices to ensure that buyer preferences do not change substantially. This non-
130
+ linear pricing problem generalizes prior work that specifically studied the case
131
+ of quadratic demand Bergemann et al. (2012a,b), and may be of independent
132
+ interest.
133
+ The menu of certifications offered by a revenue-maximizing third-party cer-
134
+ tifier can distort welfare and result in inefficient levels of trade.
135
+ That said,
136
+ we show that a certifying agent entering into the market can never lead to a
137
+ loss of welfare, no matter what menu of certification options they make avail-
138
+ able. More specifically, we can imagine that a collection of (perhaps costly)
139
+ certification options are available to the market as a baseline, perhaps provided
140
+ by a public or tightly regulated agency. If a third-party certifier then enters
141
+ the market and makes available any additional certification options into the
142
+ market, at any price, we show that the total utility of all market participants
143
+ (even excluding the new entrant) can only increase. A policy implication is that
144
+ 3
145
+
146
+ a regulatory body concerned about inefficient certification arising from profit-
147
+ seeking motives need not prevent certifiers from entering the market. Rather, it
148
+ suffices to make sure that there exists some set of certifications available, either
149
+ through third-party providers or public options, that enable an acceptable level
150
+ of welfare from trade.
151
+ 1.1
152
+ Related Literature
153
+ This work contributes to the literature on certification. The early results Mil-
154
+ grom (1981); Grossman (1981) produce unraveling type results: In equilibrium,
155
+ the quality of a good is fully revealed. The main intuition for these results in
156
+ models of certification is that certifying non-informatively will be interpreted
157
+ by the market as a sign of bad quality—adverse selection is extreme. In our
158
+ model of a revenue-maximizing certifier, there might be other reasons for certi-
159
+ fication less informatively, as the price in the competitive market may depend
160
+ on not only an individual seller’s quality, but all the sellers in the market. Later
161
+ contributions Lizzeri (1999); DeMarzo et al. (2019); Acharya et al. (2011) allow
162
+ for unsuccessful certifications, restricting the stark result obtained in Milgrom
163
+ (1981); Grossman (1981).
164
+ More broadly, our work is related to mechanism and information design in
165
+ the presence of an exogenously given game played after the design. We rec-
166
+ ommend Bergemann and Morris (2019) for a general overview of information
167
+ design. Our work is especially related to the design of information provided to
168
+ buyers and/or sellers of a good. Bergemann et al. (2017) considers the design of
169
+ information for a first-price auction, where a third party can reveal a signal cor-
170
+ related with a buyer’s valuations, and fully characterizes the achievable revenue
171
+ and payoffs. Alijani et al. (2022) extends the analysis to a scenario with mul-
172
+ tiple buyers. In a general mechanism design framework, Candogan and Strack
173
+ (2021) develop an optimal disclosure policy for action recommendations in a
174
+ game with private types and a hidden state. Dworczak (2020) designs mech-
175
+ anisms in a setting where players participate in a finite Bayesian game after
176
+ participating in the mechanism, so that game outcomes are impacted by infor-
177
+ mation revealed over the course of the mechanism. He finds a cutoff structure
178
+ of optimal mechanisms in the first stage.
179
+ For-profit certification relates to the sale of hard information, which has
180
+ been studied in the context of competitive markets. Ali et al. (2022) consider
181
+ a seller holding a good of uncertain quality. The seller can purchase a quality-
182
+ correlated signal from a revenue-maximizing intermediary before bringing the
183
+ good to market. In general the resulting equilibria are not unique and can vary
184
+ substantially, but by employing noisy signals the intermediary can robustly
185
+ guarantee high revenue. Our model differs in that any certification options are
186
+ made available to the entire market and product quality is endogenous, so the
187
+ certifying agent can impact welfare. More generally, our work relates to the
188
+ problem of how to sell payoff-relevant hard information to a potential buyer.
189
+ Bergemann et al. (2018) solve for the revenue-maximizing mechanism in binary
190
+ environments, and Bergemann et al. (2022) establish when full disclosure is
191
+ 4
192
+
193
+ approximately optimal under more general spaces of actions. In contrast, our
194
+ certifying agent is selling a signal that is valuable in that it conveys information
195
+ to other participants in a subsequent game.
196
+ In our mathematical analysis, we reduce the certifier’s problem to a pricing
197
+ problem with a non-linear valuation. The non-linear pricing literature following
198
+ Mussa and Rosen (1978) (see also treatment in Dewatripont et al. (2005) and
199
+ (B¨orgers and Krahmer, 2015, Chapter 2.3)) studies a non-linear concave valu-
200
+ ation with a quadratic cost. The functional form assumptions in these papers
201
+ allow to characterize the optimal mechanism in closed form. Typically, the op-
202
+ timal menu of offered goods is a continuum, in contrast to the linear screening
203
+ problem first studied in the influential Myerson (1981). Our analysis will show
204
+ that also the class of models we consider may feature infinite menus.
205
+ In our polynomial-time approximation scheme, we use an approximation of
206
+ a non-linear pricing model. The papers Bergemann et al. (2012a,b) consider
207
+ approximation of non-linear single- and multi-dimensional pricing environments
208
+ with finite menus (in the papers called “finite information”). The papers make
209
+ functional form assumptions similar to the ones in Mussa and Rosen (1978), and
210
+ derive rates of approximation by finite menus in these finite menus. The present
211
+ paper allows for a general class of utility functions that satisfy a single-crossing
212
+ condition, and, in addition to showing approximation by finite menus, shows
213
+ that the finite-sized menu can be computed efficiently.
214
+ Finally, our main assumption guaranteeing uniqueness of our equilibrium is
215
+ a single-crossing condition. Single-crossing conditions are important in several
216
+ domains, among them the interdependent private values auctions (Milgrom and
217
+ Weber (1982)) and social choice and voting (Saporiti and Tohm´e (2006)). A re-
218
+ cent line of work in algorithmic mechanism design has employed single-crossing
219
+ conditions to enable approximately optimal designs in interdependent value set-
220
+ tings Roughgarden and Talgam-Cohen (2013); Chawla et al. (2014). Closest to
221
+ the present paper, another implication of single-crossing is adverse selection in
222
+ markets Mirrlees (1971); Spence (1974).
223
+ 1.2
224
+ Outline
225
+ The rest of this article is structured as follows.
226
+ We formalize our model in
227
+ section 2. In section 3 we analyze the structure of equilibria given the certifier’s
228
+ offerings. Section 4 contains our main results, a reduction of revenue-maximizing
229
+ certification to a non-linear pricing problem, the FPTAS for its computation.
230
+ Section 5 contains our results on welfare maximization and explores the welfare
231
+ implications of third-party certification.
232
+ 2
233
+ Model
234
+ Market
235
+ We consider a continuum market between producers and consumers.
236
+ Producers are unit-supply and parameterized by types ψ ∈ R+ with measure G.
237
+ Consumers are unit-demand and parameterized by types φ ∈ R+ with measure
238
+ 5
239
+
240
+ F. The type measures F and G are atomless and continuous with compact
241
+ support.
242
+ Goods can be produced at different levels of quality, denoted q ∈ [0, 1].
243
+ Goods of higher quality are more valuable to consumers but more costly to
244
+ produce. We write g(q; ψ) for the cost incurred by a producer of type ψ when
245
+ producing a good of quality q. We assume g is weakly convex and non-decreasing
246
+ in q for every ψ and normalized so that g(0; ψ) = 0. We also write f(q; φ) for
247
+ the value enjoyed by a consumer of type φ for a good of quality q, where f
248
+ is weakly concave and non-decreasing in q for every φ and normalized so that
249
+ f(0; φ) = 0. We will scale valuations so that f(q; φ) ≤ 1 for all q and φ, which
250
+ is without loss for bounded values.
251
+ We will assume that costs and valuations satisfy single-crossing with respect
252
+ to the producer and consumer types, respectively. Roughly speaking, this means
253
+ that producers (consumers) of higher types have lower marginal cost (higher
254
+ marginal value) for producing higher-quality goods. More formally, for all φ1 <
255
+ φ2 and q1 < q2, we have
256
+ f(q2; φ2) − f(q1; φ2) > f(q2; φ1) − f(q1; φ1).
257
+ Likewise, for all ψ1 < ψ2 and q1 < q2, we have
258
+ g(q2; ψ2) − g(q1; ψ2) < g(q2; ψ1) − g(q1; ψ1).
259
+ Transfers between producers and consumers are permitted. We assume that
260
+ producers and consumers are risk-neutral and have quasi-linear preferences with
261
+ respect to money. That is, if consumer φ purchases a product of quality q from
262
+ producer ψ at a price of p, then the consumer enjoys utility
263
+ uC((q, p); φ) = f(q; φ) − p
264
+ and the producer’s utility is
265
+ uP ((q, p); ψ) = p − g(q; φ).
266
+ Certification
267
+ Crucially, producers and consumers cannot contract on quality.
268
+ This means that a producer cannot credibly commit to the quality of the good
269
+ they produce, and a consumer cannot verify quality at the point of trade. But
270
+ there is a third-party certifier who is able to determine the quality of a producer’s
271
+ good. This verification is costly to the certifier, with cost c ≥ 0.
272
+ After verifying the quality of a good, the certifier is able to assign to that
273
+ good a signal (or certificate) σ ∈ Σ, where Σ is an arbitrary space of potential
274
+ certificates.
275
+ This certificate is visible to all producers and consumers.
276
+ The
277
+ certifier is permitted to collect payments from producers for this service, and
278
+ these transfers can depend arbitrarily on the certificate produced.
279
+ The certifier can commit to a certification menu M, which is a collection
280
+ of certificate / transfer pairs (σ, t(σ)) along with quality requirements for each
281
+ certificate σ. Following the literature on information design and persuasion,
282
+ 6
283
+
284
+ we note that it is without loss of generality to associate each certificate signal
285
+ σ with the set of quality levels that are eligible for that certificate. We will
286
+ therefore assume without loss of generality that Σ = 2[0,1], the collection of all
287
+ subsets of [0, 1], and each σ is a subset of [0, 1]. The interpretation is that a
288
+ producer can select an item from this menu, in which case she pays the certifier
289
+ price (or transfer) t(σ), the certifier verifies the good’s quality q, and as long
290
+ as q ∈ σ the producer will receive certification σ. We will assume for technical
291
+ convenience that each σ in the certifier’s menu contains a minimum element,
292
+ meaning that inf σ ∈ σ.1
293
+ We will write σ0 = [0, 1] for the trivial signal that conveys no additional
294
+ information about quality. A good with certificate σ0 is functionally equivalent
295
+ to a good that has not been certified.
296
+ It will be notationally convenient to
297
+ assume that the certifier always offers signal σ0 at cost 0. This allows us to
298
+ think of every good as being certified, albeit possibly at the trivial level. If a
299
+ producer attempts to purchase a certificate but does not satisfy that certificate’s
300
+ requirements, they will instead be assigned certificate σ0; i.e., the certifier will
301
+ not certify the good.
302
+ The Competitive Market
303
+ All certification is assumed to occur simultane-
304
+ ously, and in advance of any trading between producers and consumers. Given
305
+ the menu M of certificates and prices offered by the certifier, the producers’
306
+ (production and certification) strategy is a mapping from producer type ψ to
307
+ a choice of quality level q and certification σ. We will denote such a strategy
308
+ Γ : ψ �→ (q, σ), and restrict attention to measurable functions Γ. In a slight
309
+ abuse of notation, we will also use Γ to denote the measure over pairs (q, σ) of
310
+ products with corresponding quality and certification that result when produc-
311
+ ers apply strategy Γ.
312
+ Goods that are assigned the same certificate are indistinguishable by the
313
+ consumers. After all certification is complete, each producer has a single unit
314
+ of a good marked with a certification σ. For any given certification σ, write Γσ
315
+ for the marginal distribution over quality q of Γ restricted to certificate σ. That
316
+ is, fixing the choices of the producers, Γσ is the distribution of levels of quality
317
+ for a product with certification σ. Then the value of a consumer of type φ for
318
+ a good with certificate σ can be evaluated as
319
+ f(σ; φ) = Eq∼Γσ[f(q; φ)].
320
+ That is, each consumer rationally evaluates the expected quality of each product
321
+ given its certification level and the choices of the producers.
322
+ 1This excludes certificates of the form “the quality q is strictly greater than 1/2,” as
323
+ opposed to “...at least 1/2.” Certificates of the former type are inconvenient because there is
324
+ no single least-cost choice of quality that satisfies the certification requirement, and hence no
325
+ utility-maximizing choice of quality for the producer. We could handle such certificates in a
326
+ straightforward but tedious way by relaxing our equilibrium notion and assuming that each
327
+ producer selects an ϵ-approximately utility maximizing choice of quality for some arbitrarily
328
+ small ϵ. For the remainder of the paper we will ignore such technical issues and simply assume
329
+ that each σ includes a minimum element.
330
+ 7
331
+
332
+ Since goods with the same certification are indistinguishable to consumers,
333
+ we can view the competitive market between producers and consumers as a
334
+ market for certificates σ.
335
+ A Walrasian (or Competitive) equilibrium of the
336
+ resulting market is an allocation x(φ) of a certificates to each consumer φ, along
337
+ with a price pσ for each certificate, such that:
338
+ • Demand satisfaction: every consumer purchases her most-preferred good.
339
+ That is, for every consumer type φ, f(x(φ); φ) − px(φ) ≥ f(σ; φ) − pσ for
340
+ every σ ∈ Σ.
341
+ • Market Clearing: every good with a positive price is sold. That is, for all
342
+ σ, the measure of consumers φ such that x(φ) = σ is at most the measure
343
+ of producers ψ who select level of certification σ. If pσ > 0 then these
344
+ measures are equal.
345
+ Since buyers (consumers) are unit-demand and hence their preferences satisfy
346
+ the gross substitutes condition, a Walrasian equilibrium is guaranteed to ex-
347
+ ist Gul and Stacchetti (2000).
348
+ We will therefore assume that trade occurs
349
+ between consumers and producers at competitive equilibrium prices given the
350
+ choices made by the producers.
351
+ Timeline
352
+ To summarize, the timing of the market with certification is as
353
+ follows:
354
+ 1. The certifier commits to a menu of certificates with corresponding prices.
355
+ 2. Each producer ψ simultaneously and privately chooses whether to produce
356
+ a good, and if so at what level of quality.
357
+ 3. Each producer that chose to produce decides whether to certify their prod-
358
+ uct, and at which certificate. These decisions are made simultaneously for
359
+ all producers.
360
+ 4. The certifier verifies the products of producers who choose to certify and
361
+ assigns certificates. Any producer who does not successfully certify re-
362
+ ceives certificate σ0.
363
+ 5. Producers and consumers trade goods in a competitive market. I.e., trade
364
+ occurs at market-clearing prices for the chosen levels of certification.
365
+ Since Walrasian equilibria are not unique in general, one might wonder if
366
+ the outcome described in the final step is well-defined. We will show in the
367
+ next section that the competitive market equilibrium described the final step
368
+ exists and its resulting allocation is unique for any certificate menu chosen by
369
+ the certifier and any choice of certification levels chosen by the producers.
370
+ 8
371
+
372
+ 3
373
+ Equilibrium Characterization
374
+ In this section we describe the market outcome that will occur for any given
375
+ menu of certificates offered by the certifier. We show that it is without loss
376
+ of generality for the certifier to restrict to offering threshold certificates that
377
+ guarantee that a product is at least a certain level of quality. We characterize
378
+ the unique Walrasian market equilibrium allocation that results from any as-
379
+ signment of such certificates to producers. We then use that characterization
380
+ to solve for each producer’s utility-maximizing choice of certificate from the
381
+ certifier’s menu, which will also be unique.
382
+ 3.1
383
+ Certifications as Minimum Quality Thresholds
384
+ A first simple observation is that since producer costs are increasing in quality
385
+ level, and since goods at different quality levels but with the same certification
386
+ are indistinguishable to consumers (and hence must sell at the same price),
387
+ a producer who is assigned certificate σ will always choose to produce at the
388
+ minimum quality level eligible for that certificate.
389
+ Observation 3.1. Fix any certifier menu M and any production and certifica-
390
+ tion strategy of the producers. Then for any producer ψ, selecting certificate σ
391
+ and producing at quality q > min σ is dominated by selecting certificate σ and
392
+ producting at quality min σ.
393
+ Given this observation, we know that for any menu M, any equilbrium
394
+ strategy Γ for the producers, and any certificate σ, the marginal distribution
395
+ over quality Γσ will be a point mass at min σ. In particular, any two certificates
396
+ with the same minimum will induce the same equilibrium beliefs over quality and
397
+ hence have indistinguishable value to all consumers. It is therefore without loss
398
+ of generality for the certifier to only offer certificates of the form σq = [q, 1]; i.e.,
399
+ certificates that are differentiated only with respect to their minimum values.
400
+ If a producer selects certificate σq, then that producer’s chosen quality level
401
+ at equilibrium will necessarily be q. Any given certification menu M therefore
402
+ reduces to a (possibly infinite) collection of quality levels in [0, 1] to certify.
403
+ Motivated by this observation, we will assume for the remainder of the paper
404
+ that all certificates are of the form [q, 1], and associate each σ = [q, 1] with its
405
+ quality threshold q. We can then think of a certification menu M as a collection
406
+ of pairs {(qi, ti)}, where qi is a quality threshold and ti is a corresponding price
407
+ for certifying that quality is at least qi.
408
+ 3.2
409
+ Uniqueness and Assortativeness of Competitive Mar-
410
+ ket Allocations
411
+ We now turn to an analysis of the competitive market outcome that will result
412
+ given the strategies of the certifier and producers. Recall that we can restrict
413
+ attention to certificates of the form σq = [q, 1] and that any good with certificate
414
+ σq will have quality q with probability 1, so for the remainder of the section we
415
+ 9
416
+
417
+ will think of a market outcome as an allocation x and prices p of quality levels.
418
+ That is, x(φ) ∈ [0, 1] for all consumers φ, and for each q ∈ [0, 1] in menu M
419
+ there is an associated market price pq. We emphasize that x is a mapping from
420
+ consumers to the certified goods they buy at market, whereas Γ is a mapping
421
+ from producers to the certificates that they choose from the certifier.
422
+ The following lemma shows that for any choice of certification menu M and
423
+ production and certification strategy Γ of the producers, all competitive market
424
+ equilibria in the resulting market have the same uniquely-determined allocation.
425
+ This allocation will be assortative, with higher-type consumers purchasing the
426
+ higher-quality certificates.
427
+ Lemma 3.2. Fix any certifier menu M and any strategy Γ of the produc-
428
+ ers. Then in every competitive market outcome (x, p), the allocation x satisfies
429
+ x(φ1) ≤ x(φ2) and px(φ1) ≤ px(φ2) for all φ1 ≤ φ2.
430
+ Proof. Since consumers are unit-demand and each producer has a single unit of
431
+ good, a Walrasian equilibrium (x, p) of the market is guaranteed to exist. By
432
+ the first welfare theorem, any such equilibrium must maximize the total welfare,
433
+
434
+ φ
435
+ f(x(φ); φ)dF(φ).
436
+ Suppose there exist types φ1 < φ2 with x(φ1) > x(φ2). By the single-crossing
437
+ condition, we have that
438
+ f(x(φ1); φ2) − f(x(φ2); φ2) > f(x(φ1); φ1) − f(x(φ2); φ1)
439
+ and hence
440
+ f(x(φ1); φ2) + f(x(φ2); φ1) > f(x(φ1); φ1) + f(x(φ2); φ2)
441
+ which contradicts the supposed welfare optimality of allocation x.
442
+ We now turn to prices. Fix any consumer types φ1 < φ2, so in particular
443
+ we know x(φ1) ≤ x(φ2), and suppose for contradiction that px(φ1) > px(φ2). By
444
+ monotonicity of the value function f, we must have f(x(φ1); φ1) ≤ f(x(φ2); φ1).
445
+ But this then means f(x(φ2); φ1)−px(φ2) > f(x(φ1); φ1)−px(φ1), which violates
446
+ the competitive market condition that consumer type φ1 is choosing her most-
447
+ preferred good. We therefore conclude that px(φ1) ≤ px(φ2), as claimed.
448
+ 3.3
449
+ Uniqueness and Assortativeness of Certificate Selec-
450
+ tion
451
+ Given that market outcomes are well-defined, we next turn to the equilibrium
452
+ choices of the producers when selecting quality levels and their corresponding
453
+ certifications. We again show that for any menu M of certificates offered, the
454
+ quality choices of producers are unique at equilibrium. Moreover, higher-type
455
+ producers will always select (weakly) higher certificates. In the Lemma 3.2, we
456
+ saw that higher-type consumers also purchase (weakly) higher certificates. As
457
+ 10
458
+
459
+ we discuss later, this means the matching in any equilibrium will be assortative
460
+ and hence constrained-efficient given the available certificates.
461
+ Recall that a producer strategy Γ is a mapping from producer type ψ to a
462
+ choice of certification and quality, which we know from will always coincide. We
463
+ will therefore write Γ(ψ) = q to mean that producer ψ produces at quality level
464
+ q and purchases certificate σq. In particular, we must have Γ(ψ) ∈ M for all ψ.
465
+ Lemma 3.3. Fix any certification menu M offered by the certifier. Then there
466
+ is a unique equilibrium strategy Γ for the producers, and Γ(ψ) is weakly increas-
467
+ ing in ψ.
468
+ Proof. Fix strategy Γ, which implies the measure of certificates chosen by the
469
+ collection of producers. Let (x, p) denote a Walrasian equilibrium in the result-
470
+ ing competitive market, and recall that x is uniquely determined.
471
+ We first show that Γ is weakly increasing is ψ. Assume for contradiction
472
+ that there exist ψ1 < ψ2 with q1 = Γ(ψ1) and q2 = Γ(ψ2) with q2 < q1. Then
473
+ by the single-crossing condition for producers, we have g(q1; ψ1) − g(q2; ψ1) >
474
+ g(q1; ψ2) − g(q2; ψ2).
475
+ But then, if we let pq1 and pq2 denote the Walrasian
476
+ equilibrium prices of q1 and q2 given Γ, we have
477
+ (pq1 − g(q1; ψ1)) + (pq2 − g(q2; ψ2)) < (pq1 − g(q1; ψ2)) + (pq2 − g(q2; ψ1))
478
+ which means that either
479
+ pq1 − g(q1; ψ1) < pq2 − g(q2; ψ1)
480
+ or
481
+ pq2 − g(q2; ψ2) < pq1 − g(q1; ψ2).
482
+ In other words, either producer ψ1 or ψ2 (or both) would strictly improve their
483
+ utility by switching their choice of quality and certification. As such a swap has
484
+ measure zero and does not influence the competitive equilibrium, this would be
485
+ an improving deviation for the producer(s), violating the assumption that Γ is
486
+ an equilibrium strategy for the producers.
487
+ We have shown that Γ is weakly increasing in ψ. On the other hand, we
488
+ know from Lemma 3.2 that the market allocation of quality levels to consumers is
489
+ weakly increasing in φ. This means that any equilibrium outcome of production
490
+ and trade is equivalent to one in which consumers and producers are matched
491
+ assortatively, with higher-type consumers trading with higher-type producers.
492
+ In other words, for any producer type ψ, there is a consumer type φ = φ(ψ) such
493
+ that ψ always trades with φ(ψ). Specifically, φ(ψ) is such that F(φ(ψ)) = G(ψ)
494
+ (treating F and G as cumulative distribution functions).
495
+ Given this, we claim that Γ(ψ), the certification selected by producer ψ
496
+ at equilibrium, will always be a certificate qi from the certifier’s menu M =
497
+ {(qi, ti)} that maximizes f(qi; φ(ψ)) − g(qi; ψ) − ti. To see why, suppose for
498
+ contradiction that the producer instead chooses some other certificate q′ at price
499
+ t′ such that f(q′; φ(ψ))−g(q′; ψ)−t′ < f(qi; φ(ψ))−g(qi; ψ)−ti−ϵ for some ϵ > 0,
500
+ 11
501
+
502
+ and sells to consumer φ(ψ) at an assumed market-clearing price pq′.2 Then, the
503
+ producer ψ could instead deviate to purchasing qi at a price of ti, and offering
504
+ it on the competitive market at a price of pq′ +g(qi; ψ)−g(q′; ψ)+(ti −t′)+ϵ/2.
505
+ Note that if consumer φ(ψ) were to purchase from producer ψ at this price,
506
+ then her utility would be
507
+ f(qi; φ(ψ)) − [pq′ + g(qi; φ(ψ)) − g(q′; φ(ψ)) + (ti − t′) + ϵ/2]
508
+ = (f(qi; φ(ψ)) − g(qi; ψ) − ti − ϵ) + (g(q′; ψ) + t′) + ϵ/2 − pq′
509
+ > f(q′; φ(ψ)) − (g(q′; ψ) + t′) + (g(q′; ψ) + t′) − pq′ + ϵ/2
510
+ > f(q′; φ(ψ)) − pq′.
511
+ But since (q′, pq′) is the most-demanded offering to consumer φ(ψ) in the market
512
+ equilibrium, this means that the offering of qi at the proposed price would be the
513
+ most-demanded offering to consumer φ(ψ) under this deviation. In particular
514
+ this means that some consumer would want to purchase qi at the suggested price,
515
+ and therefore in the adjusted market equilibrium after this proposed deviation
516
+ the price of qi must be at least this high.
517
+ We conclude that the utility of producer ψ under this deviation is at least
518
+ [pq′ + (g(qi; ψ) − g(q′; ψ) + (ti − t′) + ϵ/2] − g(qi; ψ) − ti = pq′ − g(q′; ψ) − t′ + ϵ/2
519
+ > pq′ − g(q′; ψ) − t′
520
+ and hence this deviation is strictly utility-improving for the producer, contra-
521
+ dicting the assumption that Γ is an equilibrium.
522
+ We therefore conclude that at equilibrium, each producer ψ chooses whichever
523
+ certificate qi from the menu maximizes f(qi; φ(ψ)) − g(qi; ψ) − ti. The choice of
524
+ each producer is therefore unique, up to tie-breaking on sets of measure zero.
525
+ An immediate corollary of Lemma 3.2 and Lemma 3.3 is that the equilibrium
526
+ outcome for a given menu M is not only essentially unique (up to the choice of
527
+ market-clearing prices), but also has a natural assortative interpretation. Each
528
+ producer in the market has a corresponding consumer with whom they will
529
+ always trade. The producer will select whichever certification level maximizes
530
+ the gains from trade between themselves and their partner consumer, less the
531
+ price of the certification.
532
+ Corollary 3.4. For any menu M = {(qi, ti)} of the certifier, the resulting
533
+ equilibrium market outcome has producer ψ trade with consumer φ = φ(ψ) where
534
+ G(ψ) = F(φ). The level of quality at which φ and ψ trade maximizes f(qi; φ) −
535
+ g(qi; ψ) − ti, and is weakly increasing in ψ.
536
+ 2Note that as we showed Γ is weakly increasing in ψ, the deviation can not change the
537
+ ordering of firms in terms of the certificate they purchase and hence does not change the
538
+ consumer to which they sell.
539
+ 12
540
+
541
+ 4
542
+ Revenue-Optimal Certification
543
+ In the previous section we solved for the equilibrium outcome of the game played
544
+ between producers and consumers given a certification menu M chosen by the
545
+ certifier. With that characterization in hand, we now turn to the problem faced
546
+ by a certifier who wishes to construct a revenue-maximizing slate of certificates.
547
+ A priori, it would appear that the choice of which certificates to offer in-
548
+ fluences the downstream competitive market allocation and prices. After all,
549
+ producers compete for consumers and the certificates they purchase determine
550
+ how they fare in this competition. Hence one might expect the menu of avail-
551
+ able certificates to influence how much each producer would be willing to pay
552
+ for any given certificate. However, as we will show, the assortative characteriza-
553
+ tion of market outcomes means that the certifier can reason about the behavior
554
+ of each producer type separately, without worrying about how they will jointly
555
+ interact in the competitive market. Thus, the certifier is essentially facing a
556
+ single buyer with an unknown type, and must simply maximize revenue subject
557
+ to this buyer. This leads us to define a reduction from the certifier’s problem
558
+ to that of a seller facing a (non-linear) buyer.
559
+ 4.1
560
+ Reduction to a Non-linear Pricing Problem
561
+ We will relate the certifier’s revenue-maximization problem to a non-linear pric-
562
+ ing problem between a single seller and a single buyer. The buyer seeks to buy
563
+ a perfectly divisible good. The seller may commit to a menu of quantities and
564
+ prices. If a buyer of type θ purchases quantity q ∈ [0, 1] of the good at a total
565
+ price of t, then the buyer utility is
566
+ u((q, t); θ) = v(q; θ) − t,
567
+ where v is concave in quantity q but not necessarily non-decreasing, and v(0; θ) =
568
+ 0 for all θ. The valuations v satisfy a single-crossing condition, which is that
569
+ for θ1 < θ2 and q1 < q2, we have
570
+ v(q2; θ2) − v(q1; θ2) > v(q2; θ1) − v(q1; θ1).
571
+ The principal seeks to maximize revenue subject to a constant cost of production
572
+ c > 0 for any non-zero quantity given a prior over buyer types. Given a menu
573
+ M, we will write Rev(M) to denote the expected revenue obtained from M. We
574
+ will also write OPT for the revenue of the revenue-maximizing choice of menu
575
+ M.
576
+ Proposition 4.1. The certifier’s revenue-maximization problem is equivalent
577
+ to an instance of the non-linear pricing problem described above.
578
+ Proof. The certifier’s problem is to design a certificate menu M = {(qi, ti)}
579
+ that maximizes the total revenue collected, less the certification cost c paid
580
+ to verify any non-trivial certificate qi > 0. By Corollary 3.4, given menu M,
581
+ 13
582
+
583
+ each producer ψ will purchase whichever certificate qi maximizes f(qi; φ(ψ)) −
584
+ g(qi; ψ) − ti.
585
+ For q ∈ [0, 1], we can interpret ψ as a buyer type and define valuation
586
+ function v(q; ψ) = f(qi; φ(ψ))−g(qi; ψ). Then since f and g both satisfy single-
587
+ crossing with respect to their corresponding types, valuation function v does
588
+ as well. Moreover, v is concave and v(0; ψ) = 0. By definition, the producers’
589
+ choices of certificates from menu M corresponds precisely to the buyer’s choice
590
+ of quantity when facing the same menu, interpreting each certificate quality
591
+ threshold as a quantity.
592
+ Thus the outcomes, and hence revenue, in the two
593
+ settings are equivalent.
594
+ Note that an immediate implication of Proposition 4.1, given Corollary 3.4,
595
+ is that for any menu M chosen by the seller, the choice of quantity purchased
596
+ by the buyer is weakly increasing in buyer type.3
597
+ 4.2
598
+ Optimizing Revenue in the Non-linear Pricing Prob-
599
+ lem
600
+ By Proposition 4.1, to solve the certifier’s revenue maximization problem it
601
+ suffices to optimize revenue in the non-linear pricing problem. As a first step, we
602
+ note that in the special case where the valuations v are linear in quantity (which
603
+ happens, for example, if the cost g and value f functions in our certification
604
+ problem are both linear in q), this problem reduces to a standard pricing problem
605
+ in mechanism design. A characterization due to Myerson Myerson (1981) then
606
+ immediately establishes that it is revenue-optimal to choose a menu with only
607
+ a single non-trivial item (q, p) with q = 1.
608
+ Observation 4.2. If v(q; θ) is linear in q for all θ, then there is a revenue-
609
+ optimal menu that offers only quantity q = 1 at some price p. This price p will
610
+ be chosen to maximize p × Prθ[v(1; θ) > p].
611
+ However, in general, non-linearity substantially changes the problem relative
612
+ to the linear case. In particular, it is not necessarily optimal to offer a single
613
+ menu item.
614
+ Proposition 4.3. There are problem instances in which posting any single
615
+ menu item is an arbitrarily poor approximation to the optimal revenue. The ap-
616
+ proximation factor can be as large as Ω(log(H)), where H = maxθ1,θ2
617
+ maxq v(q;θ1)
618
+ maxq v(q;θ2)
619
+ is the ratio between the highest and lowest maximum values across buyer types.
620
+ Proof. Choose c = 0 and consider the valuation function v(q; θ) = q if q ≤ θ,
621
+ and v(q; θ) = 2θ − q if q ≥ θ. That is, v is piecewise linear for each θ, with
622
+ maximum value θ occurring at q = θ. Fix some H ≥ 1 and suppose the type
623
+ distribution is such that Pr[θ > h] = 1/h for all h ∈ [1, H]. That is, the type
624
+ distribution is equal-revenue on range [1, H].
625
+ 3Alternatively, this is a direct consequence of the single-crossing condition on valuation
626
+ functions v.
627
+ 14
628
+
629
+ This valuation function is concave and satisfies v(0; θ) = 0 for all θ. More-
630
+ over, it satisfies the single-crossing condition. Indeed, for any q and any θ1 < θ2,
631
+ we note that
632
+ d
633
+ dqv(q; θ1) ≤
634
+ d
635
+ dqv(q; θ2), since for any θ this derivative is 1 for q < θ
636
+ and −1 for q > θ. Since v is also continuous in both q and θ, the single-crossing
637
+ condition is implied.4 Finally, we note that this valuation function v can in-
638
+ deed arise in our reduction from the certification problem with producers and
639
+ consumers.5
640
+ We now show the desired gap in approximation. Consider any menu M with
641
+ a single non-trivial menu item (q, p). Then the revenue achieved by the seller
642
+ is at most the welfare generated by the optimal allocation of quality level q.
643
+ Since v(q; θ) ≤ q for all θ, this is certainly at most q times the probability that
644
+ v(q; θ) > 0, which is q Pr[θ > q/2] ≤ q(2/q) = 2.
645
+ On the other hand, the seller could offer a menu that includes every quality
646
+ level q ∈ [1, H] at a price of q/2.
647
+ A buyer of type θ would then choose to
648
+ purchase quality level q = θ for a utility of θ/2, generating revenue θ/2.6 The
649
+ total revenue is then E[θ/2] = O(log H).
650
+ Posting a single menu item is therefore at best an O(log H) approximation
651
+ to the optimal revenue, as claimed.
652
+ We therefore know that any approximately revenue-optimal mechanism must
653
+ sometimes have multiple non-trivial offerings on its menu. What should such a
654
+ menu look like? Note that since buyers do not have free disposal in our setting
655
+ (i.e., valuations are non-monotone), it is not even immediately obvious that
656
+ higher quantities should be sold for higher prices in a revenue-optimal menu.
657
+ We next establish that, in fact, there is always a revenue-optimal choice of menu
658
+ for which higher quantities are sold at higher prices.
659
+ Definition 4.4. We say a menu M = {(qi, pi)} is monotone if for all (qi, pi),
660
+ (qj, pj) ∈ M such that qi < qj, we have pi ≤ pj.
661
+ Lemma 4.5. For any instance of the non-linear pricing problem there is a
662
+ revenue-optimal menu that is monotone.
663
+ Proof. Let M be a revenue-optimal menu, and write M = {(qi, pi)}i∈Λ where
664
+ Λ is some (possibly uncountable) index set. It is without loss to assume Λ is a
665
+ subset of [0, 1] such that qi < qj for all i < j (e.g., by reindexing so that the
666
+ index of qi is equal to qi). We can further assume without loss of generality that
667
+ every item in M is purchased by some buyer type, as any element that is never
668
+ purchased could be removed from M without impact. By Corollary 3.4, quality
669
+ 4Technically our construction only satisfies weak single-crossing since the inequality in
670
+ derivatives is not strict. We can make the example strict by perturbing the slope of the initial
671
+ line segment by an infinitesimal amount so that it depends on the type θ. We omit these
672
+ details for expositional clarity.
673
+ 5In particular, take φ and ψ to be supported on [1, H], define f(q; φ) = q for all φ and
674
+ g(q; ψ) = max{0, 2(q − ψ)} for all ψ. Then f is concave (in fact, linear), g is convex, the
675
+ single-crossing conditions are satisfied, and v(q; θ) = f(q; φ(θ)) − g(q; θ) as required.
676
+ 6Buying a higher quality level q′ > θ is worse for the buyer because it generates less value at
677
+ a higher price, whereas buying any quality level q′ < θ generates utility q′−q′/2 = q′/2 < θ/2.
678
+ 15
679
+
680
+ levels purchased will be monotone non-decreasing in buyer type. This means
681
+ that every item (qi, pi) is purchased by some contiguous interval of buyer types
682
+ Ii (which may have measure zero).
683
+ Suppose that menu M is not monotone. This means that either there is an
684
+ element i < sup Λ such that pi > pj for all j > i, or else there exists a pair of
685
+ menu items (qi, pi) and (qj, pj) with j > i such that (a) there exists some ℓ ∈ Λ
686
+ with i < ℓ < j, and (b) pℓ < min{pi, pj} for all i < ℓ < j.
687
+ Consider the former case, where there is an element i < sup Λ such that
688
+ pi > pj for all j > i. In particular there must exist some j ∈ Λ with j > i. Let
689
+ M ′ be the menu {(qℓ, pℓ)}ℓ∈Λ,ℓ≤i. I.e., M ′ is M with all elements with quantities
690
+ greater than qi removed. Note that for all j ≤ i, since the types Ij preferred
691
+ element (qj, pj) to any other element in M, they prefer element (qj, pj) to any
692
+ other element in M ′ as well. Moreover, since purchase decisions are monotone in
693
+ buyer type, we conclude that all types θ ∈ Iℓ with ℓ > i will purchase element
694
+ (qi, pi) from menu M ′. But since pi > pℓ for all ℓ > i, this means that the
695
+ revenue generated by menu M ′ is strictly greater than the revenue generated
696
+ by menu M, contradicting the supposed optimality of menu M.
697
+ Next consider the other case, there exists a pair of menu items (qi, pi) and
698
+ (qj, pj) such that pℓ < min{pi, pj} for all i < ℓ < j.
699
+ Let M ′ be the menu
700
+ {(qℓ, pℓ)}ℓ∈Λ,ℓ≤i ∪ {(qℓ, p��)}ℓ∈Λ,ℓ≥j. That is, M ′ is menu M with all elements
701
+ strictly between (qi, pi) and (qj, pj) removed. Then as in the previous case, for
702
+ all ℓ ≤ i and ℓ ≥ j, types Iℓ all still prefer to purchase (qℓ, pℓ). In particular,
703
+ types Ii purchase (qi, pi) and types Ij purchase (qj, pj). By monotonicity of
704
+ purchasing decisions due to Corollary 3.4, all intermediate types θ ∈ Iℓ for
705
+ i < ℓ < j must purchase either (qi, pi) or (qj, pj). As pi and pj are both larger
706
+ than the prices of the elements those types were purchasing under menu M, the
707
+ revenue of menu M ′ must be strictly larger, which is again a contradiction.
708
+ We conclude that the prices in menu M must be monotone non-decreasing
709
+ in quality levels, as claimed.
710
+ 4.3
711
+ An FPTAS for Revenue
712
+ We are now ready to consider the problem of constructing an approximately
713
+ revenue-optimal menu for the non-linear pricing problem. For this we will make
714
+ one further technical assumption, which is that the derivative of the valuations
715
+ v(q; θ) with respect to q is bounded at 0. I.e., there exists some λ > 0 such
716
+ that
717
+ d
718
+ dqv(0; θ) < λ for all θ. In other words, buyers are not infinitely sensitive
719
+ to product quantity. In the context of certification, this is implied by having
720
+ d
721
+ dqf(0; φ) < λ for all φ, meaning that consumers are not infinitely sensitive to
722
+ quality at 0.
723
+ Assumption 4.6. There exists some λ > 0 such that
724
+ d
725
+ dqv(0; θ) < λ for all θ.
726
+ Given this assumption, the following result provides an FPTAS for the op-
727
+ timal menu with k items. We also show that by taking k = 1/ϵ one can obtain
728
+ an FPTAS for the optimal (unrestricted) menu.
729
+ 16
730
+
731
+ Theorem 4.7. A menu with revenue at least OPT −λϵ can be found in time
732
+ polynomial in λ and 1/ϵ. The menu can optionally be constrained to contain at
733
+ most k quality levels, in which case OPT is the optimal revenue achievable with
734
+ at most k quality levels.
735
+ Our first step to proving Theorem 4.7 is to show that we can restrict attention
736
+ to menus with at most 1/ϵ entries at only a small loss of revenue.
737
+ Lemma 4.8. For any monotone menu M, there exists a menu M ′ of size at
738
+ most O(1/ϵ) such that Rev(M ′) ≥ Rev(M) − O(ϵ).
739
+ Proof. Fix the revenue-optimal menu M = {(qi, pi)}i∈Λ. By Lemma 4.5 we can
740
+ assume M is monotone.
741
+ We define M ′ to be the following subset of M. Let A = {ℓϵ : 0 ≤ ℓ ≤ ⌊1/ϵ⌋}.
742
+ Then for each a ∈ A, we add to M ′ the element (q, p) from M with smallest p
743
+ such that p ≥ a. Then we note that M ′ contains at most ⌈1/ϵ⌉ items. Moreover,
744
+ for each element (q, p) ∈ M there is some (q′, p′) ∈ M ′ such that p′ ≥ p − ϵ.
745
+ Any element in M ′ will still be purchased by the types that purchased that
746
+ element in M, as the menu of alternative options has only gotten smaller. Since
747
+ prices are monotone in quantity, and quantities purchased are monotone in buyer
748
+ type, we can conclude that any type that was purchasing (q, p) in the original
749
+ menu M will then purchase an element with price at least p − ϵ in menu M ′.
750
+ The total loss in revenue conditional on any given buyer type is therefore at
751
+ most ϵ, and hence the total revenue loss is at most ϵ as well.
752
+ Next, we show that we can discretize the possible (quality, price) pairs that
753
+ appear in our menu without losing too much revenue.
754
+ Lemma 4.9. For any monotone menu M of size k, there exists a menu M ′
755
+ such that
756
+ • M ′ has at most k elements,
757
+ • for each (q, p) ∈ M ′, p is a multiple of ϵ and q = ϵ(1+ϵ)ℓ for some integer
758
+ ℓ ≥ 0, and
759
+ • Rev(M ′) ≥ Rev(M) − O((k + λ)ϵ).
760
+ Proof. Let us first give the high-level idea for the construction, which is illus-
761
+ trated in Figure 1. We will round each (quantity, price) pair on menu M in
762
+ three steps. First, we will round each quantity down to an appropriate dis-
763
+ cretized grid, and then also lower the corresponding price to keep constant the
764
+ ratio of quantity to price. The concavity of the value functions will then imply
765
+ that buyer utilities cannot be reduced by too much, multiplicatively, as a result
766
+ of this change. This first step discretized the quantities; in the second step we
767
+ discretize prices by rounding each price down to an appropriate grid, which can
768
+ only increase utilities. At this point we would be almost done, except for one
769
+ complication: we must make sure that the (small) changes in buyer utility we in-
770
+ duce do not result in buyers switching from more expensive items to significantly
771
+ 17
772
+
773
+ Figure 1: Illustration of the discretization procedure from Lemma 4.9.
774
+ Red
775
+ curves denote buyer valuation functions.
776
+ Menu item (p, q) is discretized in
777
+ quantities to (˜p, q′) by shifting along a line to the origin, then discretized in
778
+ price to (ˆp, q′). A final discount is applied to obtain the adjusted menu item
779
+ (p′, q′).
780
+ cheaper items from the menu. It is here where we use the price-monotonicity of
781
+ the menu. Since the higher-quanitity items are the more expensive ones, in our
782
+ third step we will provide discounts for the higher-quantity menu items, which
783
+ will offset any utility perturbations due to discretization. These discounts are
784
+ what lead to the loss term being proportional to kϵ, rather than ϵ, in our error
785
+ bound.
786
+ We now move on to the formal construction. Fix menu M of size k, so that
787
+ M = {(q1, p1), . . . , (qk, pk)} where q1 < q2 < . . . < qk. We can assume without
788
+ loss of generality that p1 ≤ p2 ≤ . . . ≤ pk, and that each element of M is
789
+ purchased with positive probability.
790
+ We construct a new menu M ′ in the following sequence of steps. First, for
791
+ each (qi, pi) with qi ≥ ϵ, let q′
792
+ i be qi rounded down to the nearest value of ϵ(1+ϵ)ℓ
793
+ where ℓ ≥ 0 is an integer. Then define ˜pi = pi × (q′
794
+ i/qi), noting that we chose ˜pi
795
+ so that ˜pi/q′
796
+ i = pi/qi. This element (q′
797
+ i, ˜pi) corresponds to rounding quantity but
798
+ keeping the price to quantity ratio constant in our intuitive description above.
799
+ For our second step, we take ˆpi to be ˜pi rounded down to the nearest multiple
800
+ of ϵ. Finally, in our third step, we introduce our discounts. For this we define
801
+ p′
802
+ i = ˆpi − 3iϵ, noting that the difference between p′
803
+ i and ˆpi is increasing in i.
804
+ This completes our discretization procedure, so we add (q′
805
+ i, p′
806
+ i) to menu M ′. We
807
+ note that menu M ′ is not necessarily monotone, may contain elements that are
808
+ preferred by no types, and may contain elements with negative prices.
809
+ We claim that for every type θ, if θ purchased item (qi, pi) in menu M with
810
+ qi ≥ ϵ, then θ will purchase some (q′
811
+ j, p′
812
+ j) in menu M ′ such that j ≥ i. To see
813
+ why, first consider what would happen if (q′
814
+ i, ˜pi) were on the menu. What is
815
+ u((q′
816
+ i, ˜pi); θ)? Since valuation function v is concave and v(0; θ) = 0, we must
817
+ 18
818
+
819
+ price
820
+ 3E
821
+ 2e:
822
+ (p,q)
823
+ (p,q').
824
+ (p,q').
825
+ E
826
+ quantity
827
+ E E(1 +e) e(1+)2
828
+ (3 + L)3
829
+ -2E-
830
+ (p',q').have v(q′
831
+ i; θ) ≥ q′
832
+ i
833
+ qi v(qi; θ). But since q′
834
+ i ≥
835
+ 1
836
+ 1+ϵqi, this implies
837
+ u((q′
838
+ i, ˜pi); θ) = v(q′
839
+ i; θ) − ˜pi
840
+ ≥ q′
841
+ i
842
+ qi
843
+ v(qi; θ) − ˜pi
844
+ = q′
845
+ i
846
+ qi
847
+ (v(qi; θ) − pi)
848
+
849
+ 1
850
+ 1 + ϵu((qi, pi); θ)
851
+ ≥ u((qi, pi); θ) − ϵ.
852
+ Since we also know that ˜pi − ϵ ≤ ˆpi ≤ ˜pi and p′
853
+ i = ˆpi − 3iϵ, we have
854
+ u((q′
855
+ i, p′
856
+ i); θ) = u((q′
857
+ i, ˆpi); θ) + 3iϵ
858
+ ≥ u((q′
859
+ i, ˜pi); θ) + 3iϵ
860
+ ≥ u((qi, pi); θ) + (3i − 1)ϵ.
861
+ On the other hand, for any j < i, the utility of purchasing (q′
862
+ j, ˜pj) is at most
863
+ the utility of purchasing (qj, ˜pj), which is at most ϵ more than the utility of
864
+ purchasing (qj, pj) (since the ratio between pj and ˜pj is no greater than (1+ϵ)).
865
+ We therefore have
866
+ u((q′
867
+ j, p′
868
+ j); θ) = u((q′
869
+ j, ˆpj); θ) + 3jϵ
870
+ ≤ u((q′
871
+ j, ˜pj); θ) + (3j + 1)ϵ
872
+ ≤ u((qi, pi); θ) + (3j + 2)ϵ.
873
+ Since we know that u((qi, pi); θ) ≥ u((qj, pj); θ) by assumption that θ purchases
874
+ item (qi, pi) from menu M, we conclude that u((q′
875
+ i, p′
876
+ i); θ) ≥ u((q′
877
+ j, p′
878
+ j); θ) as well,
879
+ since (3j + 2) ≤ (3i − 1) for i > j.
880
+ We conclude that each type θ that purchases (qi, pi) from M with qi ≥ ϵ
881
+ will purchase a menu item (q′
882
+ j, p′
883
+ j) from M ′ such that j ≥ i. Since prices are
884
+ monotone in menu M, we conclude that the total loss in revenue can be at most
885
+ the difference in price between pi and p′
886
+ i for any i. This is at most O(kϵ).
887
+ Finally, consider a type θ that purchases (qi, pi) from M with qi < ϵ. By
888
+ Assumption 4.6, the maximum willingness to pay for any agent for quality level
889
+ ϵ is λϵ.
890
+ These types therefore generate revenue at most λϵ, thus regardless
891
+ of their purchase behavior they account for a total loss in revenue of at most
892
+ O(λϵ).
893
+ With Lemma 4.9 in hand, we can complete the proof of Theorem 4.7 by
894
+ employing dynamic programming to determine the revenue-optimal mechanism
895
+ with a given maximum-quality entry. One subtlety in the construction is that
896
+ we must be careful to account for potential cannibalization by higher-quality
897
+ elements in the menu. We handle this by insisting that the menu we construct
898
+ contains only elements that are selected by a non-zero measure of buyer types,
899
+ and we check this condition when recursively applying the dynamic program.
900
+ 19
901
+
902
+ Proof of Theorem 4.7. We show how to compute the optimal menu with qual-
903
+ ities and prices chosen from a discrete indexed set of possible options, using
904
+ dynamic programming. In general, given a quantity q and price p that lie in
905
+ our discrete set of options, we will use Q and P to denote the integer indexing
906
+ of q and p, respectively.
907
+ Given any choice of Q and P and some k ≥ 1, write M[Q, P, k] for the
908
+ optimal revenue that can be obtained using a menu with at most k elements,
909
+ of which the one with highest quality is the one indexed by Q and P. We will
910
+ also write L[Q, P, k] for the lowest type θ that purchases quality level Q in this
911
+ optimal menu. We can compute M[Q, P, k] and L[Q, P, k] recursively as follows.
912
+ If k = 1 then M[Q, P, k] is precisely p times the probability that v(q; θ) ≥ p,
913
+ and L[Q, P, k] is precisely the infimum of types θ for which v(q; θ) ≥ p.
914
+ For k > 1, we will consider all possible options for the next-highest quality
915
+ level on the menu given our discretization, say (q′, p′) with Q′ < Q. For each
916
+ choice of (Q′, P ′), we let θ(Q′, P ′) denote the type that is indifferent between
917
+ menu items (Q′, P ′) and (Q, P), if any. Recall from the single-crossing condition
918
+ that this choice of θ(Q′, P ′) is unique if it exists. If there is no such θ(Q′, P ′),
919
+ then we disqualify menu item (Q′, P ′) from consideration. Otherwise, we con-
920
+ sider L[Q′, P ′, k−1], the lowest type that purchases element (Q′, P ′) in the opti-
921
+ mal menu with highest quality level Q′ at price P ′. If L[Q′, P ′, k−1] ≥ θ(Q′, P ′),
922
+ then again we disqualify menu item (Q′, P ′) from consideration, as this means
923
+ that the optimal menu containing menu item (Q, P) does not include any types
924
+ that would purchase menu item (Q′, P ′).
925
+ Otherwise, we have that L[Q′, P ′, k−1] < θ(Q′, P ′). We can therefore calcu-
926
+ late the revenue from the optimal menu with highest and second-highest quality
927
+ levels (Q, P) and (Q′, P ′) as R(Q′, P ′) = M[Q′, P ′, k − 1] + (P − P ′)Pr[θ >
928
+ θ(Q′, P ′)]. That is, the additional revenue gain or loss due to including menu
929
+ item (Q, P) is (P − P ′)Pr[θ > θ(Q′, P ′)], the difference due to agents with type
930
+ greater than θ(Q′, P ′) switching from menu item (Q′, P ′) to menu item (Q, P).
931
+ Finally, consider also the revenue that would be obtained by using only
932
+ menu item (Q, P); call this R. If all potential choices of (Q′, P ′) were elim-
933
+ inated or if R > R(Q′, P ′) for all potential choices of (Q′, P ′), then we set
934
+ M[Q, P, k] = R and set L[Q, P, k] to be the infimum type θ such that v(Q; θ) ≥
935
+ P. This corresponds to the case that the optimal menu contains only the el-
936
+ ement (P, Q). Otherwise, let (Q′, P ′) be the choice that maximizes R(Q′, P ′),
937
+ which by assumption is larger than R. Then we take L[Q, P, k] = θ(Q′, P ′) and
938
+ M[Q, P, k] = R(Q′, P ′).
939
+ We conclude that we can fill tables M and L, with each entry taking time
940
+ proportional to ϵ−2 (the time needed to consider every possible choice (Q′, P ′)).
941
+ As there are kϵ−2 entries in total, the total time to fill the tables is at most
942
+ kϵ−4. The revenue-optimal mechanism with at most k menu items can then be
943
+ obtained by taking the maximum of M[Q, P, n] over all choices of Q and P.
944
+ Finally, by Lemma 4.8, we can take k = 1/√ϵ and our dynamic program will
945
+ obtain a menu M such that Rev(M) is at most O(λ/√ϵ) less than the optimal
946
+ revenue. An appropriate change of variables, taking k = 1/ϵ and discretizing to
947
+ multiples of ϵ2, then implies that our resulting menu is at most O(λϵ) less than
948
+ 20
949
+
950
+ that of the optimal menu.
951
+ 5
952
+ Welfare Maximization
953
+ We have studied the problem faced by a revenue-maximizing certifier. But what
954
+ about a certifier that wishes to maximize the welfare enjoyed by the consumers
955
+ and producers? We could think of such a certifier as a government agency who
956
+ is offering certification services not to generate profit, but rather to maximize
957
+ the efficiency of production and trade.
958
+ We define the welfare of a market outcome as the sum of utilities of the
959
+ consumers, the producers, and the certifier, taking into account all transfers
960
+ between parties. Given a menu M of options provided by the certifier, we will
961
+ write Wel(M) for the welfare that results in the unique market outcome resulting
962
+ from menu M.
963
+ An immediate consequence of Corollary 3.4 is that the welfare-optimal choice
964
+ of menu is to offer all possible certification levels, at the cost of verification c.
965
+ Theorem 5.1. The welfare-optimal menu of certificates offers every possible
966
+ certification level q > 0 at a cost of c, and level 0 at a cost of 0.
967
+ Proof. If all quality levels were visible, the welfare-maximizing outcome is would
968
+ be for each producer ψ to trade with consumer φ(ψ) at whichever quality level
969
+ q maximizes their gains from trade f(q; φ(ψ)) − g(q; ψ). However, since quality
970
+ levels are hidden, producers and consumers can trade at a positive level of
971
+ quality only if the cost of verification is paid. So if the maximum gains from
972
+ trade is less than c, then it is preferable to trade at level 0. However, we observe
973
+ that this is precisely the outcome implemented at equilibrium from the proposed
974
+ certification menu, so it must be welfare-optimal over all possible menus.
975
+ The welfare-optimal menu described above includes an arbitrarily large num-
976
+ ber of certificates. In practice it may be helpful to find a welfare-optimal slate of
977
+ at most k certificates. It turns out that a minor variation on the dynamic pro-
978
+ gram described in the previous section can be used to compute an approximately
979
+ welfare-optimizing slate, under Assumption 4.6.
980
+ Theorem 5.2. Let M be the welfare-maximizing menu with at most k elements.
981
+ A menu with M ′ with at most k elements and such that Wel(M ′) ≥ Wel(M) −
982
+ O(λϵ) can be found in time polynomial in 1/ϵ.
983
+ Proof. The proof is very similar to the one for Theorem 4.7, and strictly simpler,
984
+ so we only briefly describe the differences here. First, it is without loss to restrict
985
+ attention to menus that post price c for every non-trivial level of quality.
986
+ Given any such menu M, we can discretize potential levels of quality by
987
+ rounding down to the nearest multiple of ϵ. By Assumption 4.6 this reduces
988
+ welfare by at most λϵ, as the welfare from each menu item is reduced by at most
989
+ this much and each producer selects her gains-from-trade-maximizing element
990
+ from the menu.
991
+ 21
992
+
993
+ Given such a discretization, one can express the welfare-optimal menu re-
994
+ cursively via dynamic programming as in Theorem 4.7, with the simplification
995
+ that we need only index on quality rather than (quality, price) pairs (since all
996
+ prices will be set to c). Rather than defining M[Q, k] (and respectively L[Q, k])
997
+ to be the maximum revenue of a menu with k elements and maximum quality
998
+ indexed by Q, it will be the maximum welfare of a menu with k elements and
999
+ maximum quality indexed by Q. Our method of recursively computing M[Q, k]
1000
+ (and L[Q, k]) then remains nearly unchanged relative to Theorem 4.7.
1001
+ The
1002
+ only change to note is the actual welfare calculation, relative to the revenue
1003
+ calculation. In Theorem 4.7 we used that the revenue obtained when agents of
1004
+ type θ > θ′ purchase certificate q at price p is p times Pr[θ > θ′]. In contrast,
1005
+ the welfare obtained when producers of type ψ > ψ′ all purchase certificate q
1006
+ at price c can be calculated in closed form as
1007
+ � ψ′
1008
+ ψ (f(q; φ(η)) − g(q; η) − c)dη.
1009
+ Substituting this welfare calculation for the revenue calculation completes the
1010
+ necessary changes.
1011
+ Is the revenue-maximizing choice of menu also approximately welfare-maximizing?
1012
+ As it turns out, the welfare that results from the certifier’s reveneu-optimal menu
1013
+ can be an arbitrarily small fraction of optimal welfare. This is inherited from
1014
+ standard examples of monopolistic distortion in the linear settings, where a mo-
1015
+ nopolist might be incentivized to sell much less of their good (i.e., certification)
1016
+ than what would be efficient in order to inflate prices.
1017
+ Proposition 5.3. There is a sequence of instances for the problem where welfare
1018
+ in the revenue-optimal solution is an arbitrarily small fraction of first-best wel-
1019
+ fare. The approximation can be as bad as Ω(log H), where H = maxθ1,θ2
1020
+ maxq v(q;θ1)
1021
+ maxq v(q;θ2)
1022
+ is the ratio between the highest and lowest maximum values across buyer types.
1023
+ Proof. Take c = 0 and suppose f(q; φ) = φq and g(q; ψ) = 0 for all q and ψ. Our
1024
+ distribution over consumer types φ is an equal-revenue distribution supported
1025
+ on [1, H]; that is, F(φ) = 1 − 1
1026
+ φ for all φ ∈ [1, H]. Recalling Observation 4.2, it
1027
+ is revenue-optimal for the certifier to offer contract q = 1 at a price of H, for
1028
+ an expected welfare (and revenue) of 1. However, the optimal welfare, log(H),
1029
+ can be achieved by offering contract q = 1 at price 0.
1030
+ However, one implication of our equilibrium analysis is that adding addi-
1031
+ tional certification options to a menu of certificates cannot reduce the sum of
1032
+ utilities of the consumers and producers, regardless of the prices selected. One
1033
+ interpretation of this is that welfare cannot be harmed by a revenue-maximizing
1034
+ certifier entering a market for certification in which some certification options
1035
+ are already available.
1036
+ Proposition 5.4. Consider two certification menus M and M ′ with M ⊆ M ′.
1037
+ Then Wel(M ′) − Rev(M ′) ≥ Wel(M) − Rev(M).
1038
+ Proof. If producer ψ selects option (q, p) from certification menu M, then the
1039
+ welfare generated for the producer ψ and corresponding consumer φ, less the
1040
+ revenue raised by the certifier, is f(q; φ(ψ)) − g(q; ψ) − p. By Corollary 3.4,
1041
+ 22
1042
+
1043
+ producer ψ purchases precisely whichever menu item from M maximizes this
1044
+ quantity.
1045
+ Providing additional items can therefore only increase the welfare
1046
+ jointly enjoyed by producer type ψ and corresponding consumer φ(ψ). As this
1047
+ holds pointwise for every ψ, it holds in aggregate over all types as well.
1048
+ References
1049
+ Viral V Acharya, Peter DeMarzo, and Ilan Kremer. 2011. Endogenous informa-
1050
+ tion flows and the clustering of announcements. American Economic Review
1051
+ 101, 7 (2011), 2955–79.
1052
+ S Nageeb Ali, Nima Haghpanah, Xiao Lin, and Ron Siegel. 2022. How to sell
1053
+ hard information. The Quarterly Journal of Economics 137, 1 (2022), 619–
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+ 678.
1055
+ Reza Alijani, Siddhartha Banerjee, Kamesh Munagala, and Kangning Wang.
1056
+ 2022. The limits of an information intermediary in auction design. In Proceed-
1057
+ ings of the 23rd ACM Conference on Economics and Computation. 849–868.
1058
+ Dirk Bergemann, Alessandro Bonatti, and Alex Smolin. 2018. The design and
1059
+ price of information. American economic review 108, 1 (2018), 1–48.
1060
+ Dirk Bergemann, Benjamin Brooks, and Stephen Morris. 2017. First-price auc-
1061
+ tions with general information structures: Implications for bidding and rev-
1062
+ enue. Econometrica 85, 1 (2017), 107–143.
1063
+ Dirk Bergemann, Yang Cai, Grigoris Velegkas, and Mingfei Zhao. 2022.
1064
+ Is
1065
+ Selling Complete Information (Approximately) Optimal?. In Proceedings of
1066
+ the 23rd ACM Conference on Economics and Computation. 608–663.
1067
+ Dirk Bergemann and Stephen Morris. 2019.
1068
+ Information design: A unified
1069
+ perspective. Journal of Economic Literature 57, 1 (2019), 44–95.
1070
+ Dirk Bergemann, Ji Shen, Yun Xu, and Edmund Yeh. 2012a. Multi-dimensional
1071
+ mechanism design with limited information. In Proceedings of the 13th ACM
1072
+ Conference on Electronic Commerce. 162–178.
1073
+ Dirk Bergemann, Ji Shen, Yun Xu, and Edmund M Yeh. 2012b. Mechanism
1074
+ design with limited information: the case of nonlinear pricing. In International
1075
+ Conference on Game Theory for Networks. Springer, 1–10.
1076
+ Tilman B¨orgers and Daniel Krahmer. 2015. An introduction to the theory of
1077
+ mechanism design. Oxford University Press, USA.
1078
+ Ozan Candogan and Philipp Strack. 2021. Optimal Disclosure of Information
1079
+ to a Privately Informed Receiver. In EC ’21: The 22nd ACM Conference on
1080
+ Economics and Computation, Budapest, Hungary, July 18-23, 2021, P´eter
1081
+ Bir´o, Shuchi Chawla, and Federico Echenique (Eds.). ACM, 263.
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+ 23
1083
+
1084
+ Shuchi Chawla, Hu Fu, and Anna Karlin. 2014. Approximate revenue maxi-
1085
+ mization in interdependent value settings. In Proceedings of the fifteenth ACM
1086
+ conference on Economics and computation. 277–294.
1087
+ Joe
1088
+ Sandler
1089
+ Clarke
1090
+ and
1091
+ Luke
1092
+ Barratt.
1093
+ [n. d.].
1094
+ Top
1095
+ airlines’
1096
+ promises
1097
+ to
1098
+ offset
1099
+ flights
1100
+ rely
1101
+ on
1102
+ ‘phantom
1103
+ credits’.
1104
+ Unearthed
1105
+ ([n. d.]).
1106
+ https://unearthed.greenpeace.org/2021/05/04/
1107
+ carbon-offsetting-british-airways-easyjet-verra/
1108
+ Marc N Conte and Matthew J Kotchen. 2010. Explaining the price of voluntary
1109
+ carbon offsets. Climate Change Economics 1, 02 (2010), 93–111.
1110
+ Peter M DeMarzo, Ilan Kremer, and Andrzej Skrzypacz. 2019. Test design and
1111
+ minimum standards. American Economic Review 109, 6 (2019), 2173–2207.
1112
+ Mathias Dewatripont, Patrick Bolton, et al. 2005. Contract theory. Technical
1113
+ Report. ULB–Universite Libre de Bruxelles.
1114
+ Piotr Dworczak. 2020.
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+ Mechanism design with aftermarkets: Cutoff mecha-
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+ nisms. Econometrica 88, 6 (2020), 2629–2661.
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+ Daniel W Elfenbein, Raymond Fisman, and Brian McManus. 2015.
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+ Market
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+ structure, reputation, and the value of quality certification. American Eco-
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+ nomic Journal: Microeconomics 7, 4 (2015), 83–108.
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+ Patrick
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+ Greenfield.
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+ [n. d.]a.
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+ Carbon
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+ offsets
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+ used
1127
+ by
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+ flawed
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+ system,
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+ warn
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+ experts.
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+ ([n. d.]).
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+ https://www.theguardian.com/environment/2021/may/04/
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+ carbon-offsets-used-by-major-airlines-based-on-flawed-system-warn-experts
1142
+ Patrick Greenfield. [n. d.]b.
1143
+ Revealed: more than 90% of rainforest carbon
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+ offsets by biggest provider are worthless, analysis shows.
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+ The Guardian
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+ ([n. d.]).
1147
+ https://www.theguardian.com/environment/2023/jan/18/
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+ revealed-forest-carbon-offsets-biggest-provider-worthless-verra-aoe
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+ Sanford J Grossman. 1981. The informational role of warranties and private
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+ disclosure about product quality. The Journal of Law and Economics 24, 3
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+ (1981), 461–483.
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+ Alejandro Guizar-Couti˜no, Julia PG Jones, Andrew Balmford, Rachel Car-
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+ menta, and David A Coomes. 2022. A global evaluation of the effectiveness
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+ of voluntary REDD+ projects at reducing deforestation and degradation in
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+ commodities. Journal of Economic theory 92, 1 (2000), 66–95.
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+ The RAND Journal of Economics (1999), 214–231.
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+ Paul R Milgrom. 1981. Good news and bad news: Representation theorems and
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+ applications. The Bell Journal of Economics (1981), 380–391.
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+ Paul R Milgrom and Robert J Weber. 1982. A theory of auctions and com-
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+ petitive bidding. Econometrica: Journal of the Econometric Society (1982),
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+ 1089–1122.
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+ An exploration in the theory of optimum income
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+ quality certification in a “lemons market”. Economic inquiry 41, 2 (2003),
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+ 279–291.
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+ 25
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+
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1
+ Classical Algorithm for the Mean Value problem over
2
+ Short-Time Hamiltonian Evolutions
3
+ Reyhaneh Aghaei Saem1 and Ali Hamed Moosavian1
4
+ 1Phanous
5
+ Abstract
6
+ Simulating physical systems has been an important application of classical and quantum computers. In
7
+ this article we present an efficient classical algorithm for simulating time-dependent quantum mechanical
8
+ Hamiltonians over constant periods of time. The algorithm presented here computes the mean value of an
9
+ observable over the output state of such short-time Hamiltonian evolutions. In proving the performance of
10
+ this algorithm we use Lieb-Robinson type bounds to limit the evolution of local operators within a lightcone.
11
+ This allows us to divide the task of simulating a large quantum system into smaller systems that can be
12
+ handled on normal classical computers.
13
+ 1
14
+ Introduction
15
+ Understanding physical systems with quantum mechanical interactions have been an increasingly challenging
16
+ and important task in the past century. Because nature is inherently quantum, many naturally occurring or
17
+ artificially implemented phenomena cannot be explained without quantum mechanics. Some early examples
18
+ include understanding atoms [1], diatomic molecules [2], the Meissner effect in superconductors [3] and
19
+ black-body radiation [4]. With the advent of better theoretical and computational tools, the list of phenomena
20
+ that have been shown to require quantum mechanics has grown enormously in the past few decades. To name
21
+ a few, some of the more notable examples include photosynthesis [5], topological ordered phases such as
22
+ fractional quantum Hall effect [6], isomerization of diazene [7] and non-Abelian lattice gauge theories [8].
23
+ Indeed, because of the challenging nature of simulating quantum systems on classical computers, one of the
24
+ earliest motivations for quantum computers has been to use them to study other quantum systems [9].
25
+ Besides the obvious practical applications, the problem of simulating quantum systems has some critical
26
+ complexity theory value too. On the one hand, it is known that the problem of simulating several quantum
27
+ systems belongs to the Bounded-error Quantum Polynomial-time complete (BQP-complete) class [10, 11,
28
+ 12]. Also, there are specific problems where the complexity class of simulating them varies as the runtime
29
+ increases, and exhibit a dynamical phase transition [13, 14].
30
+ In this paper, we present a classical algorithm for computing the expectation value of an observable that
31
+ can be written as a tensor product of local operators acting on each qubit. In the literature this problem is
32
+ sometimes called the quantum mean value problem [15] (not to be confused with a similarly named open
33
+ problem in mathematics [16]). Our algorithm evaluates the mean value of an operator where the state is
34
+ generated by evolving a product state under a geometrically local time-dependent Hamiltonian for a short
35
+ period of time. The setup of our problem is motivated by the state of current quantum technologies. On one
36
+ 1
37
+ arXiv:2301.11420v1 [quant-ph] 26 Jan 2023
38
+
39
+ hand, analog simulators lack fault-tolerance, which means they have a limited decoherence time, and on the
40
+ other hand, we still do not have access to fault-tolerant universal quantum computers either.
41
+ Suppose that we have a bounded-norm time-dependent Hamiltonian H(t) that acts on n-qubits. The
42
+ initial state of the system is assumed to be a product state, typically |ψ(0)⟩ = |0⟩⊗n. The quantum mean value
43
+ problem wants to compute the expectation value of an operator O with respect to |ψ(T)⟩. The expectation
44
+ value is represented with µ:
45
+ µ ≡ ⟨ψ(T)|O|ψ(T)⟩ .
46
+ (1)
47
+ We consider geometrically local time-dependent Hamiltonians that are defined on 2D or 3D lattices. 1D
48
+ systems with gapped Hamiltonians have been thoroughly studied before [17, 18].
49
+ The unitary evolution operator corresponding to this Hamiltonian can be written as:
50
+ U(t) = T :
51
+
52
+ exp
53
+
54
+ − i
55
+ ¯h
56
+ � t
57
+ 0
58
+ H(t′)dt′
59
+ ��
60
+ ,
61
+ (2)
62
+ where T : [.] means the operators respect time ordering. Assuming the observable can be written as the tensor
63
+ product of single-qubit Hermitian operators Oj acting on individual sites, O = O1 ⊗ O2 ⊗ · · · ⊗ On, we
64
+ can write the mean value of the observable O as:
65
+ µ = ⟨0n| U†(T)O1 ⊗ O2 ⊗ · · · ⊗ OnU(T) |0n⟩ .
66
+ (3)
67
+ In this paper, we introduce a classical algorithm that can approximate the mean value problem for time-
68
+ dependent Hamiltonians with an additive error. The structure of the rest of this paper is as follows. In Sec. 2
69
+ we give an overview of the algorithm. Section 3 explains how the dynamics of each operator is approximately
70
+ restricted to within a lightcone. In Sec. 4 we analyze the problem of classically calculating the local unitary
71
+ operators and compare different numerical algorithms for doing the task and in Sec. 5 we conclude.
72
+ 2
73
+ Algorithm
74
+ We provide a classical algorithm for approximating µ within an additive error, δ, for the special case of the
75
+ mean value problem where the time-dependent Hamiltonian is geometrically local in 2D or 3D.
76
+ Conceptually, the algorithm can be broken into two parts. First we use a Lieb-Robinson bound [19, 20,
77
+ 21] to limit the unitary evolution corresponding to each qubit within a lightcone. This allows us to classically
78
+ compute the unitary operator. Then we follow the steps in [15] to divide the lattice into pseudo 1D slices, that
79
+ can be efficiently simulated either using Matrix Product State algorithms [22, 23, 24] (for a nice review of
80
+ these algorithms see [18]) or the algorithm ascribed to [15].
81
+ Theorem 1 Let H(t) be a bounded-norm time-dependent lattice Hamiltonian that acts on n-qubits. Suppose
82
+ the observable O is a tensor product of operators , O = O1 ⊗ O2 ⊗ · · · ⊗ On. For constant evolution times,
83
+ there exists a classical algorithm that estimates µ within an additive error δ,
84
+ |˜µ − µ| ≤ δ .
85
+ (4)
86
+ The error δ includes three different parts. The first contribution comes from the Lieb-Robinson bound where
87
+ we used it to approximately limit the evolution within the lightcone of each qubit. Second, numerical methods
88
+ such as trotterization are used to calculate the local unitary evolutions; these methods are not exact and incur
89
+ some errors. The third part comes from the additive error in simulating a constant depth quantum circuit [15].
90
+ In [15], they provide a classical algorithm for constant depth circuits in 2D or 3D which can approximate the
91
+ 2
92
+
93
+ mean value to an additive error. In our work, the short-time evolution of a geometrically local Hamiltonian
94
+ which is limited by the Lieb Robinson bound is comparable with the shallow quantum circuit with a constant
95
+ depth.
96
+ Suppose that the error of localizing the time evolution of the Hamiltonian, classical simulation of time-
97
+ dependent unitaries and simulating shallow circuits are given by εLR, εCS and εSSC respectively. Then the
98
+ total error is as follows:
99
+ |˜µ − µ| ≤ εLR + εCS + εSSC = δ .
100
+ (5)
101
+ In Eq. (10), we will see how the first error is dictated by the simulation time, the lightcone radius and the
102
+ Hamiltonian. One should note that εLR is bounded by εLR ≤ nε(LR)(L, T), where ε(LR)(L, T) is defined
103
+ later in Eq. (10) as the Lieb Robinson error for a single site. Section 4 derives the dependency of the second
104
+ error term on the simulation parameters.
105
+ Algorithm 1: High level overview of the algorithm
106
+ input :a √n × √n lattice of qubits, a geometrically local time-dependent Hamiltonian H(t), the
107
+ operator O = O1 ⊗ O2 ⊗ · · · ⊗ On where each ∥Oj∥ ≤ 1, an upper bound for error δ, a
108
+ simulation time T.
109
+ output :an approximation for µ = ⟨ψ(T)|O|ψ(T)⟩, where |ψ(0)⟩ = |0⊗n⟩ and
110
+ i¯h d
111
+ dt |ψ(t)⟩ = H(t) |ψ(t)⟩
112
+ 1 Initialization:
113
+ 2 Use T and δ to calculate the lightcone radius, L, from the Lieb-Robinson bound.
114
+ 3 As in Fig. 1 partition the lattice into 4L × √n strips twice, let us call each set {Ai}i and {Bi}i. Also,
115
+ define the sets
116
+
117
+ A0
118
+ i
119
+
120
+ i and
121
+
122
+ B0
123
+ i
124
+
125
+ i as the central part of the strips.
126
+ 4 Also, group sites into 2L × 2L super-sites to form a coarse-grained lattice.
127
+ 5 Calculations:
128
+ 6 for Ai ∈ {Ai}i do
129
+ 7
130
+ initialize an MPS.
131
+ 8
132
+ for Oj in A0
133
+ i do
134
+ 9
135
+ Use a classical ODE solver to calculate the local unitary operator Uj corresponding to Oj
136
+ which includes the terms that are in the lightcone of the jth qubit.
137
+ 10
138
+ Transform OjUj into a Matrix Product Operator.
139
+ 11
140
+ Add the necessary sites from the current super-site to the active memory and apply the MPO
141
+ on it.
142
+ 12
143
+ Measure any sites that will no longer be needed.
144
+ 13
145
+ Let us call the (not normalized) MPS outcome |ΨAi(T)⟩ .
146
+ 14 Repeat the Line 6 loop for Bjs.
147
+ 15 return ˜µ =
148
+ ��
149
+ j ⟨ΨBj(T)|
150
+ ���
151
+ i |ΨAi(T)⟩
152
+
153
+ The runtime of classical simulation of the time-dependent unitaries is related to the complexity of matrix
154
+ multiplication. We can find the lightcone radius L from Section 3 and then find the number of qubits m in
155
+ each lightcone which is O
156
+
157
+ L2�
158
+ . For most practical cases, the fastest matrix multiplication algorithm is the
159
+ famous Strasson algorithm with asymptotic complexity of O
160
+
161
+ (2m)log2 7�
162
+ [25], however, the best known
163
+ asymptotic complexity for matrix multiplication is O
164
+
165
+ (2m)2.373�
166
+ [26]. For most physical Hamiltonians
167
+ 3
168
+
169
+ Figure 1: A: It shows the relationship between simulation time and the lightcone radius. B:shows the sites
170
+ inside the lightcone for various lightcone radii. C and D:show the set of strips {Ai}i and {Bi}i as well as
171
+ the
172
+
173
+ A0
174
+ i
175
+
176
+ i and
177
+
178
+ B0
179
+ i
180
+
181
+ i sets on a 2D grid. The yellow squares represent the Oi operators that are to be applied
182
+ on either set of strips.
183
+ where we have translational symmetry in the bulk, we only need to calculate the unitary evolution matrix for
184
+ each geometrical configuration of the sites once. This means that the complexity of this part of the algorithm
185
+ would be O
186
+
187
+ 22.373m�
188
+ . But for generic Hamiltonians the number of configurations could grow linearly with
189
+ the system size and the complexity would be O
190
+
191
+ n22.373m�
192
+ .
193
+ For 2D and 3D systems with n qubits, the runtime of simulating a shallow circuit is related to the last error
194
+ term with O
195
+
196
+ nL226L2/(εSSC)2�
197
+ and O
198
+
199
+ nL326L2n1/3/(εSSC)2�
200
+ respectively [15]. We have provided a
201
+ high-level overview of the 2D algorithm in Algorithm 1. Consequently the general total complexity of the algo-
202
+ rithm for 2D and 3D systems is O
203
+
204
+ n22.373m + nL226L2/(εSSC)2�
205
+ and O
206
+
207
+ n22.373m + nL326L2n1/3/(εSSC)2�
208
+ respectively.
209
+ 4
210
+
211
+ B.
212
+ 1.00000
213
+ 0.10000
214
+ lerror
215
+ 0.01000
216
+ 0.00100
217
+ Simulation time
218
+ +0.01
219
+ + 0.02
220
+ 0.00010
221
+ +0.03
222
+ +0.04
223
+ 0.00001
224
+ 0
225
+ 2
226
+ Lightcone radius3
227
+ Local Unitary Operators
228
+ According to [20], we know that a local operator OA which is defined inside a region A, remains local after a
229
+ short-time evolution under a local Hamiltonian. Suppose that the Hamiltonian has the form
230
+ H(t) =
231
+
232
+ e
233
+ ue(t)he ,
234
+ (6)
235
+ where he acts non-trivially only on the two vertices of edge e of the graph G. Suppose that ∥ue(t)he∥ ≤ g
236
+ for 0 ≤ t ≤ T and the maximum degree of the graph to be ∆. The Hamiltonian for terms in region A and the
237
+ set of vertices in the L-boundary of it has the form
238
+ HA(t) =
239
+
240
+ e⊂A∪∂L(A)
241
+ ue(t)he .
242
+ (7)
243
+ The time evolution operator of this local Hamiltonian acts non-trivially only on the region A and its L-
244
+ boundary. The time evolution operator is given by
245
+ V(t) = T :
246
+
247
+ exp
248
+
249
+ − i
250
+ ¯h
251
+ � t
252
+ 0
253
+ HA(t′)dt′
254
+ ��
255
+ .
256
+ (8)
257
+ This is known as Dyson series and T : [.] represents time-ordering [27, p. 551]. According to [20] the
258
+ following Lieb-Robinson bound holds:
259
+ ||U†(T)OAU(T) − V†(T)OAV(T)|| ≤ ε(LR)(L, T) ,
260
+ (9)
261
+ where ε(LR)(L, T) is defined as:
262
+ ε(LR)(L, T) =
263
+
264
+ 2
265
+ π |A| ||OA|| exp
266
+
267
+ −L(log L − log T − log(4g(δ − 1))) − 1
268
+ 2 log L
269
+
270
+ .
271
+ (10)
272
+ For each local operator in O = O1 ⊗ O2 ⊗ · · · ⊗ On, we only consider the Hamiltonian terms inside the
273
+ lightcone of it. We replace the global unitary that acts on the entire system with these local unitary operators
274
+ and apply the algorithm in Sec. 2 to approximate the mean value.
275
+ The only remaining problem would be to find a classical algorithm for classically calculating the local
276
+ unitary operators V(t). We analyze different approaches for doing so in Sec. 4.
277
+ 4
278
+ Classical Simulation of the time-dependent unitaries
279
+ 4.1
280
+ Trotterization
281
+ Theorem 2 Let HA(t) be a bounded-norm time-dependent Hamiltonian and OA an observable inside region
282
+ A. Let V(t) be the unitary evoluion of HA(t). We can approximate the operator V†(T)OAV(T) with
283
+ W†(T, N)OAW(T, N) where W(t, N) is defined as
284
+ W(t, N) =
285
+ N= t
286
+ δt
287
+
288
+ j=1
289
+ exp
290
+
291
+ − i
292
+ ¯hδt HA(jδt)
293
+
294
+ ,
295
+ (11)
296
+ 5
297
+
298
+ such that
299
+ ��W†(T, N)OAW(T, N) − V†(T)OAV(T)
300
+ �� ≤ 6T 2
301
+ N¯h ∥OA∥ ∥H′
302
+ A(t∗)∥ = εCS,
303
+ (12)
304
+ where
305
+ ∥H′
306
+ A(t∗)∥ = max
307
+ 0≤t≤T ∥H′
308
+ A(t)∥.
309
+ (13)
310
+ This is a well-known textbook result that can be found in the literature. For instance see chapter IX of [28].
311
+ 4.2
312
+ Numerical Differential Equation Solvers
313
+ Another approach for approximating the unitary time evolution would be to derive a differential equation
314
+ from Schrodinger’s equation and solve that numerically by using a suitable classical algorithm.
315
+ i¯h d
316
+ dt |ψ(t)⟩ = H(t) |ψ(t)⟩ ,
317
+ i¯h d
318
+ dtU(t) |ψ(0)⟩ = H(t)U(t) |ψ(0)⟩ ,
319
+ i¯h d
320
+ dtU(t) = H(t)U(t) .
321
+ (14)
322
+ If there are m qubits inside the lightcone, the matrix representation of U and H will be 2m ×2m operators
323
+ and Eq. (14) will constitute a set of Ordinary Differential Equations (ODEs). The upside of using a classical
324
+ ODE solver is that they can consistently attain much lower errors than what is possible from Eq. (11) or a
325
+ higher order Suzuki-Trotter solution [30, 31] that is fine-tuned for a time-dependent problem [32, 33]. The
326
+ downside is that they are typically orders of magnitude slower and require more memory than the straight
327
+ forward trotterization as in Eq. (11). Assuming our lightcone is small enough, many numerical ODE solvers
328
+ will be able to handle Eq. (14). See Fig. 2 for a comparison between multiple different ODE solvers.
329
+ 5
330
+ Conclusions
331
+ To conclude we have provided a classical algorithm for the mean value problem on outcomes of short time
332
+ dependent Hamiltonian evolutions. These mean values are typically used in other algorithms such as the
333
+ Variational Quantum Algorithm [34].
334
+ The quantum mean value problem, almost by definition, belongs to the BQP-complete class for polynomial
335
+ times. So, naturally we do not expect to be able to solve the polynomial time problem on a classical computer
336
+ efficiently. Nonetheless, an important open question would be to find the minimum simulation time for getting
337
+ a quantum speedup. The current work shows us that in order to benefit from the quantum speedup we need
338
+ simulation times that are at least greater than constant [35]. This is in accordance with a wide variety of other
339
+ results that target specific problems too [15, 36, 13, 37, 38, 20]. In general the problem of mapping out the
340
+ entire dynamical complexity phase diagram is theoretically interesting.
341
+ Another direction for future research would be to improve or generalize the current algorithm. If the
342
+ algorithm cannot be further improved or generalized, then proving these limitations is another open question.
343
+ 6
344
+
345
+ Figure 2: A comparison of different ODE solvers. The top figure shows average minimum time used by each
346
+ ODE solver to solve Eq. (14) for different number of qubits. The bottom plot shows the average memory
347
+ used by each solver to solve the same differential equation. The benchmark was done on a personal PC and
348
+ each data point was repeated at least 100 times over 20 randomly generated time-dependent Hamiltonians.
349
+ All of the methods except Trotter were picked from Julia’s DifferentialEquations.jl roster of state of the art
350
+ ODE solvers [29], and compared to the Trotter method, they consistently had at least 10 orders of magnitude
351
+ less error of the form Eq. (12). The Trotter solution was also implemented in Julia, and used N = 30.
352
+ 7
353
+
354
+ 1.0×109
355
+ Solver
356
+ (su)
357
+ 1.0×108
358
+ Trotter
359
+ runtime (
360
+ 1.0×107
361
+ MMidpoint
362
+ 1.0×106
363
+ MLeapfrog
364
+ 1.0×105
365
+ 1
366
+ 2
367
+ 3
368
+ 4
369
+ 5
370
+ 6
371
+ 7
372
+ MGL4
373
+ 1.0×1010
374
+ MNC6
375
+ 1.0×109
376
+ (bytes)
377
+ MGL6
378
+ 1.0×108
379
+ memory
380
+ 1.0×107
381
+ MNC8
382
+ 1.0×106
383
+ MGL8
384
+ 1.0×105
385
+ 1
386
+ 2
387
+ 3
388
+ 4
389
+ 5
390
+ 6
391
+ 7
392
+ Number of Sites6
393
+ Acknowledgements
394
+ We would like to thank Salman Beigi for overseeing this project and commenting on the draft. We also thank
395
+ Leila Taghavi and Erfan Abedi for helpful discussions.
396
+ References
397
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+
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1
+ arXiv:2301.01513v1 [hep-th] 4 Jan 2023
2
+ HU-EP-23/01
3
+ Remarks on conformal invariants for
4
+ piecewise smooth curves and Wilson loops
5
+ Harald Dorn 1
6
+ Institut f¨ur Physik und IRIS Adlershof, Humboldt-Universit¨at zu Berlin,
7
+ Zum Großen Windkanal 6, D-12489 Berlin, Germany
8
+ Abstract
9
+ This short note is some obvious mathematical addendum to our papers on Wilson
10
+ loops on polygon-like contours with circular edges [1, 2].
11
+ Using the technique of
12
+ osculating spheres and circles we identify the conformal invariants characterising the
13
+ kinks (cusps) of generic piecewise smooth curves in 3-dimensional space.
14
+ 1dorn@physik.hu-berlin.de
15
+
16
+ 1
17
+ Introduction
18
+ The metrical invariants of smooth curves in Euclidean D-dimensional space are length,
19
+ curvature and (D −2) torsion parameters as functions along the curve. For piecewise
20
+ smooth curves with cusps 2, in addition each cusp is characterised by the Euler angles
21
+ specifying the orthogonal transformation needed to rotate the Frenet-Serret frames
22
+ on both sides of a cusp to one another.
23
+ What concerns the issue of conformal invariants, there is for smooth curves a
24
+ considerable amount of mathematical papers. In analogy to the metrical invariants
25
+ length s, curvature κ and torsion τ one finds for 3-dimensional curves, see e.g. [3],
26
+ conformal length ω,3
27
+ dω = √νds,
28
+ ν =
29
+
30
+ (κ′)2 + κ2τ 2 ,
31
+ (1)
32
+ conformal curvature Q
33
+ Q = 4(ν′′ − κ2ν)ν − 5(ν′)2
34
+ 8ν3
35
+ ,
36
+ (2)
37
+ and conformal torsion T
38
+ T = 2(κ′)2τ + κ2τ 3 + κκ′τ ′ − κκ′′τ
39
+ ν
40
+ 5
41
+ 2
42
+ .
43
+ (3)
44
+ Use of these invariants has been made in the physical literature to characterise the
45
+ boundary conditions for the treatment of minimal surfaces in AdS via Pohlmeyer
46
+ reduction [4].
47
+ But what concerns the extension to piecewise smooth curves, we did not find any
48
+ paper in the mathematical literature. Therefore, we decided to write up what one gets
49
+ by straightforward application of one of the various techniques used for the smooth
50
+ case: the kinematics of osculating spheres, see e.g. [5–8] and refs. therein.
51
+ This note is organised as follows. In the next section we find formulas expressing
52
+ the conformal invariants for cusps, with legs made of generic smooth pieces of curves,
53
+ in terms of their metrical invariants. Then in section 3 we consider the very special
54
+ curves studied in our papers [1,2], i.e. polygon-like curves with circular edges. These
55
+ curves need a separate discussion, since along the circular edges all the conformal
56
+ invariants (1),(2),(3) are zero or ill-defined.
57
+ 2
58
+ Conformal invariants of generic cusps
59
+ In 3-dimensional space the osculating circle (sphere ) at a generic point x on a smooth
60
+ curve is a circle (sphere) having contact of second (third) order with the curve at x.
61
+ Since the order of contact is preserved under conformal transformations, osculating
62
+ circles (spheres) at a given point of a curve are mapped to those for the image under
63
+ 2We follow the language in the Wilson loop literature and use the word cusp for a kink with
64
+ nonzero opening angle.
65
+ 3The prime denotes derivative with respect to s.
66
+ 1
67
+
68
+ a conformal map. Let us denote by ⃗t,⃗n,⃗b the unit tangent, normal and binormal
69
+ vectors at x. Then the center a of the osculating circle S1 is given by
70
+ a = x + 1
71
+ κ ⃗n
72
+ (4)
73
+ and the center c of the osculating sphere S2 by
74
+ c = x + 1
75
+ κ ⃗n −
76
+ κ′
77
+ τκ2 ⃗b .
78
+ (5)
79
+ We now turn to the case where x is a point of discontinuity, i.e. the tip of a cusp.
80
+ The cusp is then characterised by two osculating circles S1
81
+ −, S1
82
+ + and two osculating
83
+ spheres S2
84
+ −, S2
85
+ +. The index ” ± ” indicates the limits one gets by approaching x along
86
+ the both respective legs of the cusp.
87
+ Let us first count the number of conformal invariants we can expect for the cusp.
88
+ There are 9 metrical invariants at hand: κ±, κ′
89
+ ±, τ± and the 3 Euler angles for the
90
+ rotations from the Frenet frame {⃗t−,⃗n−,⃗b−} to {⃗t+,⃗n+,⃗b+}. The difference of the
91
+ numbers of parameters of the 3-dimensional conformal and isometry group is equal
92
+ to 4. This should result in 9 − 4 = 5 conformal parameters. Now of course two
93
+ of them are the corresponding limits of the differential of the conformal length (1).
94
+ Hence there remain 3 conformal parameters to be attributed genuinely to the cusp.
95
+ To find explicit formulas for them, we start with the conformal invariants of pairs
96
+ of spheres (Sm, Sn) of the same or of different dimension. There is a rigorous math-
97
+ ematical treatment for arbitrary dimensions in ref. [9]. Applied to our case it means
98
+ that to each of the pair (S1
99
+ −, S1
100
+ +), (S1
101
+ −, S2
102
+ +), (S2
103
+ −, S1
104
+ +), (S2
105
+ −, S2
106
+ +) belongs just one con-
107
+ formal parameter. To proceed, we first study these four invariants and will show
108
+ afterwards, that only three of them are independently.
109
+ (S1
110
+ −, S1
111
+ +)
112
+ The tangents of the osculating circles agree with those of the curve.
113
+ Therefore the related conformal invariant is
114
+ A11 = ⃗t−⃗t+ =
115
+ − cos α .
116
+ (6)
117
+ Here α is the cusp angle (understood as the opening angle, i.e. α = π in the smooth
118
+ case).
119
+ (S2
120
+ −, S2
121
+ +)
122
+ Now the conformal invariant is given by the so-called inversive product,
123
+ see e.g. [5,9]
124
+ A22 = R2
125
+ − + R2
126
+ + − (c− − c+)2
127
+ 2R−R+
128
+ = (c− − x)(c+ − x)
129
+ R−R+
130
+ .
131
+ (7)
132
+ R− and R+ are the radii of S2
133
+ − and S2
134
+ +. Obviously A22 is the cosine of the angle
135
+ between the vectors pointing from x to the centers of the two osculating spheres.
136
+ Strictly speaking, only its absolute value is invariant, since A22 changes sign under
137
+ those special conformal transformations for which the preimage of infinity is situated
138
+ inside just one of the spheres. The same comment applies to A12 and A21 below.
139
+ (S2
140
+ −, S1
141
+ +) and (S1
142
+ −, S2
143
+ +) For a sphere and an intersecting circle the conformal invariant
144
+ 2
145
+
146
+ is the scalar product of the unit vector pointing from x to the center of the sphere
147
+ with the unit tangent of the circle
148
+ A12 =
149
+ ⃗t−(c+ − x)
150
+ R+
151
+ ,
152
+ A21 = (c− − x) ⃗t+
153
+ R−
154
+ .
155
+ (8)
156
+ Using (4),(5) for both legs of the cusp we get from (7) and (8)
157
+ A22 = κ−τ−κ+τ+ ⃗n−⃗n+ + κ′
158
+ −κ′
159
+ + ⃗b−⃗b+ − κ−τ−κ′
160
+ + ⃗n−⃗b+ − κ+τ+κ′
161
+ − ⃗b−⃗n+
162
+
163
+ κ2
164
+ −τ 2
165
+ − + κ′2
166
+
167
+
168
+ κ2
169
+ +τ 2
170
+ + + κ′2
171
+ +
172
+ ,
173
+ (9)
174
+ A12
175
+ =
176
+ κ+τ+ ⃗t−⃗n+ − κ′
177
+ + ⃗t−⃗b+
178
+
179
+ κ2
180
+ +τ 2
181
+ + + κ′2
182
+ +
183
+ ,
184
+ (10)
185
+ A21
186
+ =
187
+ κ−τ− ⃗t+⃗n− − κ′
188
+ − ⃗t+⃗b−
189
+
190
+ κ2
191
+ −τ 2
192
+ − + κ′2
193
+
194
+ .
195
+ (11)
196
+ All the scalar products in the above formulas can be expressed in terms of three Euler
197
+ angles ϕ, ϑ, ψ needed to rotate the Frenet frame {n−, b−, t−} to {n+, b+, t+}. Then
198
+ we get
199
+ A11 = ⃗t−⃗t+ = cosϑ ,
200
+ i.e. ϑ = π − α ,
201
+ (12)
202
+ A22
203
+ =
204
+ 1
205
+
206
+ κ2
207
+ −τ 2
208
+ − + κ′2
209
+
210
+
211
+ κ2
212
+ +τ 2
213
+ + + κ′2
214
+ +
215
+
216
+ κ−τ−κ+τ+ (cosϕcosψ − sinϕsinψcosϑ)
217
+ +κ′
218
+ −κ′
219
+ +(cosϕcosψcosϑ − sinϕsinψ) + κ−τ−κ′
220
+ + (sinϕcosψ + cosϕsinψcosϑ)
221
+ −κ+τ+κ′
222
+ − (cosϕsinψ + sinϕcosψcosϑ)
223
+
224
+ ,
225
+ (13)
226
+ A12
227
+ =
228
+ sinϑ
229
+
230
+ κ2
231
+ +τ 2
232
+ + + κ′2
233
+ +
234
+
235
+ κ+τ+ sinϕ − κ′
236
+ + cosϕ
237
+
238
+ ,
239
+ (14)
240
+ A21
241
+ =
242
+ sinϑ
243
+
244
+ κ2
245
+ −τ 2
246
+ − + κ′2
247
+
248
+
249
+ κ−τ− sinψ + κ′
250
+ − cosψ
251
+
252
+ .
253
+ (15)
254
+ The part of A22 containing the factor cosϑ is related to the product of A12 and A21
255
+ in an obvious manner. With a bit more careful inspection one gets for the whole A22
256
+ A22(ϕ, ϑ, ψ) =
257
+ 1
258
+ sin2ϑ
259
+
260
+ A12(ϕ + π
261
+ 2 ) A21(ψ + π
262
+ 2 ) − cosϑ A12(ϕ)A21(ψ)
263
+
264
+ .
265
+ (16)
266
+ Now A12 and A21 for arguments shifted by π
267
+ 2 are not independent, but related by
268
+
269
+ A12(ϕ)
270
+ �2 +
271
+
272
+ A12(ϕ + π
273
+ 2)
274
+ �2 = (A21(ψ)
275
+ �2 +
276
+
277
+ A21(ψ + π
278
+ 2 )
279
+ �2 = sin2ϑ .
280
+ (17)
281
+ This means that a complete set of independent conformal invariants attributed to the
282
+ cusp is given by the three parameters 4
283
+ α ,
284
+ B12 = κ+τ+ sinϕ − κ′
285
+ + cosϕ
286
+
287
+ κ2
288
+ +τ 2
289
+ + + κ′2
290
+ +
291
+ ,
292
+ B21 = κ−τ− sinψ + κ′
293
+ − cosψ
294
+
295
+ κ2
296
+ −τ 2
297
+ − + κ′2
298
+
299
+ .
300
+ (18)
301
+ 4Remember ϕ, ϑ, ψ Euler angles, α = π − ϑ opening angle of the cusp.
302
+ 3
303
+
304
+ Let us add a warning. Inserting by brute force ϕ = ψ = 0 into B12 and B21 one
305
+ could reach the wrong conclusion that
306
+ κ′
307
+
308
+ κ2τ 2+κ′2 could be an invariant for smooth
309
+ curves. But since putting ϕ or ψ to zero is not a conformal invariant statement, this
310
+ conclusion is not allowed and also straightforwardly proven to be wrong.
311
+ We continue with some casual comment on the conformal invariants in 4-dimensio-
312
+ nal space. To fix all the osculating spheres from S1 to S3 at a smooth point one needs
313
+ κ, κ′, κ′′, τ1, τ ′
314
+ 1, τ2. This for the limits from both sides of the cusp, together with 6 Euler
315
+ angles, needed in 4D for the rotation of the Frenet frame, gives 18 metrical parameters.
316
+ The difference of the number of parameters oft the conformal and isometry group is
317
+ now 5, hence we tentatively reach 13 conformal parameters. Now on both sides the
318
+ limits of the differential of the conformal length and the first conformal torsion5 are
319
+ not related to the cusp. Hence we can expect 13 − 4 = 9 conformal parameters to be
320
+ attributed genuinely to the cusp.
321
+ On the other side, among the nine pairs of osculating spheres
322
+ (Si
323
+ −, Sj
324
+ +), i, j = 1, 2, 3, the pairs (S2
325
+ −, S2
326
+ +), (S2
327
+ −, S1
328
+ +), (S1
329
+ −, S2
330
+ +) have two invariants and
331
+ all other only one [9]. This yields 12 conformal parameters. We keep it as an open
332
+ question, whether there are indeed three relations of the 3D type (16),(17) to reach
333
+ the minimal number of 9 independent parameters seen in the previous paragraph.
334
+ We close this section with a comment on the cusp anomalous dimension for Wilson
335
+ loops. It is generally believed to depend on the cusp angle α only, and therefore
336
+ calculations have been done using straight edges. While in field theoretic perturbation
337
+ theory this can be justified by power counting in the corresponding Feynman integrals,
338
+ a rigorous proof for the generic situation at strong coupling is still lacking. We have
339
+ presented a proof for the planar case with generically curved edges in [10]. For full
340
+ generality in 3D, a proof of its independence from B12 and B21 is still lacking.
341
+ 3
342
+ Comments on Wilson loops on polygon-like
343
+ curves with circular edges
344
+ The polygon-like curves with circular edges, whose related Wilson loops have been
345
+ studied in our papers [1, 2], are not covered by the setting for generic curves as
346
+ presented in the previous section. Along their edges one has constant curvature κ
347
+ and zero torsion τ, resulting in zero conformal length (1) and undefined conformal
348
+ curvature and torsion. In mathematical language these edges are conformal vertices.6
349
+ Although such single edges carry no conformal data, their combination to a poly-
350
+ gon does7 It has still no extension in the sense of conformal length, but there are of
351
+ course the cusp angles at each cusp of the polygon and for more than 3 cusp points
352
+ the corresponding cross ratios. In addition, like generic curves, these polygons can
353
+ wind themselves out of a plane and exhibit torsion. In our paper [1] we have this
354
+ issue parameterised by the introduction of torsion angles βj, defined at a given cusp
355
+ 5In 3D the conformal torsion (3) cannot be build out of the metrical parameters needed to fix all
356
+ the osculating spheres at the cusp. But in 4D the osculating S3 inherits all the information.
357
+ 6See e.g. [3,5,6].
358
+ 7In a sense one could call it a conformal vertex with internal conformal substructure.
359
+ 4
360
+
361
+ point xj by
362
+ βj = ∡({xj, xj+1}, ccj) ,
363
+ (19)
364
+ where {xj, xj+1} denotes the circular edge between xj and xj+1 and ccj the circle
365
+ fixed by the three cusp points xj−1, xj, xj+1.
366
+ However, in contrast to B12 and B21 for cusps of generic curves, these torsion
367
+ angles are not attributed to the local properties at the corresponding cusp. This is
368
+ simply seen by changing in (19) the neighbouring cusp point xj+1 along the circle
369
+ of which the edge {xj, xj+1} is a part . Then the tangent of the edge at xj remains
370
+ the same, but the circle ccj and its tangent at xj changes. By this manipulation βj
371
+ changes, although the local situation at the cusp at xj remains the same as before.
372
+ We end with a remark on some setting for Wilson loops on piecewise smooth
373
+ curves intermediate between those of full generality considered in section 2 and the
374
+ polygons with circular edges in [1]. In conformal invariant gauge field theories as in
375
+ e.g. N = 4 SYM there will be an anomalous conformal Ward identity of the type
376
+ derived in [1], imposing for the Wilson loop the structure of a conformally covariant
377
+ factor depending on the distances of the tips of the cusps times a conformally invariant
378
+ remainder factor. In the generic case this remainder factor is a function of the cross
379
+ ratios formed out of cusp points and the conformal cusp parameters identified in
380
+ section 2, but in addition also a functional of the conformal invariants as functions
381
+ along the edges.
382
+ For the case of polygons with circular edges the remainder is a
383
+ function of only a finite number of conformal parameters.
384
+ To have a curve characterised by a finite number of conformal parameters but
385
+ nevertheless having nonzero conformal length, one could consider polygons with edges
386
+ which are pieces of curves with constant conformal curvature and torsion. Such curves
387
+ have been classified in [11]. Among them are loxodromes on rotational surfaces.
388
+ Acknowledgement
389
+ I thank the Quantum Field and String Theory Group at Humboldt University for
390
+ kind hospitality.
391
+ 5
392
+
393
+ References
394
+ [1] H.
395
+ Dorn,
396
+ “On
397
+ anomalous
398
+ conformal
399
+ Ward
400
+ identities
401
+ for
402
+ Wilson
403
+ loops
404
+ on
405
+ polygon-like
406
+ contours
407
+ with
408
+ circular
409
+ edges,”
410
+ JHEP
411
+ 03
412
+ (2020),
413
+ 166
414
+ doi:10.1007/JHEP03(2020)166 [arXiv:2001.03391 [hep-th]].
415
+ [2] H. Dorn, “Wilson loops for triangular contours with circular edges,” J. Phys. A
416
+ 54 (2021) no.22, 225402 doi:10.1088/1751-8121/abe311 [arXiv:2010.14822 [hep-
417
+ th]].
418
+ [3] G. Cairns, R. Sharpe, and L. Webb, “Conformal Invariants for Curves and Sur-
419
+ faces in Three Dimensional Space Forms”, Rocky Mountain J. Math. Volume 24,
420
+ Number 3 (1994), 933-959.
421
+ [4] Y. He, C. Huang and M. Kruczenski, “Minimal area surfaces in AdSn+1
422
+ and Wilson loops,”
423
+ JHEP 02 (2018),
424
+ 027 doi:10.1007/JHEP02(2018)027
425
+ [arXiv:1712.06269 [hep-th]].
426
+ [5] M.C. Romero-Fuster and E. Sanabria-Codesal,“Generalized evolutes, vertices
427
+ and conformal invariants of curves in Rn+1”, Indag. Mathem., N.S., 10 (2), 297-
428
+ 305
429
+ [6] A. Montesinos Amilibia’, M.C. Romero Fuster and E. Sanabria -Codesal, “Con-
430
+ formal curvatures of curves in Rn+1”, lndag. Mathem., N.S., 12 (3) 369-382
431
+ [7] R. Langevin, J. O’Hara and S. Sakata, “Space of subspheres and conformal
432
+ invariants of curves” arXiv:1102.0344 [math.DG]
433
+ [8] R. Langevin, J O’Hara, S Sakata, “Application of spaces of subspheres to con-
434
+ formal invariants of curves and canal surfaces”, Annales Polonici Mathematici,
435
+ January 2013 , doi: 10.4064/ap108-2-1
436
+ [9] R. Sulanke, “M¨obius Invarints for Pairs of Spheres (Sm
437
+ 1 , Sl
438
+ 2) in the M¨obius Space
439
+ Sn”, Contributions to Algebra and Geometry, Vol. 41 (2000), No.1, 233-246
440
+ [10] H. Dorn, “Wilson loops at strong coupling for curved contours with cusps,”
441
+ J. Phys. A 49 (2016) no.14, 145402 doi:10.1088/1751-8113/49/14/145402
442
+ [arXiv:1509.00222 [hep-th]].
443
+ [11] R. Sulanke, “ Submanifolds of the M¨obius space II, Frenet formulas and curves
444
+ of constant curvatures.” Mathematische Nachrichten 100.1 (1981): 235-247.
445
+ 6
446
+
G9AzT4oBgHgl3EQfjP2K/content/tmp_files/load_file.txt ADDED
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1
+ filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf,len=170
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+ page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
3
+ page_content='01513v1 [hep-th] 4 Jan 2023 HU-EP-23/01 Remarks on conformal invariants for piecewise smooth curves and Wilson loops Harald Dorn 1 Institut f¨ur Physik und IRIS Adlershof, Humboldt-Universit¨at zu Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany Abstract This short note is some obvious mathematical addendum to our papers on Wilson loops on polygon-like contours with circular edges [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
4
+ page_content=' Using the technique of osculating spheres and circles we identify the conformal invariants characterising the kinks (cusps) of generic piecewise smooth curves in 3-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
5
+ page_content=' 1dorn@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
6
+ page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
7
+ page_content='de 1 Introduction The metrical invariants of smooth curves in Euclidean D-dimensional space are length, curvature and (D −2) torsion parameters as functions along the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
8
+ page_content=' For piecewise smooth curves with cusps 2, in addition each cusp is characterised by the Euler angles specifying the orthogonal transformation needed to rotate the Frenet-Serret frames on both sides of a cusp to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
9
+ page_content=' What concerns the issue of conformal invariants, there is for smooth curves a considerable amount of mathematical papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
10
+ page_content=' In analogy to the metrical invariants length s, curvature κ and torsion τ one finds for 3-dimensional curves, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
11
+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
12
+ page_content=' [3], conformal length ω,3 dω = √νds, ν = � (κ′)2 + κ2τ 2 , (1) conformal curvature Q Q = 4(ν′′ − κ2ν)ν − 5(ν′)2 8ν3 , (2) and conformal torsion T T = 2(κ′)2τ + κ2τ 3 + κκ′τ ′ − κκ′′τ ν 5 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
13
+ page_content=' (3) Use of these invariants has been made in the physical literature to characterise the boundary conditions for the treatment of minimal surfaces in AdS via Pohlmeyer reduction [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
14
+ page_content=' But what concerns the extension to piecewise smooth curves, we did not find any paper in the mathematical literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
15
+ page_content=' Therefore, we decided to write up what one gets by straightforward application of one of the various techniques used for the smooth case: the kinematics of osculating spheres, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
16
+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
17
+ page_content=' [5–8] and refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
18
+ page_content=' therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
19
+ page_content=' This note is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
20
+ page_content=' In the next section we find formulas expressing the conformal invariants for cusps, with legs made of generic smooth pieces of curves, in terms of their metrical invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
21
+ page_content=' Then in section 3 we consider the very special curves studied in our papers [1,2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
22
+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
23
+ page_content=' polygon-like curves with circular edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
24
+ page_content=' These curves need a separate discussion, since along the circular edges all the conformal invariants (1),(2),(3) are zero or ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
25
+ page_content=' 2 Conformal invariants of generic cusps In 3-dimensional space the osculating circle (sphere ) at a generic point x on a smooth curve is a circle (sphere) having contact of second (third) order with the curve at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
26
+ page_content=' Since the order of contact is preserved under conformal transformations, osculating circles (spheres) at a given point of a curve are mapped to those for the image under 2We follow the language in the Wilson loop literature and use the word cusp for a kink with nonzero opening angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
27
+ page_content=' 3The prime denotes derivative with respect to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' 1 a conformal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Let us denote by ⃗t,⃗n,⃗b the unit tangent, normal and binormal vectors at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Then the center a of the osculating circle S1 is given by a = x + 1 κ ⃗n (4) and the center c of the osculating sphere S2 by c = x + 1 κ ⃗n − κ′ τκ2 ⃗b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (5) We now turn to the case where x is a point of discontinuity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' the tip of a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' The cusp is then characterised by two osculating circles S1 −, S1 + and two osculating spheres S2 −, S2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' The index ” ± ” indicates the limits one gets by approaching x along the both respective legs of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Let us first count the number of conformal invariants we can expect for the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' There are 9 metrical invariants at hand: κ±, κ′ ±, τ± and the 3 Euler angles for the rotations from the Frenet frame {⃗t−,⃗n−,⃗b−} to {⃗t+,⃗n+,⃗b+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' The difference of the numbers of parameters of the 3-dimensional conformal and isometry group is equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' This should result in 9 − 4 = 5 conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Now of course two of them are the corresponding limits of the differential of the conformal length (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Hence there remain 3 conformal parameters to be attributed genuinely to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' To find explicit formulas for them, we start with the conformal invariants of pairs of spheres (Sm, Sn) of the same or of different dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' There is a rigorous math- ematical treatment for arbitrary dimensions in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Applied to our case it means that to each of the pair (S1 −, S1 +), (S1 −, S2 +), (S2 −, S1 +), (S2 −, S2 +) belongs just one con- formal parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' To proceed, we first study these four invariants and will show afterwards, that only three of them are independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (S1 −, S1 +) The tangents of the osculating circles agree with those of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Therefore the related conformal invariant is A11 = ⃗t−⃗t+ = − cos α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (6) Here α is the cusp angle (understood as the opening angle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' α = π in the smooth case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (S2 −, S2 +) Now the conformal invariant is given by the so-called inversive product, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' [5,9] A22 = R2 − + R2 + − (c− − c+)2 2R−R+ = (c− − x)(c+ − x) R−R+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (7) R− and R+ are the radii of S2 − and S2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Obviously A22 is the cosine of the angle between the vectors pointing from x to the centers of the two osculating spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Strictly speaking, only its absolute value is invariant, since A22 changes sign under those special conformal transformations for which the preimage of infinity is situated inside just one of the spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' The same comment applies to A12 and A21 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (S2 −, S1 +) and (S1 −, S2 +) For a sphere and an intersecting circle the conformal invariant 2 is the scalar product of the unit vector pointing from x to the center of the sphere with the unit tangent of the circle A12 = ⃗t−(c+ − x) R+ , A21 = (c− − x) ⃗t+ R− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (8) Using (4),(5) for both legs of the cusp we get from (7) and (8) A22 = κ−τ−κ+τ+ ⃗n−⃗n+ + κ′ −κ′ + ⃗b−⃗b+ − κ−τ−κ′ + ⃗n−⃗b+ − κ+τ+κ′ − ⃗b−⃗n+ � κ2 −τ 2 − + κ′2 − � κ2 +τ 2 + + κ′2 + , (9) A12 = κ+τ+ ⃗t−⃗n+ − κ′ + ⃗t−⃗b+ � κ2 +τ 2 + + κ′2 + , (10) A21 = κ−τ− ⃗t+⃗n− − κ′ − ⃗t+⃗b− � κ2 −τ 2 − + κ′2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (11) All the scalar products in the above formulas can be expressed in terms of three Euler angles ϕ, ϑ, ψ needed to rotate the Frenet frame {n−, b−, t−} to {n+, b+, t+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Then we get A11 = ⃗t−⃗t+ = cosϑ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' ϑ = π − α , (12) A22 = 1 � κ2 −τ 2 − + κ′2 − � κ2 +τ 2 + + κ′2 + � κ−τ−κ+τ+ (cosϕcosψ − sinϕsinψcosϑ) +κ′ −κ′ +(cosϕcosψcosϑ − sinϕsinψ) + κ−τ−κ′ + (sinϕcosψ + cosϕsinψcosϑ) −κ+τ+κ′ − (cosϕsinψ + sinϕcosψcosϑ) � , (13) A12 = sinϑ � κ2 +τ 2 + + κ′2 + � κ+τ+ sinϕ − κ′ + cosϕ � , (14) A21 = sinϑ � κ2 −τ 2 − + κ′2 − � κ−τ− sinψ + κ′ − cosψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (15) The part of A22 containing the factor cosϑ is related to the product of A12 and A21 in an obvious manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' With a bit more careful inspection one gets for the whole A22 A22(ϕ, ϑ, ψ) = 1 sin2ϑ � A12(ϕ + π 2 ) A21(ψ + π 2 ) − cosϑ A12(ϕ)A21(ψ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (16) Now A12 and A21 for arguments shifted by π 2 are not independent, but related by � A12(ϕ) �2 + � A12(ϕ + π 2) �2 = (A21(ψ) �2 + � A21(ψ + π 2 ) �2 = sin2ϑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (17) This means that a complete set of independent conformal invariants attributed to the cusp is given by the three parameters 4 α , B12 = κ+τ+ sinϕ − κ′ + cosϕ � κ2 +τ 2 + + κ′2 + , B21 = κ−τ− sinψ + κ′ − cosψ � κ2 −τ 2 − + κ′2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' (18) 4Remember ϕ, ϑ, ψ Euler angles, α = π − ϑ opening angle of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' 3 Let us add a warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Inserting by brute force ϕ = ψ = 0 into B12 and B21 one could reach the wrong conclusion that κ′ √ κ2τ 2+κ′2 could be an invariant for smooth curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' But since putting ϕ or ψ to zero is not a conformal invariant statement, this conclusion is not allowed and also straightforwardly proven to be wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' We continue with some casual comment on the conformal invariants in 4-dimensio- nal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' To fix all the osculating spheres from S1 to S3 at a smooth point one needs κ, κ′, κ′′, τ1, τ ′ 1, τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' This for the limits from both sides of the cusp, together with 6 Euler angles, needed in 4D for the rotation of the Frenet frame, gives 18 metrical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' The difference of the number of parameters oft the conformal and isometry group is now 5, hence we tentatively reach 13 conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Now on both sides the limits of the differential of the conformal length and the first conformal torsion5 are not related to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Hence we can expect 13 − 4 = 9 conformal parameters to be attributed genuinely to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' On the other side, among the nine pairs of osculating spheres (Si −, Sj +), i, j = 1, 2, 3, the pairs (S2 −, S2 +), (S2 −, S1 +), (S1 −, S2 +) have two invariants and all other only one [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' This yields 12 conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' We keep it as an open question, whether there are indeed three relations of the 3D type (16),(17) to reach the minimal number of 9 independent parameters seen in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' We close this section with a comment on the cusp anomalous dimension for Wilson loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' It is generally believed to depend on the cusp angle α only, and therefore calculations have been done using straight edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' While in field theoretic perturbation theory this can be justified by power counting in the corresponding Feynman integrals, a rigorous proof for the generic situation at strong coupling is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' We have presented a proof for the planar case with generically curved edges in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' For full generality in 3D, a proof of its independence from B12 and B21 is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' 3 Comments on Wilson loops on polygon-like curves with circular edges The polygon-like curves with circular edges, whose related Wilson loops have been studied in our papers [1, 2], are not covered by the setting for generic curves as presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Along their edges one has constant curvature κ and zero torsion τ, resulting in zero conformal length (1) and undefined conformal curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' In mathematical language these edges are conformal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content='6 Although such single edges carry no conformal data, their combination to a poly- gon does7 It has still no extension in the sense of conformal length, but there are of course the cusp angles at each cusp of the polygon and for more than 3 cusp points the corresponding cross ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' In addition, like generic curves, these polygons can wind themselves out of a plane and exhibit torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' In our paper [1] we have this issue parameterised by the introduction of torsion angles βj, defined at a given cusp 5In 3D the conformal torsion (3) cannot be build out of the metrical parameters needed to fix all the osculating spheres at the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' But in 4D the osculating S3 inherits all the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' 6See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' [3,5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' 7In a sense one could call it a conformal vertex with internal conformal substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' 4 point xj by βj = ∡({xj, xj+1}, ccj) , (19) where {xj, xj+1} denotes the circular edge between xj and xj+1 and ccj the circle fixed by the three cusp points xj−1, xj, xj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' However, in contrast to B12 and B21 for cusps of generic curves, these torsion angles are not attributed to the local properties at the corresponding cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' This is simply seen by changing in (19) the neighbouring cusp point xj+1 along the circle of which the edge {xj, xj+1} is a part .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Then the tangent of the edge at xj remains the same, but the circle ccj and its tangent at xj changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' By this manipulation βj changes, although the local situation at the cusp at xj remains the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' We end with a remark on some setting for Wilson loops on piecewise smooth curves intermediate between those of full generality considered in section 2 and the polygons with circular edges in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' In conformal invariant gauge field theories as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' N = 4 SYM there will be an anomalous conformal Ward identity of the type derived in [1], imposing for the Wilson loop the structure of a conformally covariant factor depending on the distances of the tips of the cusps times a conformally invariant remainder factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' In the generic case this remainder factor is a function of the cross ratios formed out of cusp points and the conformal cusp parameters identified in section 2, but in addition also a functional of the conformal invariants as functions along the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' For the case of polygons with circular edges the remainder is a function of only a finite number of conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
109
+ page_content=' To have a curve characterised by a finite number of conformal parameters but nevertheless having nonzero conformal length, one could consider polygons with edges which are pieces of curves with constant conformal curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
110
+ page_content=' Such curves have been classified in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' Among them are loxodromes on rotational surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
112
+ page_content=' Acknowledgement I thank the Quantum Field and String Theory Group at Humboldt University for kind hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+ page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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1
+ Astronomy & Astrophysics manuscript no. Marchandetal2023
2
+ ©ESO 2023
3
+ January 5, 2023
4
+ Fast methods to track grain coagulation and ionization. III.
5
+ Protostellar collapse with non-ideal MHD
6
+ P. Marchand1, 2, U. Lebreuilly3, M.-M. Mac Low2, V. Guillet4, 5
7
+ 1 Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, UT3-PS, CNRS, CNES, 9 av. du Colonel Roche,
8
+ 31028 Toulouse Cedex 4, France
9
+ 2 Department of Astrophysics, American Museum of Natural History, 200 Central Park West, NY, NY, 10024, USA
10
+ 3 AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, 91191 Gif-sur-Yvette, France
11
+ 4 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405, Orsay, France
12
+ 5 Laboratoire Univers et Particules de Montpellier, Université de Montpellier, CNRS/IN2P3, CC 72, Place Eugène Bataillon, 34095
13
+ Montpellier Cedex 5, France
14
+ ABSTRACT
15
+ Dust grains influence many aspects of star formation, including planet formation, opacities for radiative transfer, chemistry, and
16
+ the magnetic field via Ohmic, Hall, and ambipolar diffusion. The size distribution of the dust grains is the primary characteristic
17
+ influencing all these aspects. Grain size increases by coagulation throughout the star formation process. We describe here numerical
18
+ simulations of protostellar collapse using methods described in earlier papers of this series. We compute the evolution of the grain
19
+ size distribution from coagulation and the non-ideal magnetohydrodynamics effects self-consistently and at low numerical cost. We
20
+ find that the coagulation efficiency is mostly affected by the time spent in high-density regions. Starting from sub-micron radii, grain
21
+ sizes reach more than 100 µm in an inner protoplanetary disk that is only 1000 years old. We also show that the growth of grains
22
+ significantly affects the resistivities, and indirectly the dynamics and angular momentum of the disk.
23
+ 1. Introduction
24
+ Grains play a major role during star formation. Firstly, they
25
+ are the seed of planet formation. While their characteristic size
26
+ is sub-micron in the interstellar medium (ISM) (Mathis et al.
27
+ 1977), they grow by coagulation during the collapse, and can
28
+ reach sizes larger than 10 µm in the early stages of protostel-
29
+ lar collapse (Guillet et al. 2020; Silsbee et al. 2020; Tsukamoto
30
+ et al. 2021; Vorobyov et al. 2022; Bate 2022). Observations also
31
+ suggest they reach these sizes, if not larger, in the envelopes of
32
+ Class 0-I objects (Kwon et al. 2009; Miotello et al. 2014; Le
33
+ Gouellec et al. 2019; Galametz et al. 2019). Their growth then
34
+ continues in protoplanetary disks until they eventually become
35
+ planetesimals. Variations in their size also significantly impact
36
+ non-ideal magnetohydrodynamics (MHD) effects through their
37
+ ionization and their chemical interactions with the gas, with a
38
+ direct feedback on the dynamics of the gas (Marchand et al.
39
+ 2016; Zhao et al. 2016, 2018; Marchand et al. 2020; Guillet et al.
40
+ 2020). Non-ideal MHD effects have been shown to be critical to
41
+ the regulation of magnetic field and angular momentum during
42
+ the protostellar collapse and the protoplanetary disk evolution
43
+ (Mouschovias & Paleologou 1979; Machida et al. 2006; Duf-
44
+ fin & Pudritz 2008; Mellon & Li 2009; Li et al. 2011; Tomida
45
+ et al. 2015; Wurster et al. 2016; Masson et al. 2016; Vaytet et al.
46
+ 2018; Marchand et al. 2020; Lebreuilly et al. 2021). Grains are
47
+ also the main source of opacity in protostellar environments, af-
48
+ fecting the cooling of the gas and the observations made of those
49
+ systems. Their high optical depth at densities ρ > 10−13 g cm−3
50
+ leads to the formation of the first hydrostatic core (Larson 1969).
51
+ However, numerical simulations usually do not account for
52
+ grain coagulation self-consistently due to the great cost of com-
53
+ puting a coagulation algorithm on-the-fly (although new meth-
54
+ ods are being developed, see the recent work by Lombart &
55
+ Laibe 2021). The dust evolution used to be pre-processed or
56
+ post-processed with no self-consistent feedback on the dynam-
57
+ ics (Rossi et al. 1991; Dullemond & Dominik 2005; Zhao et al.
58
+ 2016; Marchand et al. 2020). Recently, more and more studies
59
+ include the growth of grains in their hydrodynamics simulations
60
+ (Tsukamoto et al. 2021; Vericel et al. 2021; Vorobyov et al. 2022;
61
+ Bate 2022). In Marchand et al. (2021, hereafter Paper I), we
62
+ presented a simple and fast method to track coagulation self-
63
+ consistently that is particularly suited for modeling star forma-
64
+ tion. We now apply this method to non-ideal MHD protostellar
65
+ collapse simulations. It is coupled with the second method pre-
66
+ sented in Paper I: a fast calculation of the ionization and grain
67
+ charge, to obtain non-ideal MHD resistivities. These 3D simula-
68
+ tions are the first to include a self-consistent grain growth with a
69
+ direct feedback on the dynamics through the self-consistent cal-
70
+ culation of MHD resistivities.
71
+ The paper is organized as follows. In Section 2, we describe
72
+ the methods used in our study. Section 3 presents the results of
73
+ our calculations, both analytical in Section 3.1 and numerical in
74
+ Section 3.2. We compare our results to other works and discuss
75
+ the caveats in Section 4, and conclude in Section 5.
76
+ 2. Methods
77
+ We perform non-ideal MHD simulations with the RAMSES
78
+ code (Teyssier 2002). RAMSES is an Eulerian gas dynamics
79
+ code with adaptive mesh refinement (AMR) and self-gravity. It
80
+ includes a monofluid treatment of non-ideal MHD effects (Mas-
81
+ son et al. 2012; Marchand et al. 2018). We have implemented
82
+ the methods presented in Paper I to calculate the coagulation
83
+ and ionization of grains on-the-fly in a self-consistent manner.
84
+ Article number, page 1 of 11
85
+ arXiv:2301.01510v1 [astro-ph.SR] 4 Jan 2023
86
+
87
+ A&A proofs: manuscript no. Marchandetal2023
88
+ 2.1. Grain coagulation and ionization
89
+ 2.1.1. Coagulation
90
+ In Paper I, we demonstrated that the coagulation process as
91
+ described by the Smoluchowski (1916) equation is a one-
92
+ dimensional process with certain types of coagulation kernels.
93
+ Consequently, the size distribution of the coagulated grains de-
94
+ pends only on the initial size distribution and a variable χ that
95
+ encompasses the whole history of the physical conditions seen
96
+ by the grains. At a given χ that is integrated along the path of
97
+ the grains, the coagulated distribution is always the same inde-
98
+ pendently of the actual path taken. This method works for every
99
+ coagulation kernel for which the dependence on the gas variables
100
+ such as density and temperature can be separated from the grain
101
+ properties such as size and mass. In Paper I and in the present
102
+ paper, we use the turbulent kernel derived by Ormel & Cuzzi
103
+ (2007) in the intermediate coupling regime, which is suited for
104
+ star formation conditions. We show that in this case χ can be
105
+ derived from integrating
106
+ dχ = n
107
+ 3
108
+ 4
109
+ HT − 1
110
+ 4 dt,
111
+ (1)
112
+ where nH is the number density of the gas, T its temperature,
113
+ and t the time. In three-dimensional (3D) hydrodynamics simu-
114
+ lations, only the knowledge of χ is needed to track the coagula-
115
+ tion of grains.
116
+ Equation (1) is a Lagrangian derivative of χ with respect to
117
+ time along the path of the grain. We can transform it into a partial
118
+ (Eulerian) derivative
119
+ ∂χ
120
+ ∂t + u · ∇χ = n
121
+ 3
122
+ 4
123
+ HT − 1
124
+ 4 ,
125
+ (2)
126
+ where u is the velocity of the gas. We can combine equation (2)
127
+ with the mass conservation equation
128
+ ∂ρ
129
+ ∂t + ∇ · [ρu] = 0,
130
+ (3)
131
+ where ρ is the gas mass density. This yields
132
+ ∂ρχ
133
+ ∂t + ∇ · (ρχu) = ρn
134
+ 3
135
+ 4
136
+ HT − 1
137
+ 4 ,
138
+ (4)
139
+ which means that the quantity ρχ can be treated in an Eulerian
140
+ framework as a passive scalar with a source term. We exploit this
141
+ property and implement it as such in RAMSES. The value of χ
142
+ is therefore calculated self-consistently in all cells at each time-
143
+ step, as a mass-weighted average, ignoring any diffusion of dust
144
+ through the gas. We used the code Ishinisan (Marchand et al.
145
+ 2021) to pre-calculate a table containing the grain size distribu-
146
+ tion for a large number of χ values in a large relevant interval
147
+ (between χ = 1013 and χ = 1019 in cgs units). When the size
148
+ distribution is needed during the hydrodynamical simulation, it
149
+ is interpolated from the table based on the value of χ.
150
+ Our initial distribution in this paper is a Mathis et al. (1977,
151
+ MRN) distribution. The minimum and maximum radii are amin =
152
+ 5 nm and amax = 250 nm, and the slope of the distribution is
153
+ −3.5, so that the number density of grains n follows the variation
154
+ dn
155
+ da ∝ a−3.5.
156
+ (5)
157
+ The total quantity of grains is determined by the dust-to-gas
158
+ mass ratio that we assume to be 0.01. In this work, we sample the
159
+ distribution with 60 bins of size logarithmically spaced between
160
+ 10-10
161
+ 10-9
162
+ 10-8
163
+ 10-7
164
+ 10-6
165
+ 10-5
166
+ 10-4
167
+ 10-3
168
+ 10-2
169
+ 10-1
170
+ 10-3
171
+ 10-2
172
+ 10-1
173
+ 100
174
+ 101
175
+ 102
176
+ 103
177
+ Fractional mass X (ρ/nHmp)
178
+ Radius (µm)
179
+ Initial MRN
180
+ χ = 1015 cgs
181
+ χ = 1016 cgs
182
+ χ = 1017 cgs
183
+ χ = 1017.5 cgs
184
+ χ = 1018 cgs
185
+ χ = 1018.5 cgs
186
+ χ = 1019 cgs
187
+ Fig. 1. State of the grain size distribution for values of χ, between 1015
188
+ cgs and 1019 cgs. The points represent the fractional abundance of the
189
+ size-bin as function of the effective radius of the bin.
190
+ 5 nm and 5000 µm. In Figure 1, we present the coagulated MRN
191
+ size distribution at various χ. Below χ = 1017 cgs, the shift in
192
+ the size distribution is negligible. For higher values, the maxi-
193
+ mum size and the peak of the distribution are located at larger
194
+ and larger radii, while the slope of the distribution remains sim-
195
+ ilar. In all cases, the small grains are more abundant while the
196
+ large grains hold more mass. The mode of the size distribution
197
+ amax is located near the largest relevant grain size.
198
+ As grains grow, they may experience fragmentation when the
199
+ kinetic energy of the collision is high. Contrarily to what we de-
200
+ rived in paper I, we find that fragmentation does not occur in
201
+ early stages of the disk, but rather only at very high densities
202
+ ρ > 1012 cm−3 (Lebreuilly et al. 2023) for very large grains with
203
+ a > 0.08 cm. We demonstrate this result in Appendix C. We
204
+ therefore neglect fragmentation in this work. We also do not ac-
205
+ count for the grain drift with respect to the gas in this work. We
206
+ discuss the possible consequences in Section 4.2. Drift is, how-
207
+ ever, compatible with our coagulation model, and we detail the
208
+ method in Appendix B. Other limitations are discussed in Sec-
209
+ tion 4.4.
210
+ 2.1.2. Ionization and resistivities
211
+ In Paper I, we also presented a fast method to calculate the ion-
212
+ ization of the gas-grain mixture. For an arbitrary size distribu-
213
+ tion, we can calculate the average electric charge of each grain
214
+ size, the number of ions and the number of electrons, provided
215
+ the cosmic-ray (CR) ionization rate, the density and temperature
216
+ of the gas, the average atomic mass of ions µi and the sticking
217
+ probability of electrons on grains se. Here, we assume µi = 28,
218
+ which corresponds to the ion HCO+, and se = 0.6 as in Marc-
219
+ hand et al. (2016). We also assume ζ = 5 × 10−17 s−1. The den-
220
+ sity and the temperature, are taken from the hydrodynamic sim-
221
+ ulation. The calculation is performed by the Newton-Raphson
222
+ scheme described in Appendix A of Paper I. The resistivities are
223
+ computed using a similar method to Marchand et al. (2016), with
224
+ one difference. For each grain size, they sum over the contribu-
225
+ tions of the whole charge distribution (between -1 to +1 in their
226
+ case). Instead, we average the contributions using the mean elec-
227
+ tric charge. We explicit the method in more details in Appendix
228
+ A.
229
+ Article number, page 2 of 11
230
+
231
+ P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
232
+ 2.2. Star formation simulations
233
+ 2.2.1. Model
234
+ We perform four numerical simulations using the RAMSES
235
+ code. We solve the following MHD equations
236
+ ∂ρ
237
+ ∂t + ∇ · �ρu� = 0,
238
+ (6)
239
+ ∂ρu
240
+ ∂t + ∇ ·
241
+
242
+ ρuu +
243
+
244
+ P + B2
245
+ 2
246
+
247
+ I − BB
248
+
249
+ = −ρ∇Φ,
250
+ (7)
251
+ ∂B
252
+ ∂t − ∇ ×
253
+
254
+ u × B − ηΩJ − ηH
255
+ J × B
256
+ B
257
+ + ηAD
258
+ (J × B) × B
259
+ B2
260
+
261
+ = 0,
262
+ (8)
263
+ ∇ · B = 0,
264
+ (9)
265
+ where u is the velocity of the gas, P its pressure, B the mag-
266
+ netic field, J = ∇ × B the current, I the identity matrix, Φ the
267
+ gravitational potential, and ηΩ, ηH and ηAD the ohmic, Hall and
268
+ ambipolar resistivities. The temperature evolution is prescribed
269
+ by the barotropic equation
270
+ T = T0
271
+ �������1 +
272
+
273
+ ρ
274
+ 10−13 g cm−3
275
+ �γ−1������� ,
276
+ (10)
277
+ with T0 = 10 K and γ = 5/3 the adiabatic index. The initial
278
+ condition is a sphere of gas of 1 M⊙. The radius of the sphere is
279
+ controlled by the thermal over gravitational energy ratio α. We
280
+ choose α = 0.3, which sets a radius of R = 2946 au. The density
281
+ distribution has an m = 2 azimuthal perturbation
282
+ ρ(ϕ) = ρ0(1 + δρ sin ϕ),
283
+ (11)
284
+ where ρ0 is the density of a uniform sphere of same mass and
285
+ radius, and ϕ the azimuthal angle. We choose δρ = 0.05. The
286
+ computational domain outside of the sphere is filled with gas of
287
+ density ρ0/100. The sphere undergoes a solid rotation, with a
288
+ ratio of rotational to gravitational energy of β = 0.02. The mag-
289
+ netic field is initially uniform and parallel to the rotation axis.
290
+ It is defined using the mass-to-flux ratio over the critical value
291
+ (Mouschovias & Spitzer 1976)
292
+ µB =
293
+ M/ΦB
294
+ (M/ΦB)crit
295
+ ,
296
+ (12)
297
+ with
298
+ � M
299
+ ΦB
300
+
301
+ crit
302
+ = 0.53
303
+
304
+
305
+ 5
306
+ G.
307
+ (13)
308
+ Observations show that dense cores are slightly super-critical
309
+ (Crutcher 1999), although recent numerical simulations indicate
310
+ that observations may overestimate the actual strength of the
311
+ magnetic field due to projection effects (Kuznetsova et al. 2020).
312
+ We choose µB = 5 as a fiducial value. Those parameters are
313
+ used in our reference case C-3. We change the value of α to 0.4,
314
+ and the initial mass M to 5 M⊙ for two other cases called C-4
315
+ and C-3-M5, respectively, to investigate the influence of the col-
316
+ lapse time on the grain coagulation. The final run, named NC-3,
317
+ is similar to C-3 without coagulation. The grain distribution and
318
+ ionization evolve as described in sections 2.1, 2.1.2, and Paper I.
319
+ The four cases are summarized in table 1.
320
+ 2.2.2. Grid and Algorithm
321
+ The simulation box is a cube that is four times as large as the ra-
322
+ dius of the sphere, with periodic boundary conditions. The initial
323
+ grid is composed of 323 cells (level 5 of AMR) and is refined to
324
+ ensure at least 10 points per Jeans length, strongly satisfying the
325
+ Truelove et al. (1997) criterion. The maximum AMR refinement
326
+ level is 13 for C-3 and NC-3 (resolution of 1.4 au), 14 for C-4
327
+ (resolution of 0.96 au) and 16 for C-3-M5 (resolution of 0.90
328
+ au).
329
+ Simulations are performed with a 3D unsplit slope limiter
330
+ to avoid overshooting of the magnetic field at shock boundaries,
331
+ while keeping the second order convergence for the Hall effect.
332
+ We use the HLLD Riemann solver (Miyoshi & Kusano 2005)
333
+ for non-magnetic variables and the 2D HLL Riemann solver for
334
+ the magnetic field and the Hall effect (Balsara 2012; Marchand
335
+ et al. 2018). The Poisson equation is solved using the multigrid
336
+ method of Guillet & Teyssier (2011) in our periodic domain.
337
+ 3. Application to star formation
338
+ In this section, we present the results of our calculations. We first
339
+ apply our coagulation method to an analytical one-zone model in
340
+ Section 3.1, then in 3D MHD simulations in Section 3.2.
341
+ 3.1. Analytical collapse
342
+ During spherical protostellar collapse, the time evolution of the
343
+ contraction ratio of a gas cloud compared to its original radius
344
+ x = R(t)/R0 can be described by (Flower et al. 2005):
345
+ dx
346
+ dt = − π
347
+ 2τff
348
+
349
+ 1
350
+ x − 1,
351
+ (14)
352
+ where τff is the free-fall time, given by
353
+ τff =
354
+
355
+
356
+ 32Gρ0
357
+ ,
358
+ (15)
359
+ where ρ0 is the initial density. Guillet et al. (2020) showed that
360
+ assuming a uniform compression of the gas nicely reproduces
361
+ the isothermal phase of the collapse, particularly when compar-
362
+ ing the dust size distribution at the same gas density. In this case,
363
+ the density of mass scales as ρ(t) = ρ0(R0/R[t])3. The gas density
364
+ then evolves as
365
+ 1
366
+ nH
367
+ dnH
368
+ dt = 1
369
+ ρ
370
+
371
+ dt = −3
372
+ x
373
+ dx
374
+ dt .
375
+ (16)
376
+ We use a second-order Runge-Kutta scheme to numerically
377
+ integrate this equation from the beginning of the collapse at
378
+ ρ = ρ0 until the formation of the first Larson core at ρ =
379
+ 10−13 g cm−3. The evolution of χ with ρ is plotted in Figure 2,
380
+ assuming T = 10 K. The solid and dashed lines represent dif-
381
+ ferent initial densities, ρ0 = 3.8 × 10−20 g cm−3 and ρ0 =
382
+ 3.8×10−18 g cm−3, respectively. At densities nH > 10−16 g cm−3,
383
+ the value of χ is independent from ρ0 and increases as χ ∝ ρ1/4.
384
+ This evolution is expected as χ ∼ ρ3/4t and t ∼ τff ∼ ρ−1/2. At
385
+ ρ = 10−13 g cm−3, the coagulation variable reaches χ ≈ 5.6×1017
386
+ cgs, which corresponds to a peak of the size distribution of ∼ 10
387
+ µm, indicating a significant grain growth. That is consistent with
388
+ the results of Guillet et al. (2020), who use the same collapse
389
+ model, but solve coagulation on the fly.
390
+ Article number, page 3 of 11
391
+
392
+ A&A proofs: manuscript no. Marchandetal2023
393
+ Table 1. Parameters of the protostellar collapse simulations: name of the simulation, thermal over gravitational energy ratio α, radius R, mass M,
394
+ initial density ρ0 and initial magnetic field B of the initial cloud, formation time of the first hydrostatic core tfc, use of coagulation.
395
+ Name
396
+ α
397
+ R (au)
398
+ M (M⊙)
399
+ ρ0 (g cm−3)
400
+ B (µG)
401
+ tfc (kyr)
402
+ Coagulation
403
+ C-3
404
+ 0.3
405
+ 2946
406
+ 1
407
+ 5.49 × 10−18
408
+ 133
409
+ ∼ 30
410
+ Yes
411
+ C-3-M5
412
+ 0.3
413
+ 14738
414
+ 5
415
+ 2.19 × 10−19
416
+ 27
417
+ ∼ 150
418
+ Yes
419
+ C-4
420
+ 0.4
421
+ 3930
422
+ 1
423
+ 2.31 × 10−18
424
+ 75
425
+ ∼ 47
426
+ Yes
427
+ NC-3
428
+ 0.3
429
+ 2946
430
+ 1
431
+ 5.49 × 10−18
432
+ 133
433
+ ∼ 30
434
+ No
435
+ Fig. 2. Evolution of the coagulation variable χ with increasing density
436
+ nH during the isothermal collapse. The solid line represents an initial
437
+ density of ρ0 = 3.8 × 10−20 g cm−3, and the dashed line ρ0 = 3.8 ×
438
+ 10−18 g cm−3.
439
+ 3.2. Numerical collapse
440
+ The numerical simulations were run as described in Section 2
441
+ until 1000 years after the density reaches 10−13 g cm−3, the for-
442
+ mation of the first hydrostatic core at ∼ 30 kyr (Tab. 1). In refer-
443
+ ence simulation C-3, a small circumstellar disk with a radius of ≈
444
+ 20 au forms and a disk wind is launched by magneto-centrifugal
445
+ acceleration (Blandford & Payne 1982). Figure 3 shows face-on
446
+ and edge-on slices of density at the final time-step of the simu-
447
+ lation, with arrows indicating the gas velocity.
448
+ 3.2.1. Grain growth
449
+ We show in Figure 4 the value of χ as a function of the density in
450
+ simulation cells at the final time-step, for runs C-3, C-4 and C-
451
+ 3-M5. The increase is quasi unidimensional in the isothermally
452
+ collapsing envelope for ρ < 1015 g cm−3. Beyond this density,
453
+ there is a large spread of χ values of over an order of magnitude.
454
+ This spread most likely occurs in gas falling in the pseudo-disk,
455
+ then the disk and the first Larson core, or in the outflow over dif-
456
+ ferent timescales, and spending unequal times in a given density
457
+ range. The overall trend agrees well with the analytical calcula-
458
+ tion.
459
+ There is no significant difference in the χ values, and thus
460
+ dust size distributions, between the three runs, despite run C-4
461
+ needing 50% more time to collapse to the first Larson core stage,
462
+ and C-3-M5 needing 400% more time. Coagulation happening
463
+ in the isothermally collapsing envelope is therefore hardly im-
464
+ pacted by the initial conditions, as growth accelerates with in-
465
+ creasing density.
466
+ -40
467
+ -20
468
+ 0
469
+ 20
470
+ 40
471
+ x (au)
472
+ -40
473
+ -20
474
+ 0
475
+ 20
476
+ 40
477
+ y (au)
478
+ -14
479
+ -13
480
+ -12
481
+ -11
482
+ log(ρ) (g cm-3)
483
+ -40
484
+ -20
485
+ 0
486
+ 20
487
+ 40
488
+ x (au)
489
+ -40
490
+ -20
491
+ 0
492
+ 20
493
+ 40
494
+ y (au)
495
+ -40
496
+ -20
497
+ 0
498
+ 20
499
+ 40
500
+ -40
501
+ -20
502
+ 0
503
+ 20
504
+ 40
505
+ -200
506
+ -100
507
+ 0
508
+ 100
509
+ 200
510
+ x (au)
511
+ -200
512
+ -100
513
+ 0
514
+ 100
515
+ 200
516
+ z (au)
517
+ -17
518
+ -16
519
+ -15
520
+ -14
521
+ -13
522
+ -12
523
+ log(ρ) (g cm-3)
524
+ -200
525
+ -100
526
+ 0
527
+ 100
528
+ 200
529
+ x (au)
530
+ -200
531
+ -100
532
+ 0
533
+ 100
534
+ 200
535
+ z (au)
536
+ -200
537
+ -100
538
+ 0
539
+ 100
540
+ 200
541
+ -200
542
+ -100
543
+ 0
544
+ 100
545
+ 200
546
+ Fig. 3. Density slices of the C-3 simulation, with grain coagulation. The
547
+ top panel is a face-on slice of the plane z=0, and the bottom panel an
548
+ edge-on slice of the plane y=0. White arrows represent the direction of
549
+ the gas velocity. The snapshot is taken at the final time-step, 1000 years
550
+ after the formation of the first Larson core.
551
+ Figure 5 shows the mode of the size distribution as a func-
552
+ tion of density, corresponding to the distribution of χ shown in
553
+ Figure 4. Three regions are clearly demarcated, the first being
554
+ the envelope, in which grain coagulation is not efficient enough
555
+ to form large grains (ρ < 10−16 g cm−3). The second comprises
556
+ Article number, page 4 of 11
557
+
558
+ 1018
559
+ cm
560
+ -3
561
+ 16
562
+ X
563
+ 101s
564
+ 1014
565
+ 10-19
566
+ 10-13
567
+ 10-18
568
+ 10-17
569
+ 10~16
570
+ 10-14
571
+ 10-12
572
+ 10-11
573
+ 02-01
574
+ p (g cmP. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
575
+ Fig. 4. Evolution of the coagulation variable χ with increasing density
576
+ nH in the numerical collapse models C-3 (purple), C-3-M5 (light blue),
577
+ and C-4 (green). Each point corresponds to a simulation cell at the final
578
+ time-step. The red line is the analytical collapse solution for ρ0 = 3.8 ×
579
+ 10−20 g cm−3.
580
+ the pseudo-disk and the early protoplanetary disk, where grains
581
+ grow by a factor 100 from sub-micron sizes to several tens of
582
+ µm. The third region is the first Larson core, which has even
583
+ larger grains that reach 400 µm within a mere 103 yr years af-
584
+ ter its formation. That value is in line with similar recent studies
585
+ (Kawasaki et al. 2022; Lebreuilly et al. 2023). There is little dif-
586
+ ference between runs C-3, C-4 and C-3-M5, confirming that co-
587
+ agulation in the envelope does not impact large grains, as found
588
+ by previous studies (for example Silsbee et al. 2022). We discuss
589
+ observations of large grains in the envelope in Section 4.2.
590
+ The spatial distribution of those grain sizes for run C-3 is
591
+ displayed in Figure 6. Size distributions shifting significantly
592
+ from the initial MRN distribution are indeed confined to the mid-
593
+ plane, in the disk and pseudo-disk. The bottom panel also shows
594
+ moderately larger grains in the outflow, as they only traveled
595
+ through the upper layers of the pseudo-disk before being ejected
596
+ (Marchand et al. 2020). If they had been in the mid-plane of the
597
+ disk, they would have grown much more, as coagulation is irre-
598
+ versible in our model. The upper panel shows that grains reach
599
+ a radius larger than 1 µm in the outskirts of the disk, within 100
600
+ au of the center. Growth then occurs rapidly as density increases
601
+ in the inner 15 au, which is shown by the almost overlapping
602
+ contours delimiting amax = 5 µm and amax = 10 µm.
603
+ 3.2.2. Resistivities and gas dynamics
604
+ We describe here the impact of grain coagulation on non-ideal
605
+ MHD resistivities and their macroscopic effects on gas dynam-
606
+ ics. Previous studies emulated the coagulation of grains by re-
607
+ moving the very small grains (a < 0.1 µm) and redistributing
608
+ their mass to the larger end of the distribution (Zhao et al. 2016,
609
+ 2018; Marchand et al. 2020). This method leads to an increase
610
+ of resistivities, in particular the ambipolar resistivity, resulting in
611
+ weaker coupling between the magnetic field and the gas, hence
612
+ weaker magnetic braking and larger, more unstable disks. How-
613
+ ever, we observe here the exact opposite behavior.
614
+ The middle panel of figure 7 presents the volume-weighted
615
+ average resistivities as a function of density, at the final time-
616
+ step, for simulations C-3 and NC-3. A non-evolving size distri-
617
+ Fig. 5. Mode of the coagulated grain size distribution as a function of
618
+ density in simulations (purple), C-3-M5 (light blue), and C-4 (green).
619
+ The discrete values of the sizes are due to the binning of the size distri-
620
+ bution.
621
+ bution produces resistivities relatively similar in the envelope,
622
+ but two to four orders of magnitude larger at disk densities, par-
623
+ ticularly the Ohmic and ambipolar resistivities. Consequently,
624
+ the magnetic braking is weakened, and the gas retains more an-
625
+ gular momentum without the grain coagulation, forming a larger
626
+ disk, as shown in Figure 8.
627
+ This difference in regime can be quantified by the ambipolar
628
+ Elsasser number Am = B2/(ρηADΩ). In regions where Am < 1,
629
+ the ambipolar diffusion has a significant impact on the dynam-
630
+ ics of the gas. The bottom panel of Figure 7 shows the radial
631
+ profile of the ambipolar Elsasser number in runs C-3 and NC-
632
+ 3, azimuthally-averaged in the mid-plane. The higher resistivity
633
+ in run NC-3 results in Am < 1 in the inner ∼ 12 au, indicat-
634
+ ing active ambipolar diffusion and magnetic field dissipation,
635
+ while Am ≳ 104 for C-3 over the same radial range, indicat-
636
+ ing weak ambipolar diffusion. Figure 9 compares the disk size
637
+ and angular momentum in both simulations, further confirming
638
+ the lower magnetic braking in run NC-3. A second effect of the
639
+ weaker magnetic forces from the stronger ambipolar diffusion in
640
+ run NC-3 is the absence of outflow at this stage of evolution, as
641
+ shown in the lower panel of Figure 8.
642
+ The discrepancy of resistivity values between actual coagu-
643
+ lation and methods simply redistributing the mass to the large-
644
+ mass-end of the distribution originates from the lack of very
645
+ large grains (> 10 µm) in the latter. In both cases, removing
646
+ the small grains decreases the electron absorption by dust, and
647
+ therefore decreases the Ohmic resistivity. However, the ambipo-
648
+ lar resistivity is controlled by the relative abundance of ions
649
+ and charged grains in the gas. Although the small grain removal
650
+ method barely changes the abundance of ions, the dominant pos-
651
+ itive charge carriers, it reduces significantly the charged grain
652
+ population, leading to an increase of resistivity at low density. At
653
+ high density, both with a standard MRN and a truncated-MRN
654
+ distribution, the grains become the dominant charge carriers and
655
+ the abundance of ions decreases (see the dashed lines in the top
656
+ panel of Figure 7). That does not happen with coagulation be-
657
+ cause the number of grains decreases significantly, hence leading
658
+ to a higher number of ions and a lower resistivity than without
659
+ proper coagulation. This kind of method without larger grain cre-
660
+ ation is therefore inappropriate for emulating coagulation alone
661
+ at a low cost for non-ideal MHD calculations. At later times, at
662
+ Article number, page 5 of 11
663
+
664
+ 1020
665
+ C-3
666
+ C-4
667
+ C-3-M5
668
+ 1019
669
+ Analytical collapse
670
+ 1018
671
+ C1017
672
+ X
673
+ 1016
674
+ 1015
675
+ 10-T8
676
+ 10-T6
677
+ 10-T4
678
+ 10-20
679
+ 10°10
680
+ p (g cm103
681
+ C-3
682
+ C-4
683
+ C-3-M5
684
+ 10°
685
+ 10-78
686
+ 10-10
687
+ -3
688
+ p (g cmA&A proofs: manuscript no. Marchandetal2023
689
+ -100
690
+ -50
691
+ 0
692
+ 50
693
+ 100
694
+ x (au)
695
+ -100
696
+ -50
697
+ 0
698
+ 50
699
+ 100
700
+ y (au)
701
+ 0.1
702
+ 1
703
+ 10
704
+ amax (µm)
705
+ -100
706
+ -50
707
+ 0
708
+ 50
709
+ 100
710
+ x (au)
711
+ -100
712
+ -50
713
+ 0
714
+ 50
715
+ 100
716
+ y (au)
717
+ -200
718
+ -100
719
+ 0
720
+ 100
721
+ 200
722
+ x (au)
723
+ -200
724
+ -100
725
+ 0
726
+ 100
727
+ 200
728
+ z (au)
729
+ 0.1
730
+ 0.2
731
+ 0.3
732
+ 0.4
733
+ 0.5
734
+ 0.6
735
+ 0.7
736
+ 0.8
737
+ 0.9
738
+ 1
739
+ amax (µm)
740
+ -200
741
+ -100
742
+ 0
743
+ 100
744
+ 200
745
+ x (au)
746
+ -200
747
+ -100
748
+ 0
749
+ 100
750
+ 200
751
+ z (au)
752
+ Fig. 6. Slices of the C-3 simulation with coagulation showing the mode
753
+ of the grain size distribution amax, 1000 years after the first core forma-
754
+ tion. The color scale is different for each panel to yield a better contrast.
755
+ On the top panel, black contours indicate amax = 1, 5 and 10 µm, and
756
+ amax = 0.5 and 1 µm on the bottom panel.
757
+ densities ρ > 10−12 g cm−3, Lebreuilly et al. (2023) showed that
758
+ the replenishment of small grains by fragmentation would lead
759
+ to an increase in both Ohmic and ambipolar resistivities.
760
+ 4. Discussion and caveats
761
+ 4.1. Grain growth
762
+ As explained in previous sections and displayed in Figures 4 and
763
+ 5, initial gas conditions have little to no influence on grain coag-
764
+ ulation for later stages, and coagulation is ineffective in the en-
765
+ velope for growing large grains. This reinforces the idea that the
766
+ system forgets the initial conditions at the formation of the first
767
+ hydrostatic core and the disk (Vaytet & Haugbølle 2017). Con-
768
+ sequently, calculations such as those presented in this work may
769
+ provide standard initial dust grain size distributions for studies of
770
+ 10-20
771
+ 10-18
772
+ 10-16
773
+ 10-14
774
+ 10-12
775
+ 10-10
776
+ 10-8
777
+ 10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10
778
+ ne / nH, ni / nH
779
+ ρ (g cm-3)
780
+ Ions
781
+ Electrons
782
+ No coagulation
783
+ 1010
784
+ 1012
785
+ 1014
786
+ 1016
787
+ 1018
788
+ 1020
789
+ 1022
790
+ 10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10
791
+ MHD resistivities (cm2 s-1)
792
+ ρ (g cm-3)
793
+ Ohm
794
+ Ambipolar
795
+ Hall
796
+ No coagulation
797
+ 10-2
798
+ 10-1
799
+ 100
800
+ 101
801
+ 102
802
+ 103
803
+ 104
804
+ 105
805
+ 1
806
+ 10
807
+ 100
808
+ Elsasser number
809
+ r (au)
810
+ C-3 (coagulation)
811
+ NC-3 (No coagulation)
812
+ Fig. 7. Top panel: Abundances of ions (purple) and electrons (green)
813
+ as a function of density in models with coagulation (C-3; solid lines) or
814
+ without it (NC-3; dashed lines). Middle panel: Volume averaged Ohmic
815
+ (purple), ambipolar (green), and Hall (blue) resistivities for the same
816
+ models. Bottom panel: Average ambipolar Elsasser number Am in the
817
+ mid-plane as a function of radius. The thin grey line represents Am = 1.
818
+ protoplanetary disk evolution. Although other coagulation ker-
819
+ nels affect the size distribution in different ways (see Section
820
+ 4.4), it is certain that even young protoplanetary disks contain
821
+ grains significantly larger than the classical MRN distribution.
822
+ Article number, page 6 of 11
823
+
824
+ P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
825
+ -40
826
+ -20
827
+ 0
828
+ 20
829
+ 40
830
+ x (au)
831
+ -40
832
+ -20
833
+ 0
834
+ 20
835
+ 40
836
+ y (au)
837
+ -14
838
+ -13
839
+ -12
840
+ -11
841
+ log(ρ) (g cm-3)
842
+ -40
843
+ -20
844
+ 0
845
+ 20
846
+ 40
847
+ x (au)
848
+ -40
849
+ -20
850
+ 0
851
+ 20
852
+ 40
853
+ y (au)
854
+ -40
855
+ -20
856
+ 0
857
+ 20
858
+ 40
859
+ -40
860
+ -20
861
+ 0
862
+ 20
863
+ 40
864
+ -200
865
+ -100
866
+ 0
867
+ 100
868
+ 200
869
+ x (au)
870
+ -200
871
+ -100
872
+ 0
873
+ 100
874
+ 200
875
+ z (au)
876
+ -17
877
+ -16
878
+ -15
879
+ -14
880
+ -13
881
+ -12
882
+ log(ρ) (g cm-3)
883
+ -200
884
+ -100
885
+ 0
886
+ 100
887
+ 200
888
+ x (au)
889
+ -200
890
+ -100
891
+ 0
892
+ 100
893
+ 200
894
+ z (au)
895
+ -200
896
+ -100
897
+ 0
898
+ 100
899
+ 200
900
+ -200
901
+ -100
902
+ 0
903
+ 100
904
+ 200
905
+ Fig. 8. Same as figure 3 for simulation NC-3, without grain coagulation,
906
+ 1000 years after the formation of the first Larson core.
907
+ This has important implications for the dynamics of grains in the
908
+ disk. Larger grains couple differently to the gas and may trigger
909
+ the streaming instability (Youdin & Goodman 2005; Johansen
910
+ & Youdin 2007; Yang et al. 2017), which is an early step toward
911
+ planet formation. Although this regime is not reached in our sim-
912
+ ulations, the fast growth of the grains in circumstellar disks could
913
+ predict an early onset for this process.
914
+ 4.2. Large grains in envelopes and dust diffusion
915
+ Although grain coagulation is negligible in the envelope of our
916
+ simulations, large grain signatures in envelopes have been ob-
917
+ served. Galametz et al. (2019), for example, report "low and
918
+ varying dust emissivity indices" at the envelope scale for some
919
+ Class 0 and Class I protostars. This could be due to the pres-
920
+ ence of mm-size grains in low numbers. The time-scale to form
921
+ such large grains in the envelope and cold ISM cores is large
922
+ 0
923
+ 5
924
+ 10
925
+ 15
926
+ 20
927
+ 25
928
+ 30
929
+ 35
930
+ 0
931
+ 0.2
932
+ 0.4
933
+ 0.6
934
+ 0.8
935
+ 1
936
+ 1.2
937
+ 1.40
938
+ 1e+51
939
+ 2e+51
940
+ 3e+51
941
+ 4e+51
942
+ 5e+51
943
+ 6e+51
944
+ 7e+51
945
+ Disk radius (au)
946
+ Disk angular momentum (g cm2 s-1)
947
+ t (kyr) after First Core formation
948
+ Disk radius
949
+ Disk angular momentum
950
+ No coagulation
951
+ Fig. 9. Disk size (purple, left axis) and angular momentum (green, right
952
+ axis) as a function of time after the formation of the first Larson core.
953
+ Solid lines represent simulation C-3 with coagulation, and dashed lines
954
+ are simulation NC-3 without coagulation.
955
+ (> 100 Myr), so we exclude the possibility of early coagulation.
956
+ Similarly, Valdivia et al. (2019) report that their synthetic polari-
957
+ sation observations of young protostellar envelopes in RAMSES
958
+ calculations require grains larger than 10 µm to be consistent
959
+ with observations, which is also inconsistent with our findings
960
+ and those of recent studies (Silsbee et al. 2022; Lebreuilly et al.
961
+ 2023, for the latest ones).
962
+ The aerodynamic properties of the grains may cause differ-
963
+ ential velocities between grain populations and the gas, lead-
964
+ ing to varying dust-to-gas ratio throughout the cloud (Lebreuilly
965
+ et al. 2020). We note however that, as they have found, dust dif-
966
+ fusion will start to play a significant role only for very large
967
+ grains of a few hundred microns. Generally, large grains tend
968
+ to accumulate in higher density regions. Tsukamoto et al. (2021)
969
+ found that this dust diffusion can lead to what they call an ash fall
970
+ phenomenon, in which large coagulated grains (up to millimeter-
971
+ size) from the disk are ejected by an outflow, then decouple from
972
+ the gas and fall back in the envelope. This process may explain
973
+ the observations of low spectral index in Class 0 envelopes, as
974
+ the outflow fuels the envelope with large grains formed in the
975
+ central region. Eventually, those ejected grains circle back to the
976
+ outer edge of the disk, enriching the large-end of the size distri-
977
+ bution. In our case, the disk wind is fueled by the upper layers of
978
+ the pseudo-disk, in which dust only moderately grows. However,
979
+ dust-to-gas ratio enhancement in this region due to the grain dif-
980
+ ferential velocities may lead to an accelerated growth.
981
+ Further studies are needed to investigate this discrepancy
982
+ about the size of grains in envelopes between observations and
983
+ theory.
984
+ 4.3. Coagulation in the pseudo-disk
985
+ Grains in our simulations seem to grow faster than in Bate (2022)
986
+ despite their use of the same turbulent kernel as in our work. The
987
+ peaks of the distributions in that work reach a few microns at a
988
+ maximum density of nH = 1013 cm−3 (see their Figure 7), while
989
+ in our simulations the peak exceeds 100 µm at a lower maximum
990
+ density of nH ≈ 3×1012 cm−3 (or ρ ≈ 10−11 g cm−3; see our Fig-
991
+ ure 5). That discrepancy is mainly due to a different modelling
992
+ Article number, page 7 of 11
993
+
994
+ A&A proofs: manuscript no. Marchandetal2023
995
+ of the Reynolds number to calculate the turbulent coagulation
996
+ kernel. They assume a constant value of Re = 108, while we use
997
+ (Ormel et al. 2009; Guillet et al. 2020)
998
+ Re = 6.7 × 107 �
999
+ nH
1000
+ 105 cm−3
1001
+ � 1
1002
+ 2 .
1003
+ (17)
1004
+ Hence, in the central regions, our Reynolds number can be larger
1005
+ by three orders of magnitude. Their grains are then stuck in the
1006
+ tightly-coupled regime where relative grain velocities are lower
1007
+ than in the intermediate coupling-regime, which is more adapted
1008
+ to this situation (as demonstrated in Section 4.1 of paper I). The
1009
+ lower relative velocities result in lower coagulation rates, and
1010
+ therefore a slower growth rate. The poor constraints on the value
1011
+ of the Reynolds number in protostellar environments therefore
1012
+ represents a source of uncertainty for grain growth by coagula-
1013
+ tion.
1014
+ An additional explanation may involve an excess of growth
1015
+ in the pseudo-disk that forms in our simulations. The pseudo-
1016
+ disk is an over-density, typically denser than ρ ≈ 10−15 g cm−3,
1017
+ created by the convergence of gas flowing along magnetic field
1018
+ lines (Galli & Shu 1993), that takes the shape of a disk perpen-
1019
+ dicular to the magnetic field, but not supported against gravity.
1020
+ It is apparent in the bottom panels of Figures 3 and 8.
1021
+ Gas in the rotationally supported disk generally comes di-
1022
+ rectly from the pseudo-disk, and the overall efficiency of coag-
1023
+ ulation is mainly affected by the time spent in high-density re-
1024
+ gions like the pseudo-disk. This is what appears in the bottom
1025
+ panel of Figure 6, in which there is a ∼ 30 au-thick layer of large
1026
+ grains in the mid-plane. A passage of grains through the pseudo-
1027
+ disk would therefore provide an acceleration of coagulation in
1028
+ the early stage of star formation, even before arrival in the disk.
1029
+ That does not happen in Bate (2022) since they do not consider
1030
+ magnetic fields, resulting in less-coagulated size distributions as
1031
+ grains enter the disk.
1032
+ 4.4. Coagulation kernel
1033
+ Our coagulation methods works for every coagulation kernel
1034
+ where the environment variables and grain variables can be sep-
1035
+ arated, one example of which is the well-known turbulent ker-
1036
+ nel derived by Ormel & Cuzzi (2007) that we use. This kernel
1037
+ is appropriate to calculate the growth of large grains, but it has
1038
+ several limitations. We assume a steady-state Kolmogorov turbu-
1039
+ lence spectrum, and that the injection-scale of the turbulence is
1040
+ equal to the Jeans length corresponding to the local density (see
1041
+ section 2.2 of paper I), which may lead to an over-estimation of
1042
+ the grain collision rate. Other kernels may have different effects
1043
+ on the size distribution of grains. Guillet et al. (2020) showed
1044
+ that this turbulent kernel would increase the maximum size of
1045
+ the distribution, while a kernel derived from ambipolar diffusion,
1046
+ that creates a drift between charged and neutral grains, is effi-
1047
+ cient at removing small grains. This also happens in Bate (2022)
1048
+ and Lebreuilly et al. (2023), which combine several processes
1049
+ including Brownian motion and pressure gradients that generate
1050
+ drift velocities between grains and rapidly remove the smaller
1051
+ grains. These processes are not, however, efficient at producing
1052
+ larger grains without the help of the turbulent kernel. Contrary
1053
+ to fragmentation, the addition of those kernels would steepen the
1054
+ distribution at its lower end.
1055
+ 5. Conclusions
1056
+ We present here the results of numerical simulations of protostel-
1057
+ lar collapse with dust coagulation and self-consistent calculation
1058
+ of non-ideal MHD resistivities, using the methods detailed in
1059
+ Marchand et al. (2021). We performed four simulations, includ-
1060
+ ing three with coagulation and different collapse times, and one
1061
+ without coagulation for reference. Here are our main results.
1062
+ - Coagulated size distributions retain some characteristics of
1063
+ the initial MRN distribution, in particular the dominance of
1064
+ small grains in number and large grains in mass.
1065
+ - Dust coagulation is inefficient at growing larger grains in the
1066
+ envelope, even with long free-fall times. What matters is the
1067
+ time spent in high-density regions (ρ > 10−15 g cm−3) like
1068
+ the pseudo-disk. Fragmentation can also be ignored in the
1069
+ cloud-collapse phase.
1070
+ - Grain growth is extremely rapid in the disk. The peak of the
1071
+ size distribution in mass, which is also the largest relevant
1072
+ grain size of the distribution, reaches 1 µm in the pseudo-
1073
+ disk and more than 100 µm in the inner disk only 103 yr
1074
+ after its formation.
1075
+ - Grain sizes have a significant impact on non-ideal MHD re-
1076
+ sistivities. Coagulated grains result in resistivities up to four
1077
+ orders of magnitude lower than non-coagulated grains, with
1078
+ a significant impact on the dynamics of the disk. Simple re-
1079
+ distribution approximations fail to capture this effect, as it
1080
+ occurs because of the growth of the largest grains.
1081
+ Accounting for grain coagulation is therefore necessary in
1082
+ star formation and protoplanetary disk simulations, as grains
1083
+ grow rapidly to ≥ 100 µm in radius in disks. The effects of grain
1084
+ growth on chemistry and radiative transfer will be explored in
1085
+ future papers.
1086
+ Acknowledgements. P.M. acknowledges the financial support of the Kathryn W.
1087
+ Davis Postdoctoral Fellowship of the American Museum of Natural History and
1088
+ of the European Research Council (ERC) under the European Union’s Hori-
1089
+ zon 2020 research and innovation programme (ERC Starting Grant “Chemtrip”,
1090
+ grant agreement No 949278). M.-M.M.L. was partly supported by US NSF grant
1091
+ AST18-15461. U.L. acknowledges the financial support of the European Re-
1092
+ search Council (ERC) via the ERC Synergy Grant ECOGAL (grant 855130).
1093
+ Appendix A: Calculating the resistivities
1094
+ Appendix A.1: Solving for the ionization
1095
+ Let us assume a size distribution of grains divided into N bins.
1096
+ We have the following system of equations (see Paper I)
1097
+ Zk =ψτk +
1098
+ 1 − ϵ2Θ2
1099
+ 1 + ϵΘαk + ϵ2Θ2 ,
1100
+ (A.1)
1101
+ ⟨ ˜J(τk)⟩ =(1 − ψ) +
1102
+ 2
1103
+ τk
1104
+
1105
+ ϵ2Θ2 + ϵΘ
1106
+
1107
+ 1 + ϵΘαk + ϵ2Θ2 ,
1108
+ (A.2)
1109
+ ϵ =1 − ψ
1110
+ Θeψ ,
1111
+ (A.3)
1112
+ ni = −
1113
+ 1
1114
+ 1 − ϵ
1115
+
1116
+ k
1117
+ nkZk,
1118
+ (A.4)
1119
+ f(ψ) =⟨σv⟩ieϵn2
1120
+ i
1121
+ ζnH
1122
+ + nivi
1123
+ ζnH
1124
+
1125
+ k
1126
+ nkπa2
1127
+ k⟨ ˜J(τk)⟩ − 1 = 0.
1128
+ (A.5)
1129
+ ϵ = ne/ni < 1 is the ratio between the number of electrons and
1130
+ ions, ψ is the ratio between the electric potential of the grains
1131
+ and the kinetic energy of electrons, nk, Zk and ak represent the
1132
+ number density, the average charge and the radius of grains in
1133
+ bin k, ζ is the CR ionization rate, and vi = [8kBT/πµimH]1/2
1134
+ Article number, page 8 of 11
1135
+
1136
+ P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
1137
+ is the thermal velocity of ions, with T the temperature, µi the
1138
+ average atomic mass of ions and mH the mass of the pro-
1139
+ ton. We also have Θ = se[µimH/me]1/2, with se the sticking
1140
+ probability of electrons on grains and me the electron mass.
1141
+ ⟨ ˜J(τk)⟩ represents the enhancement factor for ion recombina-
1142
+ tion on grains, τk = akkBT/e2 is the reduced temperature of
1143
+ grains (Draine & Sutin 1987), and αk = [8/(πτk)]1/2. Finally,
1144
+ ⟨σv⟩ie = 2 × 10−7[T/300]−1/2 is the collision rate between ions
1145
+ and electrons.
1146
+ We solve the system of equations (A.1) - (A.5) for ψ and
1147
+ find ni, ne and Zk for all bins. A Newton-Raphson algorithm is
1148
+ described in details in the Appendix A of Paper I.
1149
+ Appendix A.2: The resistivities
1150
+ The collision time-scale of species s with neutral H2, the most
1151
+ abundant species in the gas, is given by
1152
+ τsH2 =
1153
+ 1
1154
+ asHe
1155
+ ms + mH2
1156
+ mH2
1157
+ 1
1158
+ nH2⟨σω⟩s
1159
+ .
1160
+ (A.6)
1161
+ asHe accounts for the collisions with He and is equal to 1.14
1162
+ for ions, 1.16 for electrons and 1.28 for grains (Desch &
1163
+ Mouschovias 2001). ms and mH2 are the mass of the species s
1164
+ and the H2 molecule. nH2 represents the number density of H2
1165
+ (roughly equal to the density of the gas). ⟨σω⟩s is the collision
1166
+ rate, taken from Pinto & Galli (2009) for electrons and ions
1167
+ ⟨σω⟩e = 3.16 × 10−11v1.3
1168
+ rms,e,
1169
+ (A.7)
1170
+ ⟨σω⟩i = 2.4 × 10−9v0.6
1171
+ rms,i,
1172
+ (A.8)
1173
+ where
1174
+ vrms,s =
1175
+ � 8kBT
1176
+ πµs,H2
1177
+ � 1
1178
+ 2
1179
+ ,
1180
+ (A.9)
1181
+ is in km s−1, with µs,H2 the reduced mass of species s and H2.
1182
+ These velocities have been calculated in a three-fluid formalism
1183
+ but we use them in our monofluid framework. See some related
1184
+ concerns in the appendix of Marchand et al. (2020). For grains,
1185
+ the collision rate is given by (Draine & Sutin 1987; Kunz &
1186
+ Mouschovias 2009)
1187
+ ⟨σω⟩k = πa2
1188
+ k
1189
+ �8kBT
1190
+ πmH2
1191
+ � 1
1192
+ 2 ��������1 +
1193
+ � π
1194
+ 2τk
1195
+ � 1
1196
+ 2 �������� .
1197
+ (A.10)
1198
+ We also define the conductivity σs and cyclotron frequency ωs
1199
+ of species s
1200
+ σs = nsq2
1201
+ sτsH2
1202
+ ms
1203
+ ,
1204
+ (A.11)
1205
+ ωs = qsB
1206
+ cms
1207
+ ,
1208
+ (A.12)
1209
+ where qs is the electric charge of species s, B the magnetic field
1210
+ strength and c the speed of light. The parallel, perpendicular and
1211
+ Hall conductivities are expressed as
1212
+ σ|| =
1213
+
1214
+ s
1215
+ σs,
1216
+ (A.13)
1217
+ σ⊥ =
1218
+
1219
+ s
1220
+ σs
1221
+ 1 + (ωsτsH2)2 ,
1222
+ (A.14)
1223
+ σH =
1224
+
1225
+ s
1226
+ σsωsτsH2
1227
+ 1 + (ωsτsH2)2 .
1228
+ (A.15)
1229
+ (A.16)
1230
+ Finally, the Ohmic, Hall and ambipolar resisitivities are defined
1231
+ as
1232
+ ηO = 1
1233
+ σ||
1234
+ ,
1235
+ (A.17)
1236
+ ηH =
1237
+ σH
1238
+ σ2
1239
+ ⊥ + σ2
1240
+ H
1241
+ ,
1242
+ (A.18)
1243
+ ηA =
1244
+ σ⊥
1245
+ σ2
1246
+ ⊥ + σ2
1247
+ H
1248
+ − 1
1249
+ σ||
1250
+ .
1251
+ (A.19)
1252
+ (A.20)
1253
+ Appendix B: Coagulation and dust-to-gas ratio
1254
+ Our coagulation model assumes that grains are perfectly cou-
1255
+ pled with the gas and well-mixed, so that the dust-to-gas mass
1256
+ ratio is constant. However, grains of different sizes have dif-
1257
+ ferent aerodynamic properties and may experience a significant
1258
+ drift through the gas. This characteristic is usually determined
1259
+ by their Stokes number, that is the ratio between the stopping
1260
+ time of the grain (Epstein 1924) and the dynamical timescale of
1261
+ the system. In particular, Lebreuilly et al. (2020) showed strong
1262
+ variations in the dust-to-gas mass ratio during the protostellar
1263
+ collapse. The disk and first core tend to be dust-enriched while
1264
+ the envelope becomes dust-depleted. This has strong implication
1265
+ for the coagulation, as a higher grain density promotes collisions
1266
+ and speeds up their growth (conversely for a lower density). In
1267
+ this section, we briefly explore a refinement to the coagulation
1268
+ model presented in Paper I.
1269
+ The general expression of the Smoluchowski (1916) equa-
1270
+ tion, from which we derive the expression of χ, is (Paper I eq. 5)
1271
+ dX(a, t)
1272
+ dt
1273
+ = CglocalnHI(a, X, t),
1274
+ (B.1)
1275
+ where C is a constant, glocal a function of the local properties of
1276
+ the gas (density, temperature...), and
1277
+ I(a, X, t) = −
1278
+ � ∞
1279
+ 0
1280
+ h(m, m′)X(m, t)X(m′, t)dm′
1281
+ + 1
1282
+ 2
1283
+ � m
1284
+ 0
1285
+ h(m − m′, m′)X(m − m′, t)X(m′, t)dm′.
1286
+ (B.2)
1287
+ The relative number of grains of size a at a given time t is
1288
+ X(a, t) = ngrain/nH where ngrain is the number density of grains.
1289
+ We can then write X = dgx, where dg is the dust-to-gas ratio, and
1290
+ x the normalized relative number of grains. We can then rewrite
1291
+ equation (B.1)
1292
+ dx(a, t)
1293
+ dt
1294
+ = CglocalnHdgI′(a, x, t),
1295
+ (B.3)
1296
+ and include dg in the definition of our coagulation variable, giv-
1297
+ ing it the form
1298
+ dχ′ = glocalnHdgdt.
1299
+ (B.4)
1300
+ In our case, for the Ormel & Cuzzi (2007) kernel,
1301
+ dχ′ = n
1302
+ 3
1303
+ 4
1304
+ HT − 1
1305
+ 4 dgdt.
1306
+ (B.5)
1307
+ This expression means that the dust-to-gas ratio does not change
1308
+ the evolution path of the size distribution, only the coagulation
1309
+ Article number, page 9 of 11
1310
+
1311
+ A&A proofs: manuscript no. Marchandetal2023
1312
+ speed. A dust-to-gas ratio N times higher requires a χ value N
1313
+ times lower to reach the same coagulated state. This new defini-
1314
+ tion of χ′ can be used in an environment with varying dust-to-
1315
+ gas ratio. In hydrodynamics simulations, it is possible to couple
1316
+ it with the drift of one grain size close to the peak of the mass
1317
+ density distribution, as allowed by the method proposed by Le-
1318
+ breuilly et al. (2019).
1319
+ Although it is not yet possible to associate this coagulation
1320
+ method with the differential drift of grains of different sizes, this
1321
+ is a first step towards a self-consistent treatment of dust grains
1322
+ in hydrodynamics simulations. This method will lead to an over-
1323
+ estimate of the small-to-larger grain ratio but this is probably a
1324
+ decent approximation to get the total dust-to-gas ratio. As shown
1325
+ in Lebreuilly et al. (2020), as long as the large grains dominate
1326
+ the mass in the distribution, they also dominate the differential
1327
+ gas and dust dynamics. In other words, most of the dust enrich-
1328
+ ment comes from the dynamics of large grains.
1329
+ Appendix C: The fragmentation barrier
1330
+ Ormel et al. (2009) derived criteria for the fragmentation of dust
1331
+ aggregates. They define the rolling energy Eroll that determines
1332
+ the energy needed to restructure the grain. In Section 4.2 of pa-
1333
+ per I, we misinterpreted their results by assuming the fragmen-
1334
+ tation would occur for a kinetic energy Ekin = 5Eroll. The actual
1335
+ criterion is
1336
+ Ekin > 5NtotEbr,
1337
+ (C.1)
1338
+ where Ekin is the kinetic energy, Ebr is the breaking energy and
1339
+ Ntot is the total numbers of monomers composing the two collid-
1340
+ ing grains. Ebr is defined by (Dominik & Tielens 1997)
1341
+ Ebr = Abrγ5/3
1342
+ grain
1343
+ (a0/2)4/3
1344
+ ε2/3
1345
+ ,
1346
+ (C.2)
1347
+ with Abr = 2.8 × 103, γgrain the surface energy density of the ma-
1348
+ terial, a0 the size of the monomers composing the grains, and ε
1349
+ the reduced elastic modulus. As in Ormel et al. (2009), we adopt
1350
+ a0 = 0.1 µm. Ice mantles on grains make them more resistant
1351
+ to fragmentation. Therefore, we assume bare silicates to obtain a
1352
+ lower limit for a fragmentation criterion. In this case, γgrain = 25
1353
+ erg cm−2 and ε = 2.8×1011 dyn cm−2. The kinetic energy of two
1354
+ grains of mass m and m′ reads
1355
+ Ekin = 1
1356
+ 2
1357
+ mm′
1358
+ m + m′ ∆v2,
1359
+ (C.3)
1360
+ where ∆v is the relative velocity between the two grains. This
1361
+ velocity is given by the kernel of Ormel & Cuzzi (2007) that we
1362
+ use in Paper I
1363
+ ∆v =
1364
+ � 3√
1365
+ 8
1366
+ z0[kBG]
1367
+ 1
1368
+ 2 γρs
1369
+ µmH
1370
+ � 1
1371
+ 2
1372
+ n
1373
+ − 1
1374
+ 4
1375
+ H T − 1
1376
+ 4 a
1377
+ 1
1378
+ 2 ,
1379
+ (C.4)
1380
+ where z0 = 2.97, kB and G are the Boltzmann and gravitational
1381
+ constants, γ = 5/3 the adiabatic index of the gas, ρs = 2.3 g cm−3
1382
+ the bulk density of the grains, µ = 2.3 the average atomic mass
1383
+ of the gas, mH the proton mass, and a the radius of the larger of
1384
+ the two grains. Assuming two identical grains and
1385
+ m = 4
1386
+ 3πρsa3
1387
+ 0Ntot,
1388
+ (C.5)
1389
+ equation C.1 becomes
1390
+ a > 15Ebr
1391
+ πρsa3
1392
+ 0
1393
+ � 3√
1394
+ 8
1395
+ z0[kBG]
1396
+ 1
1397
+ 2 γρs
1398
+ µmH
1399
+ �−1
1400
+ n
1401
+ 1
1402
+ 2
1403
+ HT
1404
+ 1
1405
+ 2 .
1406
+ (C.6)
1407
+ Replacing by numeric values, we find
1408
+ a > 826 µm
1409
+
1410
+ nH
1411
+ 1010 cm−3
1412
+ � 1
1413
+ 2 � T
1414
+ 10 K
1415
+ � 1
1416
+ 2
1417
+ .
1418
+ (C.7)
1419
+ This limit is much higher than the one derived in paper I, and we
1420
+ do not reach it in our simulations. The value of γgrain we use is
1421
+ taken from Ormel et al. (2009), but surface energies of different
1422
+ materials can reach 10 to 100 times this value, resulting in a
1423
+ much higher estimate of the fragmentation threshold (as Ebr ∼
1424
+ γ5/3
1425
+ grain). The value derived in equation (C.7) should then be taken
1426
+ as very conservative.
1427
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