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1
+ arXiv:2301.01690v1 [cs.LO] 4 Jan 2023
2
+ Proofs as stateful programs: A first-order logic with abstract Hoare
3
+ triples, and an interpretation into an imperative language
4
+ Thomas Powell
5
+ January 5, 2023
6
+ Abstract
7
+ We introduce an extension of first-order logic that comes equipped with additional predicates for
8
+ reasoning about an abstract state. Sequents in the logic comprise a main formula together with pre-
9
+ and postconditions in the style of Hoare logic, and the axioms and rules of the logic ensure that the
10
+ assertions about the state compose in the correct way. The main result of the paper is a realizability
11
+ interpretation of our logic that extracts programs into a mixed functional/imperative language.
12
+ All
13
+ programs expressible in this language act on the state in a sequential manner, and we make this intuition
14
+ precise by interpreting them in a semantic metatheory using the state monad. Our basic framework is
15
+ very general, and our intention is that it can be instantiated and extended in a variety of different ways.
16
+ We outline in detail one such extension: A monadic version of Heyting arithmetic with a wellfounded
17
+ while rule, and conclude by outlining several other directions for future work.
18
+ 1
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+ Introduction
20
+ The Curry-Howard correspondence lies at the heart of theoretical computer science.
21
+ Over the years, a
22
+ multitude of different techniques for extracting programs from proofs have been developed, the majority
23
+ of which translate formal proof systems into lambda calculi. As such, programs extracted from proofs are
24
+ typically conceived as pure functional programs.
25
+ Everyday programmers, on the other hand, often think and write in an imperative paradigm, in terms of
26
+ instructions that change some underlying global state. This is reinforced by the fact that many of the most
27
+ popular programming languages, including C and Python, lean towards this style. Imperative programs are
28
+ nevertheless highly complex from a mathematical perspective, and while systems such as Hoare logic [15] or
29
+ separation logic [27] have been designed to reason about them, the formal extraction of imperative programs
30
+ from proofs has received comparatively little attention.
31
+ In this paper, we propose a new idea in this direction, developing a formal system SL that enriches
32
+ ordinary first-order logic with Hoare triples for reasoning about an abstract global state. Sequents will have
33
+ the form Γ ⊢ {α · A · β}, where A is a formula and α, β assertions about the state, and proofs in the logic
34
+ will include both ordinary introduction and elimination rules for predicate logic, together with special rules
35
+ for reasoning about the state. We then construct a stateful realizability interpretation (based on Kreisel’s
36
+ modified realizability [17]) that relates formulas in SL to terms in a mixed functional/imperative language
37
+ ST. Our main result is a soundness theorem, which confirms that whenever a formula is provable in SL, we
38
+ can extract a corresponding stateful realizing term in ST. While our initial soundness theorem focuses on
39
+ pure predicate logic, we subsequently show that it can be extended to arithmetic, where in particular we are
40
+ then able to extract programs that contain both recursion and controlled while loops.
41
+ We are not the first to adapt traditional methods to extract imperative programs: A major achievement
42
+ in this direction, for example, is the monograph [22], which sets up a variant of intuitionistic Hoare logic
43
+ alongside a realizability translation into a standard imperative language. Other relevant examples include
44
+ [3, 8, 10, 13, 19, 32]. However, these and almost all other prior work in this direction tend to focus on
45
+ formal verification, with an eye towards using proof interpretations as a method for the synthesis of correct-
46
+ by-construction software. In concrete terms, this means that the formal systems tend to be quite detailed
47
+ 1
48
+
49
+ and oriented towards program analysis, while the starting point is typically a program for which we want to
50
+ construct a verification proof, rather than a proof from which we hope to extract a potentially unfamiliar
51
+ program.
52
+ Our approach, on the other hand, is much more abstract, with an emphasis on potential applications in
53
+ logic and proof theory. Our basic system SL makes almost no assumptions about the structure of the state
54
+ and what we are allowed to do with it. Rather, we focus on producing a general framework for reasoning
55
+ about ‘stateful formulas’, which can then be instantiated with additional axioms to model concrete scenarios.
56
+ The simplicity and generality of our framework is its most important feature, and we consider this work to
57
+ be a first step towards a number of potentially interesting applications. For this reason, we include not only
58
+ an extension of our system to a monadic theory of arithmetic, but conclude by sketching out some additional
59
+ ways in which we conjecture that our logic and interpretation could be used and expanded, including the
60
+ computational semantics of proofs and probabilistic logic.
61
+ We take ideas from three main sources.
62
+ The first is a case study of Berger et al.
63
+ [6], in which a
64
+ realizability interpretation is used to extract a version of in-place quicksort, and where the imperative nature
65
+ of the extracted program is presented in a semantic way using the state monad.
66
+ While their program
67
+ behaves imperatively “by-chance”, terms extracted from our logic are forced to be imperative, and thus
68
+ our framework offers one potential solution to their open problem of designing a proof calculus which only
69
+ yields imperative programs. Indeed, an implementation of the insert sort algorithm is formally extracted
70
+ in Section 6 below. Our second source of inspiration is the thesis of Birolo [7], where a general monadic
71
+ realizability interpretation is defined and then used to give an alternative, semantic presentation of learning-
72
+ based interactive realizability [2, 4]. However, our work goes beyond this in that it also involves a monadic
73
+ extension of the target logic, whereas Birolo’s applies to standard first-order logic. Finally, a number of
74
+ ideas are taken from the author’s previous work [24] on extracting stateful programs using the Dialectica
75
+ interpretation. While there the state is used in a very specific and restricted way, unlike our more general
76
+ presentation here, we use an analogous call-by-value monadic translation on terms.
77
+ It is important to stress that we do not claim that our work represents an optimal or complete method
78
+ for extracting imperative programs from proofs, nor do we claim that it is superior to alternative methods,
79
+ including the aforementioned works in the direction of verification, or, for instance, techniques based on
80
+ Krivine’s classical realizability [18], which could be viewed as imperative in nature. We simply offer what
81
+ we consider to be a new and interesting perspective that emphasises abstraction and simplicity, and propose
82
+ that our framework could prove valuable in a number of different contexts.
83
+ Overview of the paper
84
+ The main technical work that follows involves the design of three different systems, a realizability interpre-
85
+ tation that connects them, and an instantiation of this framework in the setting of first-order arithmetic,
86
+ namely:
87
+ • A novel extension SL of predicate logic with abstract Hoare triples, which can be extended with
88
+ additional axioms for characterising the state (Section 2).
89
+ • A standard calculus ST for lambda terms with imperative commands, which can again be extended
90
+ with additional constants for interacting with the state (Section 3).
91
+ • A metalanguage Sω into which both SL and ST can be embedded (Section 4), which is used to formulate
92
+ the realizability relation and prove its soundness (Section 5).
93
+ • An instantiation of SL as a theory of arithmetic, with programs extracted into an extension of ST with
94
+ recursion and while loops (Section 6).
95
+ Concrete examples are given, and potential applications surveyed in Section 7.
96
+ 2
97
+
98
+ 2
99
+ The system SL: First-order logic with state
100
+ We begin by introducing our target theory SL from which stateful programs will be extracted. This is an
101
+ extension of ordinary first-order logic in the sense that the latter can always be embedded into SL (we will
102
+ make this precise in Proposition 2.1 below). Ultimately, we are interested not so much in SL on its own, but
103
+ in theories of the form SL + ∆H + ∆S, where ∆H and ∆S are collections of (respectively non-computational
104
+ and computational) axioms that together characterise the state. Several concrete examples will be given to
105
+ illustrate this, and in Section 6 we present a variant of SL that represents a theory of first-order arithmetic
106
+ with state.
107
+ 2.1
108
+ Intuitionistic first-order logic
109
+ Before defining SL, we give a standard presentation of first-order intuitionistic predicate logic PL, which
110
+ serves as an opportunity to fix our basic style of formal reasoning. The language of PL consists of the
111
+ logical constants ∧, ∨, ⇒, ∀, ∃, ⊤, ⊥, variables x, y, z, . . ., along with function symbols f, g, h, . . . and predicate
112
+ symbols P, Q, R, . . ., each with a fixed arity. We assume the existence of at least one constant c. Terms are
113
+ built from variables and function symbols as usual, and formulas are built from prime formulas P(t1, . . . , tn),
114
+ ⊤ and ⊥ using the logical constants. We use the usual abbreviation ¬A :≡ A ⇒ ⊥. We work in a sequent
115
+ style natural deduction calculus, where sequents have the form Γ ⊢I A for some context Γ and formula A,
116
+ and a context is a set of labelled assumptions of the form Au1
117
+ 1 , . . . , Aun
118
+ n
119
+ for pairwise distinct labels ui. The
120
+ axioms and rules of PL are as in Figure 1.
121
+ Figure 1: Axioms and rules of PL
122
+ Propositional logic
123
+ Γ ⊢I A
124
+ if Au ∈ Γ for some u
125
+ Γ ⊢I ⊤
126
+ Γ ⊢I A
127
+ Γ ⊢I B
128
+ ∧I
129
+ Γ ⊢I A ∧ B
130
+ Γ ⊢I A ∧ B
131
+ ∧EL
132
+ Γ ⊢I A
133
+ Γ ⊢I A ∧ B
134
+ ∧ER
135
+ Γ ⊢I B
136
+ Γ ⊢I A
137
+ ∨IL
138
+ Γ ⊢I A ∨ B
139
+ Γ ⊢I B
140
+ ∨IR
141
+ Γ ⊢I A ∨ B
142
+ Γ ⊢I A ∨ B
143
+ Γ, Au ⊢I C
144
+ Γ, Bv ⊢I C
145
+ ∨E
146
+ Γ ⊢I C
147
+ Γ, Au ⊢I B
148
+ ⇒I
149
+ Γ ⊢I A ⇒ B
150
+ Γ ⊢I A ⇒ B
151
+ Γ ⊢I A
152
+ ⇒E
153
+ Γ ⊢I B
154
+ Γ ⊢I ⊥
155
+ ⊥E
156
+ Γ ⊢I A
157
+ Quantifier rules
158
+ Γ ⊢I A[y/x]
159
+ ∀I
160
+ Γ ⊢I ∀xA
161
+ Γ ⊢I ∀xA
162
+ ∀E
163
+ Γ ⊢I A[t/x]
164
+ Γ ⊢I A[t/x]
165
+ ∃I
166
+ Γ ⊢I ∃xA
167
+ Γ ⊢I ∃xA
168
+ Γ, A[y/x]u ⊢I C
169
+ ∃E
170
+ Γ ⊢I C
171
+ for ∀I, y ≡ x or y not free in A, and y not free in Γ
172
+ for ∃E, y ≡ x or y not free in A, and y not free in C or Γ.
173
+ 3
174
+
175
+ 2.2
176
+ Stateful first-order logic
177
+ We now define our new logical system SL, which is an extension of ordinary first-order logic with new state
178
+ propositions. To be more precise, we extend the language of PL with a ternary operation {−·−·−}, together
179
+ with special state predicate symbols p, q, r, . . ., which also have a fixed arity. Terms of SL are the same as
180
+ those of PL. On the other hand, there are two kinds of formulas in SL: state formulas and main formulas.
181
+ A state formula is defined using state predicate symbols and propositional connectives as follows:
182
+ • ⊤ and ⊥ are state formulas,
183
+ • if p a state predicate symbol of arity n and t1, . . . , tn are terms, then p(t1, . . . , tn) is a state formula,
184
+ • if α, β are state formulas, so are α ∧ β, α ∨ β, α ⇒ β.
185
+ A main formula (or just formula) of SL is now defined as:
186
+ • ⊤ and ⊥ are formulas,
187
+ • if P is an ordinary predicate symbol of arity n and t1, . . . , tn are terms, then P(t1, . . . , tn) is a formula,
188
+ • if A, B are formulas, so are A ∧ B, A ∨ B and ∃xA,
189
+ • if A, B are formulas and α, β state formulas, then A ⇒ {α · B · β} and ∀x {α · A · β} are formulas.
190
+ The notions of free and bound variables, along with substitution α[t/x] and A[t/x] can be easily defined for
191
+ both state and main formulas.
192
+ Analogous to the construction of formulas, our basic proof system uses the auxiliary notion of a state
193
+ proof in order to define a main proof. A state sequent has the form Γ ⊢H α where α is a state formula and
194
+ Γ a set of labelled state formulas. A proof of Γ ⊢H α in SL is built from the axioms and rules of classical
195
+ propositional logic i.e. the propositional axioms and rules as set out in Figure 1 plus the law of excluded
196
+ middle Γ ⊢H α ∨ ¬α, together with a set ∆H of as yet unspecified state axioms of the form Γ ⊢H α.
197
+ A main sequent of SL has the form Γ ⊢S {α · A · β}, where A is a formula and α, β state formulas, and
198
+ Γ is a set of labelled main formulas. A proof of Γ ⊢S {α · A · β} in SL uses the axioms and rules given in
199
+ Figure 2, together with a set ∆S of additional axioms.
200
+ We now make precise what we mean when we characterise SL as an extension of standard first-order
201
+ logic. The following is provable with an easy induction over derivations in PL:
202
+ Proposition 2.1. For any formula A of PL and state formula α, define the main formula Aα of SL by
203
+ • Qα := Q for Q atomic,
204
+ • (A ∧ B)α := Aα ∧ Bα, (A ∨ B)α := Aα ∨ Bα and (∃x A)α := ∃x Aα,
205
+ • (A ⇒ B)α := Aα ⇒ {α · Bα · α} and (∀x A)α := ∀x {α · Aα · α}.
206
+ Then whenever Γ ⊢I A is provable in PL, we have that Γα, ∆ ⊢S {α · Aα · α} is provable in SL, where ∆ is
207
+ arbitrary and Γα := (A1)u1
208
+ α , . . . , (An)un
209
+ α
210
+ for Γ := Au1
211
+ 1 , . . . , Aun
212
+ n .
213
+ 2.3
214
+ The intuition behind SL
215
+ The intended semantic meaning of Γ ⊢H α is that α can be inferred from the assumptions Γ for any fixed state.
216
+ More specifically, if we imagine a semantic variant [α](π) of each state formula where now the dependency
217
+ on an underlying state π is made explicit, the semantics of Γ ⊢H α is just
218
+ [Γ](π) ⇒ [α](π)
219
+ 4
220
+
221
+ Figure 2: Axioms and rules of SL
222
+ Propositional axioms and rules
223
+ Γ ⊢S {α · A · α}
224
+ if Au ∈ Γ for some u
225
+ Γ ⊢S {α · ⊤ · α}
226
+ Γ ⊢S {α · A · β}
227
+ Γ ⊢S {β · B · γ}
228
+ ∧SI
229
+ Γ ⊢S {α · A ∧ B · γ}
230
+ Γ ⊢S {α · A ∧ B · β}
231
+ ∧SEL
232
+ Γ ⊢S {α · A · β}
233
+ Γ ⊢S {α · A ∧ B · β}
234
+ ∧SER
235
+ Γ ⊢S {α · B · β}
236
+ Γ ⊢S {α · A · β}
237
+ ∨SIL
238
+ Γ ⊢S {α · A ∨ B · β}
239
+ Γ ⊢S {α · B · β}
240
+ ∨SIR
241
+ Γ ⊢S {α · A ∨ B · β}
242
+ Γ ⊢S {α · A ∨ B · β}
243
+ Γ, Au ⊢S {β · C · γ}
244
+ Γ, Bv ⊢S {β · C · γ}
245
+ ∨SE
246
+ Γ ⊢S {α · C · γ}
247
+ Γ, Au ⊢S {α · B · β}
248
+ ⇒SI
249
+ Γ ⊢S {γ · A ⇒ {α · B · β} · γ}
250
+ Γ ⊢S {α · A ⇒ {γ · B · δ} · β}
251
+ Γ ⊢S {β · A · γ}
252
+ ⇒SE
253
+ Γ ⊢S {α · B · δ}
254
+ Γ ⊢S {α · ⊥ · β}
255
+ ⊥SE
256
+ Γ ⊢S {α · A · γ}
257
+ Quantifier rules
258
+ Γ ⊢S {α[y/x] · A[y/x] · β[y/x]}
259
+ ∀SI
260
+ Γ ⊢S {γ · ∀x {α · A · β} · γ}
261
+ Γ ⊢S {α · ∀x {β · A · γ} · β[t/x]}
262
+ ∀SE
263
+ Γ ⊢S {α · A[t/x] · γ[t/x]}
264
+ Γ ⊢S {α · A[t/x] · β}
265
+ ∃SI
266
+ Γ ⊢S {α · ∃xA · β}
267
+ Γ ⊢S {α · ∃xA · β}
268
+ Γ, A[y/x]u ⊢S {β · C · γ}
269
+ ∃SE
270
+ Γ ⊢S {α · C · γ}
271
+ for ∀SI, y ≡ x or y not free in A, α, β, and y not free in Γ
272
+ for ∃SE, y ≡ x or y not free in A, and y not free in C, α, β, γ or Γ.
273
+ Basic Hoare rules
274
+ α ⊢H β
275
+ Γ ⊢S {β · A · γ}
276
+ γ ⊢H δ
277
+ cons
278
+ Γ ⊢S {α · A · δ}
279
+ ⊢H α ∨ β
280
+ Γ ⊢S {α ∧ γ �� A · δ}
281
+ Γ ⊢S {β ∧ γ · A · δ}
282
+ cond
283
+ Γ ⊢S {γ · A · δ}
284
+ Additional axioms
285
+ state axioms ∆H of the form Γ ⊢H α
286
+ main axioms ∆S of the form Γ ⊢S {α · A · β}
287
+ 5
288
+
289
+ On the other hand, the intended meaning of Γ ⊢S {α · A · β} is that from assumptions Γ, if α holds with
290
+ respect to some initial state, then we can infer that A is true and β holds with respect to some modified
291
+ state, or more precisely:
292
+ [Γ] ⇒ (∃π [α](π) ⇒ ([A] ∧ ∃π′ [β](π′)))
293
+ (1)
294
+ In particular, the computational interpretation of (1) above will be a program that takes some input state
295
+ π satisfying [α](π) and returns a realizer-state pair ⟨x, π′⟩ such that x realizes A and [β](π′) holds.
296
+ Our semantic interpretation [·] will be properly defined in Section 4. Crucially, in SL the state is implicit,
297
+ and so there are no variables or terms of state type. The state will rather be made explicit in our metatheory
298
+ Sω. The main axioms and rules of SL simply describe how this semantic interpretation propagates in a
299
+ call-by-value manner through the usual axioms and rules of first-order logic. The state itself is brought into
300
+ play through the Hoare rules along with the additional axioms ∆H and ∆S.
301
+ The two Hoare rules of SL correspond to the consequence and conditional rules of traditional Hoare
302
+ logic. The usual conditional rule falls out as a special case of ours since we assume Γ ⊢H α ∨ ¬α. Some of
303
+ the other traditional Hoare rules are derivable: The empty statement axiom corresponds to our own axiom
304
+ Γ ⊢S {α · ⊤ · α} while composition can be viewed as the special case of ∧SI for A = B = ⊤. In Section 6 we
305
+ add a controlled while loop to our logic. But for now, we illustrate our logic with some very straightforward
306
+ scenarios.
307
+ Example 2.2 (Simple read-write). Consider a very simple state, which we imagine to contain just two memory
308
+ locations (input and output), and on which we can perform the following three actions:
309
+ 1. Store any value from our domain of discourse in the input location.
310
+ 2. For the current value x in the input location, compute some y such that P(x, y) holds (where P is a
311
+ fixed binary predicate symbol of the logic), and store it in the output location.
312
+ 3. Retrieve the computed value y from the state’s output location.
313
+ We could formalise those three actions by including two unary state predicates stored and solved, where
314
+ stored(x) denotes that x is currently stored in the input location, and solved(x) denotes that some y satisfying
315
+ P(x, y) is stores in the output location. We would then add the following axioms to ∆S, which intuitively
316
+ represent each of the above actions:
317
+ 1. Γ ⊢S {α · ⊤ · stored(x)} where α ranges over all state formulas,
318
+ 2. Γ ⊢S {stored(x) · ⊤ · solved(x)}
319
+ 3. Γ ⊢S {solved(x) · ∃y P(x, y) · ⊤}
320
+ We can then, for example, derive the following in SL + ∆H + ∆S for ∆H = ∅, where α, β are any state
321
+ formulas:
322
+ ⊢S {β · ∀x {α · ∃y P(x, y) · ⊤} · β}
323
+ An example of such a derivation is, for example:
324
+ ⊢S {α · ⊤ · stored(x)}
325
+ ⊢S {stored(x) · ⊤ · solved(x)}
326
+ ∧SI
327
+ ⊢S {α · ⊤ ∧ ⊤ · solved(x)}
328
+ ∧SEL
329
+ ⊢S {α · ⊤ · solved(x)}
330
+ ⊢S {solved(x) · ∃y P(x, y) · ⊤}
331
+ ∧SI
332
+ ⊢S {α · ⊤ ∧ ∃y P(x, y) · ⊤}
333
+ ∧SEL
334
+ ⊢S {α · ∃y P(x, y) · ⊤}
335
+ ∀SI
336
+ ⊢S {β · ∀x {α · ∃y P(x, y) · ⊤} · β}
337
+ We note that while state formulas and actions are used in the proof, if we set α = β = ⊤ then the components
338
+ of the theorem itself are just formulas in ordinary first-order logic.
339
+ 6
340
+
341
+ Example 2.3 (Fixed-length array sorting). Let us now consider our state as an array of length three, and
342
+ elements in that array as having some order structure.
343
+ We formalise this in SL by introducing 1, 2, 3
344
+ as constants representing our three locations, along with two state predicates: a binary predicate ≤ for
345
+ comparing elements at locations l and l′, and a nullary predicate sorted that declares that the state is sorted.
346
+ These can be characterised by adding the following axiom schemes, but to ∆H rather than ∆S as they do
347
+ not represent state actions:
348
+ Γ ⊢H 1 ≤ 2 ∧ 2 ≤ 3 ⇒ sorted
349
+ Γ ⊢H l ≤ l′ ∨ l′ ≤ l
350
+ where l, l′ range over {1, 2, 3}
351
+ We then allow a single action on our array, namely the swapping of a pair of elements in the list. Suppose
352
+ that α is a state formula of the form
353
+ α :≡ l1 ≤ l′
354
+ 1 ∧ . . . ∧ ln ≤ l′
355
+ n
356
+ (2)
357
+ where li, li range over locations {1, 2, 3}. Now for l, l′ ∈ {1, 2, 3} let α[l ↔ l′] denote α where all instances of
358
+ l and l′ are swapped, so that if e.g. α = 3 ≤ 2 ∧ 1 ≤ 2 ∧ 1 ≤ 3 then
359
+ α[2 ↔ 3] = 2 ≤ 3 ∧ 1 ≤ 3 ∧ 1 ≤ 2
360
+ We axiomatise the swapping of the values in locations of some arbitrary pair l, l′ ∈ {1, 2, 3} by adding to ∆S
361
+ all instances of
362
+ Γ ⊢S {α · ⊤ · α[l ↔ l′]}
363
+ where α ranges over state formulas of the form (2). The statement that all arrays of length three can be
364
+ sorted is then formulated as
365
+ ⊢S {⊤ · ��� · sorted}
366
+ Let us now give a proof of this statement in SL + ∆H + ∆S. First, let α := 1 ≤ 2 ∧ 1 ≤ 3, and define D1 as
367
+ ⊢S {2 ≤ 3 ∧ α · ⊤ · 2 ≤ 3 ∧ α}
368
+ cons
369
+ ⊢S {2 ≤ 3 ∧ α · ⊤ · sorted}
370
+ 2↔3
371
+ ⊢S {3 ≤ 2 ∧ α · ⊤ · 2 ≤ 3 ∧ 1 ≤ 3 ∧ 1 ≤ 2}
372
+ cons
373
+ ⊢S {3 ≤ 2 ∧ α · ⊤ · sorted}
374
+ cond[2≤3∨3≤2]
375
+ ⊢S {α · ⊤ · sorted}
376
+ where for the left instance of cons we use 2 ≤ 3∧α ⊢H sorted, in the right that 2 ≤ 3∧1 ≤ 3∧1 ≤ 2 ⊢H sorted,
377
+ and for the final instance of cond we use ⊢H 2 ≤ 3 ∨ 3 ≤ 2. Now let D2 be defined by
378
+ 1↔2
379
+ ⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · 1 ≤ 2 ∧ 1 ≤ 3}
380
+ D1
381
+ ⊢S {1 ≤ 2 ∧ 1 ≤ 3 · ⊤ · sorted}
382
+ ∧SI
383
+ ⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ ∧ ⊤ · sorted}
384
+ ∧SEL
385
+ ⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · sorted}
386
+ Then we have D3:
387
+ D2
388
+ ⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · sorted}
389
+ {1 ≤ 2 ∧ 2 ≤ 3 · ⊤ · 1 ≤ 2 ∧ 2 ≤ 3}
390
+ cons
391
+ ⊢S {1 ≤ 2 ∧ 2 ≤ 3 · ⊤ · sorted}
392
+ cond[2≤1∨1≤2]
393
+ ⊢S {2 ≤ 3 · ⊤ · sorted}
394
+ where here cond uses ⊢H 2 ≤ 1 ∨ 1 ≤ 2, and finally
395
+ ⊢S {2 ≤ 3 · ⊤ · 2 ≤ 3}
396
+ 2↔3
397
+ ⊢S {3 ≤ 2 · ⊤ · 2 ≤ 3}
398
+ cond[2≤3∨3≤2]
399
+ ⊢S {⊤ · ⊤ · 2 ≤ 3}
400
+ D3
401
+ ⊢S {2 ≤ 3 · ⊤ · sorted}
402
+ ∧SI
403
+ ⊢S {⊤ · ⊤ ∧ ⊤ · sorted}
404
+ ∧SEL
405
+ ⊢S {⊤ · ⊤ · sorted}
406
+ 7
407
+
408
+ In contrast to Example 2.2 above, this is an example of a purely imperative proof that involves no proposi-
409
+ tional formulas other than ⊤. As we will see in Example 5.6, the proof corresponds to a purely imperative
410
+ program.
411
+ 3
412
+ The system ST: A simple functional/imperative term calculus
413
+ We now define our calculus ST + ΛS whose terms will represent realizers for proofs in SL + ∆H + ∆S. This
414
+ is a standard typed lambda calculus for mixed functional and imperative programs, and is defined to include
415
+ basic terms together with additional constants in some set ΛS, where the latter are intuitively there to realize
416
+ the axioms in ∆S. Semantics for the terms will be given via a monadic translation into the metalanguage
417
+ defined in the next section. Types are defined by the grammmar
418
+ X ::= D | C | X × X | X + X | X → X
419
+ while basic terms are defined as
420
+ e ::= skip | defaultX | c | f | x | p0(e) | p1(e) | e ◦ e | ι0(e) | ι1(e) | elim e e e | λx.e | e e | if α then e else e
421
+ where f ranges over all function symbols of SL, c are constants in ΛS, and α ranges over state formulas of
422
+ SL. Typing derivations of the form Γ ⊢ t : X are given below, where Γ is a set of typed variables. Note that
423
+ the types of constants c ∈ ΛS are also left unspecified.
424
+ Γ ⊢ f : Dn → D
425
+ where f has arity n
426
+ Γ ⊢ c : X
427
+ Γ ⊢ x : X
428
+ if x : X in Γ
429
+ Γ ⊢ skip : C
430
+ Γ ⊢ s : X
431
+ Γ ⊢ t : Y
432
+ Γ ⊢ s ◦ t : X × Y
433
+ Γ ⊢ t : X × Y
434
+ Γ ⊢ p0(t) : X
435
+ Γ ⊢ t : X × Y
436
+ Γ ⊢ p1(t) : Y
437
+ Γ ⊢ t : X
438
+ Γ ⊢ ι0(t) : X + Y
439
+ Γ ⊢ t : Y
440
+ Γ ⊢ ι1(t) : X + Y
441
+ Γ ⊢ r : X + Y
442
+ Γ ⊢ s : X → Z
443
+ Γ ⊢ t : Y → Z
444
+ Γ ⊢ elim r s t ⊢ Z
445
+ Γ, x : X ⊢ t : Y
446
+ Γ ⊢ λx.t : X → Y
447
+ Γ ⊢ t : X → Y
448
+ Γ ⊢ s : X
449
+ Γ ⊢ ts : Y
450
+ Γ ⊢ defaultX : X
451
+ Γ ⊢ s : X
452
+ Γ ⊢ t : X
453
+ x : D ∈ Γ for all free variables of α
454
+ Γ ⊢ if α then s else t : X
455
+ The type C should be interpreted as a type of commands that act on the state but don’t return any
456
+ values. It is be helpful to consider a derived operator for sequential composition:
457
+ Definition 3.1. If Γ ⊢ s : C and Γ ⊢ t : X then Γ ⊢ s ∗ t := p1(s ◦ t) : X. In particular, if Γ ⊢ t : C then
458
+ Γ ⊢ s ∗ t : C.
459
+ 4
460
+ A monadic embedding of SL and ST into a metatheory Sω
461
+ We now give a semantic interpretation of both state formulas of SL + ∆H + ∆S and terms in ST + ΛS into
462
+ a standard higher-order, many sorted logic Sω + ΛSω.
463
+ 4.1
464
+ The system Sω
465
+ This logic contains typed lambda terms along with equational axioms for reasoning about them, together
466
+ with the usual axioms and rules of many-sorted predicate logic.
467
+ Because most aspects of the logic are
468
+ completely standard, and in any case it is purely a verifying system, we are less detailed in specifying it.
469
+ Types are defined as follows:
470
+ X ::= D | 1 | Bool | S | X × X | X → X
471
+ 8
472
+
473
+ where D represents objects in the domain of SL (just as in ST), Bool a type of booleans, and states are
474
+ now explicitly represented as objects of type S. Our metatheory is an equational calculus, with an equality
475
+ symbol =X for all types. Typed terms include:
476
+ • variables x, y, z, . . . for each type, where we denote state variables by π, π1, π2, . . .
477
+ • a constant f : Dn → D for each n-ary function symbol of SL,
478
+ • additional, as yet unspecified constant symbols c : X for interpreting objects in ΛS, along with axioms
479
+ that characterise them,
480
+ • a unit element () : 1 along with the axiom x = (),
481
+ • boolean constants t and f, with the axiom x =Bool t ∨ x =Bool f,
482
+ • pairing ⟨s, t⟩ and projection proj0(t), proj1(t) operators, with the usual axioms,
483
+ • terms formed by lambda abstraction and application, with the rule (λx.t)s = t[s/x],
484
+ • for each type X a case operator case (b) (s) (t) for b : Bool and s, t : X, with axioms case f x y = x and
485
+ case t x y = y.
486
+ We sometimes write xX instead of x : X, and we use abbreviations such as ⟨x, y, z⟩ for ⟨x, ⟨y, z⟩⟩. Atomic
487
+ formulas of Sω include all ordinary predicate symbols P, Q, R, . . . of SL as atomic formulas, where an n-ary
488
+ predicate P in SL takes arguments of type Dn in Sω, along with predicates p, q, r, . . . for each state predicate
489
+ symbol of SL, but now, if p is an n-ary state predicate in SL, p takes arguments of type Dn × S in Sω.
490
+ General formulas are built using the usual logical connectives, including quantifiers for all types. The axioms
491
+ and rules of Sω include the axioms of rules of predicate logic (now in all finite types), axioms for the terms,
492
+ along with the usual equality axioms (including full extensionality). Because Sω acts as a verifying theory,
493
+ we freely use strong axioms (such as extensionality), without concerning ourselves with the minimal such
494
+ system that works.
495
+ 4.2
496
+ The embedding [·] on state formulas of SL
497
+ The main purpose of our metalanguage is to allow us to reason semantically about SL and ST. To do this,
498
+ we introduce an embedding of state formulas of SL and terms of ST into Sω. We use the same notation [·]
499
+ for both, as there is no danger of ambiguity.
500
+ Definition 4.1. For each term t of SL, there is a natural interpretation of t as a term of type D in ST, namely
501
+ x �→ x : D and f(t1, . . . , tn) �→ f(t1 ◦ · · · ◦ tn) : D. Similarly, there is a natural interpretation of t into Sω,
502
+ this time with f(t1, . . . , tn) �→ f(⟨t1, . . . , tn⟩). We use the same notation for t in each of the three systems,
503
+ as there is no risk of ambiguity.
504
+ Definition 4.2. For each state formula α of SL, we define a formula [α](π) of Sω, whose free variables are
505
+ the same as those of α (but now typed with type D) with the potential addition of a single state variable π,
506
+ as follows:
507
+ • [⊤](π) := ⊤ and [⊥](π) := ⊥,
508
+ • [p(t1, . . . , tn)](π) := p(t1, . . . , tn, π),
509
+ • [α ∧ β](π) := [α](π) ∧ [β](π), and similarly for α ∨ β and α ⇒ β.
510
+ The following Lemma is easily proven using induction over propositional derivations.
511
+ Lemma 4.3. If Γ ⊢H α in SL then [α](π) is provable in Sω from the assumptions [Γ](π), where [Γ](π) :=
512
+ [α1](π), . . . , [αn](π) for Γ := α1, . . . , αn. This extends to proofs in SL + ∆H provided that the embedding of
513
+ any axiom in ∆H is provable in Sω + ΛSω.
514
+ 9
515
+
516
+ We are now in a position to make the semantic meaning of main formulas of SL precise. While, technically
517
+ speaking, this is not necessary in what follows, neither to formulate our realizability interpretation nor to
518
+ prove our soundness theorem, for the sake of completeness we give the full definition.
519
+ Definition 4.4. For each main formula A of , we define a formula [A] of Sω, whose free variables are the
520
+ same as those of A (but now typed with type D), as follows:
521
+ • [⊤] := ⊤ and [⊥] := ⊥,
522
+ • [P(t1, . . . , tn)] := P(t1, . . . , tn),
523
+ • [A ∧ B] := [A] ∧ [B], [A ∨ B] := [A] ∨ [B] and [∃x A] := ∃xD [A],
524
+ • [A ⇒ {α · B · β}] := [A] ⇒ [{α · B · β}] and [∀x {α · A · β}] := ∀xD [{α · A · β}]
525
+ where [{α · A · β}] := ∃πS [α](π) ⇒ [A] ∧ ∃π′ [β](π′).
526
+ Similarly to Lemma 4.3, we can now prove the following by induction over derivations in SL. We omit
527
+ the proof, because it is straightforward and in any case not necessary in what follows.
528
+ Proposition 4.5. If Γ ⊢S {α · A · β} in SL then [{α · A · β}] is provable in Sω from the assumptions [Γ],
529
+ where [Γ] := [A1], . . . , [An] for Γ := A1, . . . , An. This extends to proofs in SL + ∆H + ∆S provided that the
530
+ embedding of any axiom in ∆H and ∆S is provable in Sω + ΛSω.
531
+ 4.3
532
+ The embedding [·] on terms of ST
533
+ Our translation on terms is a call-by-value monadic translation using the state monad S → X × S. We first
534
+ define a translation on types of ST as follows:
535
+ • [D] := D, [C] := 1 and [X × Y ] := [X] × [Y ],
536
+ • [X + Y ] := Bool × [X] × [Y ]
537
+ • [X → Y ] := [X] → S → [Y ] × S
538
+ Lemma 4.6. For any type X of SL, the type [X] is inhabited, in the sense that we can define a canonical
539
+ closed term 0X : [X].
540
+ Proof. Induction on types, letting 0D := c for a constant symbol which is assumed to exist in SL. The only
541
+ other nonstandard case is 0X→Y , which can be defined as λx, π . ⟨0Y , π⟩.
542
+ Finally, before introducing our translation on terms, we need to add characteristic functions to Sω
543
+ for all state formulas (analogous to the characteristic functions for quantifier-free formulas in [14]). For
544
+ any state formula α[x1, . . . , xn] of SL, where x1, . . . , xn are the free variables of α, we introduce constants
545
+ χα :Dn → S → X → X → X satisfying the axioms
546
+ [x1, . . . , xn](π) ⇒ χα ⟨x1, . . . , xn⟩ π y z = y
547
+ [¬α][x1, . . . , xn](π) ⇒ χα ⟨x1, . . . , xn⟩ π y z = z
548
+ Definition 4.7. For each term Γ ⊢ t : X of ST we define a term [Γ] ⊢ [t] : S → [X] × S of Sω as follows,
549
+ where [·] is defined on contexts as [x1 : X1, . . . , xn : Xn] := x1 : [X1], . . . , xn : [Xn]:
550
+ • [x]π := ⟨x, π⟩,
551
+ • [skip]π := ⟨(), π⟩,
552
+ • [f]π := ⟨λxDn, π . ⟨fx, π⟩, π⟩,
553
+ • [c]π is appropriately defined for each additional constant in ΛS,
554
+ 10
555
+
556
+ • [s ◦ t]π := ⟨a, b, π2⟩ where ⟨a, π1⟩ := [s]π and ⟨b, π2⟩ := [t]π1,
557
+ • [p0t]π := ⟨a, π1⟩ and [p1t]π := ⟨b, π1⟩ where ⟨a, b, π1⟩ := [t]π,
558
+ • [ι0t]π := ⟨f, a, 0Y , π1⟩ and [ι1t] := ⟨t, 0X, b, π1⟩ for ⟨a, π1⟩ := [t]π,
559
+ • [elim r s t]π := case e (faπ2) (gbπ3) for ⟨e, a, b, π1⟩ := [r]π, ⟨f, π2⟩ := [s]π1, ⟨g, π3⟩ := [t]π1,
560
+ • [λx.t]π := ⟨λx[X].[t], π⟩,
561
+ • [ts]π := faπ2 for ⟨f, π1⟩ := [t]π and ⟨a, π2⟩ := [s]π1,
562
+ • [defaultX]π := ⟨0X, π⟩,
563
+ • [if α[x1, . . . , xn] then s else t]π := χα ⟨x1, . . . , xn⟩ π ([s]π) ([t]π) where {x1, . . . , xn} are the free variables
564
+ of α.
565
+ The following lemmas will be useful when verifying our realizability interpretation in the next section.
566
+ The first is by a simple induction on terms.
567
+ Lemma 4.8. For any term t of SL, we have [t]π = ⟨t, π⟩ (cf. Definitions 4.1 and 4.7).
568
+ Lemma 4.9 (Currying in ST). Suppose that Γ, x : X, y : Y ⊢ t : Z is a term in ST, and define Γ ⊢ λ∗v.t :
569
+ X × Y → Z by λ∗v.t := λv.(λx, y.t)(p0v)(p1v) where v is not free in t. Then for any s : X × Y we have
570
+ [(λ∗v.t)s]π = [t][a/x, b/y]π1
571
+ where ⟨a, b, π1⟩ := [s]π.
572
+ Proof. By unwinding the definition of [·]. For any variable v : X × Y we have [p0v]π = ⟨proj0v, π⟩ and
573
+ [p1v]π = ⟨proj1v, π⟩, and we also have [λx, y . t]π = ⟨λx, π.⟨λy.[t], π⟩, π⟩. We therefore calculate
574
+ [(λx, y.t)(p0v)]π = (λx, π.⟨λy.[t], π⟩)(proj0v)π = ⟨λy.[t][proj0v/x], π⟩
575
+ and thus
576
+ [(λx, y.t)(p0v)(p1v)]π = (λy.[t][proj0v/x])(proj1v)π = [t][proj0v/x, proj1v/y]π
577
+ Finally, we can see that if ⟨a, b, π1⟩ := [s]π then
578
+ π = (λv.[(λx, y.t)(p0v)(p1v)])(⟨a, b⟩)π1
579
+ = (λv.[t][proj0v/x, proj1v/y])(⟨a, b⟩)π1
580
+ = [t][proj0v/x, proj1v/y][⟨a, b⟩/v]π1
581
+ = [t][a/x, b/y]π1
582
+ which completes the proof.
583
+ 5
584
+ A realizability interpretation of SL into ST
585
+ We now come to the main contribution of the paper, which is the definition of a realizability relation between
586
+ terms of ST and formulas of SL, along with a soundness theorem that shows us how to extract realizers from
587
+ proofs. Our metatheory Sω is used to define the realizability relation and prove the soundness theorem.
588
+ Definition 5.1 (Types of realizers). To each main formula A of SL we assign a type τS(A) of ST as follows:
589
+ • τS(⊤) = τS(⊥) = τS(P(t1, . . . , tn)) := C,
590
+ • τS(A ∧ B) := τS(A) × τS(B),
591
+ 11
592
+
593
+ • τS(A ∨ B) := τS(A) + τS(B),
594
+ • τS(∃x A) := D × τS(A),
595
+ • τS(A ⇒ {α · B · β}) := τS(A) → τS(B),
596
+ • τS(∀x {α · A · β}) := D → τS(A).
597
+ Definition 5.2 (Realizability relation). For each main formula A of SL we define a formula x sr A of Sω,
598
+ whose free variables are contained in those of A (now typed with type D) together with a fresh variable
599
+ x : [τS(A)], by induction on the structure of A as follows:
600
+ • x sr Q := Q for Q = ⊤, ⊥ or P(t1, . . . , tn),
601
+ • x sr A ∧ B := (proj0x sr A) ∧ (proj1x sr B),
602
+ • x sr A ∨ B := (proj0x = f ⇒ proj0(proj1x) sr A) ∧ (proj0x = t ⇒ proj1(proj1x) sr B),
603
+ • x sr ∃y A(y) := (proj1x sr A)[proj0x/y],
604
+ • f sr (A ⇒ {α · B · β}) := ∀x[τS(A)] (x sr A ⇒ fx sr {α · B · β}),
605
+ • f sr (∀x {α(x) · A(x) · β(x)}) := ∀xD (fx sr {α(x) · A(x) · β(x)}),
606
+ where for x : S → [τS(A)] × S we define
607
+ • x sr {α · A · β} := ∀πS ([α](π) ⇒ proj0(xπ) sr A ∧ [β](proj1(xπ))).
608
+ The following substitution lemma is easily proven by induction on formulas of SL.
609
+ Lemma 5.3. For any term t of SL and s : [τS(A)] we have s sr A[t/x] = (s sr A)[t/x], where x is not free
610
+ in s and on the right hand side we implicitly mean the natural interpretation of t in Sω (cf. Definition 4.1).
611
+ Theorem 5.4 (Soundness). Suppose that
612
+ Γ := Au1
613
+ 1 , . . . , Aun
614
+ n ⊢S {α · A · β}
615
+ is provable in SL. Then we can extract from the proof a term ∆, τS(Γ) ⊢ t : τS(A) of ST, where ∆ contains
616
+ the free variables of Γ and {α · A · β} (typed with type D) and τS(Γ) := x1 : τS(A1), . . . , xn : τS(An) for fresh
617
+ variables x1, . . . , xn, such that the formula
618
+ [t] sr {α · A · β}
619
+ is provable in Sω from the assumptions (x1 sr A1)u1, . . . , (xn sr An)un for xi : [τS(Ai)]. The theorem holds
620
+ more generally for proofs in SL + ∆H + ∆S, now provably in Sω + ΛSω, if:
621
+ • for any axiom Γ ⊢H α in ∆H, the corresponding axiom [Γ](π) ⇒ [α](π) is added to ΛSω,
622
+ • for any axiom in ∆S there is a term t of ST+ΛS such that [t] realizes that axiom provably in Sω +ΛSω.
623
+ Proof. Induction on the structure of derivations in SL.
624
+ In all cases, we assume as global assumptions
625
+ (x1 sr A1)u1, . . . , (xn sr An)un, and our aim is then to produce a term t such that if [α](π) holds for some
626
+ state variable π, then a sr A and [β](π1) hold for ⟨a, π1⟩ := [t]π.
627
+ • For the axiom Γ ⊢S {α · A · α}, if Au ∈ Γ we define t := x for the corresponding variable x : τS(A).
628
+ Then [x]π := ⟨x, π⟩ for x sr A and [α](π). For Γ ⊢S {α · ⊤ · α} we define t := skip and the verification
629
+ is even simpler.
630
+ 12
631
+
632
+ • (∧SI) Given terms s, t with [s] sr {α · A · β} and [t] sr {β · B · γ}, from [α](π) we can infer a sr A
633
+ and [β](π1) for ⟨a, π1⟩ := [s]π, and from [β](π1) it follows that b sr B and [γ](π2) for ⟨b, π2⟩ := [t]π1,
634
+ therefore we have shown that [s ◦ t] sr {α · A ∧ B · γ}.
635
+ • (∧SEi) If [t] sr {α · A ∧ B · β} then ⟨a, b⟩ sr A ∧ B and [β](π1) follow from [α](π), where ⟨a, b, π1⟩ :=
636
+ [t]π. But then [p0t] sr {α · A · β} and [p1t] sr {α · B · β}.
637
+ • (∨SIi) If [t] sr {α · A · β} and [α](π) holds, then a sr A and [β](π1) for ⟨a, π1⟩ := [t]π, and therefore
638
+ (b = f ⇒ a sr A) ∧ (b = t ⇒ 0τS(B) sr B)
639
+ for b := f. Thus [ι0t] sr A ∨ B. By an entirely analogous argument we can show that [ι1t] sr A ∨ B
640
+ whenever [t] sr B.
641
+ • (∨SE) Suppose that r, s(x) and t(y) are such that [r] sr {α · A ∨ B · β}, [s](x) sr {β · C · γ} assuming
642
+ x sr A, and [t](y) sr {β · C · γ} assuming y sr B. We claim that
643
+ [elim r (λx.s) (λy.t)] sr {α · C · γ}
644
+ To prove this, first note that if [α](π), we have ⟨e, a, b⟩ sr A ∨ B and [β](π1) for ⟨e, a, b, π1⟩ := [r]π.
645
+ There are now two possibilities. If e = f then
646
+ elim r (λx.s) (λy.t)tπ = faπ2
647
+ for ⟨f, π2⟩ := [λx.s]π1 = ⟨λx.[s](x), π1⟩
648
+ = (λx.[s](x))aπ1
649
+ = [s](a)π1
650
+ But since [β](π1) holds and e = f also implies that a sr A, we have c sr C and [γ](π2) for ⟨c, π2⟩ :=
651
+ [s](a)π1, which proves the main claim in the case e = f. An analogous argument works for the case
652
+ e = t.
653
+ • (⇒S I) If t(x) is such that [t](x) sr {α · B · β} whenever x sr A, then by definition we have
654
+ λx.[t] sr A ⇒ {α · B · β}
655
+ and therefore [λx.t] sr {γ · A ⇒ {α · B · β} · γ} for any γ.
656
+ • (⇒S E) Assume that [s] sr {β · A · γ} and [t] sr {α · A ⇒ {γ · B · δ} · β}. If [α](π) holds then defining
657
+ ⟨f, π1⟩ := [t]π we have [β]π1 and
658
+ x sr A ⇒ fx sr {γ · B · δ}
659
+ Similarly, defining ⟨a, π2⟩ := [s]π1, it follows that [γ](π2) and a sr A. Finally, setting ⟨b, π3⟩ := faπ2 it
660
+ follows that b sr B and [δ](π3), and we have therefore proven that [ts] sr {α · B · δ}.
661
+ • (⊥SE) If [t] sr {α · ⊥ · β} then from [α](π) we can infer a sr ⊥ and [β](π2) for ⟨a, π1⟩ := [t]π. But
662
+ a sr ⊥ = ⊥, and from ⊥ we can deduce anything, and in particular 0τS(A) sr A and [γ](π), from which
663
+ it follows that [defaultτS(A)] sr {α · A · γ}.
664
+ • (∀SI) Suppose that t(x) is such that [t](y) sr {α[y/x] · A[y/x] · β[y/x]}, where y ≡ x or y is not free in
665
+ {α · A · β}, and y is not free in Γ. Then since y is not free in any of the assumptions xi sr Ai, we can
666
+ deduce in Sω that
667
+ ∀xD [t](x) sr {α · A · β}
668
+ and therefore λx.[t] sr ∀x {α · A · β}, and thus (just as for ⇒S I) we have
669
+ [λx.t] sr {γ · ∀x {α · A · β} · γ}
670
+ for any γ.
671
+ 13
672
+
673
+ • (∀SE) Suppose that [s] sr {α · ∀x {β · A · γ} · β[t/x]} and that [α](π) holds. Then f sr ∀x {β · A · γ}
674
+ and [β][t/x](π1) for ⟨f, π⟩ := [s]π.
675
+ Now, using Lemma 4.8 we have [st]π = ftπ1 for the natural
676
+ interpretation of t in Sω, since we can prove in Sω that
677
+ ft sr {β[t/x] · A[t/x] · γ[t/x]}
678
+ it follows that a sr A[t/x] and [γ][t/x](π2) for ⟨a, π2⟩ := ftπ1, and therefore we have shown that
679
+ [st] sr {α · A[t/x] · γ[t/x]}.
680
+ • (∃SI) If [s] sr {α · A[t/x] · β} and [α](π) then a sr A[t/x] and [β](π1) for ⟨a, π1⟩ := [s]π. By Lemma
681
+ 5.3 we therefore have (a sr A)[t/x], and therefore ⟨t, a⟩ sr ∃x A. Observing (using Lemma 4.8) that
682
+ [t ◦ s]π = ⟨t, a, π1⟩, we have shown that [t ◦ s] sr {α · ∃x A · β}.
683
+ • (∃SE) Suppose that s and t(x, z) are such that [s] sr {α · ∃x A · β} and
684
+ z sr A[y/x] ⇒ [t](y, z) sr {β · C · γ}
685
+ where y ≡ x or y is not free in A, and y is also not free in C, α, β, γ or Γ. By Lemma 5.3 that
686
+ z sr A[y/x] = (z sr A)[y/x] = ⟨y, z⟩ sr ∃x A we therefore have
687
+ ⟨y, z⟩ sr ∃x A ⇒ [t](y, z) sr {β · C · γ}
688
+ Now, applying Lemma 4.9 to ∆, Γ, y : D, z : τS(A) ⊢ t : τS(C), we have
689
+ [(λ∗v.t)s]π = [t](e, a)π1
690
+ for ⟨e, a, π1⟩ := [s]π. Now, if [α](π) holds, then we have ⟨e, a⟩ sr ∃x A and [β](π1), and therefore since
691
+ [t](e, a) sr {β · C · γ}, we have c sr C and [γ](π2) for ⟨c, π2⟩ = [t](e, a)π1 = [(λ∗v.t)s]π, and thus we
692
+ have shown that [(λ∗v.t)s] sr {α · C · γ}.
693
+ • (cons) If α ⊢H β and γ ⊢H δ then by Lemma 4.3 both [α](π) ⇒ [β](π) and [γ](π) ⇒ [δ](π) are provable
694
+ in Sω (respectively Sω + ΛSω for the general version of the theorem) for any π : S. It is then easy to
695
+ show that if [t] sr {β · A · γ} then we also have [t] sr {α · A · δ}.
696
+ • (cond) Suppose that [s] sr {α ∧ γ · A · δ} and [t] sr {β ∧ γ · A · δ}. We claim that
697
+ [if α then s else t] sr {γ · A · δ}
698
+ To prove this, suppose that [γ](π) holds. Since ⊢H α ∨ β then [α](π) ∨ [β](π) is provable in Sω, and so
699
+ we consider two cases. Let {x1, . . . , xn} be the free variables of α. If [α](π) holds, then
700
+ [if α then s else t]π = χα ⟨x1, . . . , xn⟩ π ([s]π) ([t]π) = [s]π
701
+ and since then [α](π) ∧ [γ](π) we have a sr A and [δ](π1) for ⟨a, π1⟩ := [s]π. On the other hand, if
702
+ [β](π) holds, then by an analogous argument we can show that a sr A and [δ](π1) for ⟨a, π1⟩ := [t]π =
703
+ [if α then s else t]π, and we are done.
704
+ The extension of the soundness theorem to SL + ∆H + ∆S is straightforward, as the soundness proof is
705
+ modular and so any axioms along with their realizers can be added. The first condition is needed so that
706
+ Lemma 4.3 (needed for the cons rule) continues to apply.
707
+ For the free variable condition that the free variables of t are contained in those of Γ, {α · A · β} and
708
+ τS(Γ), if this were not the case, we could simply ground those variables with a canonical constant c : D and
709
+ we would still have ˜t sr {α · A · β} for the resulting term ˜t.
710
+ 14
711
+
712
+ Corollary 5.5 (Program extraction). Suppose that the sentence
713
+ ⊢S {α · ∀x {β · ∃y P(x, y) · γ(x)} · β}
714
+ is provable in SL + ∆S. Then we can extract a closed realizing term t : D → D × C in ST + ΛS such that
715
+ defining g : D → S → D × S by gxπ := ⟨a, π2⟩ for ⟨f, π1⟩ := [t]π and ⟨a, (), π2⟩ := fxπ1, we have
716
+ ∀πS([α](π) ⇒ ∀xD (P(x, proj0(gxπ)) ∧ [γ](x)(proj1(gxπ))))
717
+ provably in Sω + ΛSω.
718
+ 5.1
719
+ Simplification and removal of unit types
720
+ In presentations of modified realizability that use product types instead of type sequences, it is common to
721
+ introduce the notion of a Harrop formula (a formula that does not contain disjunction or existential quantifi-
722
+ cation in a positive position) and define realizability in a way that all Harrop formulas have unit realizability
723
+ type, so that e.g. τS(∀x (P ∧ Q)) = 1 for atomic predicates P and Q, rather than τS(∀x (P ∧ Q)) = D → 1×1
724
+ as for us. We have avoided this simplification earlier on, as it would have added additional cases and bureau-
725
+ cracy to our soundness theorem. However, we can compensate retroactively for this choice by introducing
726
+ equivalences on types that eliminate unit types, namely the closure under contexts of
727
+ 1 × X ≃ 1 ≃ X × 1
728
+ (1 → X) ≃ X
729
+ (X → 1) ≃ 1
730
+ along with corresponding equivalences on terms, also closed under contexts:
731
+ t1×X ≃ proj1(t)X
732
+ tX×1 ≃ proj0(t)X
733
+ t1→X ≃ t()
734
+ tX ≃ λx1.t
735
+ tX→1 ≃ ()
736
+ For example, in Corollary 5.5 we would then have
737
+ [t]π : (D → S → D × 1 × S) × S ≃ (D → S → D × S) × S
738
+ and
739
+ gxπ ≃ fxπ1.
740
+ For us, the equivalence relation ≃ will not play a formal role in the paper, but will be used to provide
741
+ simplified descriptions of extracted programs.
742
+ 5.2
743
+ Examples of program extraction
744
+ We now continue the short illustrative examples we outlined in Section 2.3.
745
+ Example 5.6 (Simple read-write). In Example 2.2 we considered a state where three actions were possible
746
+ (writing to the state, performing a calculation, and reading the output from the state). We can formalise these
747
+ three actions semantically in the metatheory Sω by including three constants in ΛSω, namely c1 : D → S → S,
748
+ c2 : S → S and c3 : S → D, along with the characterising axioms:
749
+ 1. stored(x, c1xπ),
750
+ 2. stored(x, π) ⇒ solved(x, c2π),
751
+ 3. solved(x, π) ⇒ P(x, c3π).
752
+ While we are able to use these constants to form terms in Sω such as λπ, π1, x . ⟨c1xπ, c2π1⟩, which could be
753
+ viewed as non-sequential in the sense that we take two input states as arguments, we can force them to be
754
+ applied in a sequential, call-by-value manner by adding three corresponding constants to our term calculus
755
+ ST, namely including write : D → C, calc : C and read : D × C in ΛS, along with the embedding rules
756
+ • [write]π := ⟨λx, π′ . ⟨(), c1xπ′⟩, π⟩ ≃ ⟨c1, π⟩,
757
+ • [calc]π := ⟨(), c2π⟩ so that [calc] ≃ c2,
758
+ 15
759
+
760
+ • [read]π := ⟨c3π, (), π⟩ ≃ ⟨c3π, π⟩.
761
+ and then restricting out attention to terms of the form [t] for t ∈ ST + {write, calc, read}. We can then prove
762
+ the following in Sω i.e. that all axioms in ∆S can be realised:
763
+ • [write(x)] sr {α · ⊤ · stored(x)},
764
+ • [calc] sr {stored(x) · ⊤ · solved(x)},
765
+ • [read] sr {solved(x) · ∃y P(x, y) · ⊤}.
766
+ and thus Theorem 5.4 applies to SL + ∆H + ∆S for ∆H = ∅. In particular, we have
767
+ [t] sr {β · ∀x {α · ∃y P(x, y) · ⊤} · β}
768
+ for t := λx . ((write(x) ∗ calc) ∗ read) where ∗ is sequential composition operator from Definition 3.1. A
769
+ formal derivation of this term from the corresponding proof given in Example 2.2 is as follows:
770
+ x : D ⊢ write(x) : C
771
+ x : D ⊢ calc : C
772
+ ∧SI
773
+ x : D ⊢ write(x) ◦ calc : C × C
774
+ ∧SEL
775
+ x : D ⊢ write(x) ∗ calc : C
776
+ x : D ⊢ read : D × C
777
+ ∧SI
778
+ x : D ⊢ (write(x) ∗ calc) ◦ read : C × D × C
779
+ ∧SEL
780
+ x : D ⊢ (write(x) ∗ calc) ∗ read : D × C
781
+ ∀SI
782
+ ⊢ λx . ((write(x) ∗ calc) ∗ read) : D → D × C
783
+ Example 5.7 (Fixed-length array sorting). In Example 2.3 we considered a situation where we are allowed a
784
+ single action on our state, namely to swap elements. Analogously to the previous example, we can formalise
785
+ this in our semantic environment Sω by adding to ΛSω constants cl,l′ : S → S for each pair l, l′ ∈ {1, 2, 3}
786
+ along with the axiom
787
+ [α](π) ⇒ [α[l ↔ l′]](cl,l′π)
788
+ ranging over state formulas α of the form (2) and locations l, l′ ∈ {1, 2, 3} of SL, together with axioms
789
+ corresponding to those of ∆H i.e.
790
+ (π) ⇒ sorted(π)
791
+ and
792
+ [l ≤ l′ ∨ l′ ≤ l](π)
793
+ Similarly, for each l, l′ ∈ {1, 2, 3} we add a term swapl,l′ : C to ΛS and define [swapl,l′]π := ⟨(), cl,l′π⟩ so that
794
+ swapl,l′ sr {α · ⊤ · α[l ↔ l′]}
795
+ A derivation of a closed term t : C of ST + {swapl,l′} such that [t] sr {⊤ · ⊤ · sorted} is given below. In
796
+ particular, we can prove in Sω that ∀πS sorted(proj1([t]π)), and so the term λπ . proj1([t]π) : S → S acts as
797
+ a sorting program for arrays of length three. For an extracted term t corresponding to the proof given in
798
+ Example 2.3, first we interpret D1 as
799
+ ⊢ skip : C
800
+ cons
801
+ ⊢ skip : C
802
+ 2↔3
803
+ ⊢ swap2,3 : C
804
+ cons
805
+ ⊢ swap2,3 : C
806
+ cond[2≤3∨3≤2]
807
+ ⊢ t1 := if (2 ≤ 3) then (skip) else (swap2,3) : C
808
+ and define t1 := if (2 ≤ 3) then (skip) else (swap2,3). Now D2 is interpreted as
809
+ 1↔2
810
+ ⊢ swap1,2 : C
811
+ D1
812
+ ⊢ t1 : C
813
+ ∧SI
814
+ ⊢ swap1,2 ◦ t1 : C × C
815
+ ∧SEL
816
+ ⊢ t2 := swap1,2 ∗ t1 : C
817
+ 16
818
+
819
+ where we define t2 := swap1,2 ∗ t1 : C. Continuing, D3 is interpreted as:
820
+ D2
821
+ ⊢ t2 : C
822
+ ⊢ skip : C
823
+ cons
824
+ ⊢ skip : C
825
+ cond[2≤1∨1≤2]
826
+ t3 := if (2 ≤ 1) then t2 else (skip) : C
827
+ where t3 := if (2 ≤ 1) then t2 else (skip), and finally
828
+ ⊢ skip : C
829
+ 2↔3
830
+ ⊢ swap2,3 : C
831
+ cond[2≤3∨3≤2]
832
+ ⊢ if (2 ≤ 3) then (skip) else (swap2,3) : C
833
+ D3
834
+ ⊢ t3 : C
835
+ ∧SI
836
+ ⊢ (if (2 ≤ 3) then (skip) else (swap2,3)) ◦ t3 : C × C
837
+ ∧SEL
838
+ ⊢ t := (if (2 ≤ 3) then (skip) else (swap2,3)) ∗ t3 : C
839
+ 6
840
+ An extension to arithmetic
841
+ We now present an extension of our framework to a stateful version of first-order intuitionistic arithmetic.
842
+ On the logic side, we will add not only a stateful induction rule, but also a Hoare-style while rule for iteration
843
+ over the natural numbers. On the computational side, these will be interpreted by stateful recursion in all
844
+ finite types, along with a controlled while loop. The addition of these constants will allow us to extract
845
+ programs that are more interesting than those obtainable from proofs in pure predicate logic, and which
846
+ can be clearly compared to well-known stateful algorithms. To exemplify this, we will present a formally
847
+ synthesised version of insertion sort, and we stress that by further extending our framework with additional
848
+ rules and terms, we would be able to extract an even richer variety of combined functional/stateful programs.
849
+ 6.1
850
+ The system SA: First-order arithmetic with state
851
+ Our system of stateful intuitionistic arithmetic SA builds on SL just as ordinary first-order Heyting arithmetic
852
+ builds on first-order predicate logic. In both cases, we introduce a constant 0, a unary successor symbol succ,
853
+ symbols for all primitive recursive functions, and our predicate symbols now include an equality relation =.
854
+ In what follows we write x + 1 instead of succ(x). The axioms and rules of SA are, in turn, analogous to
855
+ the additional axioms and rules we would require in ordinary first-order arithmetic: They include all axioms
856
+ and rules of SL (based now on the language of SA), along with a collection of additional axioms and rules.
857
+ These comprise not only basic axioms and rules for equality and the successor, and an induction rule (all now
858
+ adapted to incorporate the state), but also a new while rule for stateful iteration, which now exploits our
859
+ state and, as we will see, allows us to extract programs that contain while loops. These additional axioms
860
+ and rules are outlined in Figure 6.1.
861
+ Our formulation of stateful arithmetic follows the same basic idea as the construction of stateful predicate
862
+ logic, incorporating standard rules but keeping track of an ambient state in a call-by-value manner, and
863
+ adding new rules that explicitly correspond to stateful constructions. In particular, Proposition 2.1 clearly
864
+ extends to SA, as the usual axioms and rules of arithmetic can be embedded into those of SA:
865
+ Proposition 6.1. For any formula A of HA and state formula α, define the main formula Aα of SA as in
866
+ Proposition 2.1. Then whenever Γ ⊢I A is provable in HA, we have that Γα, ∆ ⊢S {α · Aα · α} is provable in
867
+ SA, where ∆ is arbitrary and Γ�� := (A1)u1
868
+ α , . . . , (An)un
869
+ α
870
+ for Γ := Au1
871
+ 1 , . . . , Aun
872
+ n .
873
+ We can also derive a natural extensionality rule from our stateful equality rules, which assures us that
874
+ whenever s = t in ordinary Heyting arithmetic, then we can replace s by t for stateful formulas:
875
+ Proposition 6.2. Suppose that ⊢I s = t is provable in HA. Then from Γ ⊢S {α(s) · A(s) · β(s)} we can
876
+ derive Γ ⊢S {α(t) · A(t) · β(t)} in SA.
877
+ 17
878
+
879
+ Figure 3: Additional axioms and rules of SA
880
+ Axioms and rules for equality
881
+ Γ ⊢S {α · t = t · α}
882
+ Γ ⊢S {α · s = t · β}
883
+ Γ ⊢S {α · t = s · β}
884
+ Γ ⊢S {α · r = s · β}
885
+ Γ ⊢S {β · s = t · γ}
886
+ Γ ⊢S {α · r = t · γ}
887
+ Γ ⊢S {α · s = t · β}
888
+ Γ ⊢S {β · A(s) · γ(s)}
889
+ ext
890
+ Γ ⊢S {α · A(t) · γ(t)}
891
+ Axioms and rules for arithmetical function symbols
892
+ Γ ⊢S {α · succ(t) ̸= 0 · α}
893
+ {α · succ(s) = succ(t) · β}
894
+ {α · s = t · β}
895
+ Γ ⊢S {α · l = r · α}
896
+ where l = r ranges across defining equations for prim. rec. functions
897
+ Induction rule
898
+ Γ ⊢S {α · A(0) · β(0)}
899
+ Γ, A(x) ⊢S {β(x) · A(x + 1) · β(x + 1)}
900
+ ind
901
+ Γ ⊢S {γ · ∀x {α · A(x) · β(x)} · γ}
902
+ While rule (over natural numbers)
903
+ A1
904
+ A2
905
+ A3
906
+ while
907
+ Γ, A(x) ⊢S {α(x) · B · β}
908
+ A1 := Γ, A(x + 1) ⊢S {γ(x + 1) ∧ α(x + 1) · A(x) · α(x)}
909
+ A2 := Γ, A(x + 1) ⊢S {¬γ(x + 1) ∧ α(x + 1) · B · β}
910
+ A3 := Γ, A(0) ⊢S {α(0) · B · β}
911
+ for ind and while, x is not free in Γ, and for while it is not free in B or β
912
+ Proof. By Proposition 6.1 for α := α(s) we have Γ ⊢S {α(s) · s = t · α(s)} and thus using the extensionality
913
+ rule in SA we can derive
914
+ Γ ⊢S {α(s) · s = t · α(s)}
915
+ Γ ⊢S {α(s) · A(s) · β(s)}
916
+ ext
917
+ Γ ⊢S {α(s) · A(t) · β(t)}
918
+ Since ⊢I t = s must also be provable in HA, another instance of Proposition 6.1 for α := α(t) along with the
919
+ true axiom in SA gives us
920
+ Γ ⊢S {α(t) · t = s · α(t)}
921
+ Γ ⊢S {α(t) · ⊤ · α(t)}
922
+ ext
923
+ Γ ⊢S {α(t) · ⊤ · α(s)}
924
+ Putting these together we obtain
925
+ Γ ⊢S {α(t) · ⊤ · α(s)}
926
+ Γ ⊢S {α(s) · A(t) · β(t)}
927
+ ∧SI
928
+ Γ ⊢S {α(t) · ⊤ ∧ A(t) · β(t)}
929
+ ∧SEL
930
+ Γ ⊢S {α(t) · A(t) · β(t)}
931
+ which completes the derivation.
932
+ 18
933
+
934
+ 6.2
935
+ An extended term calculus STN
936
+ In order to give derivations in SA a computation interpretation, we need to extend our term calculus ST
937
+ to include a recursor (for induction) and a controlled while loop (for the while rule). The remaining new
938
+ axioms and rules of SA are dealt with in a straightforward manner.
939
+ To be precise: the theory STN is defined to be the instance of ST for the case of arithmetic, with function
940
+ symbols for zero, successor and all primitive recursive functions. Accordingly, we rename the base type D
941
+ to Nat. In addition to the terms of ST, we add terms rec e e and whilee γ[z]e e e to our grammar, where γ[z]
942
+ ranges over state formulas of SL with a specified free variable z. The typing rules for these new terms are
943
+ Γ ⊢ s : X
944
+ Γ ⊢ t : Nat → X → X
945
+ Γ ⊢ rec s t : Nat → X
946
+ for the recursor, while for the while loop we have
947
+ Γ ⊢ r : Nat → X → X
948
+ Γ ⊢ s : Nat → X → Y
949
+ Γ ⊢ t : X → Y
950
+ Γ ⊢ u : Nat
951
+ Γ ⊢ whileu γ[z] r s t : X → Y
952
+ under the additional variable condition that z /∈ Γ, but x : Nat ∈ Γ for all free variables of γ[z] outside of
953
+ z. Note that we do not consider z a free variable of whilea γ[z] r s t, but rather a placeholder for the loop
954
+ condition. In order to give the appropriate semantics to our terms, we must add to our metatheory Sω
955
+ axioms and rules for arithmetic in all finite types, including the ability to define functions of arbitrary type
956
+ via recursion over the natural numbers, along the lines of E-HAω [30] (though as before the precise details
957
+ are not important). We then define:
958
+ • [rec s t]π := ⟨Rf, π1⟩ for ⟨f, π1⟩ := [t]π, where
959
+ Rf0π := [s]π
960
+ Rf(n + 1)π := gaπ′
961
+ 2 for ⟨a, π′
962
+ 1⟩ := Rfnπ′ and ⟨g, π′
963
+ 2⟩ := fnπ′
964
+ 1
965
+ (3)
966
+ • [whileu γ[z] r s t]π := ⟨Lf,g,hm, π4⟩ where ⟨f, π1⟩ := [r]π, ⟨g, π2⟩ := [s]π1, ⟨h, π3⟩ := [t]π2 and ⟨m, π4⟩ :=
967
+ [u]π3, where
968
+ Lf,g,h0yπ′ := hyπ′
969
+ Lf,g,h(n + 1)yπ′ :=
970
+
971
+ Lf,g,hny′π2 for ⟨a, π′
972
+ 1⟩ := fnπ′ and ⟨y′, π′
973
+ 2⟩ := ayπ′
974
+ 1
975
+ if [γ][n + 1](π′)
976
+ byπ′
977
+ 1 for ⟨b, π′
978
+ 1⟩ := gnπ′
979
+ if ¬[γ][n + 1](π′)
980
+ (4)
981
+ where in the case distinctions, we would technically speaking need to use the characteristic function
982
+ χγ⟨x1, . . . , n, . . . , xk⟩ for γ, with n substituted for the special free variable z.
983
+ 6.3
984
+ The soundness theorem for arithmetic
985
+ We now need to show that the soundness proof for stateful predicate logic also holds in the extension to
986
+ arithmetic.
987
+ Theorem 6.3. The statement of Theorem 5.4 remains valid if we replace SL by SA and ST by STN.
988
+ Proof. We need to extend the proof of Theorem 6.3 to show that the additional axioms and rules as in Figure
989
+ 6.1 can be realized by a term of the form [t] for t in STN.
990
+ • For the non-extensionality equality and arithmetic axioms this is straightforward due to the fact that
991
+ these are also true in Sω: For instance, given a realizer [s] sr {α · u = v · β} and [t] sr {β · v = w · γ},
992
+ we have that [s ◦ t] sr {α · u = v ∧ v = w · γ}, and since from u = v ∧ v = w we can infer u = w in Sω,
993
+ it follows that [p1(s ◦ t)] sr {α · u = w · γ}. The other axioms and rules are even simpler.
994
+ 19
995
+
996
+ • (ext) Extensionality is similarly simple: If [s] sr {α · u = v · β} and [t] sr {β · A(u) · γ(u)}, then [α](π)
997
+ implies that u = v and [β](π1) for ⟨. . . , π1⟩ := [s]π, and therefore a sr A(u) and [γ](u)(π2) for
998
+ ⟨a, π2⟩ := [t]π1. Now applying extensionality in Sω to the formula T (x) := a sr A(x) ∧ [γ](x)(π2),
999
+ from u = v we have a sr A(v) and [γ](v)(π2), and thus [s ◦ t] sr {α · u = v ∧ A(v) · γ(v)} and therefore
1000
+ [p2(s ◦ t)] sr {α · A(v) · γ(v)}.
1001
+ • (rec) Suppose that s and t(x, y) are such that [s] sr {α · A(0) · β(0)} and
1002
+ [t](x, y) sr {β(x) · A(x + 1) · β(x + 1)}
1003
+ assuming y sr A(x). We show that [rec s λx, y.t(x, y)] sr {γ · ∀x {α · A(x) · β(x)} · γ} for any γ. Since
1004
+ [rec s λx, y.t(x, y)]π = ⟨Rf, π⟩ for f := λx.[λy.t(x, y)] and Rf as in (3), it suffices to show that for any
1005
+ n : Nat we have
1006
+ Rfn sr {α · A(n) · β(n)}
1007
+ We prove this by induction: For the base case, we have Rf0 = [s] and the claim holds by assumption.
1008
+ For the induction step, let us assume that [α](π′) holds, and so by the induction hypothesis we have
1009
+ a sr A(n) and [β(n)](π′
1010
+ 1) for ⟨a, π′
1011
+ 1⟩ := Rfn. Since fnπ′
1012
+ 1 = ⟨g, π′
1013
+ 1⟩ for g := λy.[t](n, y), we have that
1014
+ Rf(n+1)π′ = [t](n, a)π′
1015
+ 1, and since by the property of [t] we then have b sr A(n + 1) and [β(n+1)](π′
1016
+ 2)
1017
+ for ⟨b, π′
1018
+ 2⟩ := [t](n, a)π′
1019
+ 1, we have shown that Rf(n + 1) sr {α · A(n + 1) · β(n + 1)}, which completes
1020
+ the induction.
1021
+ • (while) We suppose that
1022
+ 1. [r](x, y) sr {γ(x + 1) ∧ α(x + 1) · A(x) · α(x)} assuming that y sr A(x + 1),
1023
+ 2. [s](x, y) sr {¬γ(x + 1) ∧ α(x + 1) · B · β} assuming that y sr A(x + 1),
1024
+ 3. [t](y) sr {α(0) · B · β} assuming that y sr A(0).
1025
+ Our aim is to show that
1026
+ [(whilex γ (λx′, y′.r) (λx′, y′.s) (λy′.t))y] sr {α(x) · B · β}
1027
+ for any x, y ∈ Nat with y sr A(x). We observe, unwinding the definition, that
1028
+ [(whilex γ (λx′, y′.r) (λx′, y′.s) (λy′.t))y]π = Lf,g,hxyπ
1029
+ for f := λx′.[λy′.r(x′, y′)], g := λx′.[λy′.s(x′, y′)], h := λy′.[t](y′) and Lf,g,h as defined in (4). We now
1030
+ show by induction on n that if y sr A(n) then
1031
+ Lf,g,hny sr {α(n) · B · β}
1032
+ and then the result follows by setting n := x. The base case is straightforward since
1033
+ Lf,g,y0y = [t](y)
1034
+ and the claim follows by definition of [t]. For the induction step, suppose that y sr A(n + 1) and
1035
+ [α(n + 1)](π). There are two cases. If ¬[γ](n + 1)(π) we have
1036
+ Lf,g,h(n + 1)yπ = [s](n, y)π
1037
+ and the result holds by the property of [s]. On the other hand, if [γ](n + 1)(π) then
1038
+ Lf,g,h(n + 1)yπ = Lf,g,hny′π′
1039
+ for ⟨y′, π′⟩ := [r](n, y)π. But by the property of [r] we have y′ sr A(n) and [α(n)](π′), and therefore by
1040
+ the induction hypothesis we have b sr B and [β](π′′) for ⟨b, π′′⟩ := Lf,g,hny′π′ = Lf,g,h(n + 1)yπ, and
1041
+ so the result is proven for n + 1.
1042
+ This covers all the additional axioms and rules of SA.
1043
+ 20
1044
+
1045
+ 6.4
1046
+ Worked example: Insertion sort
1047
+ We now illustrate our extended system by synthesising a list sorting program that, intuitively, forms an
1048
+ implementation of the insertion sort algorithm. Here our state will represent the structure that is to be
1049
+ sorted, and continuing the spirit of generality that we have adhered to throughout, we characterise this
1050
+ structure through a number of abstract axioms. Instantiating the state as, say, an array of natural numbers,
1051
+ would provide a model for our theory, but our sorting algorithm can be extracted on the more abstract level.
1052
+ Crucially, the proof involves both loop iteration and induction, and the corresponding program combines an
1053
+ imperative while loop with a functional recursor.
1054
+ We begin by axiomatising our state, just as in previous examples. An intuition here is that states represent
1055
+ an infinite array of elements a0, a1, . . . possessing some total order structure ≤, and we seek to extract a
1056
+ program that, for any input n, sorts the first n elements. We use this informal semantics throughout to
1057
+ indicate the intended meaning of our axioms, but stress that none of this plays a formal role in the proof or
1058
+ resulting computational interpretation.
1059
+ We introduce three state predicates to SA, with the intuition indicated in each case:
1060
+ • sort(N)
1061
+ Sorted: The first N + 1 elements of the array i.e. [a0, . . . , aN] are sorted
1062
+ • psort(n, N)
1063
+ Partially sorted with respect to an: if n < N then the list [a0, . . . , an−1, an+1, . . . , aN] is sorted and
1064
+ an ≤ an+1. For the base cases, if n = N then the list [a0, . . . , aN−1] is sorted, and if n > N then the
1065
+ list [a0, . . . , aN] is sorted.
1066
+ • comm(n)
1067
+ Comparison: true if an ≤ an−1, and always true if n = 0
1068
+ We formalise this intuition by adding the following state independent axioms to ∆H:
1069
+ 1. Γ, sort(N) ⊢H psort(N + 1, N + 1)
1070
+ If the first N +1 elements are sorted, then they are also partially sorted with respect to the next element
1071
+ aN+1.
1072
+ 2. Γ, ¬comm(n), psort(n, N) ⊢H sort(N)
1073
+ If [a0, . . . , an−1, an+1, . . . , aN] is sorted, an ≤ an+1, but also an−1 ≤ an, then the entire segment
1074
+ [a0, . . . , aN] must be sorted.
1075
+ 3. Γ, psort(0, N) ⊢H sort(N)
1076
+ If [a1, . . . , aN] is sorted and a0 ≤ a1, then [a0, . . . , aN] is sorted.
1077
+ 4. Γ ⊢H sort(0)
1078
+ The singleton array [a0] is defined to be sorted.
1079
+ We complete the axiomatisation by adding a single state-sensitive axiom to ∆S:
1080
+ 5 Γ ⊢S {comm(n + 1) ∧ psort(n + 1, N) · ⊤ · psort(n, N)}
1081
+ If [a0, . . . , an, an+2, . . . , aN] is sorted and an+1 ≤ an+2, but an+1 ≤ an, then we can modify the state
1082
+ (i.e. swapping an and an+1 by setting ˜an := an+1 and ˜an+1 := an) so that [a0, . . . , an−1, ˜an+1, . . . , aN]
1083
+ is sorted and ˜an ≤ ˜an+1. The edge cases for n ≥ N are interpreted in a more straightforward way.
1084
+ In order to give a realizing term to this axiom, we representing element swapping semantically by adding a
1085
+ constant c : Nat → S → S to our metatheory Sω, which satisfies
1086
+ comm(n + 1, π) ∧ psort(n + 1, N, π) ⇒ psort(n, N, cnπ)
1087
+ 21
1088
+
1089
+ and a corresponding term swap : Nat → C to our term calculus, along with the embedding
1090
+ [swap]π := ⟨λn, π.⟨(), cnπ⟩, π⟩ ≃ ⟨c, π⟩
1091
+ so that we can prove
1092
+ [swap n] sr {comm(n + 1) ∧ psort(n + 1, N) · ⊤ · psort(n, N)}
1093
+ With this in place, we can now prove in SA that the first N elements of the state can be sorted, and extract
1094
+ a corresponding realizing term in STN.
1095
+ 6.4.1
1096
+ Proof of ⊢S {γ · ∀N {α · ⊤ · sort(N)} · γ} in SA
1097
+ The core of our proof begins with an instance of the while rule parametrised by N, with Γ := ∅, A(n) := ⊤,
1098
+ α(n) := psort(n, N + 1), β := sort(N + 1) and γ(n) := comm(n):
1099
+ D1
1100
+ D2
1101
+ D3
1102
+ while
1103
+ ⊤ ⊢S {psort(n, N + 1) · ⊤ · sort(N + 1)}
1104
+ ∀SI
1105
+ ⊤ ⊢S {psort(N + 1, N + 1) · ∀n {psort(n, N + 1) · ⊤ · sort(N + 1)} · psort(N + 1, N + 1)}
1106
+ ∀SE
1107
+ ⊤ ⊢S {psort(N + 1, N + 1) · ⊤ · sort(N + 1)}
1108
+ cons
1109
+ ⊤ ⊢S {sort(N) · ⊤ · sort(N + 1)}
1110
+ where the final composition inference makes use of the first state independent axiom. Here D1 represents an
1111
+ instance of the state sensitive axiom
1112
+ ⊤ ⊢S {comm(n + 1) ∧ psort(n + 1, N + 1) · ⊤ · psort(n, N + 1)}
1113
+ and D2 represents the derivation
1114
+ ⊤ ⊢S {sort(N + 1) · ⊤ · sort(N + 1)}
1115
+ cons
1116
+ ⊤ ⊢S {¬comm(n + 1) ∧ psort(n + 1, N + 1) · ⊤ · sort(N + 1)}
1117
+ where composition makes use of the second state independent axiom. Finally D3 is
1118
+ ⊤ ⊢S {sort(N + 1) · ⊤ · sort(N + 1)}
1119
+ cons
1120
+ ⊤ ⊢S {psort(0, N + 1) · ⊤ · sort(N + 1)}
1121
+ this time making use of the third state independent axiom. Finally we can prove that all lists can be sorted
1122
+ with an outer induction as follows:
1123
+ ⊢S {α · ⊤ · α}
1124
+ cons
1125
+ ⊢S {α · ⊤ · sort(0)}
1126
+ D
1127
+ ⊤ ⊢S {sort(N) · ⊤ · sort(N + 1)}
1128
+ ind
1129
+ ⊢S {γ · ∀N {α · ⊤ · sort(N)} · γ}
1130
+ where α is an arbitrary state predicate, the instance of cons uses the fourth state independent axiom, and
1131
+ D represents the derivation above.
1132
+ 6.4.2
1133
+ Program extraction
1134
+ We now extract a program that corresponds to the above proof. First of all, we note that the three premises
1135
+ of our while rule are realised by swap n, skip and skip respectively, and so our derivation D corresponds to
1136
+ the following program:
1137
+ y : C ⊢ swap n : C
1138
+ y : C ⊢ skip : C
1139
+ y : C ⊢ skip : C
1140
+ while
1141
+ y : C ⊢ t(n)y : C
1142
+ ∀SI
1143
+ y : C ⊢ λn.t(n)y : Nat → C
1144
+ ∀SE
1145
+ y : C ⊢ (λn.t(n)y)(N + 1) : C
1146
+ cons
1147
+ y : C ⊢ (λn.t(n)y)(N + 1) : C
1148
+ 22
1149
+
1150
+ where
1151
+ t(n) := whilen comm[z] (λx, y.(swap x)) (λx, y.skip) (λy.skip)
1152
+ ≃ whilen comm[z] (λx.(swap x)) (skip) (skip)
1153
+ Then our final induction generates the following program:
1154
+ ⊢ skip : C
1155
+ y : C ⊢ (λn.t(n)y)(N + 1) : C
1156
+ ind
1157
+ ⊢ rec (skip) (λx, y.((λn.t(n)y)(x + 1))) : Nat → C
1158
+ Thus our list sorting program is
1159
+ rec (skip) (λx, y.((λn.t(n)y)(x + 1)))
1160
+ ≃ rec (skip) (λx.((λn.(whilen comm[z] (λx.(swap x)) (skip) (skip)()))(x + 1)))
1161
+ which is essentially an implementation of the insertion sort algorithm, with an outer recursion that sorts
1162
+ initial segments of the list in turn, and an inner loop that inserts new elements into the appropriate place in
1163
+ the current sorted list.
1164
+ 7
1165
+ Directions for future work
1166
+ In this paper we have presented the central ideas behind a new method for extracting stateful programs
1167
+ from proofs, which include an extension of ordinary first-order logic with Hoare triples, a corresponding
1168
+ realizability interpretation, and a soundness theorem. We emphasise once again that our intention has been
1169
+ to offer an alternative approach to connecting proofs with stateful programs, one that seeks to complement
1170
+ rather than improve existing work by embracing simplicity and abstraction, and which might be well suited
1171
+ to a range of applications in proof theory or computability theory. In this spirit, we conclude with a very
1172
+ informal outline of a series interesting directions in which we anticipate that our framework could be applied.
1173
+ 7.1
1174
+ Further extensions and program synthesis
1175
+ While our main results have been presented in the neutral setting of first-order predicate logic, it would be
1176
+ straightforward to extend SL to richer logics with more complex data structures and a imperative commands.
1177
+ Already, the addition of recursion and loops over natural numbers in Section 6 has allowed us to synthesise
1178
+ a standard in-place sorting algorithm using our abstract axiomatisation of an ordered state, in a similar
1179
+ spirit to [6]. However, further extensions are naturally possible, including the addition of general fixpoint
1180
+ operators and non-controlled while loops, which would then require a Sω to be replaced by a domain theoretic
1181
+ semantics that allows for partiality.
1182
+ Looking a step further ahead, by implementing all of this in a proof assistant, we would have at our
1183
+ disposal a new technique for synthesising correct-by-construction imperative programs. While we do not
1184
+ suggest that this pipeline would directly compete with existing techniques for verifying imperative programs,
1185
+ it could be well suited to synthesising and reasoning about programs in very specific domains, where we are
1186
+ interested in algorithms for which interactions with the state have a restricted form that could be suitably
1187
+ axiomatised within our logic. For example, a more detailed axiomatisation our state as an ordered array along
1188
+ the lines of Section 6.4, with a “swap” operation and a few other ways of interacting with the state, might
1189
+ give rise to an interesting theory of in-place sort algorithms. Stateful algorithms on other data structures,
1190
+ such as graphs, could presumably also be formalised within our framework.
1191
+ 7.2
1192
+ Bar recursion and the semantics of extracted programs
1193
+ Two of the main starting points for this paper, the monadic realizability of Birolo [7] and the author’s own
1194
+ Dialectica interpretation with state [24], address the broader problem of trying to understand the operational
1195
+ semantics of programs extracted from proofs as stateful procedures (the origins and development of this
1196
+ general idea, from Hilbert’s epsilon calculus onwards, is brilliantly elucidated in Chapter 1 of Aschieri’s
1197
+ 23
1198
+
1199
+ thesis [2], who then sets out his own realizability interpretation based on learning). A number of case studies
1200
+ by the author and others [20, 21, 25, 26] have demonstrated that while terms extracted from nontrivial proofs
1201
+ can be extremely complex, they are often much easier to understand if one focuses on the way they interact
1202
+ with the mathematical environment. For example, in understanding a program extracted from a proof using
1203
+ Ramsey’s theorem for pairs [20], it could be illuminating to study the trace of the program as it queries a
1204
+ colouring at particular pairs, as this can lead to a simpler characterisation of the algorithm ultimately being
1205
+ implemented by the term.
1206
+ While the aforementioned analysis of programs has always been done in an informal way, our stateful
1207
+ realizability interpretation would in theory allow us to extract programs which store this trace formally in
1208
+ the state, where our abstract characterisation of state would allow us to implement it in whichever way is
1209
+ helpful in a given setting. For example, in the case of the Bolzano-Weierstrass theorem [21], our state might
1210
+ record information of the form xn ∈ I, collecting information about the location of sequence elements. For
1211
+ applications in algebra [26], one might instead store information about a particular maximal ideal.
1212
+ The aforementioned theorems are typically proven using some form of choice or comprehension, and that
1213
+ in itself leads to the interesting prospect of introducing both stateful recursors and while-loops that are
1214
+ computationally equivalent to variants of bar recursion [29]. In [23], several bar recursive programs that
1215
+ arise from giving a computational interpretation to arithmetical comprehension principles are formulated as
1216
+ simple while loops, and these could in principle be incorporated into our system with new controlled Hoare
1217
+ rules in the style of update recursion [5], that replace the conditions n < N and n ≥ N in the Ai above with
1218
+ e.g. n ∈ dom(f) and n /∈ dom(f), where f is some partial approximation to a comprehension function. An
1219
+ exploration of such while-loops from the perspective of higher-order computability theory might well be of
1220
+ interest in its own right.
1221
+ 7.3
1222
+ A logic for probabilistic lambda calculi
1223
+ Probabilistic functional languages are a major topic of research at present. While work in this direction dates
1224
+ back to the late 1970s [16, 28] where it typically had a semantic flavour, a more recent theme [9, 11, 12] has
1225
+ been to study simple extensions of the lambda calculus with nondeterministic choice operators ⊕, where s⊕t
1226
+ evaluates nondeterministically (or probabilistically) to either s or t. While such calculi have been extensively
1227
+ studied, corresponding logics that map under some proof interpretation to probabilistic programs are far
1228
+ more rare (although there is some recent work in this direction e.g. [1]).
1229
+ We conjecture that our framework offers a bridge between logic and probabilistic computation through
1230
+ incorporating probabilistic disjunctions into our logic SL and taking states to be streams of outcomes of
1231
+ probabilistic events together with a current ‘counter’ that increases each time an event occurs. In a simple
1232
+ setting where only two outcomes are possible with equal probability, we can axiomatise this within SL by
1233
+ adding zero and successor functions (allowing us to create numerals n), along with a unary state predicate
1234
+ count(n). We can then model probabilistic events by adding the appropriate axioms to ∆S. Suppose, for
1235
+ example, we add two predicate constants H(x) and T (x) (for heads and tails), along with constants c1, c2, . . .
1236
+ representing coins. Then flipping a coin would be represented by the axiom schema
1237
+ Γ ⊢S {count(n) · H(ci) ∨ T (ci) · count(n + 1)}
1238
+ where n ranges over numerals and ci over coin constants, the counter indicating that a probabilistic event has
1239
+ occurred. The act of reading a probability from the state could be interpreted semantically by introducing
1240
+ a constant ω : S → Bool × S to Sω, with the axiom
1241
+ count(n, π) ⇒ (e = f ⇒ H(ci)) ∧ (e = t ⇒ T (ci)) ∧ count(n + 1, π1) for ⟨e, π1⟩ := ωπ
1242
+ (alternatively, we could simply define S := Nat × (Nat → Bool) for a type of Nat natural numbers, and
1243
+ define ω⟨n, a⟩ := ⟨a(n), ⟨n + 1, a⟩⟩ and count(n, ⟨m, a⟩) := m =Nat n).
1244
+ A probabilistic choice operator ⊕ can then be added to the language of ST, along with the typing rule
1245
+ Γ ⊢ s ⊕ t : X + Y for Γ ⊢ s : X and Γ ⊢ t : Y , and the interpretation
1246
+ [s ⊕ t]π := case e ([ι0s]π1) ([ι1t]π1) where ⟨e, π1⟩ := ωπ
1247
+ 24
1248
+
1249
+ In particular, defining flip := skip ⊕ skip : C + C we would have
1250
+ [flip] sr {count(n) · H(ci) ∨ T (ci) · count(n + 1)}
1251
+ although we stress that the operator ⊕ and would allow for much more complex probabilistic disjunctions,
1252
+ potentially involving additional computational content.
1253
+ Our soundness theorem, extended to these new probabilistic axioms and terms, would then facilitate the
1254
+ extraction of probabilistic programs from proofs. For instance, including a winner predicate W(x), two player
1255
+ constant symbols p1, p2, and adding axioms H(c1), H(c2) ⊢S {α · W(p1) · α}; T (c1), T (c2) ⊢S {α · W(p1) · α};
1256
+ H(c1), T (c2) ⊢S {α · W(p2) · α} and T (c1), H(c2) ⊢S {α · W(p2) · α} for any α, we could prove
1257
+ ⊢S {count(n) · ∃x W(x) · count(n + 2)}
1258
+ expressing the fact that a winner can be determined after two flips. We can then extract a corresponding
1259
+ probabilistic term for realizing this statement, which would be isomorphic to the expected program that
1260
+ queries the state twice in order to determine the outcome of those flips, and returns either p1 or p2 as a
1261
+ realizer for ∃x W(x) depending on the content of the state.
1262
+ Of course, the details here need to be worked through carefully in order to properly substantiate the
1263
+ claim that our framework could be used to extract probabilistic programs in a natural and meaningful way.
1264
+ At the very least, it is likely that further additions to SL along with a more intricate state would be needed
1265
+ to incorporate more interesting probabilistic events, such as annotated disjunctions along the lines of [31].
1266
+ We leave such matters to future work.
1267
+ References
1268
+ [1] Antonelli, M., Dal Lago, U., and Pistone, P. Curry and Howard meet Borel. Preprint, 2022.
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+ [2] Aschieri, F. Learning, Realizability and Games in Classical Arithmetic. PhD thesis, Universit`a degli
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+ Studi di Torino and Queen Mary, University of London, 2011.
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1
+ Mono-STAR: Mono-camera Scene-level Tracking and Reconstruction
2
+ Haonan Chang1, Dhruv Metha Ramesh1, Shijie Geng1, Yuqiu Gan, Abdeslam Boularias1
3
+ Abstract— We present Mono-STAR, the first real-time 3D
4
+ reconstruction system that simultaneously supports semantic
5
+ fusion, fast motion tracking, non-rigid object deformation, and
6
+ topological change under a unified framework. The proposed
7
+ system
8
+ solves
9
+ a
10
+ new
11
+ optimization
12
+ problem
13
+ incorporating
14
+ optical-flow-based 2D constraints to deal with fast motion
15
+ and a novel semantic-aware deformation graph (SAD-graph)
16
+ for handling topology change. We test the proposed system
17
+ under various challenging scenes and demonstrate that it
18
+ significantly
19
+ outperforms
20
+ existing
21
+ state-of-the-art
22
+ methods.
23
+ Supplementary material, including videos, can be found at
24
+ https://github.com/changhaonan/Mono-STAR-demo.
25
+ I. INTRODUCTION
26
+ Real-time perception is a crucial component of modern
27
+ robotic manipulation systems. Recently, You Demonstrate
28
+ Only Once [1] has demonstrated that given the geometry
29
+ model and 6D-pose trajectory of a manipulated object during
30
+ an expert demonstration, a robot can quickly learn complex
31
+ and contact-rich manipulation skills. Such progress shows
32
+ the importance of geometric 3D reconstruction and tracking
33
+ systems for robotic manipulation.
34
+ However, a perception system that can perform both track-
35
+ ing and reconstruction simultaneously is notoriously difficult
36
+ to build because reconstruction and tracking inherently de-
37
+ pend on each other. For example, tracking algorithms usually
38
+ require geometry models, while dynamic scene reconstruc-
39
+ tion relies on accurate tracking for producing those geometry
40
+ models. Scene-level Tracking and Reconstruction (STAR) [2]
41
+ refers to a category of perception systems that generate both
42
+ the geometry and the pose of every visible object in a scene.
43
+ This problem is related to the multiple-instance dynamic
44
+ SLAM problem, where all movable objects in the scene are
45
+ assumed to be rigid so that the problem can be decomposed
46
+ into multiple dense-SLAM sub-problems. This approach was
47
+ proposed in Co-Fusion [3] and MaskFusion [4], where a
48
+ semantic neural network was employed first to decompose
49
+ the scene into multiple objects and then deal with each
50
+ object individually. This approach requires every object in
51
+ the scene to be rigid or quasi-rigid. The same problem was
52
+ investigated in MidFusion [5], where an octree was used to
53
+ improve reconstruction and tracking. However, these systems
54
+ are limited to scenes of rigid objects with slow motions.
55
+ Instead of dealing with each object individually based
56
+ on their semantic labels, one can also reconstruct all the
57
+ objects in the scene as one large non-rigid object and segment
58
+ them ulteriorly. This approach was however very challenging
59
+ 1 Authors are with the Department of Computer Science, Rutgers
60
+ University, 08854 New Brunswick, USA. This work is supported by NSF
61
+ awards 1734492, 1846043, and 2132972.
62
+ to apply until the introduction of the first real-time non-
63
+ rigid reconstruction DynamicFusion [6], where the non-
64
+ rigid reconstruction problem was decomposed into two sub-
65
+ problems, (1) building the geometry at the initial frame, and
66
+ (2) computing the deformation using an embedded defor-
67
+ mation graph, namely ED-graph. This paradigm was also
68
+ followed in OcclusionFusion [7]. Inspired by these previous
69
+ efforts, a solution to the general STAR problem was recently
70
+ proposed in STAR-no-prior [2]. In contrast to SLAM-based
71
+ methods, STAR-no-prior reverses the order of segmentation
72
+ and reconstruction. The entire scene is first reconstructed and
73
+ then segmented into different objects based on topology. By
74
+ doing so, STAR-no-prior outperforms previous state-of-the-
75
+ art methods such as [4] and MidFusion [5].
76
+ However, a major limitation of STAR-no-prior is its re-
77
+ liance on a system of multiple cameras surrounding the
78
+ scene, making it impractical for a mobile robot. To address
79
+ this shortcoming, we propose Mono-STAR, a mono-camera
80
+ STAR solution. Switching from a multi-camera system to
81
+ a mono-camera setting requires solving several non-trivial
82
+ problems. Notably, STAR-no-prior relies on the multi-camera
83
+ system to overcome the plane-based-ICP constraint that it
84
+ inherits from DynamicFusion, which supports tracking of
85
+ only slow motion along the camera view. The use of multiple
86
+ cameras can guarantee that any motion has at least one non-
87
+ zero projection to a camera view. However, the mono-camera
88
+ setting does not have such a guarantee and therefore requires
89
+ a new solution. Occlusion Fusion [7] adds a 2D constraint
90
+ using optical-flow (RAFT [8] or GMA [9]) to deal with fast
91
+ motions. Inspired by this, we propose a new 2D loss to track
92
+ motions that are perpendicular to the camera view, which not
93
+ only stabilizes tracking performance under a single view but
94
+ also improves our system’s ability to handle fast motion.
95
+ Furthermore, STAR-no-prior does not take advantage of
96
+ semantic labels. We, therefore, combine the semantic infor-
97
+ mation with the embedded deformation graph mechanism
98
+ and propose a Semantic-aware Adaptive Deformation graph,
99
+ SAD-graph, which is an extension of ED-graph. With just lit-
100
+ tle extra computation, SAD-graph can easily handle topology
101
+ changes across distinct semantic classes and assign different
102
+ levels of rigidness for each type of object. To the best of our
103
+ knowledge, Mono-STAR is the first single-view real-time 3D
104
+ reconstruction system that can simultaneously handle seman-
105
+ tic fusion, fast motion tracking, non-rigid object deformation,
106
+ and topological change under one unified framework.
107
+ II. RELATED WORKS
108
+ Simultaneous Tracking and Reconstruction. Simulta-
109
+ neous 6D tracking and 3D reconstruction was typically
110
+ arXiv:2301.13244v1 [cs.RO] 30 Jan 2023
111
+
112
+ Method
113
+ Semantic
114
+ Fast
115
+ Non-rigid
116
+ Topology
117
+ Single
118
+ motion
119
+ objects
120
+ change
121
+ view
122
+ SLAM++
123
+ 
124
+ 
125
+ 
126
+ 
127
+ 
128
+ DynamicFusion [6]
129
+ 
130
+ 
131
+ 
132
+ 
133
+ 
134
+ Volume Deform
135
+ 
136
+ 
137
+ 
138
+ 
139
+ 
140
+ SurfelWarp [10]
141
+ 
142
+ 
143
+ 
144
+ 
145
+ 
146
+ TCAFusion [11]
147
+ 
148
+ 
149
+ 
150
+ 
151
+ 
152
+ Co-fusion [3]
153
+ 
154
+ 
155
+ 
156
+ 
157
+ 
158
+ Fusion4D [12]
159
+ 
160
+ 
161
+ 
162
+ 
163
+ 
164
+ Motion2Fusion [13]
165
+ 
166
+ 
167
+ 
168
+ 
169
+ 
170
+ Functon4D [14]
171
+ 
172
+ 
173
+ 
174
+ 
175
+ 
176
+ MaskFusion [4]
177
+ 
178
+ 
179
+ 
180
+ 
181
+ 
182
+ RigidFusion [15]
183
+ 
184
+ 
185
+ 
186
+ 
187
+ 
188
+ MidFusion [5]
189
+ 
190
+ 
191
+ 
192
+ 
193
+ 
194
+ OcclusionFusion [7]
195
+ 
196
+ 
197
+ 
198
+ 
199
+ 
200
+ STAR-no-prior [2]
201
+ 
202
+ 
203
+ 
204
+ 
205
+ 
206
+ Mono-STAR
207
+ 
208
+ 
209
+ 
210
+ 
211
+ 
212
+ TABLE I: Taxonomy of the-state-of-art scene-level fusion systems.
213
+ regarded in previous works as a multiple-instance dynamic
214
+ SLAM problem. Many works such as Co-fusion [3], Mask-
215
+ Fusion [4], and RigidFusion [15] proposed to divide the
216
+ scene into multiple rigid objects and track each object
217
+ individually. More recently, STAR-no-prior [2] formalized
218
+ the STAR problem as a scene-level non-rigid reconstruction
219
+ problem. Our mono-camera system eliminates the multi-
220
+ camera requirement of STAR-no-prior by adding a new
221
+ optical-flow-based 2D constraint and a novel semantic-aware
222
+ adaptive deformation graph.
223
+ Dynamic Scene Reconstruction. Dynamic scene recon-
224
+ struction [16], [17] is the problem of reconstructing the
225
+ geometry and recording the deformation of a scene with
226
+ moving objects. DynamicFusion [6] was the first real-time
227
+ GPU-based solution for solving this problem. It adopts a
228
+ TSDF-based geometry as the canonical model and an embed-
229
+ deformation graph (ED-graph) to describe the deformation
230
+ of the whole scene. A drawback of this method is that the
231
+ combination of TSDF and ED-graph cannot handle topology
232
+ changes. Many recent techniques such as Fusion4D [12],
233
+ Motion2Fusion [13], [11], [18] have attempted to address this
234
+ problem. However, these methods require significantly more
235
+ computation or rely on expensive sensors. SurfelWarp [10]
236
+ demonstrated that a Surfel-based representation can be used
237
+ to tackle topology changes. Therefore, our proposed system
238
+ also adopts a Surfel-based representation.
239
+ III. PROBLEM FORMULATION AND BACKGROUND
240
+ A. Problem formulation
241
+ Given a sequence of RGB-D images of a given dynamic
242
+ scene taken from a single fixed camera, we consider the
243
+ problem of simultaneous tracking and reconstruction of all
244
+ the objects visible in the scene. The number of objects is
245
+ unknown. The objects can be non-rigid. Measurement, Mt
246
+ can defined as set of measurement surfels mi at time-step
247
+ t, generated from the RGB-D input. mi = (vi,ni,ci), where
248
+ v,n,c are 3D coordinates, normal and color respectively.
249
+ The proposed system returns at each time-step t a Surfel-
250
+ based geometry St (the reconstructed scene) for the entire
251
+ scene and its corresponding deformation graph Gt. Surfel-
252
+ based geometry St is a set of surfels si. si = (vi,ni,ci,ri,lsi),
253
+ where vi,ni,ci,ri,lsi are respectively the 3D coordinates,
254
+ normal, color, radius and semantic label of surfel si ∈ St. We
255
+ assume that there is a maximum of H pre-defined different
256
+ semantic categories {1,2,...,H}. If a surfel does not belong
257
+ to any pre-defined category, it will be labeled as H +1 (i.e.,
258
+ unrecognized). Deformation graph Gt is defined by a set of
259
+ nodes {gi}. Each node gi has a semantic label lgi, and is
260
+ connected to its nearest-neighbor nodes, denoted as NG(gi),
261
+ in the 3D space. Deformation graph Gt is associated with
262
+ a warp field Wt, defined as W = {[pi ∈ R3,δi ∈ R+,Ti ∈
263
+ SE(3)]}, wherein i is the index of a node in Gt, pi is the 3D
264
+ point that corresponds to node gi, δi is the node’s radius of
265
+ influence, and Ti is the 6D transformation defined on node
266
+ gi. Ti is represented by a dual quaternion qi for smooth
267
+ interpolation [19]. Warp field W describes the deformation
268
+ between two consecutive time steps. For each surfel s =
269
+ (v,n,c,r,l) ∈ S, we compute its 6D transformation
270
+ ¯W(s)
271
+ based on warp field W,
272
+ ¯W(s) = normalize( ∑
273
+ k∈NG(s)
274
+ w(v, pk)qk),
275
+ (1)
276
+ wherein NG(s) denotes the neighbors nodes of surfel s,
277
+ w(s) is an interpolation parameter, defined as w(s) =
278
+ exp
279
+
280
+ ∥v− pk∥2
281
+ 2 /(2δ 2
282
+ k )
283
+
284
+ , and v is the 3D position of surfel
285
+ s. The local transformation ¯W(s) is then used to describe the
286
+ deformation of surfel s as follows:
287
+ vwarp = ¯W(s)v
288
+ nwarp = rotation
289
+ � ¯W(s)
290
+
291
+ n.
292
+ (2)
293
+ Here, v,n are the vertex and normal of s before warping, and
294
+ vwarp,nwarp are the vertex and normal after the deformation.
295
+ Symbol
296
+ Meaning
297
+ Definition
298
+ Mt
299
+ Measurement at time t.
300
+ III-A, IV-A.1
301
+ St−1
302
+ Surfel geometry from t-1.
303
+ III-A
304
+ Ra
305
+ t−1
306
+ 2D maps rendered from St−1.
307
+ IV-B.1
308
+ Swarp
309
+ t−1
310
+ Warped geometry after non-rigid alignment. IV-B.4
311
+ Rg
312
+ t−1
313
+ 2D maps rendered from Swarp
314
+ t−1 .
315
+ IV-B.1
316
+ TABLE II: Notation sheet.
317
+ IV. PROPOSED APPROACH
318
+ An overview of the proposed method is shown in Fig. 1.
319
+ Mono-STAR uses two parallel threads, a measurement thread,
320
+ and a geometry thread. The first thread is responsible for
321
+ loading measurements Mt and generating semantic labels
322
+ Lm
323
+ t . The geometry thread uses this measurement Mt and the
324
+ alignment rendering Ra
325
+ t−1 to compute an optical-flow OFt.
326
+ Then, Mt,Ra
327
+ t−1,OFt are given to the optimization module that
328
+ then computes the non-rigid deformation Wt. After the non-
329
+ rigid alignment, previous geometry St−1 is warped to Swarp
330
+ t−1 ,
331
+ and the geometry rendering Rg
332
+ t−1 is generated. Finally, Rg
333
+ t−1,
334
+ deformation Wt, warped geometry Swarp
335
+ t−1 , and semantic labels
336
+ Lm
337
+ t are combined to generate the latest geometry St.
338
+ Noticeably, loading Mt takes less time than updating
339
+ geometry St−1 and rendering Ra
340
+ t−1. Semantic segmentation is
341
+ also faster than the combined process of generating optical-
342
+ flow OFt and the non-rigid alignment. Thus, the geometry
343
+ thread fully hides the latency of the measurement thread.
344
+
345
+ Fig. 1: Overview of the proposed system. The system runs in two parallel threads, one for measurement and one for geometry. In each
346
+ time-step t, the measurement thread loads a measurement Mt from images or a camera buffer. Then, a segmentation network generates a
347
+ set of semantic labels Lmt . Once the measurement is loaded on the GPU memory, Mt and previous alignment rendering Ra
348
+ t−1 are fed into
349
+ an optical-flow network to generate the optical-flow OFt from previous geometry St−1 to measurement Mt. Optical-flow OFt, geometry
350
+ rendering Rt and measurement Mt are used to compute warp-field Wt with non-rigid alignment. After the alignment, previous geometry
351
+ St−1 will be warped to Swarp
352
+ t−1 . The fusion rendering map Rg
353
+ t−1 is then rendered from Swarp
354
+ t−1 . Rg
355
+ t−1, Swarp
356
+ t−1
357
+ and semantic labels Lmt are used
358
+ to generate the updated geometry St, deformation graph Gt and the surfel semantic label Lst .
359
+ A. Measurement Thread
360
+ 1) Measurement: We use one Intel RealSense-415 camera
361
+ to collect RGB-D images. Depth images are denoised with a
362
+ Gaussian filter. The maximum frame rate for this module is
363
+ limited to 20 fps to coordinate with other modules. We use
364
+ a double-buffer strategy to hide latency. Specifically, we use
365
+ two buffers B0 and B1 to store measurements. When B0 is
366
+ used by other threads, B1 can read images simultaneously.
367
+ The filtered images are used to construct three maps, V m
368
+ t ,
369
+ Nm
370
+ t , Cm
371
+ t , storing 3D coordinates vm
372
+ i , normal nm
373
+ i , and color cm
374
+ i ,
375
+ respectively, for surfel mi of each pixel in the measurement.
376
+ 2) Segmentation: The segmentation module receives the
377
+ color map Cm
378
+ t
379
+ and returns a semantic label map Lm
380
+ t
381
+ of H
382
+ pre-defined semantic classes. Here, we use two different
383
+ segmentation models, a transformer-based Segmenter Mask
384
+ [20], and a more traditional MaskRCNN [21]. The two
385
+ models are pre-trained on two different datasets, ADE20K
386
+ [22], [23] and COCO-Stuff [24] respectively. We do not
387
+ further train these models on any other dataset. We select
388
+ which one to use based on the types of objects in the scene.
389
+ B. Geometry Thread
390
+ 1) Geometry Rendering: The input of the geometry ren-
391
+ dering pipeline is a geometry S, and the output is the
392
+ rasterized rendering R for geometry S from the current
393
+ camera view. The rendering process to generate R follows
394
+ the classical point cloud rasterizing process [25], where every
395
+ surfel is projected to its nearest pixel position on the camera
396
+ plane based on its 3D coordinates. Each rendering map R
397
+ is composed of four 2D maps. Rt = {Ct,Vt,Nt,It}, where
398
+ Vt,Nt,Ct,It are respectively the vertex map, the normal map,
399
+ the color map and the index map. These maps store the 3D
400
+ coordinates vi, the normal ni, the color ci and the surfel index
401
+ i of the projected surfel si at each pixel.
402
+ At each time-step, the geometry rendering pipeline is
403
+ called twice; once to generate Ra
404
+ t−1 from previous geometry
405
+ St−1 for non-rigid alignment, and once to get Rg
406
+ t−1 with
407
+ warped geometry Swarp
408
+ t−1 for updating the geometry. Rendering
409
+ Rg
410
+ t−1 used for updating the geometry operates on surfel-
411
+ level granularity, whereas Ra
412
+ t−1 used for geometry alignment
413
+ rendering operates on deformation node granularity. Another
414
+ difference between Rg
415
+ t−1 and Ra
416
+ t−1 is resolution, Rg
417
+ t−1 is up-
418
+ sampled by 4 × 4 compared to Ra
419
+ t−1 to prevent different
420
+ surfels from being projected onto the same pixel. Rg
421
+ t−1
422
+ requires a higher resolution for accurate geometry update
423
+ and Ra
424
+ t−1 has a lower resolution for faster optimization.
425
+ 2) Optical Flow:
426
+ The optical flow module receives
427
+ Ca
428
+ t−1,V a
429
+ t−1 from geometry rendering Ra
430
+ t−1, and Cm
431
+ t−1,V m
432
+ t−1 from
433
+ measurement Mt, and generates an optical-flow map OFt.
434
+ OFt predicts the optical-flow from previous geometry St−1 to
435
+ the latest measurement Mt. This prediction is later used for
436
+ registration through non-rigid alignment. We generate OFt
437
+ using a neural network based on the RAFT architecture [8],
438
+ along with additional global motion features as performed
439
+ in GMA [9]. The global motion features provide stability
440
+ for predicting motion features, even in occluded scenes.
441
+ Both RAFT and GMA models were originally trained using
442
+ only RGB images. The optical flow model used in [7]
443
+ shows that using RGB-D images for training provides a
444
+ far more stable flow, even with motion blurring. Thus, our
445
+ model is also trained on RGB-D images from the datasets
446
+ FlyingThings3D [26], Monkaa [26] and Sintel [27], [28].
447
+ 3) Non-rigid alignment:
448
+ Non-rigid alignment is per-
449
+ formed in order to compute non-rigid deformation Wt. This
450
+ step solves a massive optimization problem to warp the
451
+ previous geometry St−1 to a geometry Swarp
452
+ t−1 that fits current
453
+ measurement Mt. We use a Gauss-Seidel solver implemented
454
+ with CUDA to solve this problem, which is summarized as
455
+ minWEtotal(W) = wpicpEpicp(W)+w2DE2D(W)
456
+ +waregEareg(W),
457
+
458
+ Mt
459
+ Mt
460
+ Lm
461
+ Input
462
+ Output
463
+ Sync
464
+ Sync
465
+ Measurement
466
+ St
467
+ (St Gt L)
468
+ -warp
469
+ Lm
470
+ -warp
471
+ R
472
+ Geometry
473
+ St-1
474
+ St-1
475
+ warp
476
+ Ls
477
+ Gt
478
+ Normal
479
+ Vertex
480
+ Vertex
481
+ Normal
482
+ va-1
483
+ Ng
484
+ Sync
485
+ Sync
486
+ t-1
487
+ Color
488
+ Index
489
+ Color
490
+ Index
491
+ Cf-1
492
+ 19-1where wpicp,wof ,wareg are the weights of terms Epicp(W),
493
+ E2D(W) and Eareg(W), explained in the following.
494
+ Registration. Let u = (x,y) be a pixel in measurement map
495
+ u, and let mi = M(u) bet its associated surfel. Let (mi,sΠ(i))
496
+ denote a pair of registered measurement and geometry surfel.
497
+ Π(i) is defined as Π(i) := Ia
498
+ t−1(x − of x
499
+ t ,y − of y
500
+ t ), wherein
501
+ (x,y) = u,(o f x
502
+ t ,o f y
503
+ t ) = OFt(u). Ia
504
+ t−1 ∈ Ra
505
+ t−1 is the index map
506
+ of the rendered geometry.
507
+ PICP Loss. Point-to-point ICP loss is sensitive to dis-
508
+ turbance and outliers, which limits its utility in real-world
509
+ applications. Instead, we use a plane-based ICP (PICP) loss
510
+ to align the differences along the depth direction as follows,
511
+ Epicp(W) = ∑
512
+ mi∈M
513
+ nm
514
+ i ·( ¯W(sΠ(i))vs
515
+ Π(i) −vm
516
+ i ),
517
+ (3)
518
+ wherein vs
519
+ Π(i) is the 3D coordinates of surfel sΠ(i), vm
520
+ i ,nm
521
+ i are
522
+ the 3D coordinates and normal of measurement surfel mi. ¯W
523
+ is defined in Eq. 2.
524
+ 2D Loss. One limitation of the PICP loss is that it cannot
525
+ correctly capture motions within the same plane, such as
526
+ the moving calendar shown in Fig. 6. We thus add to the
527
+ objective function a 2D loss E2D defined as follows,
528
+ P =
529
+
530
+ 1
531
+ 0
532
+ 0
533
+ 0
534
+ 1
535
+ 0
536
+
537
+ (4)
538
+ E2D(W) = ∑
539
+ mi∈M
540
+ ∥P( ¯W(sΠ(i))vs
541
+ Π(i) −vm
542
+ i )∥2.
543
+ (5)
544
+ Here, P is a projection matrix, projecting the 3D difference
545
+ to the camera X-Y plane. This term constrains mi and sΠ(i)
546
+ to be as close as possible on the camera X-Y plane. It is
547
+ worth noting that our proposed 2D loss is different from the
548
+ one proposed in OcclusionFusion [7], where pixel differences
549
+ are used to calculate the 2D loss. The influence of pixel
550
+ differences scales with the distance to the camera, which
551
+ makes the optimization parameters harder to tune.
552
+ Semantic-aware Adaptive Deformation Graph. The tra-
553
+ ditional Embedded Deformation graph (ED-graph) has been
554
+ widely used in non-rigid tracking and non-rigid reconstruc-
555
+ tion. It can describe complicated warping fields with a simple
556
+ data structure and an interpolation strategy. Moreover, the as-
557
+ rigid-as-possible (ASAP) regulation term defined on defor-
558
+ mation nodes provides a continuity guarantee for neighboring
559
+ nodes. However, the ED-graph cannot handle the topology
560
+ changes of different nodes. For example, if we use an ED
561
+ graph to describe a cup being lifted up from a table, as shown
562
+ in Fig. 2 (a), the motion of the cup’s nodes also influences
563
+ and propagates to the table’s nodes. Many previous works
564
+ have attempted to overcome this limitation of the ED-graph
565
+ by proposing a dual deformation graph [18] or a level-
566
+ set-based TSDF fusion mechanism [11]. However, these
567
+ approaches are too complicated or introduce too much over-
568
+ head computation. We propose the Semantic-aware Adaptive
569
+ Deformation Graph (SAD-graph) to address this issue of
570
+ topological changes among objects with different semantic
571
+ classes. More importantly, the proposed algorithm is intuitive
572
+ and requires little extra computation compared with ED-
573
+ graph. Another advantage is that existing ED-graph based
574
+ approaches can be easily upgraded to support SAD-graph.
575
+ The core idea of SAD-graph is that instead of imposing a
576
+ uniform regulation continuity constraint on all deformation
577
+ nodes, constraints of varying adaptive strengths are imposed
578
+ on different edges. A variable weight ωi, j is associated with
579
+ the constraint (defined in Eq. 7) between neighboring nodes
580
+ gi and gj, and the strength of the constraint is systematically
581
+ adjusted. Constraint weight ωgi,gj is a function of lgi,lg j, the
582
+ semantic labels of nodes gi and gj. It is defined as follows,
583
+ ωgi,g j(lgi,lg j) =
584
+
585
+ 0.1,
586
+ if lgi ̸= lg j
587
+ δ k,
588
+ if lgi = lg j = k,k ∈ [1,...,H +1]
589
+ (6)
590
+ where δ k is a constant describing the average rigidness of
591
+ objects belonging to semantic category k, e.g., δtable = 1.0,
592
+ δ human = 0.3, etc. For example, in Fig. 2 (b), since the
593
+ internal rigidity constraint within the cup or the table is much
594
+ larger than the constraint between them, their geometries can
595
+ be accurately reconstructed during the topology separation.
596
+ It is important to note that these constraints are not hard
597
+ because the semantic labels obtained from a neural network
598
+ detector are error-prone.
599
+ Fig. 2: Illustration of the proposed Semantic-aware Adaptive De-
600
+ formation Graph (SAD-graph). The scene describes a cup (brown)
601
+ being lifted up from the table (green). Black edges indicate strong
602
+ continuity constraints, while grey edges indicate weak constraints.
603
+ Adaptive Regulation Loss. We introduce a deformation
604
+ graph, SAD-graph, and a new regulation term, adaptive
605
+ regulation Eareg(W). A semantic-related connection weight
606
+ ω is used to adjust the regulation strength among and within
607
+ different semantic classes as follows,
608
+ Eareg(W) = ∑
609
+ g j∈G
610
+
611
+ gi∈NG(g j)
612
+ ωgi,gj
613
+ ��Tjpj −Tipi
614
+ ��2
615
+ 2 ,
616
+ (7)
617
+ wherein G is the deformation graph, NG(gj) refers to the set
618
+ of neighbors of node gj in the deformation graph, Tj and Ti
619
+ are the transformations defined on nodes gi and gj. pi and pj
620
+ are the 3D coordinate of gi and gj, and ωgi,g j is the weight
621
+ of the connection between nodes gi and gj, defined in Eq. 6.
622
+ 4) Geometry and Graph Update: Once the non-rigid
623
+ deformation is computed, the geometry update process of
624
+ Mono-STAR is similar to SurfelWarp [10]. Thus, we only
625
+ briefly describe that process and we focus on the semantic
626
+ update. This step returns the updated geometry St (the
627
+ reconstructed scene) and the updated graph Gt, both of which
628
+ are needed for processing the scene in the next time-step.
629
+ Updating the Geometry. The previous geometry St−1 is
630
+ warped to Swarp
631
+ t−1 after the non-rigid alignment step. Although
632
+ Swarp
633
+ t−1
634
+ is already close enough to measurement Mt, there
635
+
636
+ Before deformation
637
+ After deformation
638
+ Before deformation
639
+ After deformation
640
+
641
+ X
642
+
643
+ (a) ED-graph
644
+ (b) SAD-graphstill exists a discrepancy between them due to measurement
645
+ noises, emerging surfaces, topology changes, or even track-
646
+ ing failures. The geometry is updated to address this gap
647
+ between the warped geometry Swarp
648
+ t−1
649
+ and measurement Mt.
650
+ There are four steps in total in this process.
651
+ 1. Registration: A projective registration is made between
652
+ measurement Mt and warped geometry Swarp
653
+ t−1
654
+ according to
655
+ rendering map Rg
656
+ t−1.
657
+ 2. Fusion: If a surfel mi ∈ Mt is mapped to s j ∈ Swarp
658
+ t−1
659
+ in
660
+ the registration, mi is merged into sj to average measurement
661
+ noises. The semantic label ls j of sj is defined as a probability
662
+ distribution psj. When mi is fused into sj, ps j is also updated
663
+ by lmi. The update formula for psj is:
664
+ psj(k) = (psj(k)+δm)/∑
665
+ k′
666
+ psj(k′),if k = lmi
667
+ (8)
668
+ psj(k) = ps j(k)/∑
669
+ k′
670
+ psj(k′),otherwise.
671
+ (9)
672
+ Here, δm is the confidence of the measurement.
673
+ 3. Append: If there are no surfels in Swarp
674
+ t−1
675
+ that can be
676
+ registered to mi, mi must belong to a newly observed surface
677
+ or be noise. In the first case, mi will be appended to Swarp
678
+ t−1 .
679
+ The semantic label distribution ps
680
+ i of mi is initialized as:
681
+ ps
682
+ i(k) = δm,if k = lmi; ps
683
+ i(k) = 0,otherwise,
684
+ (10)
685
+ 4. Removal: After each mi ∈ Mt is either fused or ap-
686
+ pended, some surfels sj ∈ Swarp
687
+ t−1
688
+ are left with no correspon-
689
+ dence. A geometry violation test is performed on the remain-
690
+ ing surfels, and those that fail the test are removed [10].
691
+ After the four steps given above, we get the updated
692
+ geometry St for time-step t.
693
+ Updating the Graph. The update of the SAD-graph is
694
+ identical to the update of the traditional ED-graph. The
695
+ update appends new nodes but does not remove existing ones.
696
+ Let Sappend be the set of the appended surfels during the
697
+ geometry update. We first compute the distances between
698
+ every surfel s ∈ Sappend and every node g ∈ Gt−1. Let
699
+ D(s,G) = ming∈G distance(s, g). A surfel s is said to be
700
+ unsupported if D(s,G) > σ, for some threshold σ. We per-
701
+ form a spatially uniform sampling from all the unsupported
702
+ surfels. Sampled surfels are appended to graph G as new
703
+ nodes. The semantic label of node gi, lgi is updated according
704
+ to the semantic labels of NS(gi), neighbor surfels of gi.
705
+ lgi = argmaxk{∑sj∈Ns(gi) δ(lsj,k)}. Here δ(ls j,k) = 1,if lsj =
706
+ k;δ(ls
707
+ j,k) = 0,otherwise.
708
+ V. EXPERIMENTS
709
+ We
710
+ test
711
+ our
712
+ technique
713
+ on
714
+ a
715
+ dataset
716
+ we
717
+ collected
718
+ and a public dataset VolumeDeform [29]. An ablation
719
+ study
720
+ and
721
+ comparisons
722
+ with
723
+ SoTA
724
+ methods
725
+ such
726
+ as
727
+ STAR-no-prior [2] and MaskFusion [4] on challenging
728
+ scenes are presented in this section. Since collecting
729
+ ground-truth geometry and deformation for non-rigid ob-
730
+ jects is extremely challenging, experiments and compar-
731
+ isons in this area are limited to qualitative results [29].
732
+ Supplementary results and resources can be found at
733
+ https://github.com/changhaonan/Mono-STAR-demo.
734
+ A. Performance
735
+ We tested our system on a desktop machine with a
736
+ GeForce RTX 3090 and an AMD-Ryzen 9 5900X. On aver-
737
+ age, measurement loading takes 4 ms and segmentation costs
738
+ 10 ms (Segmenter Mask [20]). The optimization module uses
739
+ 20 ms. The geometry update uses 7 ms. The major bottleneck
740
+ is the optical-flow network, which takes 60 ms. Since the
741
+ latency for the measurement thread is fully hidden by the
742
+ geometry thread, our entire system runs in 11 Hz. If the
743
+ optical-flow runs on a separate graphic card, it would take
744
+ only 26 ms [7], which would double the speed of our system.
745
+ B. Qualitative Results
746
+ 1) Soft objects: Fig. 3 illustrates the non-rigid deforma-
747
+ tion ability of MaskFusion and Mono-STAR. We can clearly
748
+ see that MaskFusion fails to track the deformations of the
749
+ pillow and umbrella, while Mono-STAR correctly captures
750
+ both of them in the reconstructed model, which shows the
751
+ advantage of our technique over MaskFusion in handling
752
+ non-rigid deformation.
753
+ Fig. 3: Comparison on deformable objects with MaskFusion on
754
+ our recorded dataset (top) and VolumeDeform dataset (bottom).
755
+ 2) Fast Motion: Fig. 4 demonstrates Mono-STAR’s abil-
756
+ ity to handle fast motions. The top scene in Fig. 4 shows
757
+ an accident that was recorded during our data collection.
758
+ While we were pushing a cup on the table, the cup hit a
759
+ bump and fell down. The bottom scene is about passing a
760
+ basketball between two hands. Objects in both scenes moved
761
+ very fast. One is 18 frames, and the other is 30 frames.
762
+ Significant motion blur can be observed in both middle
763
+ images. However, Mono-STAR can still capture these fast
764
+ motions and correctly reconstruct the objects at each frame.
765
+
766
+ =
767
+ =
768
+ 150
769
+ Measurement
770
+ MaskFusion
771
+ OursFig. 4: Experiment on fast motion. Pushing down a coffee cup
772
+ (top). Passing a basketball between two hands (bottom). The second
773
+ and fourth row are our 3D scene reconstruction results.
774
+ 3) Resilience to Semantic Segmentation Noises: Fig. 5
775
+ shows how our proposed method can resist noise in semantic
776
+ segmentation. The figures on the left are the RGB measure-
777
+ ment from the beginning and the end frames. The right side
778
+ compares the segmentation from the measurement and the
779
+ segmentation from our reconstruction. Although the ground-
780
+ truth measurement suffers from major segmentation errors,
781
+ where the cup label is completely lost for t > 0, Mono-STAR
782
+ still maintains the correct semantic labels in its reconstruction
783
+ result through semantic fusion.
784
+ Fig. 5: Resilience to semantic label noise. The top sequence is the
785
+ segmentation map Lmt of the measurement. The bottom sequence is
786
+ the segmentation map Lst from our reconstruction technique.
787
+ C. Ablation Study
788
+ 1) 2D Loss: We test Mono-STAR with and without the 2D
789
+ loss on the “adventcalender” dataset from VolumeDeform.
790
+ Fig. 6 shows that the proposed 2D loss E2D can efficiently
791
+ track the motions within a plane. In contrast, tracking without
792
+ E2D fails in this type of motion, which clearly shows the
793
+ effectiveness of the proposed 2D loss.
794
+ 2) SAD-graph: In Fig. 7, we compare the ED-graph with
795
+ the topology-aware ED-graph (STAR-no-prior) and the SAD-
796
+ graph (Mono-STAR). We can see that the ED-graph fails
797
+ Fig. 6: Ablation study on the 2D loss.
798
+ to support the topology change that results from lifting the
799
+ object from the table. Topology-aware ED-graph can separate
800
+ the topology, but it also generates many outliers on the table.
801
+ With the help of the proposed SAD-graph, Mono-STAR can
802
+ conduct a smoother and cleaner separation.
803
+ Fig. 7: Comparing ED-graph (left), topology-aware ED-graph
804
+ (middle, STAR-no-prior [2]), and SAD-graph (right, ours). The
805
+ scene shows a plushy toy being lifted up from the table.
806
+ D. Discussion of Limitations
807
+ Although Mono-STAR shows great potential in many dif-
808
+ ferent aspects, it still has two limitations. First, it relies on the
809
+ optical flow to track fast motions. However, even the state-of-
810
+ art optical flow detector GMA [9] is not always accurate, es-
811
+ pecially when the motion is too fast and the tracked surfaces
812
+ are heavily occluded. Our system can tolerate some noise
813
+ from the GMA optical-flow module. However, if the optical
814
+ flow provides inaccurate predictions for multiple consecutive
815
+ frames, the tracking of the corresponding object may still
816
+ fail. Another drawback of our system is the incompleteness
817
+ of the reconstructed geometry. Our reconstructed geometries
818
+ usually have holes and are not as smooth as TSDF-based
819
+ geometry. The reason is that Surfel-based geometry, unlike
820
+ TSDF-based geometry, is discrete by default. Therefore,
821
+ it is difficult to maintain the smoothness of Surfel-based
822
+ geometry in highly dynamic scenes. These two challenges
823
+ can be addressed in future works.
824
+ VI. CONCLUSION
825
+ We presented Mono-STAR, a single-view solution for
826
+ the semantic-aware STAR problem. Mono-STAR uses a
827
+ novel semantic-aware and adaptive deformation graph for
828
+ simultaneous tracking and reconstruction, and can handle
829
+ topology changes as well as semantic fusion. Experiments
830
+ show that Mono-STAR achieves promising results in non-
831
+ rigid object reconstruction, while resisting to semantic seg-
832
+ mentation errors, and capturing fast motions on various chal-
833
+ lenging scenes. We believe that this system can inspire and
834
+ boost more future research on imitation learning, dexterous
835
+ manipulation, and many other relevant robotics problems.
836
+
837
+ t = 47
838
+ t = 58
839
+ t = 65
840
+ t = 15
841
+ t = 26
842
+ t = 45Human
843
+ Cup
844
+ Background
845
+ Measurement
846
+ t=0
847
+ t=0
848
+ t = 102
849
+ t = 162
850
+ Reconstruction
851
+ t = 162
852
+ 0=1
853
+ t = 102
854
+ t = 162t= 1
855
+ t = 41
856
+ t = 41
857
+ Initial
858
+ without 2D Loss
859
+ Ourst =31
860
+ t =31
861
+ t = 38
862
+ t = 38
863
+ ED-graph
864
+ STAR-no-prior
865
+ OursREFERENCES
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+ Available:
989
+ http:
990
+ //arxiv.org/abs/1603.08161
991
+
29FQT4oBgHgl3EQfGTVE/content/tmp_files/load_file.txt ADDED
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1
+ arXiv:2301.00522v1 [math.RT] 2 Jan 2023
2
+ Irreducible module decompositions of rank 2
3
+ symmetric hyperbolic Kac-Moody Lie algebras by
4
+ sl2 subalgebras which are generalizations of
5
+ principal sl2 subalgebras
6
+ TSURUSAKI Hisanori∗
7
+ Abstract
8
+ There exist principal sl2 subalgebras for hyperbolic Kac-Moody Lie
9
+ algebras.
10
+ In the case of rank 2 symmetric hyperbolic Kac-Moody Lie
11
+ algebras, certain sl2 subalgebras are constructed. These subalgebras are
12
+ generalizations of principal sl2 subalgebras.
13
+ We show that the rank 2
14
+ symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly
15
+ decomposed under the action of this sl2 subalgebras. Furthermore, we
16
+ classify irreducible components of the decomposition. In particular, we
17
+ obtain multiplicities of unitary principal series and complementary series.
18
+ 1
19
+ Introduction
20
+ A nilpotent orbit in a finite dimensional simple Lie algebra g0 is an orbit ob-
21
+ tained by acting on the nilpotent element x of g0 by inner automorphisms. In
22
+ [Dyn57], these are classified by weighted Dynkin diagrams. From the Jacobson-
23
+ Morozov theorem, for a nilpotent element x of g0, we can construct a sl2-triple
24
+ with x as a nilpositive element ([CM93, Theorem 3.3.1]). This makes it equiva-
25
+ lent to classify nilpotent orbits of g0 and to classify sl2 triples in g0 up to inner
26
+ automorphisms. Among the nilpotent orbits of a finite dimensional simple Lie
27
+ algebra, the one whose dimension as an algebraic variety is maximal is called the
28
+ principal nilpotent orbit. Correspondingly, we can construct a principal SO(3)
29
+ subalgebra that is compatible with compact involution ([Kos59]).
30
+ Kac-Moody Lie algebras are generalizations of finite-dimensional simple Lie
31
+ algebras.
32
+ They are classified into three types: finite type, affine type, and
33
+ indefinite type. The finite type Kac-Moody Lie algebras are finite dimensional
34
+ simple Lie algebras. Within indefinite Kac-Moody Lie algebra, there is a class
35
+ called hyperbolic Kac-Moody Lie algebra. A hyperbolic Kac-Moody Lie algebra
36
+ ∗Graduate
37
+ School
38
+ of
39
+ Mathematical
40
+ Sciences,
41
+ University
42
+ of
43
+ Tokyo
44
+ ,
45
+ htsu-
46
+ rusaki1929@gmail.com
47
+ 1
48
+
49
+ is an indefinite type Kac-Moody Lie algebra such that any true subdiagram of
50
+ its Dynkin diagram is of finite or affine type.
51
+ By analogy with the above theory, in [NO01], for a hyperbolic Kac-Moody
52
+ Lie algebra, its principal SO(1, 2) subalgebra was constructed. Note that [GOW02]
53
+ shows that it is possible to construct a principal SO(1, 2) subalgebra for certain
54
+ indefinite Kac-Moody Lie algebra that is not hyperbolic.
55
+ Corresponding to this principal SO(1, 2) subalgebra, we can construct a
56
+ principal sl2-subalgebra in a hyperbolic Kac-Moody Lie algebra. In [Tsu], for
57
+ the rank 2 symmetric hyperbolic Kac-Moody Lie algebras g, the following re-
58
+ sult is obtained. Let the space that the positive real root vectors span be Rg.
59
+ we consider sl2 subalgebras whose nilpositive element exists in Rg. Then we
60
+ can construct certain sl2 subalgebras. These subalgebras are generalizations of
61
+ principal sl2 subalgebra.
62
+ In this paper, for an sl2 subalgebra of rank 2 symmetric hyperbolic Kac-
63
+ Moody Lie algebra g constructed in [Tsu], we show g is decomposed into irre-
64
+ ducible sl2-modules by its action on g.
65
+ We are going to more details. Let s be an sl2 subalgebra constructed in
66
+ [Tsu]. Let H, X, Y be an sl2 triple and assume that s is spanned by H, X, Y .
67
+ Let the Chevalley generators of g be ei, fi, hi, (i = 0, . . . , n − 1). Let hR be the
68
+ R-span of Chevalley generators. From [Kac90, Theorem 2.2], g has a C-valued
69
+ nondegenerate invariant symmetric bilinear form (· | ·) called the standard form.
70
+ An antilinear automorphism ω0 of g, called compact involution, is determined
71
+ by
72
+ ω0(ei) = −fi,
73
+ ω0(fi) = −ei
74
+ (i = 0, . . . , n − 1),
75
+ ω0(h) = −h
76
+ (h ∈ hR).
77
+ From [Kac90, §2.7], we can determine a nondegenerate Hermitian form (· | ·)0
78
+ on g with (x | y)0 = −(ω0(x) | y).
79
+ s-module V is called unitarizable if the following conditions are satisfied.
80
+ (1) (· | ·)0 on V is positive definite.
81
+ (2) for v1, v2 ∈ V , the conditions as follows are satisfied.
82
+ ([X, v1], v2)0 = −(v1, [Y, v2])0,
83
+ ([H, v1], v2)0 = −(v1, [H, v2])0.
84
+ Theorem 1.1 (Theorem 4.5). g can be decomposed into a direct sum of irre-
85
+ ducible s-modules such that s itself is one of the direct summand. All of these
86
+ modules except for s are unitarizable.
87
+ Also, we classify how many highest weight modules, lowest weight modules,
88
+ and modules that are neither highest weight module nor lowest weight module
89
+ appear in this decomposition. We regard a root sα1 + tα2 as a point (s, t) in
90
+ xy-plane, and We define a region L, −L in xy-plane in §5. If a root α satisfies
91
+ α(H) ∈ (0, 2), α ∈ L. If a root α satisfies α(H) ∈ (−2, 0), α ∈ −L.
92
+ 2
93
+
94
+ Theorem 1.2 (Theorem 7.1). We consider an irreducible decomposition of g
95
+ by the action of s.
96
+ (1) Let M is an irreducible component of decomposition of g, which contain
97
+ a root space for a real root in L. Then, M is an unitary principal or
98
+ complementary series representation.
99
+ (2) (cf. [Tsu, Proposition 7.3]) There is an unitary principal series represen-
100
+ tation containing an 1-dimensional space in h.
101
+ (3) g is decomposed into a direct sum of s-submodules described in (1) and (2)
102
+ above, s itself, irreducible lowest weight modules, and irreducible highest
103
+ weight modules.
104
+ We also discuss how to calculate multiplicities of irreducible highest or low-
105
+ est modules (§7). Furthermore, we classified irreducible components which are
106
+ neither highest weight modules nor lowest weight modules, as either unitary
107
+ principal or complementary series representations.
108
+ Theorem 1.3 (Theorem 8.11). We consider irreducible components which are
109
+ neither highest weight modules nor lowest weight modules and contain root
110
+ vectors about real roots in L, obtained by Theorem 7.1. The irreducible com-
111
+ ponents are complementary series representations, except those described in
112
+ Lemma 8.5 and Lemma 8.10. For the exceptions, the irreducible components
113
+ are unitary principal series representations.
114
+ 2
115
+ General theory of Kac-Moody Lie algebras
116
+ Let g be a symmetrizable Kac-Moody Lie algebra on C. Let A be the Cartan
117
+ matrix of g and let A be an n × n matrix. Let h be a Cartan subalgebra of g.
118
+ Let the Chevalley generators of g be ei, fi, hi, (i = 0, . . . , n − 1). Let hR be the
119
+ R-span of Chevalley generators.
120
+ From [Kac90, Theorem 2.2], g has a C-valued nondegenerate invariant sym-
121
+ metric bilinear form (· | ·) called the standard form.
122
+ An antilinear automorphism ω0 of g, called compact involution, is deter-
123
+ mined by
124
+ ω0(ei) = −fi,
125
+ ω0(fi) = −ei
126
+ (i = 0, . . . , n − 1),
127
+ ω0(h) = −h
128
+ (h ∈ hR).
129
+ From [Kac90, §2.7], we can determine a nondegenerate Hermitian form (· | ·)0
130
+ on g with (x | y)0 = −(ω0(x) | y).
131
+ Write n+ for a subalgebra of g generated by ei’s and n− for a subalgebra of
132
+ g generated by fi’s.
133
+ We can construct a 3-dimensional subalgebra of g which is spanned by three
134
+ non-zero elements J+ ∈ n+, J− ∈ n−, J3 ∈ h. J+, J− and J3 satisfy
135
+ [J3, J±] = ±J±,
136
+ 3
137
+
138
+ [J+, J−] = −J3.
139
+ This subalgebra is called SO(1, 2) subalgebra of g.
140
+ A representation of SO(1, 2) subalgebra is called unitary if the representation
141
+ space V has a Hermitian scalar product (·, ·) and the following two conditions
142
+ are satisfied.
143
+ (1) The actions of J+ and J− are adjoint each other, and the action of J3 is
144
+ self-adjoint. That is, for x, y ∈ V , we have
145
+ ([J+, x], y) = (x, [J−, y]),
146
+ ([J3, x], y) = (x, [J3, y]).
147
+ (2) Hermitian scalar product (·, ·) is positive definite.
148
+ When considering the adjoint action of an SO(1, 2) subalgebra of g to g, from
149
+ [Tsu, Lemma 3.1, Lemma 3.2], we can see that the adjoint action satisfying the
150
+ condition (1) to be unitary and J− = −ω0(J+) are equivalent. In [NO01], prin-
151
+ cipal SO(1, 2) subalgebras for hyperbolic Kac-Moody Lie algebras are studied.
152
+ Principal SO(1, 2) subalgebra satisfies that J− = −ω0(J+).
153
+ When three non-zero elements X ∈ n+, Y ∈ n−, H ∈ h of g satisfy
154
+ [H, X] = 2X,
155
+ [H, Y ] = −2Y,
156
+ [X, Y ] = H,
157
+ these three elements are called sl2-triple of g. A g-subalgebra that these elements
158
+ span is called sl2 subalgebra. The SO(1, 2) subalgebras and the sl2 subalgebras
159
+ can be converted by
160
+ J+ =
161
+ 1
162
+
163
+ 2X,
164
+ J− = − 1
165
+
166
+ 2Y,
167
+ J3 = 1
168
+ 2H.
169
+ The condition J− = −ω0(J+) in SO(1, 2) subalgebra is converted to Y =
170
+ ω0(X) in sl2 subalgebra. In the following paper, we consider sl2 subalgebra
171
+ that satisfies Y = ω0(X).
172
+ 4
173
+
174
+ 3
175
+ sl2-triples of rank 2 hyperbolic symmetric Lie
176
+ algebra that is compatible to compact involu-
177
+ tion
178
+ Let a be an integer that satisfies a ≥ 3, and let g be a hyperbolic Kac-Moody
179
+ Lie algebra on C such that the Cartan matrix of g is
180
+
181
+ 2
182
+ −a
183
+ −a
184
+ 2
185
+
186
+ .
187
+ Let α0, α1 be the simple roots of g.
188
+ Let {Fn} be the sequence of numbers
189
+ determined by F0 = 0, F1 = 1, Fk+2 = aFk+1 − Fk.
190
+ Lemma 3.1 ([KM95, Proposition 4.4]). The real positive roots of g are of the
191
+ form
192
+ α = Fk+1α0 + Fkα1
193
+ or
194
+ β = Fkα0 + Fk+1α1.
195
+ We distinguish these roots as type α and type β, and we also distinguish
196
+ root vectors belonging to each root as type α and type β (cf. [Tsu, §4]).
197
+ Let X be an element of the space which real positive root vectors span. Then
198
+ X can be written as
199
+ X =
200
+
201
+ k
202
+ ckEk,
203
+ (k ∈ {0, . . ., nX − 1}, ck ∈ C, ck ̸= 0, Ek ∈ gβk, Ek ̸= 0)
204
+ where βk (k ∈ {0, . . . , nX − 1}) are distinct real roots and nX is a positive
205
+ integer.
206
+ We call this nX the length of X. Then the following holds.
207
+ Lemma 3.2 ([Tsu, Theorem 5.8]). Let X be an element in the space which
208
+ real positive root vectors span.
209
+ (1) When the length of X is 1 or more than 3, X, Y = ω0(X), H = [X, Y ]
210
+ do not form sl2-triple.
211
+ (2) Suppose the length of X is 2 and E0, E1 are real positive root vectors
212
+ of different types (in the sense of α-type and β-type). Then, taking the
213
+ appropriate c0, c1 ∈ C, X = c0E0 + c1E1, Y = ω0(X), and H = [X, Y ]
214
+ form sl2-triple. In particular, c0, c1 can be chosen so that c0, c1 ∈ R.
215
+ Lemma 3.3 ([Tsu, Theorem 6.4]). Take ⟨H, X, Y ⟩ in Lemma 3.2, (2).
216
+ Let
217
+ X = c0E0 +c1E1, where E0 is type α and E1 is type β. From Lemma 3.1, using
218
+ integers i, j ≥ 0, we can write E0 ∈ gFi+1α0+Fiα1, E1 ∈ gFjα0+Fj+1α1. If and
219
+ only if i = j − 1, j, j + 1, H is dominant.
220
+ 5
221
+
222
+ 4
223
+ Irreducible decomposition of g as an sl2 mod-
224
+ ule
225
+ In this section, we consider an sl2-subalgebra s = ⟨H, X, Y ⟩ of g, which satisfies
226
+ the following conditions.
227
+ (1) H ∈ h and H is dominant.
228
+ (2) X is in the space which is spanned by positive root vectors.
229
+ (3) Y = ω0(X).
230
+ We show that g is decomposed to irreducible modules by the action of s.
231
+ s-module V is called unitarizable if the following conditions are satisfied.
232
+ (1) (· | ·)0 on V is positive definite.
233
+ (2) for v1, v2 ∈ V , the conditions as follows are satisfied.
234
+ ([X, v1], v2)0 = −(v1, [Y, v2])0,
235
+ ([H, v1], v2)0 = −(v1, [H, v2])0.
236
+ When Y = ω0(X), the condition (2) are automatically satisfied.
237
+ Therefore,
238
+ (· | ·)0 is positive definite on V if and only if V is unitarizable.
239
+ First, we put
240
+ U = {x ∈ g | ∀y ∈ s (x | y)0 = 0}.
241
+ U is closed under the action of s, and g = s ⊕ U.
242
+ Lemma 4.1. (· | ·)0 is positive definite on U.
243
+ Proof. From [Kac90, Theorem 11.7], (· | ·)0 is positive definite on n+ ⊕ n−. The
244
+ sign of (· | ·)0 on h is (n − 1, 1). Since s itself is not unitarizable, when we write
245
+ h = s ⊕ h′, (· | ·)0 is not positive definite on s. Therefore, (· | ·)0 is positive
246
+ definite on h′. Since U = h′ ⊕ n+ ⊕ n−, (· | ·)0 is positive definite on U.
247
+ Lemma 4.2. Consider a subspace V of U that is closed under the action of H.
248
+ Let V ⊥ be the subspace of U orthogonal to V with respect to the Hermitian
249
+ form (· | ·)0. Then U = V ⊕ V ⊥.
250
+ Proof. We consider the eigenspace decomposition of U by H. Let Uλ be the
251
+ eigenspace for λ and write
252
+ U =
253
+
254
+ λ∈C
255
+ Uλ.
256
+ Since H is a Hermitian operator on (· | ·)0, Uλ and Uµ are orthogonal with
257
+ respect to this inner product if λ ̸= µ.
258
+ Since H is dominant, Uλ is finite-
259
+ dimensional. For each λ, V also inherits the eigenspace decomposition of U.
260
+ 6
261
+
262
+ Let Vλ be an eigenspace of V for λ, and V can be written as a direct sum of
263
+ Vλ’s. Let
264
+ V ′
265
+ λ = {v ∈ Vλ | ∀x ∈ Vλ (v | x)0 = 0},
266
+ and
267
+ V ′ =
268
+
269
+ λ∈C
270
+ V ′
271
+ λ.
272
+ Vλ is finite dimensional. From Lemma 4.1, (· | ·)0 is positive definite on U. Thus
273
+ we have Uλ = Vλ ⊕ V ′
274
+ λ. Therefore, we have U = V ⊕ V ′ and V ′ = V ⊥.
275
+ In the following, we show that U can be decomposed into irreducible modules
276
+ by the action of s.
277
+ Lemma 4.3. Any non-zero sl2-submodule V of U includes an irreducible sub-
278
+ module.
279
+ Proof. Take the eigenspace decomposition of U by the action of H. V is also
280
+ decomposed into eigenspaces with this decomposition, and each eigenspace of
281
+ V is finite-dimensional. We regard H as a linear transform on V and take some
282
+ eigenvalue λ of H on V . Let U(sl2) be an universal enveloping algebra of sl2.
283
+ Considering the Casimir element C of U(sl2), it preserves Vλ. Since Vλ is finite-
284
+ dimensional, there exists an eigenvector of C. Let v denote this. Consider the
285
+ sl2-submodule generated by v, which includes an irreducible submodule.
286
+ Theorem 4.4. U can be decomposed into direct sum of irreducible s-modules,
287
+ and all of these modules are unitarizable.
288
+ Proof. We consider a set of irreducible submodules of U such that these sub-
289
+ modules are orthogonal to each other with respect to (· | ·)0. Let T be the
290
+ set. We order the elements of T by inclusion. Then T is non-empty and in-
291
+ ductively ordered. Therefore, from Zorn’s lemma, T has a maximal element.
292
+ Take a maximal element of T and denote it by M. Consider the direct sum
293
+ of all submodules belonging to M. Let M denote this sum. Suppose U ̸= M,
294
+ we derive the contradiction. Since M is a subspace of U which is closed by the
295
+ action of H, from Lemma 4.2, we have U = M ⊕ M ⊥. Since M ⊥ is non-zero
296
+ sl2 submodule of U, from Lemma 4.3, M ⊥ includes an irreducible submodule.
297
+ Let W denote this. we have M ∪ {W} ∈ T , that is contradict the maximality
298
+ of M. Therefore, we have U = M, and U can be decomposed into direct sum
299
+ of irreducible submodules. Combining this with Lemma 4.1, we can also see the
300
+ unitarizability of the modules.
301
+ Theorem 4.5. g can be decomposed into direct sum of irreducible s-modules,
302
+ which consists s itself. All of these modules except for s are unitarizable.
303
+ 7
304
+
305
+ 5
306
+ sl2 modules in g
307
+ In the following, we consider what kind of modules appear when g is decom-
308
+ posed into irreducible s-modules. In particular, we consider how many unitary
309
+ principal or complementary series representations.
310
+ For a lie algebra a, Let U(a) be the universal enveloping algebra of a. Let
311
+ V be an irreducible s-module which is an irreducible component of g.
312
+ The
313
+ Casimir element C of U(s) acts on V by constant multiplication. Let µ be this
314
+ constant. From [HT92, Chapter II, Corollary 1.1.11], for an eigenvalue λ0 ∈ C
315
+ of H on V , some interval I ⊂ Z exists, and V can be expressed as a direct sum
316
+ of 1-dimensional eigenspaces such that the eigenvalues of H are λk = λ0 + 2k
317
+ (k ∈ I). From [HT92, Chapter II, Theorem 1.1.13], for an eigenvalue λ of H on
318
+ V , we define s1(k) for an integer k as
319
+ s1(k) = 8µ − (λ + 2k − 1)2 + 1
320
+ 4
321
+ .
322
+ (A)
323
+ We take an element vk of the eigenspace of V with respect to an eigenvalue
324
+ λ + 2k. Then we have X(Y vk) = s1(k)vk. If k ∈ Z such that s1(k) = 0 does
325
+ not exist, then V is an irreducible module that is neither highest weight module
326
+ nor lowest weight module. If there exists a k ∈ Z such that s1(k) = 0, V is a
327
+ highest weight module or a lowest weight module.
328
+ Let W be the Weyl group of g. Using Lemma 3.2, we may write H, X, Y in
329
+ s as follows.
330
+ X = c0w0(ep) + c1w1(eq)
331
+ (c0, c1 ∈ R, w0, w1 ∈ W, (p, q) ∈ {(0, 1), (0, 0), (1, 1)}),
332
+ Y = −c0w0(fp) − c1w1(fq),
333
+ H = −c0w0(hp) − c1w1(hq).
334
+ Let ks, ls, ms, ns be real numbers such that c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈
335
+ gmsα0+nsα1. From Lemma 3.3, we can write ks = Fi+1, ls = Fi, ms = Fj, ns =
336
+ Fj+1 with integers i, j ≥ 0, and furthermore, i ∈ {j − 1, j, j + 1}.
337
+ When we take the root vector E ∈ gsα0+tα1 with s, t ∈ Z, we want to find
338
+ out which of the three types of modules E generates under the action of s.
339
+ We define L in the xy-plane as follows. L is a region satisfying x ≥ 0, y ≥
340
+ 0, (x, y) ̸= (0, 0), x2 − axy + y2 ≤ 1 and the following conditions.
341
+ x < ks = Fi+1,
342
+ (when i = j − 1)
343
+ x + y < ks + ls = Fi + Fi+1,
344
+ (when i = j)
345
+ y < ls = Fi.
346
+ (when i = j + 1)
347
+ If we take the root sα0 + tα1 with s, t ∈ Z, then from [KM95, Cor 4.3], the
348
+ point in the xy-plane given by (s, t) is in the interior or on the boundary of the
349
+ hyperbola x2 − axy + y2 = 1. Let hC be this hyperbola. Let λ ∈ R as the value
350
+ for which HE = λE. We have λ = (sα0 + tα1)(H). λ ∈ (0, 2) if and only if
351
+ (s, t) ∈ L. In the following, we regard a root sα0 + tα1 as a point (s, t) in the
352
+ xy-plane.
353
+ 8
354
+
355
+ Figure
356
+ 1: Imaginary roots and real roots in L, a = 3, X = c0r0r1(e0) +
357
+ c1r1r0(e1)
358
+ 0
359
+ 2
360
+ 4
361
+ 6
362
+ 8
363
+ 0
364
+ 2
365
+ 4
366
+ 6
367
+ 8
368
+ x
369
+ y
370
+ imaginary roots
371
+ real roots
372
+ Lemma 5.1. We consider the hyperbola hC on the xy-plane.
373
+ The hC was
374
+ represented by x2 − axy + y2 = 1. Let lb be a line represented by the function
375
+ y = −x+ b with some real number b ≥ 0. There are two intersections of hC and
376
+ lb. Let p1 and p2 be these points. Let db be a distance between p1 and p2. db is
377
+ strictly monotonically increasing with respect to b ≥ 0. The same result holds
378
+ when lb is a line represented by y = b or x = b.
379
+ Proof. First, we consider the case where lb is represented by y = −x + b. Cal-
380
+ culating the y-coordinates of p1, p2 gives
381
+ y = (a + 2)b ±
382
+
383
+ (a + 2)(a − 2)b2 + 4(a + 2)
384
+ 2(a + 2)
385
+ .
386
+ Therefore, we have
387
+ db =
388
+
389
+ 2 ·
390
+
391
+ (a + 2)(a − 2)b2 + 4(a + 2)
392
+ a + 2
393
+ .
394
+ This db is strictly monotonically increasing with respect to b ≥ 0.
395
+ Next, we consider the case where lb is represented by y = b. Calculating the
396
+ x-coordinates of p1, p2 gives
397
+ x = ab ±
398
+
399
+ (a2 − 4)b2 + 4
400
+ 2
401
+ .
402
+ Therefore, we have
403
+ db =
404
+
405
+ (a2 − 4)b2 + 4.
406
+ This db is strictly monotonically increasing with respect to b ≥ 0. The same
407
+ argument is presented when lb is a line represented by x = b.
408
+ 9
409
+
410
+ Let R be the interior of hC and hC itself. For s, t ∈ Z, (s, t) is a root if and
411
+ only if (s, t) ∈ R.
412
+ Lemma 5.2. If (x, y) ∈ L ∪ −L, then neither (x + ks − ms, y + ls − ns) nor
413
+ (x − ks + ms, y − ls + ns) are roots.
414
+ Proof. First we assume (x, y) ∈ L. The points (ks, ls) and (ms, ns) are on the
415
+ hyperbola hC. Let l1 be the line connecting these two points. Using some real
416
+ number b > 0, l1 is represented by y = −x+ b when i = j, y = b when i = j − 1,
417
+ and x = b when i = j + 1. Let l2 be a line parallel to l1 and passing through
418
+ (x, y). Using some real number 0 < b′ < b, l2 is represented by y = −x + b′
419
+ when i = j, y = b′ when i = j − 1, and x = b′ when i = j + 1. Let p11, p12
420
+ be intersections of hC and l1. Let d1 be the distance between p11 and p12. Let
421
+ p21, p22 be intersections of hC and l2. Let d2 be the distance between p21 and p22.
422
+ From Lemma 5.1, we have d1 > d2. The distance between (ks, ls) and (ms, ns)
423
+ is d1. The distance between (x, y) and (x+ks −ms, y+ls −ns) is also d1. These
424
+ two points are on l2. The length of the part of l2 that is inside the hyperbola
425
+ is d2 < d1. From the fact that (x, y) is inside hC, (x + ks − ms, y + ls − ns) is
426
+ outside the hyperbola. Therefore, (x + ks − ms, y + ls − ns) is not in R. The
427
+ same argument for (x − ks + ms, y − ls + ns) shows that it is not in R. From
428
+ symmetry, the case when (x, y) ∈ −L is also shown.
429
+ Lemma 5.3. For a point (s, t) ∈ L corresponding to the root, we consider the
430
+ root vector E ∈ gsα0+tα1. Then [X, [Y, E]] ∈ gsα0+tα1.
431
+ Proof. We have X = c0w0(ep) + c1w1(eq), Y = −c0w0(fp) − c1w1(fq). Also we
432
+ have c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈ gmsα0+nsα1. Then we have
433
+ [X, [Y, E]] ∈ gsα0+tα1 + g(s−ks+ms)α0+(t−ls+ns)α1 + g(s−ms+ks)α0+(t−ns+ls)α1.
434
+ Since (s, t) is a root, from Lemma 5.2, (s − ks + ms, t − ls + ns) and (s − ms +
435
+ ks, t − ns + ls) are not roots. Therefore, we have g(s−ks+ms)α0+(t−ls+ns)α1 +
436
+ g(s−ms+ks)α0+(t−ns+ls)α1 = 0, and [X, [Y, E]] ∈ gsα0+tα1.
437
+ We consider the Casimir element C of U(s). We can write C = 1
438
+ 8H2 − 1
439
+ 4H +
440
+ 1
441
+ 2XY .
442
+ Lemma 5.4. C acts on a root space as endomorphism. The action is diago-
443
+ nalizable.
444
+ Proof. From Lemma 5.3, C acts on the root spaces as endomorphism. Since g
445
+ is completely reducible as an s-modules, the action on the root space is diago-
446
+ nalizable.
447
+ Lemma 5.5. For a point (s, t) ∈ L corresponding to the root, we can take the
448
+ root vector E ∈ gsα0+tα1 such that E is an eigenvector of the Casimir element
449
+ C, and E generates an irreducible s-module.
450
+ Proof. From Lemma 5.4, we have the lemma.
451
+ 10
452
+
453
+ From Lemma 5.5, if we decompose g by the action of s, the decomposition
454
+ is compatible with the root space decomposition in the root in L.
455
+ We consider how many unitary principal or complementary series represen-
456
+ tations appear in the decomposition of g. Since the set of eigenvalues of unitary
457
+ principal or complementary series representations is {λ + 2k | k ∈ Z} for some
458
+ λ, such a module must contain an eigenspace such that its eigenvalue lie on
459
+ [0, 2). Therefore, we consider the root vector of H such that the eigenvalue λ of
460
+ H satisfies λ ∈ [0, 2).
461
+ If λ = 0, i.e., s = t = 0, Since the dimension of h is 2, there are two
462
+ irreducible components of V which have 0-eigenspace (cf. [Tsu, §7]). Since one
463
+ is sl2 itself, we consider the other module. The casimir element C acts on this
464
+ module by a constant multiple (let µ times). If k satisfies s1(k) = 0, we get
465
+ 8µ+ 1 = (2k − 1)2. Since µ < −1 from [Tsu, Proposition 7.3], the left hand side
466
+ is less than 0. Therefore, there is no integral solution to s1(k) = 0, and this is
467
+ an irreducible module that is neither highest weight module nor lowest weight
468
+ module. In particular, this module is an unitary principal series representation.
469
+ In the following, we consider the case of λ ∈ (0, 2). In this case, (s, t) is a
470
+ root in L. We compute [X, [Y, E]]. Since Y = −c0w0(fp) − c1w1(fq), we have
471
+ [Y, E] = [−c0w0(fp), E] + [−c1w1(fq), E].
472
+ We have also
473
+ [−c0w0(fp), E] ∈ g(s−ks)α0+(t−ls)α1,
474
+ [−c1w1(fq), E] ∈ g(s−ms)α0+(t−ns)α1.
475
+ If [−c0w0(fp), E] and [−c1w1(fq), E] are not 0, then the eigenvalue of H for them
476
+ must be in the (−2, 0) interval. we consider root vectors which the eigenvalue
477
+ of H are in the (−2, 0). Since R = −R, the roots with respect to these root
478
+ vectors are −L. From Lemma 5.2, if we take two points such that the difference
479
+ is (ks − ms, ls − ns) and one of which is a root in −L, then the other is not a
480
+ root. Now we have ((s − ms) − (s − ks), (t − ns) − (t − ls)) = (ks − ms, ls − ns).
481
+ Therefore, we know that at least one of [−c0w0(fp), E], [−c1w1(fq), E] is zero.
482
+ When both of these are 0, we have [Y, E] = 0 and from the fact that C =
483
+ 1
484
+ 8H2 − 1
485
+ 4H + 1
486
+ 2XY , we can write 8µ = λ2 − 2λ.
487
+ When [−c0w0(fp), E] ̸= 0, i.e., (s − ks, t − ls) ∈ R, we have
488
+ [X, [Y, E]] = [c0w0(ep), [−c0w0(fp), E]]
489
+ = [E, [−c0w0(fp), c0w0(ep)]] + [−c0w0(fp), [c0w0(ep), E]].
490
+ We define ps ∈ C by [−c0w0(fp), [c0w0(ep), E]] = psE, then we have
491
+ [X, [Y, E]] = [E, c2
492
+ 0w0(hp)] + psE.
493
+ When ps = 0, we have
494
+ [X, [Y, E]] = −[c2
495
+ 0w0(hp), E].
496
+ 11
497
+
498
+ Therefore in this case, if we let −[c2
499
+ 0w0(hp), E] = k0E, then we have 8µ =
500
+ λ2 − 2λ + 4k0.
501
+ When [c0w0(ep), E] = 0, i.e., (s + ks, t + ls) ̸∈ R, we have ps = 0.
502
+ To summarize the above, we take an irreducible decomposition of g by s. let
503
+ sα0 +tα1 be a root in L. Let E ∈ gsα0+tα1 such that E generates an irreducible
504
+ component of g. Let C be the Casimir element of U(s), and Let µ be a complex
505
+ number such that CE = µE. Let k0 and ps be complex numbers satisfying
506
+ [−c2
507
+ 0w0(hp), E] = k0E,
508
+ [−c0w0(fp), [c0w0(ep), E]] = psE.
509
+ If (s − ms, t − ns) ̸∈ R, we have
510
+ 8µ =
511
+
512
+
513
+
514
+
515
+
516
+ λ2 − 2λ
517
+ ((s − ks, t − ls) ̸∈ R) ,
518
+ λ2 − 2λ + 4k0
519
+ ((s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R) ,
520
+ λ2 − 2λ + 4k0 + ps
521
+ ((s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R) .
522
+ If (s − ks, t − ls) ̸∈ R and not necessarily (s − ms, t − ns) ̸∈ R, we have
523
+ 8µ =
524
+
525
+
526
+
527
+
528
+
529
+ λ2 − 2λ
530
+ ((s − ms, t − ns) ̸∈ R) ,
531
+ λ2 − 2λ + 4k0
532
+ ((s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R) ,
533
+ λ2 − 2λ + 4k0 + ps
534
+ ((s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R) .
535
+ Solving
536
+ s1(k) = 8µ − (λ + 2k − 1)2 + 1
537
+ 4
538
+ = 0
539
+ for k on R, we obtain that
540
+ k =
541
+
542
+
543
+
544
+
545
+
546
+
547
+
548
+
549
+
550
+
551
+
552
+
553
+
554
+
555
+
556
+
557
+
558
+
559
+
560
+
561
+
562
+
563
+
564
+
565
+
566
+ 0, 1 − λ
567
+ ((s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R) ,
568
+ 1 − λ ±
569
+
570
+ (λ − 1)2 + 4k0
571
+ 2
572
+
573
+
574
+
575
+ (s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R
576
+ or
577
+ (s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R
578
+
579
+
580
+  ,
581
+ 1 − λ ±
582
+
583
+ (λ − 1)2 + 4k0 + ps
584
+ 2
585
+
586
+
587
+
588
+ (s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R
589
+ or
590
+ (s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R
591
+
592
+
593
+  .
594
+ When (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R, since (s, t) ∈ L, we have
595
+ 1 − λ ∈ (−1, 1). Therefore, we know that the only integral solution of s1(k) = 0
596
+ is 0. In this case E belongs to an irreducible lowest weight module.
597
+ 6
598
+ Classification by roots
599
+ Based on the previous section, we classify the root (s, t) in L. We define the
600
+ types of roots as follows.
601
+ 12
602
+
603
+ (1) We say that (s, t) is of type A when (s−ks, t−ls) ̸∈ R and (s−ms, t−ns) ̸∈
604
+ R.
605
+ (2) We say that (s, t) is of type B when
606
+
607
+
608
+
609
+
610
+
611
+ (s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R
612
+ or
613
+ (s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R
614
+
615
+
616
+
617
+
618
+
619
+ .
620
+ (3) We say that (s, t) is of type C when
621
+
622
+
623
+
624
+
625
+
626
+ (s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R
627
+ or
628
+ (s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R
629
+
630
+
631
+
632
+
633
+
634
+ .
635
+ All roots belong to one of the above types. We put f(x, y) = x2 − axy + y2 for
636
+ x, y ∈ R. From [KM95, Cor 4.3], for s, t ∈ Z, (s, t) ̸= (0, 0), (s, t) is a real root if
637
+ and only if f(s, t) = 1, and (s, t) is an imaginary root if and only if f(s, t) < 1.
638
+ Lemma 6.1. For x, y, x′, y′ ∈ R, if there exists w ∈ W such that (x′, y′) =
639
+ w(x, y), then f(x′, y′) = f(x, y).
640
+ Proof. It is sufficient to check the case w = r0 and the case w = r1. From the
641
+ symmetry, it is sufficient to check the case w = r0. In this case, from the fact
642
+ that x′ = ay − x and y′ = y, we have
643
+ f(x′, y′) = f(ay − x, y)
644
+ = (ay − x)2 − ay(ay − x) + y2
645
+ = x2 − axy + y2
646
+ = f(x, y).
647
+ First, we know the following results on real roots.
648
+ Lemma 6.2. If (s, t) is a real root in L and s > t, then f(s − ks, t − ls) ≤ 0.
649
+ Also, If (s, t) is a real root in L and s < t, then f(s − ms, t − ns) ≤ 0.
650
+ Proof. From symmetry, it is sufficient to show f(s − ks, t − ls) ≤ 0 when s > t.
651
+ We can write s = Fc+1, t = Fc with c ≥ 0 being an integer. Since ks = Fi+1
652
+ and ls = Fi, we have c < i. Let dic = i − c. From Lemma 6.1, by acting r0 and
653
+ r1 on (s − ks, t − ls), we know that
654
+ f(s − ks, t − ls) = f(Fc+1 − Fi+1, Fc − Fi)
655
+ = f(r0(Fc+1 − Fi+1, Fc − Fi))
656
+ = f(Fc−1 − Fi−1, Fc − Fi)
657
+ = f(r1(Fc−1 − Fi−1, Fc − Fi))
658
+ = f(Fc−1 − Fi−1, Fc−2 − Fi−2)
659
+ = · · ·
660
+ 13
661
+
662
+ =
663
+
664
+ f(F1 − Fdic+1, F0 − Fdic)
665
+ (when c is even)
666
+ f(F0 − Fdic, F1 − Fdic+1)
667
+ (when c is odd)
668
+ = f(F1 − Fdic+1, F0 − Fdic).
669
+ Since F1 = 1, F0 = 0, we have
670
+ f(s − ks, t − ls) = f(1 − Fdic+1, −Fdic)
671
+ = 2 − aFdic + 2Fdic−1
672
+ < 2 − 2(Fdic − Fdic−1)
673
+ ≤ 0.
674
+ Lemma 6.3. If (s, t) is a real root in L, then (s, t) is of type B.
675
+ Proof. First we show that (s, t) is not of type A. From the fact that (s, t) is
676
+ a real root and from symmetry, we can write s = Fc+1, t = Fc with c ≥ 0
677
+ being an integer. From ks = Fi+1, ls = Fi, we have c < i. From Lemma 6.2,
678
+ f(s − ks, t − ls) ≤ 0. Therefore, (s − ks, t − ls) ∈ R and so we know that (s, t)
679
+ is not of type A.
680
+ Next, we show that (s, t) is of type B. To show this, we need to show that
681
+ (s + ks, t + ls) ̸∈ R. We show f(s + ks, t + ls) > 1. Let dic = i − c. From
682
+ Lemma 6.1, by acting r0 and r1 on (s + ks, t + ls), we have
683
+ f(s + ks, t + ls) = f(Fc+1 + Fi+1, Fc + Fi)
684
+ = f(r0(Fc+1 + Fi+1, Fc + Fi))
685
+ = f(Fc−1 + Fi−1, Fc + Fi)
686
+ = f(r1(Fc−1 + Fi−1, Fc + Fi))
687
+ = f(Fc−1 + Fi−1, Fc−2 + Fi−2)
688
+ = · · ·
689
+ =
690
+
691
+ f(F1 + Fdic+1, F0 + Fdic)
692
+ (when c is even)
693
+ f(F0 + Fdic, F1 + Fdic+1)
694
+ (when c is odd)
695
+ = f(F1 + Fdic+1, F0 + Fdic)
696
+ = f(1 + Fdic+1, Fdic)
697
+ = 2 + aFdic − 2Fdic−1
698
+ > 2 + 2(Fdic − Fdic−1)
699
+ > 4.
700
+ This shows that (s, t) is of type B.
701
+ We classify also for imaginary roots in L.
702
+ Lemma 6.4. If (s, t), (s′, t′) are imaginary roots, then (s + s′, t + t′) is also
703
+ imaginary root.
704
+ 14
705
+
706
+ Proof. Since f(s, t) ≤ 0, for any r ∈ R, we have f(rs, rt) = r2f(s, t) ≤ 0. It
707
+ shows that the line connecting the origin and (s, t) is inside the asymptotes of
708
+ the hyperbola x2 − axy + y2 = 1. Similarly, the line connecting the origin and
709
+ (s′, t′) is also inside the asymptotes.
710
+ Since (s+ s′, t+ t′) is the midpoint of (2s, 2t) and (2s′, 2t′), this point is also
711
+ inside the asymptotes. Therefore, (s + s′, t + t′) is an imaginary root.
712
+ Lemma 6.5. Let (u, v) ∈ L (u > v) be a real root such that (uα0+vα1)(H) ̸= 0.
713
+ Put (s, t) = (ks−u, ls−v). Then (s, t) is a type C imaginary root in L. Similarly,
714
+ let (u′, v′) ∈ L (u′ < v′) be a real root such that (u′α0 + v′α1)(H) ̸= 0. Put
715
+ (s′, t′) = (ms − u′, ns − v′). Then (s′, t′) ∈ L and (s′, t′) is the imaginary root
716
+ of type C. The other imaginary roots in L are of type A.
717
+ Proof. From Lemma 6.2, f(−s, −t) = f(s, t) ≤ 0.
718
+ It shows that (s, t) is a
719
+ imaginary root. We also see that the eigenvalue of H for (s, t) is in the range
720
+ (0, 2). Therefore, (s, t) ∈ L is shown.
721
+ We show that (s, t) is of type C. To show this, we show that f(s + ks, t +
722
+ ls) ≤ 1. Using c ∈ Z, we can write (u, v) = (Fc+1, Fc). Together this with
723
+ s + ks = 2ks − u, t + ls = 2ls − v, we have
724
+ f(s + ks, t + ls) = f(2Fi+1 − Fc+1, 2Fi − Fc).
725
+ Let dic = i−c > 0. From Lemma 6.1, acting r0, r1 on (s+ks, t+ls), i−c = λ ≥ 1,
726
+ we have
727
+ f(2Fi+1 − Fc+1, 2Fi − Fc) = f(r0(2Fi+1 − Fc+1, 2Fi − Fc))
728
+ = f(2Fi−1 − Fc−1, 2Fi − Fc)
729
+ = f(r1(2Fi−1 − Fc−1, 2Fi − Fc))
730
+ = f(2Fi−1 − Fc−1, 2Fi−2 − Fc−2)
731
+ = · · ·
732
+ =
733
+
734
+ f(2Fdic+1 − F1, 2Fdic − F0)
735
+ (when c is even)
736
+ f(2Fdic − F0, 2Fdic+1 − F1)
737
+ (when c is odd)
738
+ = f(2Fdic+1 − F1, 2Fdic − F0)
739
+ = f(2Fdic+1 − 1, 2Fdic)
740
+ = −2aFdic + 4Fdic−1 + 5
741
+ < −6Fdic + 4Fdic−1 + 5
742
+ = (−4Fdic + 4Fdic−1) + (−2Fdic + 5)
743
+ < −4 − 2Fdic + 5
744
+ ≤ −1.
745
+ This shows that f(s+ ks, t+ ls) ≤ −1 and that (s, t) is type C. From symmetry,
746
+ we also know that (s′, t′) is in L and is the imaginary root of type C.
747
+ Finally, we show the other imaginary roots in L are of type A. Let (s′′, t′′) ∈ L
748
+ be such an imaginary root. We show (s′′−ms, t′′−ns) ̸∈ R and (s′′−ks, t′′−ls) ̸∈
749
+ 15
750
+
751
+ R. If (s′′ − ms, t′′ − ns) ∈ R or (s′′ − ks, t′′ − ls) ∈ R, (s′′ − ms, t′′ − ns) ∈ −L or
752
+ (s′′−ks, t′′−ls) ∈ −L. Since (s′′−ms, t′′−ns)−(s′′−ks, t′′−ls) = (ks −ms, ls−
753
+ ns), from Lemma 5.2, we know (s′′ − ms, t′′ − ns) ̸∈ R or (s′′ − ks, t′′ − ls) ̸∈ R.
754
+ From symmetry, it is sufficient to consider when (s′′ − ms, t′′ − ns) ̸∈ R.
755
+ Under this assumption, (s′′ − ks, t′′ − ls) is an imaginary root or not a root. If
756
+ (s′′ − ks, t′′ − ls) is imaginary root, then (ks − s′′, ls − t′′) is also imaginary root
757
+ from the symmetry of R. We consider that (ks, ls) = (s′′, t′′) + (ks − s′′, ls − t′′).
758
+ The left hand side is real root and the right hand side is the sum of imaginary
759
+ roots, which contradicts Lemma 6.4. Therefore, (s′′ − ks, t′′ − ls) is not a root
760
+ and (s′′, t′′) is of type A.
761
+ The contents of this section can be summarized as follows.
762
+ Theorem 6.6.
763
+ (1) A real roots in L is of type B.
764
+ (2) We consider an imaginary root that can be written as (ks − s, ls − t) or
765
+ (ms − s, ns − t) where (s, t) is a real root. Such an imaginary root is of
766
+ type C.
767
+ (3) The other imaginary roots are of type A.
768
+ We now summarize the irreducible s-modules through type A and type C.
769
+ For s-modules through type A, we have the following.
770
+ Lemma 6.7. An irreducible s-module containing a root vector about a root of
771
+ type A in L is a lowest weight module which the root vector is the lowest weight
772
+ element.
773
+ Proof. Since (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R for the root (s, t) of
774
+ type A, we know that acting Y on the type A root vector will result in 0. This
775
+ shows the lemma.
776
+ Lemma 6.8. Let M be an irreducible s-module containing a root vector (say
777
+ v) with respect to type C root in L. Then one of the following conditions (1),
778
+ (2), or (3) is hold.
779
+ (1) M is a lowest weight module such that v is a lowest element.
780
+ (2) M is a highest weight module such that v is a highest element.
781
+ (3) M contains a real root vector with respect to a real root in −L.
782
+ Proof. The type C root (s, t) can be written with some real root (sr, tr) that
783
+ (ks − sr, ls − tr) or (ms − sr, ns − tr). Therefore, the root vector E of type C
784
+ becomes either zero or a real root vector when Y act on it. If E becomes 0
785
+ under the action of Y , then E generates an irreducible lowest weight module.
786
+ If E becomes a real root vector, then the real root for this vector is in −L, and
787
+ this lemma is shown.
788
+ 16
789
+
790
+ We also give the type A, B, C distinction to the root of −L by defining
791
+ Theorem 6.6.
792
+ Then, if there is a unitary principal or complementary series
793
+ representation that passes through a root vector of type C in L, −L, it will also
794
+ pass through the root vector of type B in −L, L. Therefore, We have only to
795
+ classify the modules that contains a type B root space.
796
+ Figure 2: a = 3, X = c0r0(e1) + c1r1(e0)
797
+ 0
798
+ 0.5
799
+ 1
800
+ 1.5
801
+ 2
802
+ 2.5
803
+ 3
804
+ 0
805
+ 1
806
+ 2
807
+ 3
808
+ x
809
+ y
810
+ type A
811
+ type B
812
+ type C
813
+ roots of X
814
+ Figure 3: a = 3, X = c0r0(e1) + c1r1r0(e1)
815
+ 0
816
+ 0.5
817
+ 1
818
+ 1.5
819
+ 2
820
+ 2.5
821
+ 3
822
+ 0
823
+ 2
824
+ 4
825
+ 6
826
+ 8
827
+ x
828
+ y
829
+ type A
830
+ type B
831
+ type C
832
+ roots of X
833
+ 17
834
+
835
+ 7
836
+ Irreducible modules which contains a root space
837
+ with respect to a type B root
838
+ We consider an irreducible decomposition of g by s, and we consider an irre-
839
+ ducible component M containing a type B root space. The multiplicity of a real
840
+ root space is 1. We can take 0 < λ < 2 such that {λ + 2k′ | k′ ∈ Z} is the set
841
+ of the eigenvalues of H in M. We consider the H eigenspace of M such that
842
+ the eigenvalue is λ. We assume this eigenspace is gsα0+tα1 such that (s, t) ∈ L,
843
+ and (s, t) is real root. We consider k such that s1(k) = 0 in (A) in §5. We show
844
+ that it is not an integer.
845
+ Let e0, e1, f0, f1, h0 and h1 be Chevalley generators. Using some c0, c1 ∈
846
+ R, w0, w1 ∈ W, and (p, q) ∈ {(0, 1), (0, 0), (1, 1)}, let X = c0w0(ep) + c1w1(eq).
847
+ Suppose s > t. We take the root vector E with respect to the root sα0+tα1. We
848
+ define λ by HE = λE, and define k0 by [−c2
849
+ 0w0(hp), E] = k0E. Thus s1(k) = 0
850
+ implies
851
+ k = 1 − λ ±
852
+
853
+ (λ − 1)2 + 4k0
854
+ 2
855
+ .
856
+ We put
857
+ k+ = 1 − λ +
858
+
859
+ (λ − 1)2 + 4k0
860
+ 2
861
+ ,
862
+ k− = 1 − λ −
863
+
864
+ (λ − 1)2 + 4k0
865
+ 2
866
+ and we show that k± ̸∈ R or 0 < k± < 1.
867
+ When (λ − 1)2 + 4k0 < 0 or (λ − 1)2 + 4k0 ̸∈ R, k± are imaginary numbers.
868
+ Therefore we can assume (λ − 1)2 + 4k0 ≥ 0. From 0 < λ < 1, it is clear that
869
+ k+ > 0 and k− < 1. To show k+ < 1, we need to show
870
+ 1 − λ +
871
+
872
+ (λ − 1)2 + 4k0 < 2.
873
+ we can easily show that it is reduced to k0 < λ. Also, to show that k− > 0, we
874
+ need to show
875
+ 1 − λ −
876
+
877
+ (λ − 1)2 + 4k0 > 0.
878
+ we can easily show that it is reduced to k0 < 0. In summary, we have only to
879
+ show that k0 < 0.
880
+ First, consider the case (s, t) = (1, 0), i.e., E ∈ gα0.
881
+ In this case, from
882
+ Lemma 3.3, we have c0w0(ep) ∈ gFi+1α0+Fiα1 and i ≥ 1. Since
883
+ k0E = [−c2
884
+ 0r0r1r0 . . . r1−p(hp), E]
885
+ = [−c2
886
+ 0(Fi+1h0 + Fih1), E]
887
+ = −c2
888
+ 0(2Fi+1 − aFi)E
889
+ = −c2
890
+ 0(Fi+1 + Fi−1)E,
891
+ 18
892
+
893
+ we have k0 < 0. When (s, t) = (0, 1), we can show that k0 < 0 by replacing i
894
+ with j, p with q and making the same argument.
895
+ If (s, t) is general and s > t, we can write (s, t) = (Fi′+1, Fi′). Let p′ be 0 or
896
+ 1, we can write E = r0r1r0 . . . r1−p′(ep′). From this, we have
897
+ [−c2
898
+ 0w0(hp), E] = −c2
899
+ 0[r0r1r0 . . . r1−p(hp), r0r1r0 . . . r1−p′(ep′)]
900
+ = −c2
901
+ 0r0r1r0 . . . r1−p′[rp′r1−p′rp′ . . . r1−p(hp), ep′].
902
+ We consider k0 and c0 when i is replaced by i−i′, and rewrite them as k′
903
+ 0 and c′
904
+ 0.
905
+ Considering (s, t) = (1, 0) or (0, 1) cases, we have [rp′r1−p′rp′ . . . r1−p(hp), ep′] =
906
+ − k′
907
+ 0
908
+ c′
909
+ 0 ep′. That is, k0 = c2
910
+ 0
911
+ c′2
912
+ 0 k′
913
+ 0. Since k′
914
+ 0 < 0, we have k0 < 0. When s < t, we can
915
+ show that k0 < 0 as well.
916
+ From the above, it can be shown that k0 < 0 in any case, i.e., k is not an
917
+ integer. From this, we can see the following.
918
+ Theorem 7.1. We consider an irreducible decomposition of g by the action of
919
+ s.
920
+ (1) Let M is an irreducible component of decomposition of g, which contain
921
+ a root space for a type B root sα0 + tα1. Then, M is an unitary principal
922
+ or complementary series representation.
923
+ (2) (cf. [Tsu, Proposition 7.3]) There is an unitary principal series represen-
924
+ tation containing an 1-dimensional space in h.
925
+ (3) g is decomposed into a direct sum of s-submodules described in (1) and (2)
926
+ above, s itself, irreducible lowest weight modules, and irreducible highest
927
+ weight modules.
928
+ From [KM95, §3], the multiplicity of each root of g is calculated. Using
929
+ this, we can find how many modules appear such that the following condition is
930
+ satisfied: the modules are highest or lowest modules, and eigenvalues of H for
931
+ root vectors with the highest or the lowest roots are certain value.
932
+ First, the modules which contain root spaces in L and −L can be seen from
933
+ previous contents. Among the positive root spaces not in L, those with the
934
+ smallest eigenvalue in H are considered together. Let λH be their eigenvalue
935
+ and dH be their dimensions. Suppose pH modules which contain space with
936
+ eigenvalue λH that also contain the root spaces already obtained. Then there
937
+ are dH−pH lowest weight modules with the root with eigenvalue λH as the lowest
938
+ root. The multiplicities of modules can be obtained inductively by replacing λH
939
+ with the next smallest eigenvalue of H and performing the same calculation.
940
+ Negative root spaces can be classified by the same calculation.
941
+ 8
942
+ Unitary principal series representation and com-
943
+ plementary series representation
944
+ In this section, we consider a module (say M) that is neither highest weight
945
+ module nor lowest weight module containing a root vector about the root of type
946
+ 19
947
+
948
+ B. We compute whether the module is a unitary principal series representation
949
+ or a complementary series representation. First, we state the following lemma.
950
+ Lemma 8.1. If 8µ ≤ −1, then M is a unitary principal series representation.
951
+ If 8µ > −1, then M is a complementary series representation.
952
+ Proof. From [HT92, §II 1.2], M is isomorphic to U(ν+, ν−). U(ν+, ν−) is a
953
+ sl2-module with H eigenvectors {vn | n ∈ Z} as a basis of linear space, such
954
+ that
955
+ Hvn = (ν+ − ν− + 2j)vn
956
+ (n ∈ Z),
957
+ e+vn = (ν+ + n)vn+1,
958
+ e−vn = (ν− − n)vn−1,
959
+ 8µ = (ν+ + ν− − 1)2 − 1.
960
+ From [HT92, §III Theorem 1.1.3], if ν+ + ν− = 1, U(ν+, ν−) is a unitary
961
+ principal series representation. When 8µ ≤ −1, from
962
+ λ = ν+ − ν− ∈ R,
963
+ 8µ = (ν+ + ν− − 1)2 − 1 < −1,
964
+ using b ∈ R we can write
965
+ ν+ − ν− = λ,
966
+ ν+ + ν− = 1 + bi.
967
+ (i =
968
+
969
+ −1)
970
+ In this case, we have
971
+ ν+ + ν− = λ + 1
972
+ 2
973
+ + b
974
+ 2i + −λ + 1
975
+ 2
976
+ − b
977
+ 2i
978
+ = 1.
979
+ Therefore, M is a unitary principal series representation.
980
+ Consider the case when 8µ > −1.
981
+ From [HT92, §III Theorem 1.1.3], if
982
+ ν± ∈ R and if ν− − 1 and −ν+ are both contained in the interval (l − 1, l) with
983
+ some l ∈ Z, then U(ν+, ν−) is a complementary series representation. From
984
+ 8µ > −1, we have
985
+ ν+ + ν− = 1 ±
986
+
987
+ 8µ + 1,
988
+ ν+ − ν− = λ.
989
+ Therefore, we have
990
+ −ν+, ν− − 1 = −λ − 1 ± √8µ + 1
991
+ 2
992
+ .
993
+ We show that they are in (−1, 0).
994
+ 20
995
+
996
+ We show first that 0 < λ < 1. Let n, m be integers such that n > m ≥ 0.
997
+ We can write
998
+ λ = 2(Fm+1 + Fm)
999
+ Fn+1 + Fn
1000
+ .
1001
+ It is clear that λ > 0. From a ≥ 3, for integer z ≥ 0, we have
1002
+ Fz+2 = aFz+1 − Fz
1003
+ > (a − 1)Fz+1
1004
+ ≥ 2Fz+1.
1005
+ Hence we have
1006
+ Fm+1 + Fm
1007
+ Fn+1 + Fn
1008
+ < 1
1009
+ 2.
1010
+ Therefore, we have λ < 1. We show that
1011
+ −1 < −λ − 1 + √8µ + 1
1012
+ 2
1013
+ .
1014
+ From λ < 1, we have −1 < −λ−1
1015
+ 2
1016
+ . Therefore, this inequality is shown. Next we
1017
+ show
1018
+ −λ − 1 + √8µ + 1
1019
+ 2
1020
+ < 0.
1021
+ We have 8µ = λ(λ − 2) + 4k0. From 0 < λ < 1, we have λ(λ − 2) < 0. Also,
1022
+ since k0 < 0, we have 8µ < 0. Therefore, we have √8µ + 1 < 1. Using 0 < λ
1023
+ again, we know that
1024
+ −λ − 1 + √8µ + 1
1025
+ 2
1026
+ < 0.
1027
+ For
1028
+ −λ − 1 − √8µ + 1
1029
+ 2
1030
+ < 0,
1031
+ this is clear from λ > 0. Finally, we show
1032
+ −1 < −λ − 1 − √8µ + 1
1033
+ 2
1034
+ .
1035
+ From k0 < 0 and 8µ = λ2 − 2λ + 4k0, we have λ2 − 2λ > 8µ. From this and
1036
+ λ < 1 we get 1 − λ > √8µ + 1, which can be transformed to
1037
+ −1 < −λ − 1 − √8µ + 1
1038
+ 2
1039
+ .
1040
+ From the above, −λ−1±√8µ+1
1041
+ 2
1042
+ are both in (−1, 0). Therefore, M is a comple-
1043
+ mentary series representation.
1044
+ 21
1045
+
1046
+ Hereafter, we want to determine when M is complementary series. First, we
1047
+ consider the case where i = j. we have
1048
+ 8µ = λ2 − 2λ + 4k0,
1049
+ λ = 2(Fn+1 + Fn)
1050
+ Fi+1 + Fi
1051
+ ,
1052
+ k0 =
1053
+ −2(2Fi+1 − aFi)
1054
+ a(F 2
1055
+ i + F 2
1056
+ i+1) − 4FiFi+1 − 2,
1057
+ (*)
1058
+ where n is an integer such that i > n ≥ 0. That is, 8µ is determined by i, n,
1059
+ and a. We show that 8µ is greater than −1 with finite exceptions.
1060
+ Lemma 8.2. We assume i = j. If we consider 8µ to be a function of n by (*),
1061
+ 8µ is monotonically decreasing with respect to n.
1062
+ Proof. k0 is independent on n. λ is monotonically increasing with respect to n.
1063
+ Since 8µ = λ(λ − 2) + 4k0 and 0 < λ < 1, we know that 8µ is monotonically
1064
+ decreasing with respect to n.
1065
+ To show that 8µ is greater than −1 with finite exceptions, we need to examine
1066
+ when n is large.
1067
+ Lemma 8.3. We assume i = j, n = i − 1. If we consider 8µ to be a function
1068
+ of i by (*), 8µ is monotonically increasing with respect to i.
1069
+ Proof. First we write {Fi} explicitly as follows. The real solutions of x2 − ax +
1070
+ 1 = 0 are x = a±
1071
+
1072
+ a2−4
1073
+ 2
1074
+ . As α = a−
1075
+
1076
+ a2−4
1077
+ 2
1078
+ , β = a+
1079
+
1080
+ a2−4
1081
+ 2
1082
+ , we can write
1083
+ Fi = βi − αi
1084
+ β − α .
1085
+ From n = i − 1, we have
1086
+ λ = 2(Fi + Fi−1)
1087
+ Fi+1 + Fi
1088
+ ,
1089
+ k0 =
1090
+ −2(2Fi+1 − aFi)
1091
+ a(F 2
1092
+ i + F 2
1093
+ i+1) − 4FiFi+1 − 2.
1094
+ Let t be a real variable. We define the functions Λ and K0 as follows.
1095
+ Λ(t) = 2(βt − αt + βt−1 − αt−1)
1096
+ βt+1 − αt+1 + βt − αt
1097
+ ,
1098
+ K0(t) =
1099
+ −2(β − α)(2(βt+1 − αt+1) − a(βt − αt))
1100
+ a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2,
1101
+ We have λ = Λ(i) and k0 = K0(i). Using these function, we can calculate as
1102
+ follows.
1103
+ d
1104
+ dtΛ =
1105
+ 4 log β(a + 2)(β − α)
1106
+ (βt+1 − αt+1 + βt − αt)2
1107
+ 22
1108
+
1109
+ d
1110
+ dt(Λ2 − 2Λ) = 8 log β(a + 2)(β − α)
1111
+
1112
+ (1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1�
1113
+ (βt+1 − αt+1 + βt − αt)3
1114
+ d
1115
+ dtK0 =
1116
+ 2(β − α)(a2 − 4) log β
1117
+ (a2 − 4)2(β2t+1 + α2t+1 − 2)2
1118
+ ·
1119
+
1120
+ 2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1
1121
+ +(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt�
1122
+ d
1123
+ dt(Λ2 − 2Λ + 4K0) = 8 log β(β − α) ·
1124
+
1125
+ (a + 2)
1126
+
1127
+ (1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1�
1128
+ (βt+1 − αt+1 + βt − αt)3
1129
+ +
1130
+ 1
1131
+ (a2 − 4)(β2t+1 + α2t+1 − 2)2
1132
+ ·
1133
+
1134
+ 2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1
1135
+ +(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt�
1136
+
1137
+ Clearing the denominator, we can calculate as follows.
1138
+ (βt+1 − αt+1 + βt − αt)3(a − 2)(β2t+1 + α2t+1 − 2)2
1139
+ 8(a + 2) log β(β − α)
1140
+ · d
1141
+ dt(Λ2 − 2Λ + 4K0)
1142
+ = (β6t+3 − α6t+3) − (β6t+2 − α6t+2)
1143
+ + (a − 2)(1 − a)(β5t+2 − α5t+2) + 3(a − 2)(β5t+1 − α5t+1)
1144
+ + 2(β4t+3 − α4t+3) + (a − 4)(β4t+2 − α4t+2)
1145
+ + (11 − 2a)(a − 2)(β3t+2 − α3t+2) − (8a + 1)(a − 2)(β3t+1 − α3t+1)
1146
+ + (−11a + 5)(β2t+1 − α2t+1) + 17(β2t − α2t)
1147
+ + (8a − 14)(a − 2)(βt+1 − αt+1) + (12a − 6)(a − 2)(βt − αt)
1148
+ + 14(β − α)
1149
+ The coefficient on the left hand side is positive. Using the fact that βt − αt
1150
+ is monotonically increasing, we can calculate that the right hand side is also
1151
+ positive. This shows that 8µ = (Λ2 − 2Λ + 4K0)(i) is monotonically increasing
1152
+ with respect to i.
1153
+ From Lemma 8.3, we consider the case when i = 1, n = 0.
1154
+ Lemma 8.4. We assume i = j = 1 and n = 0. If we consider 8µ to be a
1155
+ function of a by (*), 8µ is monotonically increasing with respect to a.
1156
+ Proof. Under this assumption, we have
1157
+ 8µ =
1158
+ −4a2
1159
+ a3 − 3a − 2.
1160
+ Differentiating this as a function of the real variable a, from a ≥ 3, we know
1161
+ that 8µ is monotonically increasing with respect to a.
1162
+ 23
1163
+
1164
+ Lemma 8.5. When i = j, we consider s-modules of g that are neither a highest
1165
+ weight module nor a lowest weight module containing a root vector about the
1166
+ root of type B obtained by Theorem 7.1. The modules are complementary series
1167
+ representations, except for the following five types. For these exceptions, the
1168
+ modules are unitary principal series representations.
1169
+ (a, i, n) = (4, 1, 0), (3, 1, 0), (3, 2, 1), (3, 3, 2), (3, 4, 3)
1170
+ Proof. We use Lemma 8.6, Lemma 8.3, and Lemma 8.4.
1171
+ First, when a = 5, i = 1, n = 0, we have 8µ = − 25
1172
+ 27 > −1. Therefore, when
1173
+ a ≥ 5, for any i, n, the module for a, i, n is a complementary series representa-
1174
+ tion.
1175
+ Next, when a = 4, i = 1, n = 0, we have 8µ = − 32
1176
+ 25 < −1. Hence the module
1177
+ for this is a unitary principal series representation. On the other hand, when
1178
+ a = 4, i = 2, n = 1, we have 8µ > −1. Therefore, when a = 4, the module for
1179
+ a, i, n is a complementary series representation except when i = 1, n = 0.
1180
+ Finally, when a = 3, 8µ < −1 when i = 1, 2, 3, 4 and n = i − 1, and in
1181
+ these four cases the module is a unitary principal series representation. When
1182
+ n = i − 2 or i = 5, we have 8µ > −1. Therefore, we know that the module is a
1183
+ complementary series representation in other cases.
1184
+ From the above, with five exceptions, neither a highest weight module nor
1185
+ a lowest weight module containing a root vector about the root of type B is a
1186
+ complementary series representation.
1187
+ Next, we consider the case i = j − 1 or i = j + 1. From symmetry, it is
1188
+ sufficient to consider the case i = j − 1. In this case, λ can be written
1189
+ λ = 2Fn
1190
+ Fi+1
1191
+ with n as an integer such that i ≥ n ≥ 0. On the other hand, for k0, we have
1192
+ k0 =
1193
+ −2(2Fi+1 − aFi)
1194
+ a(F 2
1195
+ i + F 2
1196
+ i+1) − 4FiFi+1 − 2
1197
+ as for i = j. As with i = j, 8µ is determined by i, n and a. The next lemma is
1198
+ obvious.
1199
+ Lemma 8.6. We assume i = j − 1. If we consider 8µ to be a function of n by
1200
+ (*), 8µ is monotonically decreasing with respect to n.
1201
+ In the following, we consider whether 8µ is monotonically increasing with
1202
+ respect to i when n = i. In this case, we have
1203
+ λ = 2Fi
1204
+ Fi+1
1205
+ .
1206
+ When i = j, n = i − 1, we already know
1207
+ λ = 2(Fi + Fi−1)
1208
+ Fi+1 + Fi
1209
+ .
1210
+ 24
1211
+
1212
+ We rewrite as
1213
+ λ1 = 2Fi
1214
+ Fi+1
1215
+ ,
1216
+ λ2 = 2(Fi + Fi−1)
1217
+ Fi+1 + Fi
1218
+ .
1219
+ Let t be a real variable. We define Λ1, Λ2 and K0 as follows.
1220
+ Λ1(t) = 2(βt − αt + βt−1 − αt−1)
1221
+ βt+1 − αt+1 + βt − αt
1222
+ ,
1223
+ Λ2(t) = 2(βt − αt)
1224
+ βt+1 − αt+1 ,
1225
+ K0(t) =
1226
+ −2(β − α)(2(βt+1 − αt+1) − a(βt − αt))
1227
+ a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2.
1228
+ We have λ1 = Λ1(i), λ2 = Λ2(i), k0 = K0(i), and 8µ = (Λ2
1229
+ 1 − 2Λ1 + 4K0)(i).
1230
+ When i = j, 8µ = (Λ2
1231
+ 2 − 2Λ2 + 4K0)(i). We compare
1232
+ d
1233
+ dtΛ1 and
1234
+ d
1235
+ dtΛ2. Since
1236
+ d
1237
+ dt(Λ2
1238
+ 1 − 2Λ1 + 4K0) = 2(Λ1 − 1) d
1239
+ dtΛ1 + 4 d
1240
+ dtk0 and 0 < Λ1 − 1 < 1, the smaller
1241
+ the value of
1242
+ d
1243
+ dtΛ1, the larger the value of
1244
+ d
1245
+ dt(Λ2
1246
+ 1 − 2Λ1 + 4K0). We know that
1247
+ d
1248
+ dt(Λ2
1249
+ 2−2Λ2+4K0) > 0. If we show d
1250
+ dtΛ1 > d
1251
+ dtΛ2, we also know (Λ2
1252
+ 2−2Λ2+4K0)
1253
+ is monotonically increasing with respect to t. Therefore, we know 8µ is also
1254
+ monotonically increasing with respect to i when i = j − 1, n = i.
1255
+ Lemma 8.7.
1256
+ d
1257
+ dtΛ1 > d
1258
+ dtΛ2.
1259
+ Proof. we show that
1260
+ d
1261
+ dtΛ1 − d
1262
+ dtΛ2 > 0. we have
1263
+ d
1264
+ dtΛ1 − d
1265
+ dtΛ2 =
1266
+ 4 log β(a + 2)(β − α)
1267
+ (βt+1 − αt+1 + βt − αt)2 − 4 log β(β − α)
1268
+ (βt+1 − αt+1)2 .
1269
+ Calculating this, we have
1270
+ (βt+1 − αt+1 + βt − αt)2(βt+1 − αt+1)2
1271
+ 4 log β(β − α)
1272
+ � d
1273
+ dtΛ1 − d
1274
+ dtΛ2
1275
+
1276
+ =(β2t+3 + β2t+3) + (β2t+2 + α2t+2) − (β2t+1 + α2t+1) − (β2t + α2t).
1277
+ The coefficient on the left hand is positive. We can easily calculate to know
1278
+ that βt + αt is monotonically increasing with respect to t. From this, we know
1279
+ the right hand side is also positive. Therefore,
1280
+ d
1281
+ dtΛ1 > d
1282
+ dtΛ2.
1283
+ Lemma 8.8. We assume i = j − 1, n = i. If we consider 8µ to be a function
1284
+ of i by (*), 8µ is monotonically increasing with respect to i.
1285
+ Lemma 8.9. We assume i = 0, j = 1, and n = 0. If we consider 8µ to be a
1286
+ function of a by (*), 8µ is monotonically increasing with respect to a.
1287
+ 25
1288
+
1289
+ Proof. Under this assumption,
1290
+ 8µ =
1291
+ −4
1292
+ a − 2.
1293
+ This is monotonically increasing with respect to a ≥ 3.
1294
+ Lemma 8.10. When i = j −1, We consider s-modules containing a root vector
1295
+ about the root of type B that are neither highest weight modules nor lowest
1296
+ weight modules obtained by Theorem 7.1.
1297
+ The modules are complementary
1298
+ series representations, except for the following 23 types. For these exceptions,
1299
+ the modules are unitary principal series representations.
1300
+ (a, i, n) =(a′, 0, 0) (6 ≤ a′ ≤ 18),
1301
+ (5, 0, 0), (5, 1, 1),
1302
+ (4, 0, 0), (4, 1, 1),
1303
+ (3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 2, 2), (3, 3, 3), (3, 4, 4)
1304
+ Proof. We use Lemma 8.6, Lemma 8.8, and Lemma 8.9.
1305
+ First, when a =
1306
+ 18, i = 0, n = 0, 8µ = −1. Therefore, when a ≥ 18, the modules for a, i, n
1307
+ are complementary series representations except when (a, i, n) = (18, 0, 0).
1308
+ Then, when 6 ≤ a ≤ 17, i = 0, n = 0, from 8µ = − 4
1309
+ 3 > −1, the module for
1310
+ this pair is a unitary principal series representation. On the other hand, when
1311
+ 6 ≤ a ≤ 17, 8µ < −1 except when (a, i, n) = (a, 0, 0), i.e., the module about
1312
+ a, i, n is a complementary series representation.
1313
+ When a = 5, if (a, i, n) = (5, 0, 0), (5, 1, 1), then the modules are unitary
1314
+ principal series representations, and the others are complementary series repre-
1315
+ sentations.
1316
+ When a = 4, if (a, i, n) = (4, 0, 0), (4, 1, 1), then the modules are unitary
1317
+ principal series representations, and the others are complementary series repre-
1318
+ sentations.
1319
+ When a = 3, if (a, i, n) = (3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 2, 2), (3, 3, 3), (3, 4, 4),
1320
+ then the modules are unitary principal series representations, and the others
1321
+ are complementary series representations.
1322
+ From the above, 23 unitary principal series representations are obtained, and
1323
+ the rest are all complementary series representations.
1324
+ Theorem 8.11. We consider modules obtained by (1) of Theorem 7.1. The
1325
+ modules are neither highest weight modules nor lowest weight modules and
1326
+ contain root vectors about roots of type B. The modules are complementary
1327
+ series representations, except those enumerated by Lemma 8.5 and Lemma 8.10.
1328
+ For the exceptions, the modules are unitary principal series representations.
1329
+ Proof. It can be shown from Lemma 8.5 and Lemma 8.10.
1330
+ 26
1331
+
1332
+ Acknowledgements
1333
+ I would like to express my appreciation to my supervisor, Prof. Hisayosi Matu-
1334
+ moto for his thoughtful guidance.
1335
+ References
1336
+ [CM93] D. H. Collingwood, W. M. McGovern, Nilpotent Orbits in Semisimple
1337
+ Lie Algebras, Van Nostrand Reinhold, 1993
1338
+ [Dyn57] E. Dynkin, Semisimple subalgebras of simple Lie algebras, American
1339
+ Mathematical Society Translations: Series 2, 6, 1957, pp. 111–245
1340
+ [GOW02] M. R. Gaberdiel, et al., A class of Lorentzian Kac-Moody algebras,
1341
+ Nuclear Physics B, 645, 2002, pp. 403–437
1342
+ [Kac90] V. G. Kac, Infinite dimensional Lie algebras 3rd edition, Cambridge
1343
+ university press, 1990
1344
+ [KM95] S-J. Kang, D. J. Melville, Rank 2 symmetric hyperbolic Kac-Moody
1345
+ algebras, Nagoya Mathematical Journal, 140, 1995, pp. 41–75
1346
+ [Kos59] B. Kostant, The principal three-dimensional subgroup and the Betti
1347
+ numbers of a complex simple Lie group, American Journal of Mathemat-
1348
+ ics, 81, 1959, pp. 973–1032
1349
+ [HT92] R. Howe, E. C. Tan, Non-Abelian Harmonic Analysis, Springer-Verlag,
1350
+ 1992
1351
+ [NO01] H. Nicolai, D. I. Olive, The Principal SO(1, 2) Subalgebra of a Hy-
1352
+ perbolic Kac Moody Algebra, Letters in Mathematical Physics, 2001, pp.
1353
+ 141–152
1354
+ [Tsu] H. Tsurusaki, sl2 triples whose nilpositive elements are in a space which is
1355
+ spanned by the real root vectors in rank 2 symmetric hyperbolic Kac-Moody
1356
+ Lie algebras, Publications of the Research Institute for Mathematical Sci-
1357
+ ences, to appear
1358
+ 27
1359
+
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1
+ Application of Causal Inference Techniques to the
2
+ Maximum Weight Independent Set Problem
3
+ Jianfeng Liu †§
4
+ Sihong Shao‡∗
5
+ Chaorui Zhang§
6
+ January 16, 2023
7
+ Abstract
8
+ A powerful technique for solving combinatorial optimization problems is
9
+ to reduce the search space without compromising the solution quality by ex-
10
+ ploring intrinsic mathematical properties of the problems. For the maximum
11
+ weight independent set (MWIS) problem, using an upper bound lemma which
12
+ says the weight of any independent set not contained in the MWIS is bounded
13
+ from above by the weight of the intersection of its closed neighbor set and the
14
+ MWIS, we give two extension theorems — independent set extension theorem
15
+ and vertex cover extension theorem. With them at our disposal, two types
16
+ of causal inference techniques (CITs) are proposed on the assumption that a
17
+ vertex is strongly reducible (included or not included in all MWISs) or re-
18
+ ducible (contained or not contained in a MWIS). One is a strongly reducible
19
+ state-preserving technique, which extends a strongly reducible vertex into a
20
+ vertex set where all vertices have the same strong reducibility. The other, as a
21
+ reducible state-preserving technique, extends a reducible vertex into a vertex
22
+ set with the same reducibility as that vertex and creates some weighted pack-
23
+ ing constraints to narrow the search space. Numerical experiments show that
24
+ our CITs can help reduction algorithms find much smaller remaining graphs,
25
+ improve the ability of exact algorithms to find the optimal solutions and help
26
+ heuristic algorithms produce approximate solutions of better quality. In par-
27
+ ticular, detailed tests on 12 representative graphs generated from datasets in
28
+ †Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
29
+ ‡CAPT, LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, P.R.
30
+ China
31
+ §Theory Lab, Central Research Institute, 2012 Labs, Huawei Technologies Co., Ltd.
32
+ ∗Author to whom correspondence should be addressed: sihong@math.pku.edu.cn
33
+ 1
34
+ arXiv:2301.05510v1 [math.OC] 13 Jan 2023
35
+
36
+ Network Data Repository demonstrate that, compared to the state-of-the-art
37
+ algorithms, the size of remaining graphs is further reduced by more than 32.6%,
38
+ and the number of solvable instances is increased from 1 to 5.
39
+ AMS subject classifications: 05C69; 68W40; 90C06; 90C27; 90C57
40
+ Keywords: maximum weight independent set; independent set extension;
41
+ vertex cover extension; causal inference techniques; reduction algorithm; exact
42
+ algorithm; heuristic algorithm; Network Data Repository.
43
+ 1
44
+ Introduction
45
+ Let G = (V, E, w) be an undirected vertex-weighted graph, where each vertex v ∈ V
46
+ is associated with a weight w(v) ∈ R+. A subset I ⊆ V is called an independent set
47
+ if its vertices are pairwise non-adjacent, and the vertex cover of graph G is a subset
48
+ of vertices V C ⊆ V such that every edge e ∈ E is incident to at least one vertex
49
+ in subset V C. Independent set and vertex cover are two complementary concepts
50
+ in graph and can be transformed into each other on demand [29]. The maximum
51
+ weight independent set (MWIS) problem is to find the independent set of largest
52
+ weight among all possible independent sets and the weight of a MWIS of graph G is
53
+ denoted by αw(G), while the minimum weight vertex cover (MWVC) problem asks for
54
+ the vertex cover with the minimum weight. Furthermore, if subset I ⊆ V is a MWIS,
55
+ then subset V C = V \I is a MWVC, and vice versa [6, 29]. The MWIS problem is an
56
+ extension of the maximum independent set (MIS) problem, which is a classic NP-hard
57
+ problem [13, 9]. It can be applied to various real-world problems, such as information
58
+ retrieval [4], computer vision [12], combinatorial auction problem [29] and dynamic
59
+ map labeling problem [17].
60
+ Due to its wide range of practical applications, the
61
+ research on efficient algorithms for computing the MWIS is of great significance.
62
+ Most previous work are focused on heuristic algorithms to find near-optimal solutions
63
+ in reasonable time [24, 20, 6, 18], while exact algorithms, usually referring to Branch-
64
+ and-Bound (B&B) methods [3, 26, 2, 22], become infeasible when the size of problem
65
+ increases.
66
+ Recently, it has been well demonstrated that reduction rules (a.k.a. kerneliza-
67
+ tion) are very effective in practice for solving the MIS problem [25]. These rules mine
68
+ the structural properties of underlying graph and reduce the search space by such
69
+ as removing vertices, contracting subgraphs, restricting the set of independent sets,
70
+ etc., to produce a smaller kernel graph such that the MIS of the original graph can be
71
+ recovered from the MIS of the kernel. After integrating them, some state-of-the-art
72
+ exact solvers are able to solve the MIS problem on many large real networks [11].
73
+ 2
74
+
75
+ These solvers can be usually divided into two types: One performs the kernelization
76
+ only once and runs the B&B algorithm [23, 16] on the kernelized instance, while
77
+ the other joins hands with the Branch-and-Reduce (B&R) algorithm [19] and per-
78
+ forms reduction in every branch of the search tree. As for those instances that can’t
79
+ be solved exactly, high-quality solutions can be found by combining kernelization
80
+ with local search [8, 10]. Moreover, when a vertex is selected for branching in the
81
+ branching process of the B&R algorithm, if it is assumed to be in all MISs, then
82
+ its satellite set will also be in all MISs [14], while its mirror set will be removed
83
+ directly from the graph, if it is assumed not to be in all MISs [13]. Further, a conflict
84
+ analysis on the assumption that a vertex is in all MISs can be also plugged in to
85
+ find some contradictions and the concept of “unconfined/confined vertices” was in-
86
+ troduced [28]. Later, an auxiliary constraint called packing constraint was proposed
87
+ to accelerate the B&R algorithm by simply exploring branches that satisfy all pack-
88
+ ing constraints [1]. The central idea behind all these attempts for the MIS problem
89
+ involves a state-preserving technique which starts from a vertex, named the starting
90
+ vertex for convenience, and then finds a vertex set with the same state as the starting
91
+ vertex to reduce the search space, thereby implying that some subsequent operations
92
+ can be implemented on the resulting vertex set instead of only on the starting vertex.
93
+ For the MWIS problem, similar state-preserving techniques are rarely used except
94
+ for a recent work using unconfined/confined vertices [27], though some simple and
95
+ fast reduction rules have been used in B&R algorithms [15, 27]. To this end, we
96
+ devote ourselves into developing state-preserving techniques for the MWIS problem
97
+ in this work. The state of the starting vertex we consider can be
98
+ • strongly reducible, meaning that the vertex is included in all MWISs/MWVCs;
99
+ or
100
+ • reducible, meaning that the vertex is contained in a MWIS/MWVC.
101
+ Considering that the assumed state of the starting vertex must be used to analyze its
102
+ local structure to obtain inference results, these targeted state-preserving techniques
103
+ are called causal inference techniques (CITs). Inspired by their success in solving the
104
+ MIS problem, we will systematically develop CITs to solve the MWIS problem by
105
+ analyzing intrinsic mathematical properties of underlying graph. More specifically,
106
+ our main contributions are in three aspects as follows.
107
+ First, by virtue of the upper bound lemma, i.e., the weight of any independent set
108
+ not contained in the MWIS is bounded from above by the weight of the intersection
109
+ of its closed neighbor set with the MWIS, two extension theorems are developed.
110
+ With them, we propose a series of CITs which have been rarely used previously in
111
+ 3
112
+
113
+ the MWIS problem. According to the state of the starting vertex, our CITs can be
114
+ divided into two categories. The first type is a strongly reducible state-preserving
115
+ technique. We first assume that the starting vertex is strongly reducible, and then
116
+ try to extend this vertex to obtain a vertex set with the same strong reducibility.
117
+ If the upper bound lemma is not satisfied in this process, then this contradicts
118
+ the assumption, and the starting vertex can be removed from the graph directly.
119
+ Otherwise, combined with the state-preserving result obtained from the previous
120
+ process, we continue to search for a set called the simultaneous set, which is either
121
+ included in a MWIS or contained in a MWVC. The second type is a reducible state-
122
+ preserving technique. Under the assumption that the starting vertex is reducible, a
123
+ vertex set with the same reducibility can be obtained by extending from this vertex.
124
+ Moreover, if this vertex is selected for branching in the B&R algorithm, with the
125
+ upper bound lemma, an inequality constraint called weight packing constraint will
126
+ be created to restrict subsequent searches.
127
+ Next, according to the characteristics of the proposed CITs, we integrate them
128
+ into the existing algorithmic framework. The first type of CIT can be used to design
129
+ reduction rules to simplify graph.
130
+ These reduction rules are integrated into the
131
+ existing reduction algorithm. In the B&R algorithm, when a vertex is selected to
132
+ branch, a vertex set and a weight packing constraint depending on the assumed
133
+ state of the vertex can be obtained from state-preserving results of two types of
134
+ CITs. The vertex set is used to further simplify the corresponding branch, while we
135
+ can prune branches that violate constraints and simplify the graph by maintaining all
136
+ created weight packing constraints. During the local search process of the heuristic
137
+ algorithm, when the state of a vertex needs to be changed, all vertex states in the
138
+ vertex set obtained by the second type of CIT will also be modified to be the same
139
+ as that vertex, which expands the area of local search and improves the ability of
140
+ local search to find better local optima.
141
+ Numerical experiments on 12 representative graphs generated from datasets in
142
+ Network Data Repository show that the performance of various algorithms is greatly
143
+ improved after integrating our CITs. The size of the kernel obtained by the resulting
144
+ reduction algorithm is greatly reduced. In addition, compared to the state-of-the-art
145
+ exact algorithm, the number of solvable instances have been increased from 1 to 5.
146
+ And the ability of the heuristic algorithm to find better local optimal solutions is
147
+ significantly improved. These experimental results form the third major contribution
148
+ of this paper.
149
+ Relevant notations used in this work are given in Table 1 and the rest of the paper
150
+ is organized as follows. We present two extension theorems in Section 2 and detail
151
+ CITs in Section 3. How the CITs are combined with existing algorithmic frameworks
152
+ 4
153
+
154
+ G = (V,E,w)
155
+ an undirected vertex-weight graph G with vertex set V , edge set E and vertex weight function w : V → R+
156
+ N(v) = {u ∈ V |{u,v} ∈ E}
157
+ the neighbor set of vertex v
158
+ N[v] = N(v) ∪ {v}
159
+ the closed neighbor set of vertex v
160
+ N(S) = (
161
+
162
+ v∈S
163
+ N(v))\S
164
+ the open neighbor set of set S
165
+ N[S] = N(S) ∪ {S}
166
+ the closed neighbor set of set S
167
+ |S|
168
+ the size of set S
169
+ w(S) =
170
+
171
+ v∈S
172
+ w(v)
173
+ the weight of all vertices in set S
174
+ d(v)
175
+ the degree of a vertex v
176
+ dist(u,v)
177
+ the minimum number of edges in the path from vertex u to vertex v
178
+ Nl(v) = {u| dist(u,v) = l}
179
+ the set of vertices at distance l from vertex v,
180
+ in particular, N1(v) = N(v)
181
+ G[S] = (S,ES,w),
182
+ ∀e = {u,v} ∈ ES, u,v ∈ S
183
+ the subgraph induced by a non-empty vertex subset S of V
184
+ α(G)
185
+ the size of a MIS of unweight graph G
186
+ αw(G)
187
+ the weight of a MWIS of graph G
188
+ AI
189
+ the set of all MWISs in graph G
190
+ AC
191
+ the set of all MWVCs in graph G
192
+ S ◀ AI
193
+ set S is an independent set and is included in all MWISs
194
+ C ◁ AC
195
+ set C is contained in all MWVCs
196
+ vertex v is strongly reducible
197
+ vertex v is included in all MWISs/MWVCs
198
+ vertex v is reducible
199
+ vertex v is contained in a MWIS/MWVC
200
+ vertex v is strongly inclusive
201
+ vertex v is included in all MWISs
202
+ vertex v is strongly sheathed
203
+ vertex v is contained in all MWVCs
204
+ vertex v is inclusive
205
+ vertex v is included in a MWIS
206
+ vertex v is sheathed
207
+ vertex v is contained in a MWVC
208
+ set S is strongly inclusive
209
+ set S is an independent set and is included in all MWISs
210
+ set S is strongly sheathed
211
+ set S is contained in all MWVCs
212
+ set S is inclusive
213
+ set S is an independent set and is included in a MWIS
214
+ set S is sheathed
215
+ set S is contained in a MWVC
216
+ independent set S is strongly exclusive
217
+ independent set S is not contained in all MWIS
218
+ independent set S is exclusive
219
+ independent set S is not contained in a MWIS
220
+ a set S called a simultaneous set
221
+ set S is either included in a MWIS or contained in a MWVC
222
+ Table 1: Notations used throughout the paper.
223
+ is described in Section 4. Extensive numerical tests are carried out in Section 5 to
224
+ verify the performance improvement of integrating our CITs into existing algorithmic
225
+ frameworks in terms of efficiency and accuracy. The paper is concluded in Section 6
226
+ with a few remarks.
227
+ 2
228
+ Two Extension Theorems
229
+ The theoretical cornerstones of CITs in this paper are two extension theorems: inde-
230
+ pendent set extension theorem and vertex cover extension theorem. Before delineat-
231
+ ing them, we need to have a deep understanding of the local structure of the MWIS
232
+ and first give the upper bound lemma.
233
+ Lemma 2.1 (upper bound lemma). Let set IC be an independent set in the graph.
234
+ (a) Suppose there is an Iw ∈ AI such that IC ̸⊆ Iw, then w(Iw ∩ N[IC]) ⩾ w(IC)
235
+ holds.
236
+ (b) Assume that IC ̸⊆ I, ∀I ∈ AI holds, then it satisfies: w(IC) < w(I∩N[IC]), ∀I ∈
237
+ AI.
238
+ Proof. Proof We first prove (a) by contradiction. If not, we can obtain an indepen-
239
+ dent set I′
240
+ w = (Iw\N[IC])∪(IC) such that w(I′
241
+ w) = w(Iw)+w(IC)−w(Iw ∩N[IC]) >
242
+ w(Iw), a contradiction.
243
+ Next, we consider (b). If there is an I1 ∈ AI such that w(IC) ⩾ w(I1 ∩ N[IC])),
244
+ we can construct an independent set I′
245
+ 1 = (I1\N[IC])∪IC satisfying w(I′
246
+ 1) = w(I1)+
247
+ w(IC) − w(I1 ∩ N[IC]) ⩾ w(I1).
248
+ Then I′
249
+ 1 ∈ AI and IC ⊆ I′
250
+ 1, which leads to a
251
+ contradiction.
252
+ 5
253
+
254
+ The upper bound lemma describes such a property: For any independent set that
255
+ is (strongly) exclusive, the weight of the intersection of its closed neighbor set with
256
+ the MWIS is the upper bound on its weight. With it, the independent set extension
257
+ theorem can be introduced as follows.
258
+ Theorem 2.2 (Independent Set Extension Theorem). Let sets IS and S be two
259
+ independent sets in the graph.
260
+ (a) Assume that there exists an Iw ∈ AI such that IS ⊆ Iw.
261
+ If there is an
262
+ independent set IS′ ⊆ N(IS) such that w(IS′) > w(IS ∩ N(IS′)), then
263
+ there exists an independent set IS′′ ⊆ N(IS′)\N[IS] satisfying the inequal-
264
+ ity: w(IS′) ⩽ w(IS ∩ N(IS′)) + w(IS′′). In addition, IS ∪ IS′′ ⊆ Iw if such
265
+ IS′′ is unique.
266
+ (b) Suppose S ◀ AI, then for any independent set S′ ⊆ N(S), there is an indepen-
267
+ dent set S′′ ⊆ N(S′)\N[S] such that w(S′) < w(S ∩ N(S′)) + w(S′′). Besides,
268
+ if such S′′ is unique, then S ∪ S′′ ◀ AI.
269
+ Proof. Proof We first consider the proof of (a), and it is obvious that IS′ ̸⊆ Iw. In
270
+ view of the fact that the relationship between Iw and N[IS′] satisfies: Iw ∩ N[IS′] =
271
+ Iw∩N(IS′) = (IS∩N(IS′))∪(Iw∩(N(IS′)\N[IS])) and by the upper bound lemma,
272
+ we can get: w(IS ∩ N(IS′)) + w(Iw ∩ (N(IS′)\N[IS])) = w(Iw ∩ N(IS′)) = w(Iw ∩
273
+ N[IS′]) ⩾ w(IS′). Thus, the existence of such IS′′ is proved. Furthermore, assuming
274
+ that such IS′′ is unique, then IS′′ = Iw ∩ (N(IS′)\N[IS]) and IS ∪ IS′′ ⊆ Iw.
275
+ Similar ideas can be used to prove (b). Obviously S′ ̸⊆ I, ∀I ∈ AI holds, so from
276
+ the upper bound lemma, it can be directly obtained: ∀I ∈ AI, w(I ∩N[S′]) > w(S′).
277
+ Further, by considering that the relationship between I and N[S′] satisfies: I ∩
278
+ N[S′] = I ∩N(S′) = (S∩ N(S′))∪(I ∩(N(S′)\N[S])), we prove the existence of such
279
+ S′′. Also, if such S′′ is unique, the following result holds: S′′ = I∩(N(S′)\N[S]), ∀I ∈
280
+ AI, and then S ∪ S′′ ◀ AI.
281
+ The independent set extension theorem gives a method for extending independent
282
+ set that is (strongly) inclusive: Try to find an independent set to add to the extended
283
+ independent set, and that independent set is the only one that guarantees that the
284
+ upper bound lemma is satisfied in the local structure of the extended independent
285
+ set.
286
+ Next, with the help of the upper bound lemma, the vertex cover extension
287
+ theorem is given below.
288
+ Theorem 2.3 (Vertex Cover Extension Theorem). Let sets IC and C be two vertex
289
+ subsets in the graph.
290
+ 6
291
+
292
+ (a) Suppose set IC ⊆ V Cw, then the vertices in IC have the property: ∀p ∈ IC,
293
+ w(p) ⩽ αw(G[N(p)\IC]). Also, for a vertex v ∈ IC and a vertex u ∈ N 2(v),
294
+ IC ∪ {u} ⊆ V Cw holds if the inequality w(v) > αw(G[N(v)\(IC ∪ N(u))]) is
295
+ satisfied.
296
+ (b) Assume that set C ◁ AC, then ∀p ∈ C, w(p) < αw(G[N(p)\C]) is always
297
+ satisfied. In addition, if there exists a vertex v ∈ C and a vertex u ∈ N 2(v)
298
+ such that w(v) ⩾ αw(G[N(v)\(C ∪ N(u))]), then C ∪ {u} ◁ AC.
299
+ Proof. Proof We first consider (a) and let set Iw = V \V Cw. From the upper bound
300
+ lemma, these results can be directly obtained: ∀p ∈ IC, w(p) ⩽ w(Iw ∩ N[p]) =
301
+ w(Iw ∩ N(p)) ⩽ αw(G[N(p)\IC]). Also, based on the assumption about u in (a),
302
+ if u ∈ Iw, then w(v) ⩽ w(Iw ∩ N[v]) = w(Iw ∩ N(v)) ⩽ αw(G[N(v)\(IC ∪ N(u))]),
303
+ which leads to a contradiction.
304
+ Similar methods can be used to prove (b).
305
+ First, ∀p ∈ C, ∀I ∈ AI, w(p) <
306
+ w(I ∩N[p]) = w(I ∩N(p)) ⩽ αw(G[N(p)\C]) can be obtained from the upper bound
307
+ lemma. Besides, under given conditions about u in (b), if there is an I∗ ∈ AI such
308
+ that u ∈ I∗, a contradiction is deduced from w(p) < w(I∗ ∩ N[p]) = w(I∗ ∩ N(p)) ⩽
309
+ αw(G[N(p)\(C ∪ N(u))]).
310
+ The vertex cover extension theorem describes how to expand a set that is (strongly)
311
+ sheathed: Attempt to find a vertex that satisfies the condition that after removing
312
+ its neighbor set, the upper bound lemma is not satisfied in the local structure of the
313
+ expanded set. If such a vertex is found, it is directly added to the expanded set.
314
+ 3
315
+ Causal Inference Techniques
316
+ In this section, with the help of the upper bound lemma and two extension theorems,
317
+ we give the CITs used in this paper. Our CITs can be divided into two types: The
318
+ first type is a strongly reducible state-preserving technique introduced in Section 3.1,
319
+ while the second type is a reducible state-preserving technique shown in Section 3.2.
320
+ 3.1
321
+ Strongly reducible state-preserving technique
322
+ The strongly reducible state-preserving technique exploits the assumption that a
323
+ vertex is strongly reducible, and the assumed state of the vertex can be divided into
324
+ two cases: The vertex is assumed to be strongly inclusive or is assumed to be strongly
325
+ sheathed. We first consider the assumption that a vertex is strongly inclusive and
326
+ give the following definition.
327
+ 7
328
+
329
+ Definition 3.1. Let set S be an independent set in the graph. If a vertex u ∈ N(S)
330
+ such that w(u) ⩾ w(S ∩ N(u)), we call it a child of set S. A child u is called an
331
+ extending child if and only if there exists a unique independent set S∗ ⊆ N(u)\N[S]
332
+ such that w(u) < w(S ∩ N(u)) + w(S∗) and vertex set S∗ is called a satellite set of
333
+ set S.
334
+ On the basis of Definition 3.1, with the assumption that a vertex is strongly inclu-
335
+ sive, the concept of ‘confined/unconfined vertices’ is given by the following conflict
336
+ analysis process:
337
+ Definition 3.2. Let v be a vertex in the graph. Suppose set S := {v} ◀ AI, repeating
338
+ (i) until (ii) or (iii) holds:
339
+ (i) As long as set S has an extending child in N(S), set S is extended by including
340
+ the corresponding satellite set into set S.
341
+ (ii) If a child u such that w(u) ⩾ w(S ∩N(u))+αw(G[N(u)\N[S]]) could be found,
342
+ that is, the upper bound lemma is not satisfied in the local structure of set S,
343
+ then halt and vertex v is called an unconfined vertex.
344
+ (iii) If any child is not an extending child, then halt and return set Sv = S. In this
345
+ case, vertex v is called a confined vertex and the set Sv is called the confining
346
+ set of vertex v.
347
+ Some examples of unconfined vertex are given in Figure 1. By means of the con-
348
+ flict analysis process in Definition 3.2, vertices a and h can be found to be unconfined
349
+ vertices. It is also worth noting that, by the definition of unconfined vertex given
350
+ in [27], in Figure 1, only vertex a can be found to be an unconfined vertex. The
351
+ reason for this is that we further generalize the concept of confined/unconfined ver-
352
+ tices in this work. Compared with the definition of extending child u in [27], which
353
+ requires |N(u)\N[S]| = 1 and w(u) < w(N(u)\N(S)), we can consider the more
354
+ general case where N(u)\N[S] is an independent set rather than a single vertex,
355
+ helping us find more unconfined vertices.
356
+ Next, we will explore the properties of confined/unconfined vertices.
357
+ By the
358
+ conflict analysis process in Definition 3.2 and the independent set extension theorem,
359
+ set S can be extended under the assumption: set S := {v} ◀ AI, and set S ◀ AI is
360
+ always satisfied. If vertex v is a unconfined vertex, then the upper bound lemma is not
361
+ satisfied in the local structure of set S, which contradicts set S ◀ AI. Thus, vertex v
362
+ is sheathed. Otherwise, then there is a state-preserving result, i.e., the corresponding
363
+ confining set Sv ◀ AI holds. Furthermore, suppose two confined vertices u, v and the
364
+ corresponding confining sets Su, Sv such that u ∈ Sv and v ∈ Su. If {v} ◀ AI, then
365
+ 8
366
+
367
+ Figure 1:
368
+ Some examples of unconfined vertices, and a MWIS in this graph
369
+ is {b, d, g, i}.
370
+ Let set S := {a}, from Definition 3.1, vertex b is an extend-
371
+ ing child of set S and set {c} is a satellite set of set S.
372
+ Thus, set S can be
373
+ extended as:
374
+ {a, c}.
375
+ At this time, it can be found that a child d such that
376
+ w(d) ⩾ w(S ∩ N(d)) + αw(G[N(d)\N[S]]), then halt and conclude that vertex a
377
+ is an unconfined vertex.
378
+ Similarly, let set S := {h}, then it can be found that
379
+ vertex g is an extending child of set S and set {f, l} is a satellite set of set S.
380
+ So set S can be further expanded as: {h, l, f}.
381
+ After that, the child i satisfied:
382
+ w(i) ⩾ w(S ∩ N(i)) + αw(G[N(i)\N[S]]), hence, vertex h is an unconfined vertex.
383
+ obviously {u} ◀ AI holds. If not, vertex v is sheathed in graph G. Since v ∈ Su,
384
+ then vertex v is included in the satellite set of an intermediate state set S′ of Su,
385
+ which means that in graph G[V \{v}], the upper bound lemma is not satisfied in the
386
+ local structure of set S′. Thus, by Definition 3.2, vertex u is an unconfined vertex
387
+ of graph G[V \{v}] and is sheathed in this graph. From these analysis results and
388
+ the symmetry of the relationship between vertex v and vertex u, we can know that
389
+ vertex set {u, v} is a simultaneous set. Therefore, the following properties can be
390
+ obtained:
391
+ Corollary 3.3. Let v is a vertex in the graph.
392
+ (a) If vertex v is an unconfined vertex, then it is sheathed and after deleting it from
393
+ the graph, the weight of the MWIS in the remaining graph remains unchanged.
394
+ (b) Suppose vertex v is a confined vertex, then either it is sheathed or the corre-
395
+ sponding confining set Sv ◀ AI. Moreover, if a vertex u ∈ Sv is also a confined
396
+ vertex with the corresponding confining set Su and v ∈ Su, then vertex set {u, v}
397
+ is a simultaneous set.
398
+ From Corollary 3.3, it can be known that the conflict analysis process in Defini-
399
+ tion 3.2 can be used to find the vertex that is sheathed or a simultaneous set. These
400
+ 9
401
+
402
+ h
403
+ a
404
+ :
405
+ 4
406
+ 3
407
+ b
408
+ 6
409
+ 11
410
+ 2
411
+ g
412
+ d
413
+ 9
414
+ 10
415
+ m
416
+ c
417
+ 5
418
+ 5
419
+ k
420
+ 5
421
+ f
422
+ 3
423
+ 8
424
+ 4
425
+ eCITs will be used to design reduction rules in Section 4.1. In addition, by the prop-
426
+ erty of confined vertex, a fact is obvious: If confined vertex v such that {v} ◀ AI,
427
+ then the corresponding confining set Sv ◀ AI. We will exploit this state-preserving
428
+ result in the B&R algorithm to design a branching rule to search for a solution in
429
+ Section 4.2.
430
+ Next, we proceed to consider the assumption that a vertex is strongly sheathed.
431
+ In the MIS problem, the notion of mirror is given by means of such an assumption
432
+ and is very useful in practice [1]. We will generalize the notion of mirror to the
433
+ MWIS problem: For a vertex v ∈ V , a mirror of vertex v is a vertex u ∈ N 2(v) such
434
+ that w(v) ⩾ αw(G[N(v)\N(u)]).
435
+ Remark 3.4. When the weight of all vertices in the graph is 1, then α(G[N(v)\N(u)]) =
436
+ αw(G[N(v)\N(u)]) ⩽ w(v) = 1. This means that N(v)\N(u) induces a clique or is
437
+ an empty set, and this is exactly the definition that vertex u is the mirror of vertex
438
+ v in the MIS problem.
439
+ To make the concept of mirror more practical, we further generalize it to the case
440
+ of set, which leads to the following definitions:
441
+ Definition 3.5. Let set C be a vertex subset in the graph. If a vertex v ∈ C satisfies
442
+ the inequality: w(v) < αw(G[N(v)\C]), we call it a father of set C. Furthermore,
443
+ if there exists a vertex u ∈ N 2(v) such that w(v) ⩾ αw(G[N(v)\(C ∪ N(u))]), then
444
+ the father v is called an extending father of set C and vertex u is called a mirror of
445
+ vertex v. We use M(v) to denote the set of mirrors of vertex v.
446
+ By means of Definition 3.5, and under the assumption that a vertex is strongly
447
+ sheathed, the concept of ‘covered/uncovered vertices’ is given by the following conflict
448
+ analysis process:
449
+ Definition 3.6. Let v be a vertex in the graph. At the beginning, suppose set C :=
450
+ {v} ◁ AC and repeating (i) until (ii) or (iii) are met:
451
+ (i) When set C has an extending father, extend set C by including the correspond-
452
+ ing set of mirrors to set C.
453
+ (ii) If there is a vertex u ∈ C such that w(u) ⩾ αw(G[N(u)\C]), in this case, the
454
+ upper bound lemma is not satisfied, then halt and vertex v is called an uncovered
455
+ vertex.
456
+ (iii) If set C has no extending father, then halt and return set Cv = C. In this case,
457
+ vertex v is called a covered vertex and vertex set Cv is called the covering set
458
+ of vertex v.
459
+ 10
460
+
461
+ Figure 2: An example of uncovered vertex and a MWIS of this graph is {a, e, g, h, j, l}.
462
+ Starting with set C := {a}, from Definition 3.5, it can be seen that vertex a is an
463
+ extending father of set C and set {e, g, h} is the mirrors set of vertex a. Thus, set C
464
+ can be extended to: {a, e, g, h}. Then, vertex h is also an extending father of set C
465
+ and set {j, k, l} is the mirrors set of vertex h. So set C can be further expanded as:
466
+ {a, e, g, h, j, k, l}. At this time, we find that w(l) ⩾ αw(G[N(l)\C]), then halt and
467
+ conclude that vertex a is uncovered.
468
+ An example of uncovered vertex is given in Figure 2 and we find that vertex a
469
+ is an uncovered vertex. In addition, the properties of uncovered/covered vertices
470
+ are worth further study. From the vertex cover extension theorem, in the conflict
471
+ analysis process of Definition 3.6, for any extending father f of set C, ∀u ∈ M(f),
472
+ if set C ◁ AC, set C ∪ {u} ◁ AC always holds. Thus, under the assumption set
473
+ C := {v} ◁ AC, if vertex v is not an uncovered vertex, then a state-preserving
474
+ result can be obtained: The corresponding covering set Cv ◁ AC. Otherwise, the
475
+ upper bound lemma is not satisfied in the local structure of set C, which contradicts
476
+ hypothesis set C ◁ AC. So vertex v is inclusive. Also, assume that the two covered
477
+ vertices u, v and the corresponding covering set Cu, Cv satisfy: v ̸∈ N(u), u ∈ Cv and
478
+ v ∈ Cu. If vertex v is inclusive, we first remove N[v] from graph G. Since v ∈ Cu,
479
+ then vertex v is a mirror of an extending father of an intermediate state set C′ of
480
+ set Cu and the upper bound lemma cannot be satisfied in graph G[V \N[v]] at this
481
+ time. Thus, vertex u is an uncovered vertex of graph G[V \N[v]] and is inclusive in
482
+ this graph. So there exists a MWIS in graph G containing both vertex v and vertex
483
+ u. Moreover, if {v} ◁ AC, {u} ◁ AC is clearly satisfied. Thus, from the symmetry
484
+ of the relationship between vertex u and vertex v, it can be known that vertex set
485
+ {u, v} is a simultaneous set. These properties are summarized as follows.
486
+ Corollary 3.7. Let v be a vertex in the graph G.
487
+ (a) If vertex v is an uncovered vertex, then it is inclusive. After deleting N[v] from
488
+ the graph, the weight of the MWIS in the remaining graph satisfies: αw(G) =
489
+ 11
490
+
491
+ h
492
+ d
493
+ b
494
+ a
495
+ J
496
+ 6
497
+ 3
498
+ 5
499
+ 3
500
+ 4
501
+ 3
502
+ g
503
+ c
504
+ k
505
+ e
506
+ 2
507
+ 8
508
+ 6
509
+ 9
510
+ 4
511
+ fαw(G[V \N[v]]) + w(v).
512
+ (b) If vertex v is a covered vertex. Then, either vertex v is inclusive or the cor-
513
+ responding covering set Cv ◁ AC. Also, if another covered vertex u with the
514
+ corresponding covering set Cu satisfies: v ̸∈ N(u), u ∈ Cv and v ∈ Cu, then
515
+ vertex set {u, v} is a simultaneous set.
516
+ Corollary 3.7 gives the following results: The conflict analysis process in Defi-
517
+ nition 3.6 can be applied to find the vertex that is inclusive or a simultaneous set.
518
+ In Section 4.1, we will use these CITs to design reduction rules. Besides, by the
519
+ property of covered vertex in (b) of Corollary 3.7, we can know a state-preserving
520
+ result: if the covered vertex v such that {v} ◁ AC, then the corresponding covering
521
+ set Cv ◁ AC.
522
+ 3.2
523
+ Reducible state-preserving technique
524
+ Similar to the first type of CIT, the reducible state-preserving technique utilizes the
525
+ assumption that a vertex is reducible, that is, assumes that a vertex is inclusive or
526
+ sheathed. With these assumptions, we can give state-preserving results similar to
527
+ the first type of CIT. Before that, we give the following definition.
528
+ Definition 3.8. Let sets IS and IC be two vertex subsets in the graph and set IS
529
+ is an independent set.
530
+ (a) A vertex u ∈ N(IS) is called an inferred child of set IS if it holds that
531
+ w(u) > w(IS ∩ N(u)).
532
+ Further, if there is only a unique independent set
533
+ IS∗ ⊆ N(u)\N[IS] that satisfies the inequality: w(u) ⩽ w(IS∩N(u))+w(IS∗),
534
+ we call the inferred child u an inferred extending child of set IS and vertex set
535
+ IS∗ is called an inferred satellite set of set IS.
536
+ (b) A vertex v ∈ IC is called an inferred father of set IC if it holds that w(v) ⩽
537
+ αw(G[N(v)\IC]). An inferred father v is called an inferred extending father of
538
+ set IC if there exists a vertex u ∈ N 2(v) such that w(v) > αw(G[N(v)\(IC ∪
539
+ N(u))]) and vertex u is called an inferred mirror of vertex v. Also, IM(v) is
540
+ used to denote its set of inferred mirrors.
541
+ By virtue of Definition 3.8 and the assumption that a vertex is inclusive or
542
+ sheathed, we can directly give the definitions of inferred confining set and inferred
543
+ covering set accordingly.
544
+ Definition 3.9. Suppose there are no unconfined vertex in the graph. Let v be a
545
+ vertex in the graph. Beginning with the assumption set IS := {v} ⊆ Iw.
546
+ 12
547
+
548
+ (i) Only if set IS has an inferred extending child in N(IS), set IS can be extended
549
+ by including the corresponding inferred satellite set to set IS.
550
+ (ii) The above process halts if set IS has no inferred extending child in N(IS) and
551
+ return set ISv = IS. We call vertex set ISv is the inferred confining set of
552
+ vertex v.
553
+ Definition 3.10. We assume that there are no uncovered vertex in graph. Let v be
554
+ a vertex in the graph. Starting with the assumption set IC := {v} ⊆ V Cw.
555
+ (i) While set IC has an inferred extending father, extend set IC by including the
556
+ corresponding set of inferred mirrors to set IC.
557
+ (ii) The above process halts if set IC has no inferred extending father and return
558
+ set ICv = IC. We call vertex set ICv is the inferred covering set of vertex v.
559
+ Examples of inferred confining set and inferred covering set are given in Fig-
560
+ ure 3.
561
+ By the process in Definition 3.9, we can find the inferred confining set
562
+ ISa = {a, c, e, j, g, h, k} of vertex a.
563
+ Similarly, according to the process in Defi-
564
+ nition 3.10, we can find the inferred covering set ICd = {b, d, f, i, l} of vertex d.
565
+ Moreover, from the independent set extension theorem and the vertex cover exten-
566
+ sion theorem, we can directly obtain the following Corollary:
567
+ Corollary 3.11. Let v be a vertex in the graph.
568
+ (a) If {v} ⊆ Iw, then the corresponding inferred confining set ISv ⊆ Iw.
569
+ (b) Suppose {v} ⊆ V Cw, then the corresponding inferred covering set ICv ⊆ V Cw.
570
+ From (a) of Corollary 3.11, under the premise {v} ⊆ Iw, the state-preserving
571
+ result can be obtained: ISv ⊆ Iw. We will integrate this result into the local search
572
+ process of heuristic algorithm in Section 4.3. In addition, (b) of Corollary 3.11 also
573
+ gives a similar state-preserving result result: If {v} ⊆ V Cw, then the corresponding
574
+ inferred covering set ICv ⊆ V Cw. This result can be used to design a branching rule
575
+ to search for a solution in Section 4.2.
576
+ Furthermore, during the branching process of the B&R algorithm, it is assumed
577
+ that a vertex v is selected for branching. Inspired by the successful application of
578
+ packing constraints in the MIS problem, we extend them to the MWIS problem and
579
+ propose the concept of “weight packing constraint”.
580
+ When assuming that vertex v is inclusive, if ∃u ∈ N(v) such that w(u) ⩾ w(v),
581
+ let N +(u) = N(u)\N[v]. To avoid obtaining another MWIS by adding vertex u to
582
+ the independent set and removing vertices in N(u) from the independent set, by the
583
+ 13
584
+
585
+ Figure 3: Examples of inferred confining set and inferred covering set. A MWIS for
586
+ this graph is {a, c, e, g, h, j, k}. We first search for the inferred confining set ISa of
587
+ vertex a. Let set IS := {a}, it can be seen from (a) of Definition 3.8 that vertex b
588
+ is an inferred extending child of set IS and set {c} is an inferred satellite set of set
589
+ IS. Thus, set IS can be extended to: {a, c}. Further, vertex d is also an inferred
590
+ extending child of set IS and set {e, j} is the corresponding inferred satellite set. So
591
+ set IS can be further extended to: {a, c, e, j}. At this time, it can be found that both
592
+ vertex f and vertex i are inferred extending children of set IS. Then, both vertex
593
+ set {g, h} and vertex set {k} are the corresponding inferred satellite sets. Finally,
594
+ the inferred confining set of vertex a can be found as: ISa = {a, c, e, j, g, h, k}.
595
+ Furthermore, we continue to search the inferred covering set ICd of vertex d. Let
596
+ set IC := {d}, according to (b) of Definition 3.8, vertex d is an extending father of
597
+ set IC and set {b, f, i} is its inferred mirrors set. Then, set IC can be extended as:
598
+ {b, d, f, i}. Next, it can be found that vertex b is an extending father of set IC and set
599
+ {l} is its inferred mirrors set. Thus, set IC can be further extended as: {b, d, f, i, l}.
600
+ Finally, the inferred covering set of vertex d can be found as: ICd = {b, d, f, i, l}.
601
+ .
602
+ upper bound lemma, the following state-preserving result needs to be guaranteed to
603
+ hold:
604
+ w(v) +
605
+
606
+ z∈N+(u)
607
+ w(z)(1 − xz) > w(u).
608
+ The 0-1 integer variable xz is used to indicate whether vertex z ∈ N +(u) is in the
609
+ independent set, and xz = 0 means it is in the independent set, otherwise it is not.
610
+ Thus, a weight packing constraint can be created as shown below:
611
+
612
+ z∈N+(u)
613
+ w(z)xz <
614
+
615
+ z∈N+(u)
616
+ w(z) − (w(u) − w(v)).
617
+ (3.1)
618
+ When assuming that vertex v is sheathed, to avoid that a MWIS containing it can
619
+ be found by modifying its state, by means of the upper bound lemma, the following
620
+ 14
621
+
622
+ d
623
+ C
624
+ e
625
+ 9
626
+ b
627
+ 9
628
+ 3
629
+ 3
630
+ 11
631
+ 4
632
+ a
633
+ 8
634
+ 10
635
+ 5
636
+ 3
637
+ 6
638
+ 3
639
+ 10
640
+ h
641
+
642
+ kstate-preserving result needs to be satisfied:
643
+
644
+ z∈N(v)
645
+ w(z)(1 − xz) > w(v).
646
+ So a weight packing constraint can also be created as follows:
647
+
648
+ z∈N(v)
649
+ w(z)xz <
650
+
651
+ z∈N(v)
652
+ w(z) − w(v).
653
+ (3.2)
654
+ These constraints will be kept and managed while the algorithm is searching for
655
+ a solution, and we only need to search all branches satisfying these constraints, since
656
+ no better solution exists in the remaining branches, thus narrowing the search space.
657
+ Let �
658
+ z∈S
659
+ w(z)xz < k be a weight packing constraint such that set S is non-empty.
660
+ When a vertex z is found to be inclusive, for each constraint that includes variable
661
+ xz, we delete the variable on the left side of the constraint and keep the right side
662
+ of the constraint unchanged. When a vertex z is inferred to be sheathed, for each
663
+ constraint that contains variable xz, we delete the variable on the left side of the
664
+ constraint and decrease the weight of vertex z on the right side of the constraint.
665
+ In the process of keeping and managing these constraints, some properties of causal
666
+ inference are mined, which can be divided into the following three cases.
667
+ (a) When there is a constraint whose right-hand term k is less than or equal to 0,
668
+ then we can directly prune subsequent searches from the current branch vertex.
669
+ (b) When there is a constraint whose right-hand term k is less than or equal to
670
+ the weight of any vertex in set S, if this set is not an independent set, we can
671
+ prune subsequent searches from the current branch vertex. If not, the vertices
672
+ in set S will be included in the independent set.
673
+ In addition, some new weight packing constraints can also be introduced. Sup-
674
+ pose there is a vertex p ∈ N(S) such that w(p) ⩾ w(N(p) ∩ S), let N +(p) =
675
+ N(p)\N[S], by the upper bound lemma, the following state-preserving result
676
+ needs to be guaranteed:
677
+ w(N(u) ∩ S) +
678
+
679
+ z∈N+(u)
680
+ w(z)(1 − xz) > w(u).
681
+ Therefore, we can introduce the following weight packing constraint:
682
+
683
+ z∈N+(p)
684
+ w(z)xz <
685
+
686
+ z∈N+(p)
687
+ w(z) − (w(p) − w(N(p) ∩ S)).
688
+ (3.3)
689
+ 15
690
+
691
+ (c) When there is a constraint whose right-hand term k > 0 and there is vertex
692
+ u ∈ N(S) such that
693
+
694
+ z∈N(u)∩S
695
+ w(z) ⩾ k, it can be inferred that vertex u is
696
+ sheathed to ensure that this constraint holds. In addition, in order to ensure
697
+ that the current state-preserving result is valid, similar to constraint (3.2), the
698
+ following constraint needs to be introduced:
699
+
700
+ z∈N(u)
701
+ w(z)xz <
702
+
703
+ z∈N(u)
704
+ w(z) − w(u).
705
+ (3.4)
706
+ The above properties of causal inference provide new pruning search techniques for
707
+ the B&R algorithm and can simplify the graph. We will integrate these techniques
708
+ into B&R algorithm in Section 4.2.
709
+ 4
710
+ Integrate CITs into Existing Algorithmic Frame-
711
+ works
712
+ We next describe in detail how CITs in Section 3 are integrated into the existing
713
+ algorithmic frameworks. Section 4.1 introduces how to apply the first type of CIT
714
+ to the reduction algorithm. Further, integrating the resulting reduction algorithm
715
+ and the state-preserving results of two types of CITs into B&R algorithm will be
716
+ presented in Section 4.2, and Section 4.3 will introduce the application of the state-
717
+ preserving results of the second type of CIT to the local search process of heuristic
718
+ algorithm.
719
+ 4.1
720
+ The Causal Reduce
721
+ We first introduce how to design reduction rules with the first type of CIT and
722
+ how to integrate them into the existing reduction algorithm. From the property of
723
+ unconfined vertex in Corollary 3.3 and the property of uncovered vertex in Corol-
724
+ lary 3.7, the following reduction rules that can directly determine whether a vertex
725
+ is reducible are given first:
726
+ • Rule I: Check whether a vertex v is unconfined or confined by the procedure
727
+ in Definition 3.2, and if it is unconfined, remove vertex v directly from the
728
+ graph.
729
+ • Rule II: Use the procedure in Definition 3.6 to check whether a vertex v is cov-
730
+ ered or uncovered, and if it is uncovered, include vertex v into the independent
731
+ set and remove N[v] from the graph.
732
+ 16
733
+
734
+ Before further introducing how to utilize the first type of CIT to design reduction
735
+ rules, we first give an important property about simultaneous set mentioned in [27]:
736
+ A simultaneous set S can be contracted by removing all vertices in set S from the
737
+ graph and introducing a vertex v∗ such that it is adjacent to all vertices in N(S)
738
+ with weight w(v∗) = w(S), while the weight of the MWIS in the remaining graph
739
+ remain unchanged.
740
+ Next, we will design reduction rules on simultaneous set through the first type of
741
+ CIT, and give the following definitions by the results of the simultaneous set given
742
+ in (b) of Corollary 3.3 and (b) of Corollary 3.7.
743
+ Definition 4.1. Let u, v be two vertices in the graph.
744
+ (a) Suppose vertices u and v be two confined vertices with confining set Su and Sv.
745
+ If u ∈ Sv and v ∈ Su, then set {u, v} is called a confining simultaneous set.
746
+ (b) Assume that vertices u and v be two covered vertices with covering set Cu and
747
+ Cv. Set {u, v} is called a covering simultaneous set if u ∈ Cv and v ∈ Cu.
748
+ From Definition 4.1, we have the following rules:
749
+ • Rule III: If there are two confined vertices that constitute a confining simul-
750
+ taneous set, then merge them.
751
+ • Rule IV: Merge two covered vertices u and v if they form a covering simulta-
752
+ neous set.
753
+ Next, we will describe how to integrate our reduction rules into an existing re-
754
+ duction algorithm—Reduce proposed by [27]. Reduce consists of seven steps. The
755
+ reduction rules used in these steps exploit the sufficient conditions that a vertex is
756
+ reducible. It executes these steps incrementally, which means that the next step is
757
+ only executed when all previous steps are no longer applicable. Thus, if the graph
758
+ is changed, it will go back to the first step. Notably, our reduction rules I and III
759
+ are further generalization of the reduction rules used in step 5 of Reduce. So, we
760
+ can combine our reduction rules I and III into one step to replace step 5 in Reduce
761
+ and label this step as Remove Unconfined & Contract Confining. Similarly,
762
+ we can also integrate our reduction rules II and IV into another new step in the
763
+ reduction algorithm, called Remove Uncovered & Contract Covering.
764
+ • Remove Unconfined & Contract Confining: Check whether a vertex is
765
+ unconfined or confined. If it is confined, apply Rule I to remove it; If not, use
766
+ Rule III to contract the corresponding confining simultaneous set when it can
767
+ be found.
768
+ 17
769
+
770
+ • Remove Uncovered & Contract Covering: If a vertex is checked to be
771
+ uncovered, use Rule II to reduce it. Otherwise, if the corresponding covering
772
+ simultaneous set can be found, use Rule IV to merge it.
773
+ Figure 4: Casual Reduce: Given an input graph G, each step of the algorithm
774
+ is executed sequentially and the graph changes, immediately go back to the first
775
+ step. When all steps are completed and the graph no longer changes, return to the
776
+ remaining graph kernel.
777
+ Thus, a new reduction algorithm called Causal Reduce can be obtained by using
778
+ Remove Unconfined & Contract Confining to replace step 5 of Reduce and
779
+ adding Remove Uncovered & Contract Covering between Remove Uncon-
780
+ fined & Contract Confining and step 6 of Reduce, which is shown in Figure 4.
781
+ We will use Causal Reduce(G) = (K, c) to represent the processing of this algo-
782
+ rithm on a given input graph G. The processing result of this algorithm consists of
783
+ two parts: One is the remaining graph called kernel K and the other is the weight of
784
+ the vertex set contained in the MWIS obtained by inference. It’s worth noting that
785
+ the reduction algorithm Causal Reduce may not resolve all instances directly, but
786
+ it can be used as a preprocessing for heuristic and exact algorithm.
787
+ 4.2
788
+ The Causal B&R Solver
789
+ Before introducing how to integrate our CITs into B&R algorithm, we briefly intro-
790
+ duce the state-of-the-art exact algorithm Solve proposed by [27]. Solve is based on
791
+ the idea of B&R algorithm, which first apply reduction algorithm Reduce to reduce
792
+ the instance. Then, apply branching rule by virtue of the property of the confining
793
+ set and perform reduction algorithm Reduce in every branch of the search tree to
794
+ 18
795
+
796
+ Step 4 of Reduce
797
+ Remove Unconfined & Contract Confining
798
+ Step 3 of Reduce
799
+ Remove Uncovered& Contract Covering
800
+ Graph Change
801
+ Step 2 of Reduce
802
+ Step 6 of Reduce
803
+ Step 1 of Reduce
804
+ Step 7 of Reduce
805
+ G = (V, E, w)
806
+ Kernel Kfind a solution. During the searching, it uses a standard technique based on finding
807
+ upper and lower bounds to prune the search tree and take the best solution weight
808
+ Wb currently found in the algorithm as the lower bound. Initially, let Wb be the
809
+ weight of the solution obtained by heuristic algorithm on the kernel K, and update
810
+ Wb once a better solution is obtained in the algorithm. The heuristic algorithm,
811
+ denoted by Greedy(G), is a greedy algorithm that iteratively selects a vertex in
812
+ order of some measure and removes its closed neighbor set from the graph. In each
813
+ searching branch, it uses a heuristic method to find an upper bound Wub of the op-
814
+ timal solution weight of the current graph, which is based on weight clique covers
815
+ and is denoted by UpperBound(G). If the current best solution weight Wb is not
816
+ smaller than Wub, then there is no better solution in this searching branch and it can
817
+ be discarded directly.
818
+ Algorithm 1 The Causal B&R Solver(G)
819
+ Require: A vertex weight graph G = (V, E, w);
820
+ Ensure: The weight of a MWIS of G.
821
+ 1: Initialization of global variable Wb: Wb ← 0;
822
+ 2: if weight packing constraints have been created then
823
+ 3:
824
+ while True do
825
+ 4:
826
+ (K, c) ← Causal Reduce(G);
827
+ 5:
828
+ check constraints();
829
+ 6:
830
+ if existence constraints are not satisfied then
831
+ 7:
832
+ return Wb;
833
+ 8:
834
+ else if graph is simplified then
835
+ 9:
836
+ continue;
837
+ 10:
838
+ else
839
+ 11:
840
+ break;
841
+ 12: else
842
+ 13:
843
+ (K, c) ← Causal Reduce(G);
844
+ 14: Wb ← max{Wb, c + Greedy(K)};
845
+ 15: if c + UpperBound(G) ⩽ Wb then
846
+ 16:
847
+ return Wb;
848
+ 17: Pick up a vertex v of maximum degree and compute the confining set Sv and the inferred covering set ICv;
849
+ 18: create weight packing constraint (3.1) and Wb ← max{Wb, c + w(Sv) + Causal B&R Solver(K − N[Sv])};
850
+ 19: create weight packing constraint (3.2) and Wb ← max{Wb, c + Causal B&R Solver(K − ICv)};
851
+ 20: return Wb;
852
+ Our CITs will be integrated into two parts of Solve, resulting in a new exact al-
853
+ gorithm called Causal B&R Solver. The first part is that we will use our reduction
854
+ algorithm Causal Reduce to reduce the instance to get the kernel K, and perform
855
+ the reduction algorithm on each branch of the search tree. The second part is that
856
+ we will make use of the state-preserving results of two types of CITs during the
857
+ branching process. Similar to the idea of Solve in [27], using property of confining
858
+ set to the branching process, when choosing a vertex with the maximum degree to
859
+ branch, the state-preserving results of (b) of Corollary 3.11 and (b) of Corollary 3.3
860
+ 19
861
+
862
+ will be used in this part. This means that during branching, we either remove the
863
+ inferred covering set of the branching vertex from the graph or include the confining
864
+ set of the branching vertex into the independent set. Furthermore, we will create
865
+ weight packing constraint (3.2) while removing the inferred covering set of branching
866
+ vertex. Similarly, we will also create weight packing constraint (3.1) when includ-
867
+ ing the confining set of branching vertex into the independent set. We will keep and
868
+ manage these weight packing constraints when searching for solutions in each branch
869
+ of the search tree. Specifically, another step called check constraints is added after
870
+ the last step of Casual Reduce. In this step, for each weight packing constraint, we
871
+ will check whether the constraint holds and whether the graph can be simplified by
872
+ the causal inference properties of that constraint. If any constraint is violated, the
873
+ searching branch will be skipped. If the graph can be simplified, Causal Reduce
874
+ will continue to execute after reducing the graph. If none of the above conditions are
875
+ met, the subsequent process will be performed. The main steps of Causal B&R
876
+ Solver are listed in Algorithm 1.
877
+ 4.3
878
+ The Causal Search
879
+ After taking our reduction algorithm Causal Reduce as a preprocessing, we ap-
880
+ ply the state-preserving result of second type of CIT to the local search process
881
+ of heuristic algorithm DynWVC2 [6] to solve the complementary problem of the
882
+ MWIS problem—the MWVC problem, which leads to a new algorithm called Causal
883
+ Search.
884
+ Algorithm 2 The basic framework of DynWVC2 algorithm.
885
+ Require: A vertex weight graph G = (V, E, w), the cutoff time of the running T;
886
+ Ensure: A vertex cover of G.
887
+ 1: V C ← Construct();
888
+ 2: V C∗ ← V C;
889
+ 3: while elapsed time<T do
890
+ 4:
891
+ R ← RemoveVertices(VC)
892
+ 5:
893
+ while some edge is uncovered by V C do
894
+ 6:
895
+ choose a vertex v from N(R);
896
+ 7:
897
+ V C ← V C ∪ {v};
898
+ 8:
899
+ remove redundant vertices from V C;
900
+ 9:
901
+ if w(V C)<w(V C∗) then
902
+ 10:
903
+ V C∗ ← V C
904
+ The DynWVC2 algorithm proposed by [6], is the state-of-the-art heuristic algo-
905
+ rithm for solving MWVC problem. The basic framework of this algorithm is shown in
906
+ Algorithm 2. The local search process of this algorithm mainly consists of a removing
907
+ phase and an adding phase, and the specific process can be found in [6].
908
+ 20
909
+
910
+ Our CITs will be considered in the removing phase of the algorithm — Re-
911
+ moveVertices function. In this function, there are two scoring functions loss and
912
+ valid score used to select the vertices to remove from the vertex cover V C. The
913
+ specific definition of these two scoring functions can be seen in [6]. The loss and
914
+ valid score functions have fundamentally different effects on the behavior of the al-
915
+ gorithm. Vertex selection using loss function is an “exploratory” selection; in other
916
+ words, it is quite possible that such a chosen vertex is good for the quality of the
917
+ solution, but this cannot be determined. Different from “exploratory” vertex selec-
918
+ tion, valid score is a “deterministic” selection, that is, we can determine whether
919
+ removing a vertex will have a positive impact on the quality of the solution. For
920
+ example, if a vertex has a negative valid score value, this means that after removing
921
+ this vertex and adding its adjacent uncovering vertices, a vertex cover with lower
922
+ weight than the current vertex cover can be obtained [6].
923
+ In removing phase, the vertex with the minimum loss is removed from vertex
924
+ cover V C first, and then the second removed vertex is selected by a dynamic vertex
925
+ selection strategy. The details of dynamic vertex selection strategy can be learned
926
+ in [6]. After removing the two vertices, if the total degree of the removed vertices
927
+ does not reach a predetermined value (which is set to 2 times average degree of
928
+ the graph), another vertex to be selected with the BMS strategy [5], which samples
929
+ t (t = 50) vertices from vertex cover V C and chooses the one with the minimum
930
+ loss, will be removed to expand the search region. In this way, it solves the problem
931
+ that when removing two vertices the resulting search area is too small and limits the
932
+ ability of the adding phase to find better local optima. If the search area obtained
933
+ by removing two vertices is large enough, in order to balance the search time and
934
+ search quality, the third vertex will not be selected for removing.
935
+ The state-preserving result of second type of CIT will be applied to the dynamic
936
+ vertex selection strategy for selecting the second vertex to be removed. The dynamic
937
+ vertex selection strategy consists of a primary vertex scoring function valid score
938
+ and a secondary scoring function loss. When the removed vertex v is selected by
939
+ valid score function, it can be seen from the nature of the valid score function:
940
+ There is a high probability that there exists a MWIS I containing it. If the vertex v
941
+ is indeed included in I, by (a) of Corollary 3.11, the corresponding inferred confining
942
+ set ISv also contained in I. Inspired by this result, when selecting the second removed
943
+ vertex v by scoring function valid score, we will remove the vertices in the inferred
944
+ confining set ISv from the vertex cover V C. In this way, the search region can be
945
+ expanded and the number of times to continue to use the third removed vertex to
946
+ expand the search area is reduced, which means that the ability of local search to
947
+ find better local optima is improved. An example of our CITs applied to the vertex
948
+ 21
949
+
950
+ removing process is presented in Figure 5.
951
+ Figure 5: Example of our CITs applied to the vertex removing process: When we uti-
952
+ lize valid score to select the removed vertex c from the vertex cover V C = {a, c, d},
953
+ we can compute the corresponding inferred confining set ISc = {a, c, d} of vertex c
954
+ and remove set ISc from vertex cover V C.
955
+ In addition, it can be seen from the calculation process of Definition 3.9 about
956
+ the inferred confining set: the computational complexity of ISv for each vertex v is
957
+ O(|N(ISv)||ISv|). This means that in the actual application process, since the size
958
+ of the generally obtained inferred confining set is relatively small, its computational
959
+ cost is very small. Thus, our CITs is helpful for improving the performance of local
960
+ search process.
961
+ 5
962
+ Experiments
963
+ We will conduct four experiments to verify the effect of integrating our CITs into
964
+ current algorithmic frameworks. The first experiment is used to analyze the impact
965
+ of our CITs for the reduction algorithm. The examination of the performance gain
966
+ of our CITs in the B&R algorithm is shown in the second experiment. The third
967
+ experiment is used to test the ability of our Causal Reduce as a preprocessing
968
+ to improve the performance of the heuristic algorithm. The last experiment is con-
969
+ ducted to verify the effect of adding our CITs to the local search process of heuristic
970
+ algorithm.
971
+ Experiment environment Setup. All of our algorithms are implemented in C++,
972
+ and compiled by g++ with ‘-O3’ option. All experiments are run on a platform with
973
+ 128G RAM and one Intel(R) Xeon(R) Gold 5117 CPU @ 2.00GHz.
974
+ Compared Algorithms. In previous studies, most of them only use some sim-
975
+ ple rules as preprocessing to reduce problem instances, and do not pay attention to
976
+ the performance of preprocessing. Two recent papers [15, 27] have studied in depth
977
+ the reduction rules for the MWIS and analyzed their performance. Since the algo-
978
+ rithm Reduce in [27] outperforms the algorithm in [15] and our Causal Reduce
979
+ 22
980
+
981
+ a
982
+ a
983
+ 6
984
+ Compute the corresponding
985
+ 5
986
+ inferred confining set
987
+ 12
988
+ 12
989
+ 5
990
+ b
991
+ C
992
+ b
993
+ 4
994
+ e
995
+ eis obtained by integrating our CITs into Reduce, in this paper, we only use it as a
996
+ baseline to analyze the impact of our CITs for the reduction algorithm. Additionally,
997
+ in order to fully understand the role of different CITs on the reduction algorithm, we
998
+ control the application of CITs in Reduce and conduct comparative experiments.
999
+ Similar to the Causal Reduce shown in Figure 4, we use Re-Confin to represent
1000
+ the algorithm obtained after replacing the step 5 of Reduce with Remove Uncon-
1001
+ fined & Contract Confining and Re-Cover to denote the algorithm obtained
1002
+ by adding Remove Uncovered & Contract Covering between step 5 and 6 of
1003
+ Reduce.
1004
+ On the basis of the reduction algorithm Reduce, the authors of [27] also de-
1005
+ veloped a fast exact algorithm Solve, which is the state-of-art exact algorithm in
1006
+ previous work, and our Causal B&R Solver is obtained by applying our CITs
1007
+ into it, so it will be used as a baseline to verify the performance improvement of
1008
+ our CITs for the B&R algorithm. Furthermore, we use Solve-CR to identify the
1009
+ algorithm obtained by replacing Reduce with Causal Reduce in Solve, Solve-
1010
+ CR-IC refers to the algorithm obtained by further simplifying the branch by using
1011
+ the inferred covering set of the branching vertex in the branching process on the basis
1012
+ of Solve-CR, and Solve-Packing to represent the algorithm obtained by applying
1013
+ our weight packing constraints to the branching process of Solve. We will conduct
1014
+ comparative experiments on these algorithms to clarify the impact of different CITs
1015
+ on the B&R algorithm.
1016
+ Two state-of-the-art heuristic algorithms FastWVC (Fast) [7] and DynWVC2 (Dyn) [6]
1017
+ will be used to verify that our Causal Reduce as preprocessing improves the per-
1018
+ formance of the heuristic algorithm. We will use Causal Re + Fast and Causal
1019
+ Re + Dyn to denote applying our Causal Reduce as preprocessing before execut-
1020
+ ing FastWVC and DynWVC2. In addition, to further verify the superiority of our
1021
+ Causal Reduce as preprocessing for improving the performance of the heuristic al-
1022
+ gorithm, we also conduct comparative experiments using Reduce as a preprocessing
1023
+ of the heuristic algorithm. Likewise, we use Re + Fast and Re + Dyn to indicate
1024
+ the application of the previous reduction algorithm Reduce before FastWVC and
1025
+ DynWVC2 are executed. Moreover, our Causal Search is obtained by integrating
1026
+ CITs into the local search process of DynWVC2. Therefore, we can verify the effect
1027
+ of this operation by comparing DynWVC2 with Causal Search.
1028
+ Instances. We evaluate all algorithms on six real graphs which are most representa-
1029
+ tive and most difficult graphs from different domains. These graphs are downloaded
1030
+ from Network Data Repository [21]. All of them have 100 thousands to millions of
1031
+ vertices, and dozens of millions of edges. These instances become popular in recent
1032
+ works for the MWIS problem. Statistics of these graphs are shown in Table 2. In our
1033
+ 23
1034
+
1035
+ experiment, the weight of each vertex in the graph will have two random allocation
1036
+ mechanisms ∗, which are commonly used in previous work [15, 27, 6, 7]. The first
1037
+ allocation mechanism is that the weight of each vertex in the graph is obtained from
1038
+ [1, 200] uniformly at random, we will number the six datasets with 1−6. The second
1039
+ allocation mechanism is that the weight of each vertex in the graph follows a random
1040
+ uniform distribution of [20, 100], and 7 − 12 will be used to number the six datasets.
1041
+ inf-road-usa
1042
+ soc-livejournal
1043
+ sc-ldoor
1044
+ tech-as-skitter
1045
+ sc-msdoor
1046
+ inf-roadNet-CA
1047
+ Vertices
1048
+ 23947347
1049
+ 4033137
1050
+ 952203
1051
+ 1694616
1052
+ 415863
1053
+ 1957027
1054
+ Edges
1055
+ 28854312
1056
+ 27933062
1057
+ 20770807
1058
+ 11094209
1059
+ 9378650
1060
+ 2760388
1061
+ NO.
1062
+ 1, 7
1063
+ 2, 8
1064
+ 3, 9
1065
+ 4, 10
1066
+ 5, 11
1067
+ 6, 12
1068
+ Table 2: All graphs are sorted in descending order regarding the number of edges.
1069
+ In the row headed by “NO.”, each number is used to represent the corresponding
1070
+ number of the dataset generated by the graph according to the corresponding vertex
1071
+ weight allocation mechanism.
1072
+ 5.1
1073
+ Impact of CITs on Reduction Algorithm
1074
+ We first analyze the impact of our CITs for the reduction algorithm and evaluate the
1075
+ performance of all reduction algorithms by measuring the running time, the size of
1076
+ the remaining graphs (kernel size), and the ratio of the kernel size to the number of
1077
+ vertices in the original graph (We simply refer to it here as the ratio for convenience.).
1078
+ The experimental results of all algorithms are output in Table 3. We can know
1079
+ that all reduction algorithms can significantly simplify the graph, and even reduce
1080
+ the graph to less than 0.1% of the original size. Besides, we can see that our Causal
1081
+ Reduce achieves best reduction effect in all datasets, that is, our Causal Reduce
1082
+ results in a much smaller kernel size than other algorithms. Moreover, compared
1083
+ with Reduce, Re-Confin can achieve better reduction effect in all datasets, while
1084
+ Re-Cover has basically no performance improvement. This shows that replacing the
1085
+ step 5 of Reduce with Remove Unconfined & Contract Confining plays a key
1086
+ role in improving the performance of the reduction algorithm, and combined with
1087
+ Remove Uncovered & Contract Covering, the performance of the reduction
1088
+ ∗All datasets obtained through these two random assignment mechanisms can be found at http:
1089
+ //lcs.ios.ac.cn/~caisw/graphs.html.
1090
+ 24
1091
+
1092
+ Reduce
1093
+ Re-Confin
1094
+ Re-Cover
1095
+ Causal Reduce
1096
+ NO.
1097
+ |V |
1098
+ Time(S)
1099
+ Kernel Size
1100
+ Ratio(%)
1101
+ Time(S)
1102
+ Kernel Size
1103
+ Ratio(%)
1104
+ Time(S)
1105
+ Kernel Size
1106
+ Ratio(%)
1107
+ Time(S)
1108
+ Kernel Size
1109
+ Ratio(%)
1110
+ 1
1111
+ 23947347
1112
+ 72.00
1113
+ 431891
1114
+ 1.80
1115
+ 71.66
1116
+ 428137
1117
+ 1.79
1118
+ 74.03
1119
+ 431888
1120
+ 1.80
1121
+ 55.88
1122
+ 275082
1123
+ 1.15
1124
+ 2
1125
+ 4033137
1126
+ 12.93
1127
+ 7273
1128
+ 0.18
1129
+ 16.57
1130
+ 5960
1131
+ 0.15
1132
+ 16.20
1133
+ 7261
1134
+ 0.18
1135
+ 17.92
1136
+ 3281
1137
+ 0.08
1138
+ 3
1139
+ 952203
1140
+ 3.08
1141
+ 6492
1142
+ 0.68
1143
+ 3.89
1144
+ 2780
1145
+ 0.29
1146
+ 3.89
1147
+ 6447
1148
+ 0.68
1149
+ 3.84
1150
+ 1682
1151
+ 0.18
1152
+ 4
1153
+ 1694616
1154
+ 1.78
1155
+ 5904
1156
+ 0.35
1157
+ 2.32
1158
+ 5613
1159
+ 0.33
1160
+ 2.26
1161
+ 5909
1162
+ 0.35
1163
+ 5.03
1164
+ 3974
1165
+ 0.23
1166
+ 5
1167
+ 415863
1168
+ 1.66
1169
+ 6570
1170
+ 1.58
1171
+ 1.84
1172
+ 3166
1173
+ 0.76
1174
+ 2.04
1175
+ 6570
1176
+ 1.58
1177
+ 1.86
1178
+ 2162
1179
+ 0.52
1180
+ 6
1181
+ 1957027
1182
+ 76.14
1183
+ 305470
1184
+ 15.61
1185
+ 63.65
1186
+ 300135
1187
+ 15.34
1188
+ 76.57
1189
+ 305470
1190
+ 15.61
1191
+ 41.12
1192
+ 202885
1193
+ 10.37
1194
+ 7
1195
+ 23947347
1196
+ 117.54
1197
+ 437993
1198
+ 1.83
1199
+ 94.52
1200
+ 434606
1201
+ 1.81
1202
+ 113.07
1203
+ 438004
1204
+ 1.83
1205
+ 58.02
1206
+ 235243
1207
+ 0.98
1208
+ 8
1209
+ 4033137
1210
+ 11.68
1211
+ 7620
1212
+ 0.19
1213
+ 15.90
1214
+ 6371
1215
+ 0.16
1216
+ 15.65
1217
+ 7576
1218
+ 0.19
1219
+ 23.38
1220
+ 3806
1221
+ 0.09
1222
+ 9
1223
+ 952203
1224
+ 8.05
1225
+ 29116
1226
+ 3.06
1227
+ 5.08
1228
+ 7906
1229
+ 0.83
1230
+ 8.88
1231
+ 29116
1232
+ 3.06
1233
+ 5.06
1234
+ 4628
1235
+ 0.49
1236
+ 10
1237
+ 1694616
1238
+ 1.94
1239
+ 6999
1240
+ 0.41
1241
+ 2.55
1242
+ 6623
1243
+ 0.39
1244
+ 2.53
1245
+ 6991
1246
+ 0.41
1247
+ 5.19
1248
+ 4429
1249
+ 0.26
1250
+ 11
1251
+ 415863
1252
+ 4.09
1253
+ 24736
1254
+ 5.95
1255
+ 2.75
1256
+ 10098
1257
+ 2.43
1258
+ 4.24
1259
+ 24736
1260
+ 5.95
1261
+ 2.66
1262
+ 7081
1263
+ 1.70
1264
+ 12
1265
+ 1957027
1266
+ 19.01
1267
+ 131498
1268
+ 6.72
1269
+ 17.00
1270
+ 128637
1271
+ 6.57
1272
+ 19.54
1273
+ 131498
1274
+ 6.72
1275
+ 8.64
1276
+ 65124
1277
+ 3.33
1278
+ Table 3: Impact of CITs for the the reduction algorithm. The bold and underlined
1279
+ numbers are the minimum kernel size and shortest running time, respectively.
1280
+ algorithm will be greatly improved, but only adding Remove Uncovered & Con-
1281
+ tract Covering can hardly improve the performance of the reduction algorithm.
1282
+ More notably, our Causal Reduce take less time than other algorithms on half
1283
+ of the datasets. On the rest of the datasets, our Causal Reduce only takes a few
1284
+ seconds longer than other algorithms. These phenomena show that integrating our
1285
+ CITs into the reduction algorithm can significantly improve the performance of the
1286
+ algorithm, but the increase in time cost is very small, and they can even reduce the
1287
+ time cost.
1288
+ 5.2
1289
+ Performance Gain of CITs on the B&R Algorithm
1290
+ We will examine the performance gain of our CITs for B&R algorithm. The running
1291
+ time bound is set as 1, 000 seconds for all algorithms, and if the algorithm cannot
1292
+ find the optimal solution within the time bound, the best solution found in all search
1293
+ branches is output.
1294
+ We output the numerical results and running times of all algorithms in Table 4. It
1295
+ can be seen from Table 4 that Solve-CR and Solve-CR-IC, like our Causal B&R
1296
+ Solver, can obtain the optimal solution in five data sets, while Solve-Packing,
1297
+ like Solve, can only obtain the optimal solution in one data set. In addition, on
1298
+ those datasets where the optimal solution cannot be solved within 1000 seconds, our
1299
+ Causal B&R Solver can basically obtain better numerical solutions than Solve-
1300
+ CR-IC, and Solve-CR-IC can obtain numerical results that are slightly better than
1301
+ Solve-CR, while Solve-Packing can generally get better numerical solutions than
1302
+ Solve. These results demonstrate that our reduction algorithm, Causal Reduce,
1303
+ is critical for the B&R algorithm to obtain optimal solutions on more datasets.
1304
+ 25
1305
+
1306
+ Solve
1307
+ Solve-CR
1308
+ Solve-Packing
1309
+ Solve-CR-IC
1310
+ Causal B&R Solver
1311
+ NO.
1312
+ Time(S)
1313
+ Result
1314
+ Time(S)
1315
+ Result
1316
+ Time(S)
1317
+ Result
1318
+ Time(S)
1319
+ Result
1320
+ Time(S)
1321
+ Result
1322
+ 1
1323
+ 1000
1324
+ 1380579565
1325
+ 1000
1326
+ 1380810330
1327
+ 1000
1328
+ 1380579506
1329
+ 1000
1330
+ 1380980673
1331
+ 1000
1332
+ 1380980749
1333
+ 2
1334
+ 1000
1335
+ 232813323
1336
+ 28.5058
1337
+ 232828253
1338
+ 1000
1339
+ 232814520
1340
+ 28.8413
1341
+ 232828253
1342
+ 25.4889
1343
+ 232828253
1344
+ 3
1345
+ 3.8108
1346
+ 10303506
1347
+ 5.3508
1348
+ 10303506
1349
+ 4.7170
1350
+ 10303506
1351
+ 4.1507
1352
+ 10303506
1353
+ 3.9148
1354
+ 10303506
1355
+ 4
1356
+ 1000
1357
+ 124020452
1358
+ 1000
1359
+ 124020466
1360
+ 1000
1361
+ 124020706
1362
+ 1000
1363
+ 124021474
1364
+ 1000
1365
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1366
+ 5
1367
+ 1000
1368
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1369
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1370
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1371
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1372
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1380
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1381
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+ 1000
1387
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1388
+ 7
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1400
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+ 1000
1405
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1406
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1407
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1408
+ 32.1868
1409
+ 134621271
1410
+ 9
1411
+ 1000
1412
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1413
+ 6.8162
1414
+ 7273973
1415
+ 1000
1416
+ 7237411
1417
+ 6.8372
1418
+ 7273973
1419
+ 6.9285
1420
+ 7273973
1421
+ 10
1422
+ 1000
1423
+ 71945454
1424
+ 1000
1425
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1426
+ 1000
1427
+ 71946049
1428
+ 1000
1429
+ 71944343
1430
+ 1000
1431
+ 71945241
1432
+ 11
1433
+ 1000
1434
+ 2707746
1435
+ 1000
1436
+ 2743962
1437
+ 1000
1438
+ 2707846
1439
+ 1000
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+ 1000
1442
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1443
+ 12
1444
+ 1000
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1446
+ 1000
1447
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1448
+ 1000
1449
+ 61702794
1450
+ 1000
1451
+ 61819495
1452
+ 1000
1453
+ 61818234
1454
+ Table 4: Performance gain of CITs on B&R algorithm. The bold and underlined
1455
+ numbers are the best numerical results of all algorithms and the shortest running
1456
+ time of all algorithms to find the optimal solution, respectively.
1457
+ Moreover, both the inferred covering set of the branching vertex and the weighted
1458
+ packing constraints can help B&R algorithm find more promising branches and find
1459
+ better solutions.
1460
+ 5.3
1461
+ Causal Reduce’s Improvement on Heuristic Algorithm
1462
+ Next, we will verify the superiority of our Causal Reduce as a preprocessing for
1463
+ improving the heuristic algorithm.
1464
+ Table 5 presents the running time (including
1465
+ preprocessing time) and numerical results. We find that the preprocessed heuristic
1466
+ algorithm with Causal Reduce usually stop execution after running for a short
1467
+ time, while the rest of the heuristic algorithms are allowed to run for 1000 seconds.
1468
+ Meanwhile, it can be observed from Table 5 that adding the reduction algorithm as
1469
+ preprocessing is obvious for improving the performance of the heuristic algorithm,
1470
+ and our Causal Reduce helps heuristics find better solutions on all instances in less
1471
+ time (essentially within 100 seconds) than Reduce. Thus, although our Causal Re-
1472
+ duce takes no more than 12 seconds longer than Reduce on half of the datasets (as
1473
+ can be known from the numerical results in Section 5.1), it can further reduce the size
1474
+ of remaining graph by more than 32.6%, which is critical for subsequent processing
1475
+ of the problem (also be mentioned in Section 5.2), so such processing time cost is
1476
+ worth it!
1477
+ 26
1478
+
1479
+ Fast
1480
+ Re + Fast
1481
+ Causal Re + Fast
1482
+ Dyn
1483
+ Re + Dyn
1484
+ Causal Re + Dyn
1485
+ NO.
1486
+ Time(S)
1487
+ Result
1488
+ Time(S)
1489
+ Result
1490
+ Time(S)
1491
+ Result
1492
+ Time(S)
1493
+ Result
1494
+ Time(S)
1495
+ Result
1496
+ Time(S)
1497
+ Result
1498
+ 1
1499
+ 1000
1500
+ 1308864893
1501
+ 1000
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+ 250
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+ 2
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1622
+ 1000
1623
+ 71459210
1624
+ 1000
1625
+ 71947516
1626
+ 6
1627
+ 71947590
1628
+ 11
1629
+ 1000
1630
+ 2682982
1631
+ 1000
1632
+ 2748925
1633
+ 25
1634
+ 2748945
1635
+ 1000
1636
+ 2743648
1637
+ 1000
1638
+ 2748982
1639
+ 10
1640
+ 2749005
1641
+ 12
1642
+ 1000
1643
+ 59619787
1644
+ 1000
1645
+ 61850413
1646
+ 50
1647
+ 61852628
1648
+ 1000
1649
+ 60802433
1650
+ 1000
1651
+ 61855209
1652
+ 75
1653
+ 61857313
1654
+ Table 5: A comparative experiment of the effect of Causal Reduce on improving
1655
+ the heuristic algorithm. The numbers in bold and underlined are the correspond-
1656
+ ing experimental results and running time (including preprocessing time) of each
1657
+ heuristic algorithm using our Causal Reduce as preprocessing, respectively.
1658
+ 5.4
1659
+ Comparative Experiment on Causal Search
1660
+ On the basis of preprocessing the input graph with Causal Reduce, we will compare
1661
+ our Causal Search with DynWVC2 to verify the effect of adding CITs to the local
1662
+ search process of DynWVC2 algorithm. The running time for both algorithms (in-
1663
+ cluding pre-processing time) is set to 1000 seconds. To avoid randomness, we run
1664
+ each instance 5 times and record the mean and maximum values. Furthermore, in
1665
+ order to estimate the gap between the results obtained by these two algorithms and
1666
+ the MWIS, we need to calculate the upper bound of each instance. The upper bound
1667
+ for the 2nd, 3rd, 5th, 8th, 9th instance is nothing but the weight of the optimal so-
1668
+ lution obtained by Causal B&R Solver, and for the rest of the instances, it is
1669
+ obtained by applying the weighted clique cover method mentioned in Section 4.2 to
1670
+ the remaining graph obtained by Causal Reduce. Table 6 outputs the numerical
1671
+ results and the estimated gap. The small gaps there demonstrate that after prepro-
1672
+ cessing with our Causal Reduce, both algorithms can obtain numerical results very
1673
+ close to the optimal solution. In particular, for those instances where the optimal
1674
+ solution is obtained, their gap can basically reach 10−6 ∼ 10−7, and in the remaining
1675
+ instances, the estimated gap can basically reach 10−4 ∼ 10−2. Besides, from the
1676
+ mean and maximum values, our Causal Search can basically achieve better perfor-
1677
+ mance than DynWVC2, thereby implying that our CITs can help local search find
1678
+ better local optima.
1679
+ 27
1680
+
1681
+ Dyn
1682
+ Causal Search
1683
+ NO.
1684
+ Upper Bound
1685
+ Mean
1686
+ Gap
1687
+ Max
1688
+ Gap
1689
+ Mean
1690
+ Gap
1691
+ Max
1692
+ Gap
1693
+ 1
1694
+ 1384376268
1695
+ 1381464698.6
1696
+ 2.103 × 10−3
1697
+ 1381467306
1698
+ 2.101 × 10−3
1699
+ 1381466033.6
1700
+ 2.102 × 10−3
1701
+ 1381470750
1702
+ 2.099 × 10−3
1703
+ 2
1704
+ 232828253
1705
+ 232828153.4
1706
+ 4.278 × 10−7
1707
+ 232828171
1708
+ 3.522 × 10−7
1709
+ 232828159.2
1710
+ 4.029 × 10−7
1711
+ 232828188
1712
+ 2.792 × 10−7
1713
+ 3
1714
+ 10303506
1715
+ 10303485.2
1716
+ 2.019 × 10−6
1717
+ 10303491
1718
+ 1.456 × 10−6
1719
+ 10303488.6
1720
+ 1.689 × 10−6
1721
+ 10303494
1722
+ 1.165 × 10−6
1723
+ 4
1724
+ 124076790
1725
+ 124026438.6
1726
+ 4.058 × 10−4
1727
+ 124026451
1728
+ 4.057 × 10−4
1729
+ 124026444.8
1730
+ 4.058 × 10−4
1731
+ 124026449
1732
+ 4.057 × 10−4
1733
+ 5
1734
+ 3916599
1735
+ 3916582.2
1736
+ 4.289 × 10−6
1737
+ 3916583
1738
+ 4.085 × 10−6
1739
+ 3916582.6
1740
+ 4.187 × 10−6
1741
+ 3916584
1742
+ 3.830 × 10−6
1743
+ 6
1744
+ 103562461
1745
+ 101846521.0
1746
+ 1.657 × 10−2
1747
+ 101848242
1748
+ 1.655 × 10−2
1749
+ 101847524.8
1750
+ 1.656 × 10−2
1751
+ 101849650
1752
+ 1.654 × 10−2
1753
+ 7
1754
+ 800748442
1755
+ 799264479.2
1756
+ 1.853 × 10−3
1757
+ 799265827
1758
+ 1.852 × 10−3
1759
+ 799266278.8
1760
+ 1.851 × 10−3
1761
+ 799267573
1762
+ 1.849 × 10−3
1763
+ 8
1764
+ 134621271
1765
+ 134621255.0
1766
+ 1.189 × 10−7
1767
+ 134621257
1768
+ 1.040 × 10−7
1769
+ 134621256.8
1770
+ 1.055 × 10−7
1771
+ 134621265
1772
+ 4.457 × 10−6
1773
+ 9
1774
+ 7273973
1775
+ 7273939.8
1776
+ 4.564 × 10−6
1777
+ 7273945
1778
+ 3.849 × 10−6
1779
+ 7273936.8
1780
+ 4.977 × 10−6
1781
+ 7273947
1782
+ 3.574 × 10−6
1783
+ 10
1784
+ 71978922
1785
+ 71947636.0
1786
+ 4.347 × 10−4
1787
+ 71947639
1788
+ 4.346 × 10−4
1789
+ 71947636.8
1790
+ 4.346 × 10−4
1791
+ 71947642
1792
+ 4.346 × 10−4
1793
+ 11
1794
+ 2819343
1795
+ 2749008.4
1796
+ 2.495 × 10−2
1797
+ 2749009
1798
+ 2.495 × 10−2
1799
+ 2749010.8
1800
+ 2.495 × 10−2
1801
+ 2749019
1802
+ 2.494 × 10−2
1803
+ 12
1804
+ 62276875
1805
+ 61860510.4
1806
+ 6.686 × 10−3
1807
+ 61860616
1808
+ 6.684 × 10−3
1809
+ 61860549.0
1810
+ 6.685 × 10−3
1811
+ 61860717
1812
+ 6.682 × 10−3
1813
+ Table 6: Compare our Causal Search with the DynWVC2 algorithm. The bold
1814
+ and underlined numbers are better maximum and average values, respectively. In
1815
+ the column headed by “Upper Bound”, each number is the upper bound of the
1816
+ MWIS of the corresponding instance.
1817
+ 6
1818
+ Conclusion and Outlook
1819
+ In this paper, we propose a series of causal inference techniques (CITs) for the max-
1820
+ imum weight independent set (MWIS) problem by fully exploiting the upper bound
1821
+ property of MWIS. After integrating our CITs, the performance of various existing
1822
+ algorithms, including the Branch-and-Reduce (B&R) algorithm and some heuristic
1823
+ algorithms, is significantly improved. We are now conducting theoretical analysis
1824
+ to find some guarantees on solution quality, developing strategies to help the B&R
1825
+ algorithm analyze the causes of conflicts and perform more efficient backtracking
1826
+ searches, and generalizing the proposed CITs to other combinatorial optimization
1827
+ problems.
1828
+ Acknowledgements
1829
+ This research was supported by the National Key R&D Program of China (Nos.
1830
+ 2020AAA0105200, 2022YFA1005102) and the National Natural Science Foundation
1831
+ of China (Nos. 12288101, 11822102). SS is partially supported by Beijing Academy
1832
+ of Artificial Intelligence (BAAI). The authors would like to thank Professor Hao Wu
1833
+ for his useful discussions and valuable suggestions.
1834
+ 28
1835
+
1836
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+ sparse graph, INFORMS Journal on Computing, 10 (1998), pp. 438–447.
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+ Springer, 2016, pp. 345–356.
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+ Computer Science, 469 (2013), pp. 92–104.
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+
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1
+ ⋆ manuscript No.
2
+ (will be inserted by the editor)
3
+ A quasi-energy function for Pixton diffeomorphisms
4
+ defined by generalized Mazur knots
5
+ Timur Medvedev · Olga Pochinka
6
+ Received: date / Accepted: date
7
+ Abstract In this paper we give a lower estimate for the number of critical points
8
+ of the Lyapunov function for Pixton diffeomorphisms (i.e. Morse-Smale diffeo-
9
+ morphisms in dimension 3 whose chain recurrent set consists of four points: one
10
+ source, one saddle and two sinks). Ch. Bonatti and V. Grines proved that the class
11
+ of topological equivalence of such diffeomorphism f is completely defined by the
12
+ equivalency class of the Hopf knot Lf that is the knot in the generating class of
13
+ the fundamental group of the manifold S2 × S1. They also proved that there are
14
+ infinitely many such classes and that any Hopf knot can be realized by a Pixton
15
+ diffeomorphism. D. Pixton proved that diffeomorphisms defined by the standard
16
+ Hopf knot L0 = {s} × S1 have an energy function (Lyapunov function) whose set
17
+ of critical points coincide with the chain recurrent set whereas the set of critical
18
+ points of any Lyapunov function for Pixton diffeomorphism with nontrivial (i.e.
19
+ non equivalent to the standard) Hopf knot is strictly larger than the chain recur-
20
+ rent set of the diffeomorphism. The Lyapunov function for Pixton diffeomorphism
21
+ with minimal number of critical points is called the quasi-energy function. In this
22
+ paper we construct a quasi-energy function for Pixton diffeomorphisms defined by
23
+ a generalized Mazur knot.
24
+ Keywords Hopf knot · Mazur knot · Pixton diffeomorphism · quasi-energy
25
+ function
26
+ Mathematics Subject Classification (2020) 37C15 · 37D15
27
+ ⋆ The research was done with the support of Russian National Foundation (project 21-11-
28
+ 00010) except construction of the quasi-energy function which was supported by International
29
+ Laboratory of Dynamical Systems and Applications of National Research University Higher
30
+ School of Economics, grant of Government of Russian Federation 075-15-2022-1101.
31
+ T. Medvedev
32
+ Laboratory of Algorithms and Technologies for Network Analysis; HSE University
33
+ 136 Rodionova Street, Niznhy Novgorod, Russia
34
+ E-mail: mtv2001@mail
35
+ O. Pochinka
36
+ International Laboratory of Dynamical Systems and Applications; HSE University, 25/12 Bol-
37
+ shaya Pecherckaya Street, Niznhy Novgorod, Russia
38
+ arXiv:2301.02405v1 [math.DS] 6 Jan 2023
39
+
40
+ 2
41
+ Timur Medvedev, Olga Pochinka
42
+ 1 Introduction and the main results
43
+ Let M n be a smooth closed n-manifold with a metric d and let f : M n →
44
+ M n be a diffeomorphism. For two given points x, y ∈ M n a sequence of points
45
+ x = x0, . . . , xm = y is called an ε-chain of length m ∈ N connecting x to y if
46
+ d(f(xi−1), xi) < ε for 1 ⩽ i ⩽ m (Fig. 1).
47
+ x
48
+ x
49
+ =
50
+ 0
51
+ f x( )
52
+ 0
53
+ f x( )
54
+ 1
55
+ f x( )
56
+ 2
57
+ f x(
58
+ )
59
+ m-1
60
+ f x(
61
+ )
62
+ m 2
63
+ -
64
+ x1
65
+ x2
66
+ x3
67
+ xm-1
68
+ y x
69
+ =
70
+ m
71
+
72
+
73
+
74
+
75
+
76
+ Fig. 1 An ε-chain of length m ∈ N
77
+ A point x ∈ M n is called chain recurrent for the diffeomorphism f if for every
78
+ ε > 0 there is an ε-chain of length m connecting x to itself for some m (m depends
79
+ on ε > 0). The chain recurrent set, denoted by Rf, is the set of all chain recurrent
80
+ points of f. Define the equivalence on Rf by the rule: x ∼ y if for every ε > 0
81
+ there is are ε-chains connecting x to y and y to x. This equivalence relation defines
82
+ equivalence classes called chain components.
83
+ If the chain recurrent set of a diffeomorphism f is finite then it consists of
84
+ periodic points. A periodic point p ∈ Rf of period mp is said to be hyperbolic if
85
+ absolute values of all the eigenvalues of the Jacobian matrix
86
+
87
+ ∂f mp
88
+ ∂x
89
+
90
+ |p are not
91
+ equal to 1. If absolute values of all these eigenvalues are greater (less) than 1 then
92
+ p is called a sink (a source). Sinks and sources are called knots. If a periodic point
93
+ is not a knot then it is called a saddle.
94
+ Let p be a hyperbolic periodic point of a diffeomorphism f whose chain recur-
95
+ rent set is finite. The Morse index of p, denoted by λp, is the number of eigen-
96
+ values of Jacobian matrix whose absolute values are greater than 1. The stable
97
+ manifold W s
98
+ p = {x ∈ M n :
99
+ lim
100
+ k→+∞ d(f kmp(x), p) = 0} and the unstable manifold
101
+ W u
102
+ p = {x ∈ M n :
103
+ lim
104
+ k→+∞ d(f −kmp(x), p) = 0} of p are smooth manifolds diffeo-
105
+ morphic to Rλp and Rn−λp, respectively. Stable and unstable manifolds are called
106
+ invariant manifolds. A connected component of the set W u
107
+ p \ p (W s
108
+ p \ p) is called
109
+ a unstable (stable) separatrice of p.
110
+ A diffeomorphism f : M n → M n is called a Morse-Smale diffeomorphism if
111
+ 1. its chain recurrent set Rf consists of finite number of hyperbolic points;
112
+ 2. for any two points p, q ∈ Rf the manifolds W s
113
+ p , W u
114
+ q intersect transversally.
115
+
116
+ Quasi-energy function for Pixton diffeomorphisms
117
+ 3
118
+ C Conley in [3] gave the following definition: a Lyapunov function for a Morse-
119
+ Smale diffeomorphism f : M n → M n is a continuous function ϕ : M n → R
120
+ satisfying
121
+ – ϕ(f(x)) < ϕ(x) if x /∈ Rf;
122
+ – ϕ(f(x)) = ϕ(x) if x ∈ Rf.
123
+ Notice that every Morse-Smale diffeomorphism f has a Morse-Lyapunov func-
124
+ tion 1, i.e. a Lyapunov function ϕ : M n → R which is a Morse function such that
125
+ each periodic point p ∈ Rf is its non-degenerate critical point of index λp with
126
+ Morse coordinates (Vp, φp : y ∈ Vp �→ (x1(y), . . . , xn(y)) ∈ Rn and
127
+ φ−1
128
+ p (Ox1 . . . xλp) ⊂ W u
129
+ p , φ−1
130
+ p (Oxλp+1 . . . xn) ⊂ W s
131
+ p .
132
+ (∗)
133
+ If the function ϕ has no critical points outside Rf then following [15] we call it
134
+ the energy function for the Morse-Smale diffeomorphism f.
135
+ The proof of existence of an energy Morse function for a Morse-Smale diffeo-
136
+ morphism of the circle is an easy exercise. D. Pixton [15] in 1977 proved that
137
+ every Morse-Smale diffeomorphism of a surface has an energy function. There he
138
+ also constructed an example of a Morse-Smale diffeomorphism on the 3-sphere
139
+ which admits no energy function. The obstacle to existence of an energy function
140
+ in his example was the wild embedding of the saddle separatrices in the ambient
141
+ manifold (i.e. the closure of the separatrice is not a submanifold of the ambient
142
+ space). From [11] it follows that there are Morse-Smale diffeomorphisms with no
143
+ energy function on manifolds of any dimension n > 2. Therefore, following [7]
144
+ for a Morse-Smale diffeomorphism f we call a Morse-Lyapunov function with the
145
+ minimal number of critical points (denote it by ρf ) a quasi-energy function. Notice
146
+ that ρf is a topological invariant, i.e. if two diffeomorphisms f, f ′ : M n → M n
147
+ are topologically conjugate (that is there is a diffeomorphism h : M n → M n such
148
+ that h ◦ f = f ′ ◦ h) then ρf = ρf′ .
149
+ In this paper we give a lower estimate of ρf for Pixton diffeomorphisms. The
150
+ class of Pixton diffeomorphisms P is defined in the following way. Every diffeo-
151
+ morphism f ∈ P is a Morse-Smale 3-diffeomorphism whose chain recurrent set
152
+ consists of four points: one source, one saddle and two sinks (for details see section
153
+ 2). Notice that Pixton’s example is a diffeomorphism of this class. According to [2]
154
+ the class of topological conjugacy of a diffeomorphism f ∈ P is completely defined
155
+ by the equivalence class of the Hopf knot Lf, i.e. the knot in the generating class
156
+ of the fundamental group of the manifold S2 × S1 (see Proposition 1). Moreover,
157
+ any Hopf knot can be realized as a Pixton diffeomorphism.
158
+ Recall that a knot in S2 × S1 is a smooth embedding γ : S1 → S2 × S1 or the
159
+ image of this embedding L = γ(S1). Two knots L, L′ are said to be equivalent if
160
+ there is a homeomorphism h : S2 × S1 → S2 × S1 such that h(L) = L′. Two knots
161
+ 1 This function can be constructed, for example, by suspension. Consider the topological
162
+ flow ˆft on the manifold Mn × R defined by ˆft(x) = x + t. Define the diffeomorphism g :
163
+ Mn × R → Mn × R by g(x, τ) = (f(x), τ − 1) and let G = {gk , k ∈ Z} and W = (Mn × R)/G.
164
+ Denote by pW : Mn × R → W the natural projection and denote by ft the flow on W defined
165
+ by ft(x) = pW ( ˆft(p−1
166
+ W (x))). The flow ft is called the suspension over f. By construction the
167
+ chain recurrent set of ft consists of the finite number of periodic orbits δi = pW (Oi × R), i ∈
168
+ {1, . . . , kf} and this means that the suspension ft is a Morse-Smale flow. A Lyapunov function
169
+ for these flows is constructed in [12]. Then the restriction of this function on M is the desired
170
+ Lyapunov function for f.
171
+
172
+ 4
173
+ Timur Medvedev, Olga Pochinka
174
+ γ, γ′ are smoothly homotopic if there exists a smooth map Γ : S1 ×[0, 1] → S2 ×S1
175
+ such that Γ(s, 0) = γ(s) and Γ(s, 1) = γ′(s) for every s ∈ S1. If Γ|S1×{t} is an
176
+ embedding for every t ∈ [0, 1] then the knots are said to be isotopic.
177
+ Any Hopf knot L ⊂ S2 × S1 is smoothly homotopic to the standard Hopf
178
+ knot L0 = {s} × S1 (see, for example, [9]) but generally it is neither isotopic
179
+ nor equivalent to it. B. Mazur [10] constructed the Hopf knot LM which we call
180
+ the Mazur knot and which is non-equivalent and non-isotopic to L0 (see Fig. 2).
181
+ It follows from the results of [1] that there exists a countable family of pairwise
182
+ Fig. 2 Two non-isotopic and non equivalent Hopf knots L0 and LM: a) the standard Hopf
183
+ knot L0; b) the Mazur knot LM
184
+ non-equivalent Hopf knots Ln, n ∈ N which are generalized Mazur knots (Fig. 3).
185
+ Fig. 3 A generalized Mazur knot Ln
186
+ According to [6] a Pixton diffeomorphism f admits an energy Morse function
187
+ if and only if the knot Lf is trivial (i.e. equivalent to the standard one). If the
188
+
189
+ Quasi-energy function for Pixton diffeomorphisms
190
+ 5
191
+ knot Lf is not trivial then the number ρf of the critical points of a quasi-energy
192
+ Morse function of f is evidently even and
193
+ ρf ⩾ 6.
194
+ The main result of this paper is the proof of Theorem 1.
195
+ Theorem 1 Let f be a Pixton diffeomorphism (f ∈ P) and let Ln, n ∈ N be
196
+ its knot. Then the number ρf of critical points of a quasi-energy function of f is
197
+ calculated by2
198
+ ρf = 4 + 2n.
199
+ 2 Construction of Pixton diffeomorphisms
200
+ In dynamics a wild Artin-Fox arc was for the first time introduced by D. Pixton
201
+ in [15] where he constructed a Morse-Smale diffeomorphism on the 3-sphere with
202
+ the unique saddle whose invariant manifolds form an Artin-Fox arc. We give the
203
+ modern construction of these diffeomorphisms following Ch. Bonatti and V. Grines
204
+ [2] where Pixton diffeomorphisms were also classified (see also [8], [11]).
205
+ For x = (x1, . . . , xn) ∈ Rn denote ||x|| =
206
+
207
+ x2
208
+ 1 + · · · + x2n. Let h : R3 → R3
209
+ be the diffeomorphism defined by h(x1, x2, x3) =
210
+ � x1
211
+ 2 , x2
212
+ 2 , x3
213
+ 2
214
+
215
+ . Define the map
216
+ p : R3 \ O → S2 × S1 by
217
+ p(x1, x2, x3) =
218
+ � x1
219
+ ||x||, x2
220
+ ||x||, log2(||x||)
221
+ (mod 1)
222
+
223
+ .
224
+ Let L ⊂ (S2 ×S1) be a Hopf knot and let U(L) be its tubular neighborhood. Then
225
+ the set ¯L = p−1(L) is the h-invariant arc in R3 and U(¯L) = p−1(U(L)) is its
226
+ h-invariant neighborhood diffeomorphic to D2 × R1 (Fig. 4).
227
+ Let C = {(x1, x2, x3) ∈ R3
228
+ : x2
229
+ 2 + x2
230
+ 3 ⩽ 4} and let gt : C → C be the flow
231
+ defined by
232
+ gt(x1, x2, x3) = (x1 + t, x2, x3).
233
+ Then there is a diffeomorphism ζ : U(L) → C that conjugates h|U(L) and g = g1|C.
234
+ Define the flow φt on C by:
235
+
236
+
237
+
238
+
239
+
240
+
241
+
242
+
243
+
244
+
245
+
246
+
247
+
248
+
249
+
250
+
251
+
252
+
253
+
254
+
255
+
256
+
257
+
258
+
259
+
260
+
261
+
262
+
263
+
264
+ ˙x1 =
265
+
266
+ 1 − 1
267
+ 9(x2
268
+ 1 + x2
269
+ 2 + x2
270
+ 3 − 4)2,
271
+ x2
272
+ 1 + x2
273
+ 2 + x2
274
+ 3 ⩽ 4
275
+ 1,
276
+ x2
277
+ 1 + x2
278
+ 2 + x2
279
+ 3 > 4
280
+ ˙x2 =
281
+
282
+
283
+
284
+
285
+
286
+ x2
287
+ 2
288
+
289
+ sin
290
+ � π
291
+ 2
292
+
293
+ x2
294
+ 1 + x2
295
+ 2 + x2
296
+ 3 − 3
297
+ ��
298
+ − 1
299
+
300
+ ,
301
+ 2 < x2
302
+ 1 + x2
303
+ 2 + x2
304
+ 3 ⩽ 4
305
+ −x2,
306
+ x2
307
+ 1 + x2
308
+ 2 + x2
309
+ 3 ⩽ 2
310
+ 0,
311
+ x2
312
+ 1 + x2
313
+ 2 + x2
314
+ 3 > 4
315
+ ˙x3 =
316
+
317
+
318
+
319
+
320
+
321
+ x3
322
+ 2
323
+
324
+ sin
325
+ � π
326
+ 2
327
+
328
+ x2
329
+ 1 + x2
330
+ 2 + x2
331
+ 3 − 3
332
+ ��
333
+ − 1
334
+
335
+ ,
336
+ 2 < x2
337
+ 1 + x2
338
+ 2 + x2
339
+ 3 ⩽ 4
340
+ −x3,
341
+ x2
342
+ 1 + x2
343
+ 2 + x2
344
+ 3 ⩽ 2
345
+ 0,
346
+ x2
347
+ 1 + x2
348
+ 2 + x2
349
+ 3 > 4.
350
+ By construction the diffeomorphism φ = φ1 has two fixed points: the saddle
351
+ P(1, 0, 0) and the sink Q(−1, 0, 0) (Fig. 5), both being hyperbolic. One unstable
352
+ 2 For n = 1 Theorem 1 is proved in [7].
353
+
354
+ 6
355
+ Timur Medvedev, Olga Pochinka
356
+ L
357
+ L
358
+ _
359
+ U(L)
360
+ _
361
+ Fig. 4 Suspension of a Hopf knot
362
+ 1
363
+ -1
364
+ 2
365
+ O
366
+ Fig. 5 Trajectories of the flow φt
367
+ separatrice of the saddle P coincides with the open interval
368
+
369
+ (x1, x2, x3) ∈ R3 : |x1| < 1, x2 = x3 = 0
370
+
371
+ in the basin of the sink Q while the other coincides with the ray
372
+
373
+ (x1, x2, x3) ∈ R3 : x1 > 1, x2 = x3 = 0
374
+
375
+ .
376
+ Notice that φ coincides with the diffeomorphism g = g1 outside the ball {(x1, x2, x3) ∈
377
+ C : x2
378
+ 1 + x2
379
+ 2 + x2
380
+ 3 ⩽ 4}.
381
+ Define the diffeomorphism ¯fL : R3 → R3 so that ¯fL coincides with h outside
382
+ U(L) and it coincides with ζ−1φζ on U(L). Then ¯fL has in U(L) two fixed points:
383
+ the sink ζ−1(Q) and the saddle ζ−1(P), both being hyperbolic. The unstable
384
+ separatrice of the saddle ζ−1(P) lies in L (Fig. 6).
385
+ Now project the dynamics onto the 3-sphere. Denote by N(0, 0, 0, 1) the North
386
+ Pole of the sphere S3 = {x = (x1, x2, x3, x4) : ||x|| = 1}. For every point x ∈ (S3 \
387
+ {N}) there is the unique line passing through N and x in R4. This line intersects
388
+
389
+ Quasi-energy function for Pixton diffeomorphisms
390
+ 7
391
+ R
392
+ Fig. 6 The phase portrait of the diffeomorphism ¯fL
393
+ R3 in exactly one point ϑ+(x) (Fig. 7). The point ϑ+(x) is the stereographic
394
+ projection of the point x. One can easily check that
395
+ ϑ+(x1, x2, x3, x4) =
396
+
397
+ x1
398
+ 1 − x4 ,
399
+ x2
400
+ 1 − x4 ,
401
+ x3
402
+ 1 − x4
403
+
404
+ .
405
+ Thus, the stereographic projection ϑ+ : S3 \ {N} → R3 is a diffeomorphism.
406
+ N
407
+ x
408
+ ��(x)
409
+ Fig. 7 The stereographic projection.
410
+ By construction ¯fL coincides with h in some neighborhood of the point O and
411
+ in some neighborhood of the infinity. Therefore, it induces on S3 the Morse-Smale
412
+ fL(x) =
413
+
414
+ ϑ−1
415
+ + ( ¯fL(ϑ+(x))), x ̸= N;
416
+ N, x = N
417
+ .
418
+ It follows directly from the construction that the non-wandering set of fL consists
419
+ of exactly four fixed hyperbolic points: two sinks ω = ϑ−1
420
+ + (ζ−1(Q)), S, one saddle
421
+ σ = ϑ−1
422
+ + (ζ−1(P)) and one source N. We say the constructed diffeomorphism to
423
+ be model and it is of Pixton class.
424
+
425
+ 8
426
+ Timur Medvedev, Olga Pochinka
427
+ Proposition 1 ([2])
428
+ – Any diffeomorphism f ∈ P is topologically conjugate to some model diffeomor-
429
+ phism fL.
430
+ – Two model diffeomorphisms fL, fL′ are topologically conjugate if and only if
431
+ their knots L, L′ are equivalent.
432
+ 3 Genus of Hopf knot
433
+ In this section we introduce the notion of genus for a Hopf knot and use it to
434
+ estimate the number of critical points of the quasi-energy function of the Pixton
435
+ diffeomorphism defined by this knot.
436
+ Let L be a Hopf knot and let ¯L = p−1(L) be its cover in R3\O. We say a closed
437
+ orientable surface Σ ⊂ S2 × S1 to be a secant surface of the knot L if it intersects
438
+ L in a unique point and there is an h-compressible 3-manifold QΣ ⊂ R3 (that is
439
+ h(QΣ) ⊂ int QΣ) with the boundary ¯Σ such that Σ = p( ¯Σ) and the intersection
440
+ ¯L∩ ¯Σ is the unique point ¯y. The minimally possible genus gL of the secant surface
441
+ is called the genus of the knot L. The secant surface of L of genus gL is said to be
442
+ minimal.
443
+ Lemma 1 If Σ is a minimal secant surface of the knot L then the surface ¯Σ \ ¯y
444
+ is non-compressible in R3 \ (O ∪ ¯L), i.e. any simple closed curve c ⊂ int ( ¯Σ \ ¯y) is
445
+ contractible on ¯Σ\ ¯y if it bounds a smoothly embedded 2-disk D ⊂ int (R3\(O∪ ¯L))
446
+ such that D ∩ ( ¯Σ \ ¯y) = ∂D = c.
447
+ Proof Let Σ be a minimal secant surface of L and let ¯y be the unique point of
448
+ the intersection ¯L ∩ ¯Σ. Assume the opposite: the surface ¯Σ \ ¯y is compressible in
449
+ R3 \ (O ∪ ¯L). Then there is a non-contractible simple closed curve c ⊂ int ( ¯Σ \ ¯y)
450
+ and there is the smoothly embedded 2-disk D ⊂ int (R3 \ (O ∪ ¯L)) such that
451
+ D ∩ ( ¯Σ \ ¯y) = ∂D = c (see, for example, [14]). Then we have two possibilities:
452
+ (int D) ∩
453
+ � �
454
+ k∈Z
455
+ hk( ¯Σ)
456
+
457
+ = ∅,
458
+ (1)
459
+ (int D) ∩
460
+ � �
461
+ k∈Z
462
+ hk( ¯Σ)
463
+
464
+ ̸= ∅.
465
+ (2)
466
+ In case (1) two subcases are possible: (1a) D ⊂ QΣ, (1b) D ⊂ (R3 \ int QΣ). For
467
+ case 1a) let N(D) ⊂ QΣ be a tubular neighborhood of the disk D. Then exactly
468
+ one connected component of the set QΣ \ int N(D) intersects ¯L. According to (1)
469
+ this component is h-compressible and its boundary intersects ¯L at a unique point.
470
+ The projection of this boundary into S2 × S1 is, therefore, the secant surface of
471
+ L of genus less than gL. This contradicts the fact that the surface Σ is minimal.
472
+ In case 1b) let N(D) ⊂ (R3 \ int QΣ) be a tubular neighborhood of D. Then due
473
+ to (1) the set QΣ ∪ N(D) is h-compressible and its boundary intersects ¯L at a
474
+ unique point. The projection of this boundary into S2 ×S1 is, therefore, the secant
475
+ surface of L of genus less than gL and we have the same contradiction.
476
+
477
+ Quasi-energy function for Pixton diffeomorphisms
478
+ 9
479
+ In case (2) without loss of generality assume the intersection int D∩( �
480
+ k∈Z
481
+ hk( ¯Σ))
482
+ to be transversal and denote it by Γ. Let γ be a curve from Γ. We say the curve
483
+ γ to be innermost if it is the boundary of the disk Dγ ⊂ D such that int Dγ
484
+ contains no curves of Γ. Consider this innermost curve γ ⊂ f k(Σ). There are
485
+ two subcases: a) γ is essential on f k(Σ) and b) γ is contractible on f k(Σ). In
486
+ case a) the arguments of the case (1) apply for the body f k(QΣ) and the disk
487
+ Dγ and we get the contradiction to the minimality of the surface Σ. In case b)
488
+ denote by dγ ⊂ f k(Σ) the 2-disk bounded by γ and denote by Bγ ⊂ (R3 \ O)
489
+ the 3-ball bounded by the 2-sphere Dγ ∪ dγ. Consider: b1) Bγ ⊂ f k(QΣ) and b2)
490
+ Bγ ⊂ (R3 \ int f k(QΣ)). For b1) let N(Bγ) ⊂ f k(QΣ) be a tubular neighborhood
491
+ of Bγ. Then the set QΣ \ int N(Bγ) is h-compressible because the curve γ lies in
492
+ its interior and the boundary of QΣ \ int N(Bγ) intersects ¯L at a unique point.
493
+ The projection of this boundary into S2 ×S1 is, therefore, the secant surface of the
494
+ knot L of genus gL for which the number of connected components of the set Γ is
495
+ less. We get the same result for b2) for the set QΣ ∪ N(Bγ). Thus, iterating the
496
+ process we come either to the case a) or to the case (1) and get a contradiction.
497
+ Lemma 2 For any diffeomorphism f ∈ P the following estimation holds
498
+ ρf ⩾ 4 + 2gLf .
499
+ (3)
500
+ Proof Since Proposition 1 is true and since the number ρf of the critical points of a
501
+ quasi-energy function of f ∈ P is invariant, from now on we consider model Pixton
502
+ diffeomorphismsfL with the Hopf knot L. Denote by ℓ the non-stable separatrice
503
+ of the saddle σ lying in the basin of the sink S. Let
504
+ pS : W s
505
+ S \ S → S2 × S1
506
+ be the natural projection sending a point w ∈ (W s
507
+ S\S) to the point p(f kw(w)), f kw(w) ∈
508
+ VS. Since the diffeomorphism fL coincides with the homothety h in some neigh-
509
+ borhood VS of S, the natural projection pS is well defined and pS(ℓ) = L by
510
+ construction.
511
+ Consider an arbitrary Morse-Lyapunov function ϕ : S3 → R of the diffeomor-
512
+ phism fL. To be definite let ϕ(S) = 0, ϕ(σ) = 1 and ϕ(N) = 3. From the definition
513
+ of the Morse-Lyapunov function it follows that ϕ|ℓ monotonically decreases in some
514
+ neighborhood of the saddle σ. Therefore, there is ε1 ∈ (0, 1) such that the interval
515
+ (1 − ε1, 1) contains no critical values of ϕ and the connected component ¯Σ1 of the
516
+ level set ϕ−1(1 − ε1) intersects the separatrice ℓ at the unique point. Denote this
517
+ point by w1.
518
+ Let ¯Q1 be the connected component of the set ϕ−1([0, 1 − ε1]) which contains
519
+ the segment [w1, S] of the closure of the separatrice ℓ. Since ϕ decreases along
520
+ the trajectories of f, the values of ϕ on W s
521
+ σ are greater than 1. Therefore, the
522
+ manifold ¯Q1 lies in the manifold W s
523
+ S diffeomorphic to R3. Let the function ϕ| ¯
524
+ Q1
525
+ have kq, q ∈ {0, . . . , 3} critical points of index q. Due to [5, Theorem 6.1] on the
526
+ manifold ¯Q1 there exists a self-indexing Morse function ψ (the value of the function
527
+ in a critical point equals the index of this point) which has kq critical points of
528
+ index q and which is constant on ∂ ¯Q1. Thus, the manifold ¯Q1 is the surface ˜Q1 of
529
+
530
+ 10
531
+ Timur Medvedev, Olga Pochinka
532
+ genus g1 = 1 + k1 − k0 with attached handles of indexes 2 and 3. Then the genus
533
+ of any surface of the set ∂ ¯Q1 cannot be greater than g1.
534
+ On the other hand, the number of critical points of ϕ| ¯
535
+ Q1 is not less than k0+k1.
536
+ If k0 ⩾ 1 and g1 = 1 + k1 − k0 then one gets k0 + k1 = g1 + 2k0 − 1 ⩾ g1 + 1. Thus,
537
+ ϕ| ¯
538
+ Q1 has at least g1 + 1 critical points.
539
+ Denote by ¯Σ1 the connected component of ∂ ¯Q1 which intersects the separatrice
540
+ ℓ. Then the surface ¯Σ1 divides the manifold W s
541
+ S ∼= R3 into two parts, one of which
542
+ Q1 being an h-compressible body. This means that Σ1 = pS( ¯Σ1) is the secant
543
+ surface of L and, therefore,
544
+ g1 ⩾ gL.
545
+ Analogously, there is ε2 ∈ (0, 1) for which the interval (1, 1 + ε2) contains no
546
+ critical points of ϕ and the connected component ¯Q2 of the level set ϕ−1([0, 1+ε2)]
547
+ contains cl(W u
548
+ σ ) in its interior while the intersection ¯Q2 with W s
549
+ σ is the unique
550
+ 2-disk. Due to construction the function ϕ| ¯
551
+ Q2 has at least g1 + 3 critical points
552
+ and genus of the connected components of ∂ ¯Q2 is less or equals g1. Denote by ¯Σ2
553
+ the connected component of ∂ ¯Q2 which intersects W s
554
+ σ and denote by g2 its genus.
555
+ The surface ¯Σ2 divides the manifold W u
556
+ N ∼= R3 into two parts, one of which Q2
557
+ being an h−1-compressible body. Arguing as above one comes to conclusion that
558
+ the number of critical points of ϕ|Q2 is at least g2 +1. Therefore, the total number
559
+ of critical points of ϕ is greater or equal to
560
+ g1 + 3 + g2 + 1 ⩾ 4 + 2g1 ⩾ 4 + 2gLf .
561
+ 4 The generalized Mazur knot Ln
562
+ In this section we show that the genus gLn of a generalized Mazur knot equals n.
563
+ At first we give a detailed description of construction of Ln.
564
+ 4.1 Construction of the generalized Mazur knot Ln
565
+ Recall that h : R3 → R3 is the homothety defined by
566
+ h(x1, x2, x3) =
567
+ �x1
568
+ 2 , x2
569
+ 2 , x3
570
+ 2
571
+
572
+ and p : R3 \ O → S2 × S1 is the natural projection defined by
573
+ p(x1, x2, x3) =
574
+ � x1
575
+ ||x||, x2
576
+ ||x||, log2(||x||)
577
+ (mod 1)
578
+
579
+ .
580
+ Consider the annulus
581
+ K =
582
+
583
+ (x1, x2, x3) ∈ R3 : 1
584
+ 4 ≤ x2
585
+ 1 + x2
586
+ 2 + x2
587
+ 3 ≤ 1
588
+
589
+ bounded by the spheres
590
+ S2 =
591
+
592
+ (x1, x2, x3) ∈ R3 : x2
593
+ 1 + x2
594
+ 2 + x2
595
+ 3 = 1
596
+
597
+ , h(S2).
598
+
599
+ Quasi-energy function for Pixton diffeomorphisms
600
+ 11
601
+ Pick on the circle
602
+ S1 =
603
+
604
+ (x1, x2, x3) ∈ R3 : x2
605
+ 1 + x2
606
+ 2 = 1, x3 = 0
607
+
608
+ pairwise distinct points α1, . . . , α2n+1 numbered in counter-clockwise order (Fig. 8).
609
+ Let ai, i ∈ {1, . . . , 2n} be the arc of the circle S1 bounded by αi, αi+1 whose inte-
610
+ rior contains no points of {α1, . . . , α2n+1}. Let B, Ai ⊂ int K, i ∈ {1, . . . , 2n} be
611
+ pairwise disjoint smooth arcs such that:
612
+ 1. the boundary points of B are α2n+1, h(α1); the boundary points of A2j−1
613
+ are α2j−1, α2j and the boundary points of A2j are h(α2j), h(α2j+1) for j ∈
614
+ {1, . . . , n};
615
+ 2. the closed curves c2j−1 = cl(a2j−1 ∪ A2j−1), c2j = cl(h(a2j) ∪ A2j) bound the
616
+ 2-disks d2j−1, d2j, the transversal intersection of these disks being the arc lj
617
+ with the boundary points b2j−1 = d2j−1 ∩ A2j and b2j = d2j ∩ A2j−1;
618
+ 3. the arc cl(h(A1) ∪ A2 ∪ · · · ∪ h(A2n−1) ∪ A2n ∪ B) is smooth.
619
+ Fig. 8 Construction of the knot Ln
620
+ Let
621
+ ¯Ln =
622
+
623
+ k∈Z
624
+ hk(B ∪ A1 ∪ · · · ∪ A2n), Ln = p(¯Ln).
625
+ 4.2 The genus of the knot Ln
626
+ Lemma 3 The genus gLn of the knot Ln equals n.
627
+
628
+ 12
629
+ Timur Medvedev, Olga Pochinka
630
+ Fig. 9 A secant surface of Ln of genus n
631
+ Proof Since there is a secant surface of Ln of genus n, we have gLn ⩽ n (Fig. 9).
632
+ Now we show that gLn ⩾ n. To that end we prove that for Ln there exists a
633
+ minimal secant surface Σ such that ¯Σ ⊂ K and ¯Ln ∩ ¯Σ = h(α1).
634
+ Indeed, let Σ0 be some minimal secant surface of Ln. Then there exists the
635
+ connected component ¯Σ0 of p−1(Σ0) such that it intersects the curve ¯Ln at the
636
+ point ¯y0 situated on ¯Ln between α1, h(α1) and that bounds the h-compressible
637
+ body QΣ0. Without loss of generality let ¯y0 = h(α1) (otherwise the desired surface
638
+ is constructed by removing the tubular neighborhood of the arc [¯y0, h(α1)] ⊂ ¯Ln
639
+ from QΣ0).
640
+ Denote by k+, k− ⩾ 0 the maximal integers for whichf k( ¯Σ0)∩ ¯Σ0 ̸= ∅, f −k( ¯Σ0)∩
641
+ ¯Σ0 ̸= ∅, k ⩾ 0, respectively. If k+ = k− = 0 then ¯Σ0 is the desired surface. Other-
642
+ wise we show the way to decrease by 1 the number k+ > 0 (for k− the arguments
643
+ are the same) using isotopy of the secant surface.
644
+ Notice that ¯Σ0 ∩ f k+(c2j−1) = ∅, j ∈ {1, . . . , n}. Without loss of generality let
645
+ the intersection Γ =
646
+ n�
647
+ j=1
648
+ f k+(d2j−1) ∩ ¯Σ0 be transversal. Let γ be a curve from
649
+ Γ. Then γ bounds the unique disk Dγ ⊂ f k+(d2j−1). There are two possibilities:
650
+ 1) b2j−1 /∈ Dγ, 2) b2j−1 ∈ Dγ. In case 1) we say the curve γ to be innermost
651
+ if it bounds the disk Dγ ⊂ f k+(d2j−1) such that int Dγ contains no curves of
652
+ Γ. Consider this innermost curve γ. Due to Lemma 1 the surface ¯Σ0 \ ¯y0 is non-
653
+ compressible in R3 \ (O ∪ ¯Ln) and, therefore, there exists the disk dγ ⊂ ( ¯Σ0 \ ¯y0)
654
+ bounded by γ. Denote by Bγ ⊂ (R3 \(O∪ ¯Ln)) the 3-ball bounded by the 2-sphere
655
+ Dγ ∪ dγ. Consider two subcases: 1a) Bγ ⊂ QΣ0 and 1b) Bγ ⊂ (R3 \ int QΣ0).
656
+ In case 1a) let N(Bγ) ⊂ QΣ0 be a tubular neighborhood of the ball Bγ. Then
657
+ the set QΣ \ int N(Bγ) is h-compressible because the curve γ lies in its interior
658
+ and its boundary intersects ¯Ln at a unique point. The projection of this boundary
659
+
660
+ Quasi-energy function for Pixton diffeomorphisms
661
+ 13
662
+ to S2 × S1 is, therefore, a secant surface of Ln of the same genus as Σ0. For it the
663
+ number of the connected components of the set Γ is less. One gets the same result
664
+ in case 1b) for the set QΣ0 ∪ N(Bγ).
665
+ If we continue this process then we get the secant surface of Ln of the same
666
+ genus as Σ0 and for which the set Γ contains no curves of type 1). Denote the
667
+ resulting surface again by Σ0. Now the set Γ consists only of the curves γ bounding
668
+ the disk Dγ ⊂ b2j−1 which contains the point b2j−1. Since (b2j−1 ⊔ c2j−1) ⊂
669
+ (R3 \QΣ0), the number of these curves on the disk d2j−1 is even. Since the surface
670
+ ¯Σ0\¯y0 is non-compressible in R3\(O∪ ¯Ln), all these curves are pairwise homotopic
671
+ on ¯Σ0 \ ¯y0 and, therefore, they lie in the annulus κ ⊂ ( ¯Σ0 \ ¯y0) bounded by the
672
+ pair of these curves γ1, γ2. Denote by ˜κ ⊂ d2j−1 the annulus bounded by the same
673
+ curves on the disk d2j−1. Let ˜Σ0 = ¯Σ0 \ κ ∪ ˜κ. Due to construction the surface
674
+ ˜Σ0 is of the same genus as the surface ¯Σ0 and it bounds an h-compressible body.
675
+ Having removed a tubular neighborhood of the annulus ˜κ from this body we get a
676
+ h-compressible body whose boundary does not intersect the disk d2j−1 and whose
677
+ projection to S2 ×S1 is the secant surface of the knot Ln of the same genus as Σ0.
678
+ If we continue this process then we get a secant surface of Ln of the same
679
+ genus as Σ0 and for which the set Γ is not empty. Denote this surface again by
680
+ Σ0. Without loss of generality let the intersections of the surface ¯Σ0 with the
681
+ spheres f k(S2) be transversal. Denote by F the set of the connected components
682
+ of the intersection f k+(K) ∩ ¯Σ0. Now we show the way to reduce by 1 the number
683
+ of the components in F using isotopy of the secant surface.
684
+ Denote by Q the set obtained by removal from the annulus f k+(K) of the
685
+ tubular neighborhoods of the disks d2j−1 as well as the tubular neighborhoods of
686
+ the curves A2j, j ∈ {1, . . . , n}. Then Q is homeomorphic to the direct product
687
+ of the 2-sphere with 2n + 1 deleted points and the segment. Since Q ∩ ¯Σ0 =
688
+ f k+(K)∩ ¯Σ0 and since ¯Σ0 \ ¯y0 is non-compressible in R3 \(O∪ ¯Ln), each connected
689
+ component of F ∈ F is non-compressible in Q. Due to [16, Corollary 3.2] there
690
+ exists a surface ˜F ⊂ f k+−1(S2) diffeomorphic to F for which ∂F = ∂ ˜F and the
691
+ surface F ∪ ˜F bounds in Q the body ∆ diffeomorphic to the direct product F ×[0, 1].
692
+ Then we replace the part F of ¯Σ0 with ˜F. If we continue the process we get the
693
+ desired secant surface Σ ⊂ K.
694
+ Notice (see, for instance, [4, Exercise 2.8.1]) that the fundamental group π1(K\
695
+ ¯Ln) has 2n generators γ1, . . . , γ2n, each of which γi, i ∈ {1, . . . , 2n} being the
696
+ generator of the punctured disk di \bi (Fig. 10). Since b2j−1 ∈ int QΣ and c2j−1 ∩
697
+ QΣ = ∅, there exists the connected component of ˜d2j−1 of the intersection d2j−1 ∩
698
+ QΣ which contains the point b2j−1. This component is the 2-disk bounded by the
699
+ curve ˜γ2j−1 ⊂ ( ¯Σ \ h(α1)) with holes and the curves γ2j−1, ˜γ2j−1 are homotopic
700
+ on the punctured disk d2j−1 \ b2j−1. In the same way one finds the curves ˜γ2j ⊂
701
+ ( ¯Σ \ h(α1)) homotopic to the curves γ2j on the punctured disk d2j \ b2j (Fig. 10).
702
+ Due to Lemma 1 the surface ¯Σ \ h(α1) is non-compressible in K \ ¯Ln. Then the
703
+ curves ˜γ1, . . . , ˜γ2n are pairwise non-homotopic to the generators on the surface
704
+ ¯Σ \ h(α1). Therefore, the genus of the surface ¯Σ cannot be less than n.
705
+
706
+ 14
707
+ Timur Medvedev, Olga Pochinka
708
+ Fig. 10 Generators of the group π1(K \ ¯Ln)
709
+ 5 Construction of a quasi-energy function for a Pixton diffeomorphism
710
+ with the Hopf knot Ln
711
+ Let f be a Pixton diffeomorphism constructed for a generalized Mazur knot Ln.
712
+ Then its non-wandering set Ωf consists of four points: two sinks ω, S, a source N
713
+ and a saddle σ. Then W u
714
+ σ \σ consists of two separatrices ℓω, ℓS respective closures
715
+ of which contain the sinks ω, S, the separatrice ℓω being tame while ℓS being wild.
716
+ Let ¯Σ be the surface of genus n bounding the handle-body QΣ of the same genus.
717
+ Now we construct for f a Morse-Lyapunov function with 6 + 2n critical points.
718
+ Our construction of a quasi-energy function is analogous to the construction
719
+ of an energy function in [7].
720
+ 1. Choose an energy function ϕp : Up → R in the neighborhood of each fixed
721
+ point p of f so that ϕp(p) = dim W u
722
+ p . Let Bω, BS be the 3-balls which are the
723
+ level sets of respective functions ϕω, ϕS and such that BS ⊂ int QΣ. Choose a
724
+ tubular neighborhood Tσ of the arc W u
725
+ σ \ (Bω ∪ QΣ) so that the handle-body
726
+ Bω ∪ QΣ ∪ Tσ of genus n is f-compressible and its intersection with W s
727
+ σ is
728
+ the 2-disk. Denote by P + the smoothing of this body by addition of a small
729
+ exterior collar.
730
+ 2. Due to [7, Section 4.3] there exists an energy function ϕ : P + \ int QΣ whose
731
+ value on ∂P + is 4/3, whose value on ¯Σ is 2/3 and which has exactly two critical
732
+ points ω, σ of respective Morse indexes 0, 1. The disks d1, . . . , d2n−1 cut the
733
+ handle-body QΣ making the 3-ball. Denote by BΣ the smoothing of this ball
734
+ by removal of the interior collar. The results of the classic Morse theory (see,
735
+ for example, [13]) allow to extend the function ϕ to the set QΣ \ int BΣ in
736
+ such way that it has n critical points of Morse index 1, one point lying on
737
+ each disk d1, . . . , d2n−1, while the value of ϕ on ∂BΣ is 1/3. Due to [7, Lemma
738
+
739
+ Quasi-energy function for Pixton diffeomorphisms
740
+ 15
741
+ 4.2] the function ϕ can be extended to the ball BΣ by an energy function
742
+ with the unique critical point S of Morse index 0. Since f(QΣ) ⊂ int BΣ, the
743
+ constructed function decreases along the trajectories of the diffeomorphism f.
744
+ 3. It follows from the definition of the knot Ln that P − = S3 \ int P + is the
745
+ handle-body of genus n. Moreover, the disks d2, . . . , d2n cut P − making the
746
+ 3-ball. Denote by B− smoothing of this ball by removal of the interior collar.
747
+ The results of the classic Morse theory (see, for example, [13]) allow extension
748
+ of the function ϕ to the set P − \ int B− in such way that it has n critical
749
+ points of Morse index 2, one point lying on each disk d2, . . . , d2n, while the
750
+ value of ϕ on ∂B− is 5/3. According to [7, Lemma 4.2] the function ϕ can be
751
+ extended to the ball B− by an energy function with unique critical point N
752
+ of Morse index 3. Since f(B−) ⊂ int P −, the constructed function decreases
753
+ along the trajectories of the diffeomorphism f and, therefore, it is the desired
754
+ quasi-energy function.
755
+ Conflict of interest
756
+ The authors declare that they have no conflict of interest.
757
+ References
758
+ 1. Akhmetiev, P., Medvedev, T., Pochinka, O.: On the number of the classes of topological
759
+ conjugacy of Pixton diffeomorphisms. Qualitative Theory of Dynamical Systems 20(3),
760
+ 1–15 (2021)
761
+ 2. Bonatti, C., Grines, V.: Knots as topological invariants for gradient-like diffeomorphisms
762
+ of the sphere S3. Journal of Dynamical and Control Systems 6(4), 579–602 (2000)
763
+ 3. Conley, C.: Isolated invariant sets and the morse index. American Mathematical Society,
764
+ CBMS, Providence, RI 38 (1978)
765
+ 4. Daverman, R.J., Venema, G.: Embeddings in manifolds, vol. 106. American Mathematical
766
+ Soc. (2009)
767
+ 5. Fomenko, A.: Differential Geometry and Topology: Additional Chapters. Moscow Univer-
768
+ sity Press (1983)
769
+ 6. Grines, V., Laudenbach, F., Pochinka, O.: The energy function for gradient-like diffeomor-
770
+ phisms on 3-manifolds. Doklady Mathematics 78(2), 702–704 (2008)
771
+ 7. Grines, V.Z., Laudenbach, F., Pochinka, O.V.: Quasi-energy function for diffeomorphisms
772
+ with wild separatrices. Mathematical Notes 86(1), 163–170 (2009)
773
+ 8. Grines, V.Z., Medvedev, T.V., Pochinka, O.V.: Dynamical Systems on 2- and 3-Manifolds,
774
+ Developments in Mathematics, vol. 46. Springer International Publishing (2016). DOI
775
+ 10.1007/978-3-319-44847-3
776
+ 9. Kirk, P., Livingston, C.: Knot invariants in 3-manifolds and essential tori. Pacific Journal
777
+ of Mathematics 197(1), 73–96 (2001)
778
+ 10. Mazur, B.: A note on some contractible 4-manifolds. Annals of Mathematics 79(1), 221–
779
+ 228 (1961)
780
+ 11. Medvedev, T.V., Pochinka, O.V.: The wild Fox-Artin arc in invariant sets of dynamical
781
+ systems. Dynamical Systems 33(4), 660–666 (2018). DOI 10.1080/14689367.2017.1421903.
782
+ URL https://doi.org/10.1080/14689367.2017.1421903
783
+ 12. Meyer, K.R.: Energy functions for morse smale systems. American Journal of Mathematics
784
+ 90(4), 1031–1040 (1968). URL http://www.jstor.org/stable/2373287
785
+ 13. Milnor, J.: Morse theory.(am-51), volume 51.
786
+ In: Morse Theory.(AM-51), Volume 51.
787
+ Princeton university press (2016)
788
+ 14. Neumann, W.D.: Notes on geometry and 3-manifolds. Citeseer (1996)
789
+ 15. Pixton, D.: Wild unstable manifolds.
790
+ Topology 16, 167–172 (1977).
791
+ DOI 10.1016/
792
+ 0040-9383(77)90014-3
793
+ 16. Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Annals of Math-
794
+ ematics pp. 56–88 (1968)
795
+
8dE0T4oBgHgl3EQffgB5/content/tmp_files/load_file.txt ADDED
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1
+ filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf,len=476
2
+ page_content='⋆ manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
3
+ page_content=' (will be inserted by the editor) A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots Timur Medvedev · Olga Pochinka Received: date / Accepted: date Abstract In this paper we give a lower estimate for the number of critical points of the Lyapunov function for Pixton diffeomorphisms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
4
+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
5
+ page_content=' Morse-Smale diffeo- morphisms in dimension 3 whose chain recurrent set consists of four points: one source, one saddle and two sinks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
6
+ page_content=' Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
7
+ page_content=' Bonatti and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
8
+ page_content=' Grines proved that the class of topological equivalence of such diffeomorphism f is completely defined by the equivalency class of the Hopf knot Lf that is the knot in the generating class of the fundamental group of the manifold S2 × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
9
+ page_content=' They also proved that there are infinitely many such classes and that any Hopf knot can be realized by a Pixton diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
10
+ page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
11
+ page_content=' Pixton proved that diffeomorphisms defined by the standard Hopf knot L0 = {s} × S1 have an energy function (Lyapunov function) whose set of critical points coincide with the chain recurrent set whereas the set of critical points of any Lyapunov function for Pixton diffeomorphism with nontrivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
12
+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
13
+ page_content=' non equivalent to the standard) Hopf knot is strictly larger than the chain recur- rent set of the diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
14
+ page_content=' The Lyapunov function for Pixton diffeomorphism with minimal number of critical points is called the quasi-energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
15
+ page_content=' In this paper we construct a quasi-energy function for Pixton diffeomorphisms defined by a generalized Mazur knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
16
+ page_content=' Keywords Hopf knot · Mazur knot · Pixton diffeomorphism · quasi-energy function Mathematics Subject Classification (2020) 37C15 · 37D15 ⋆ The research was done with the support of Russian National Foundation (project 21-11- 00010) except construction of the quasi-energy function which was supported by International Laboratory of Dynamical Systems and Applications of National Research University Higher School of Economics, grant of Government of Russian Federation 075-15-2022-1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
17
+ page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
18
+ page_content=' Medvedev Laboratory of Algorithms and Technologies for Network Analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
19
+ page_content=' HSE University 136 Rodionova Street, Niznhy Novgorod, Russia E-mail: mtv2001@mail O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
20
+ page_content=' Pochinka International Laboratory of Dynamical Systems and Applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
21
+ page_content=' HSE University, 25/12 Bol- shaya Pecherckaya Street, Niznhy Novgorod, Russia arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
22
+ page_content='02405v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
23
+ page_content='DS] 6 Jan 2023 2 Timur Medvedev, Olga Pochinka 1 Introduction and the main results Let M n be a smooth closed n-manifold with a metric d and let f : M n → M n be a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
24
+ page_content=' For two given points x, y ∈ M n a sequence of points x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
25
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
26
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
27
+ page_content=' , xm = y is called an ε-chain of length m ∈ N connecting x to y if d(f(xi−1), xi) < ε for 1 ⩽ i ⩽ m (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
28
+ page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
29
+ page_content=' x x = 0 f x( ) 0 f x( ) 1 f x( ) 2 f x( ) m-1 f x( ) m 2 x1 x2 x3 xm-1 y x = m � � � � � Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
30
+ page_content=' 1 An ε-chain of length m ∈ N A point x ∈ M n is called chain recurrent for the diffeomorphism f if for every ε > 0 there is an ε-chain of length m connecting x to itself for some m (m depends on ε > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The chain recurrent set, denoted by Rf, is the set of all chain recurrent points of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Define the equivalence on Rf by the rule: x ∼ y if for every ε > 0 there is are ε-chains connecting x to y and y to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' This equivalence relation defines equivalence classes called chain components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
34
+ page_content=' If the chain recurrent set of a diffeomorphism f is finite then it consists of periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' A periodic point p ∈ Rf of period mp is said to be hyperbolic if absolute values of all the eigenvalues of the Jacobian matrix � ∂f mp ∂x � |p are not equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' If absolute values of all these eigenvalues are greater (less) than 1 then p is called a sink (a source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Sinks and sources are called knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' If a periodic point is not a knot then it is called a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let p be a hyperbolic periodic point of a diffeomorphism f whose chain recur- rent set is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
40
+ page_content=' The Morse index of p, denoted by λp, is the number of eigen- values of Jacobian matrix whose absolute values are greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The stable manifold W s p = {x ∈ M n : lim k→+∞ d(f kmp(x), p) = 0} and the unstable manifold W u p = {x ∈ M n : lim k→+∞ d(f −kmp(x), p) = 0} of p are smooth manifolds diffeo- morphic to Rλp and Rn−λp, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Stable and unstable manifolds are called invariant manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' A connected component of the set W u p \\ p (W s p \\ p) is called a unstable (stable) separatrice of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' A diffeomorphism f : M n → M n is called a Morse-Smale diffeomorphism if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' its chain recurrent set Rf consists of finite number of hyperbolic points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
47
+ page_content=' for any two points p, q ∈ Rf the manifolds W s p , W u q intersect transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Quasi-energy function for Pixton diffeomorphisms 3 C Conley in [3] gave the following definition: a Lyapunov function for a Morse- Smale diffeomorphism f : M n → M n is a continuous function ϕ : M n → R satisfying – ϕ(f(x)) < ϕ(x) if x /∈ Rf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
49
+ page_content=' – ϕ(f(x)) = ϕ(x) if x ∈ Rf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Notice that every Morse-Smale diffeomorphism f has a Morse-Lyapunov func- tion 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' a Lyapunov function ϕ : M n → R which is a Morse function such that each periodic point p ∈ Rf is its non-degenerate critical point of index λp with Morse coordinates (Vp, φp : y ∈ Vp �→ (x1(y), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
53
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
54
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
55
+ page_content=' , xn(y)) ∈ Rn and φ−1 p (Ox1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
56
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
57
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
58
+ page_content=' xλp) ⊂ W u p , φ−1 p (Oxλp+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
59
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
60
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' xn) ⊂ W s p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' (∗) If the function ϕ has no critical points outside Rf then following [15] we call it the energy function for the Morse-Smale diffeomorphism f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The proof of existence of an energy Morse function for a Morse-Smale diffeo- morphism of the circle is an easy exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Pixton [15] in 1977 proved that every Morse-Smale diffeomorphism of a surface has an energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' There he also constructed an example of a Morse-Smale diffeomorphism on the 3-sphere which admits no energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The obstacle to existence of an energy function in his example was the wild embedding of the saddle separatrices in the ambient manifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' the closure of the separatrice is not a submanifold of the ambient space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' From [11] it follows that there are Morse-Smale diffeomorphisms with no energy function on manifolds of any dimension n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Therefore, following [7] for a Morse-Smale diffeomorphism f we call a Morse-Lyapunov function with the minimal number of critical points (denote it by ρf ) a quasi-energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Notice that ρf is a topological invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' if two diffeomorphisms f, f ′ : M n → M n are topologically conjugate (that is there is a diffeomorphism h : M n → M n such that h ◦ f = f ′ ◦ h) then ρf = ρf′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' In this paper we give a lower estimate of ρf for Pixton diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The class of Pixton diffeomorphisms P is defined in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Every diffeo- morphism f ∈ P is a Morse-Smale 3-diffeomorphism whose chain recurrent set consists of four points: one source, one saddle and two sinks (for details see section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Notice that Pixton’s example is a diffeomorphism of this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' According to [2] the class of topological conjugacy of a diffeomorphism f ∈ P is completely defined by the equivalence class of the Hopf knot Lf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' the knot in the generating class of the fundamental group of the manifold S2 × S1 (see Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Moreover, any Hopf knot can be realized as a Pixton diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Recall that a knot in S2 × S1 is a smooth embedding γ : S1 → S2 × S1 or the image of this embedding L = γ(S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Two knots L, L′ are said to be equivalent if there is a homeomorphism h : S2 × S1 → S2 × S1 such that h(L) = L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Two knots 1 This function can be constructed, for example, by suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Consider the topological flow ˆft on the manifold Mn × R defined by ˆft(x) = x + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Define the diffeomorphism g : Mn × R → Mn × R by g(x, τ) = (f(x), τ − 1) and let G = {gk , k ∈ Z} and W = (Mn × R)/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Denote by pW : Mn × R → W the natural projection and denote by ft the flow on W defined by ft(x) = pW ( ˆft(p−1 W (x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The flow ft is called the suspension over f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' By construction the chain recurrent set of ft consists of the finite number of periodic orbits δi = pW (Oi × R), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' , kf} and this means that the suspension ft is a Morse-Smale flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' A Lyapunov function for these flows is constructed in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then the restriction of this function on M is the desired Lyapunov function for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 4 Timur Medvedev, Olga Pochinka γ, γ′ are smoothly homotopic if there exists a smooth map Γ : S1 ×[0, 1] → S2 ×S1 such that Γ(s, 0) = γ(s) and Γ(s, 1) = γ′(s) for every s ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' If Γ|S1×{t} is an embedding for every t ∈ [0, 1] then the knots are said to be isotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Any Hopf knot L ⊂ S2 × S1 is smoothly homotopic to the standard Hopf knot L0 = {s} × S1 (see, for example, [9]) but generally it is neither isotopic nor equivalent to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Mazur [10] constructed the Hopf knot LM which we call the Mazur knot and which is non-equivalent and non-isotopic to L0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' It follows from the results of [1] that there exists a countable family of pairwise Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 2 Two non-isotopic and non equivalent Hopf knots L0 and LM: a) the standard Hopf knot L0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' b) the Mazur knot LM non-equivalent Hopf knots Ln, n ∈ N which are generalized Mazur knots (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 3 A generalized Mazur knot Ln According to [6] a Pixton diffeomorphism f admits an energy Morse function if and only if the knot Lf is trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' equivalent to the standard one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' If the Quasi-energy function for Pixton diffeomorphisms 5 knot Lf is not trivial then the number ρf of the critical points of a quasi-energy Morse function of f is evidently even and ρf ⩾ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The main result of this paper is the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Theorem 1 Let f be a Pixton diffeomorphism (f ∈ P) and let Ln, n ∈ N be its knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then the number ρf of critical points of a quasi-energy function of f is calculated by2 ρf = 4 + 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 2 Construction of Pixton diffeomorphisms In dynamics a wild Artin-Fox arc was for the first time introduced by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Pixton in [15] where he constructed a Morse-Smale diffeomorphism on the 3-sphere with the unique saddle whose invariant manifolds form an Artin-Fox arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' We give the modern construction of these diffeomorphisms following Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Bonatti and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Grines [2] where Pixton diffeomorphisms were also classified (see also [8], [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' For x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' , xn) ∈ Rn denote ||x|| = � x2 1 + · · · + x2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let h : R3 → R3 be the diffeomorphism defined by h(x1, x2, x3) = � x1 2 , x2 2 , x3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Define the map p : R3 \\ O → S2 × S1 by p(x1, x2, x3) = � x1 ||x||, x2 ||x||, log2(||x||) (mod 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let L ⊂ (S2 ×S1) be a Hopf knot and let U(L) be its tubular neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then the set ¯L = p−1(L) is the h-invariant arc in R3 and U(¯L) = p−1(U(L)) is its h-invariant neighborhood diffeomorphic to D2 × R1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let C = {(x1, x2, x3) ∈ R3 : x2 2 + x2 3 ⩽ 4} and let gt : C → C be the flow defined by gt(x1, x2, x3) = (x1 + t, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then there is a diffeomorphism ζ : U(L) → C that conjugates h|U(L) and g = g1|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Define the flow φt on C by: � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ˙x1 = � 1 − 1 9(x2 1 + x2 2 + x2 3 − 4)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
131
+ page_content=' x2 1 + x2 2 + x2 3 ⩽ 4 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
132
+ page_content=' x2 1 + x2 2 + x2 3 > 4 ˙x2 = � � � � � x2 2 � sin � π 2 � x2 1 + x2 2 + x2 3 − 3 �� − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
133
+ page_content=' 2 < x2 1 + x2 2 + x2 3 ⩽ 4 −x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
134
+ page_content=' x2 1 + x2 2 + x2 3 ⩽ 2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
135
+ page_content=' x2 1 + x2 2 + x2 3 > 4 ˙x3 = � � � � � x3 2 � sin � π 2 � x2 1 + x2 2 + x2 3 − 3 �� − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
136
+ page_content=' 2 < x2 1 + x2 2 + x2 3 ⩽ 4 −x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
137
+ page_content=' x2 1 + x2 2 + x2 3 ⩽ 2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
138
+ page_content=' x2 1 + x2 2 + x2 3 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' By construction the diffeomorphism φ = φ1 has two fixed points: the saddle P(1, 0, 0) and the sink Q(−1, 0, 0) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 5), both being hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' One unstable 2 For n = 1 Theorem 1 is proved in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 6 Timur Medvedev, Olga Pochinka L L _ U(L) _ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 4 Suspension of a Hopf knot 1 1 2 O Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 5 Trajectories of the flow φt separatrice of the saddle P coincides with the open interval � (x1, x2, x3) ∈ R3 : |x1| < 1, x2 = x3 = 0 � in the basin of the sink Q while the other coincides with the ray � (x1, x2, x3) ∈ R3 : x1 > 1, x2 = x3 = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Notice that φ coincides with the diffeomorphism g = g1 outside the ball {(x1, x2, x3) ∈ C : x2 1 + x2 2 + x2 3 ⩽ 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Define the diffeomorphism ¯fL : R3 → R3 so that ¯fL coincides with h outside U(L) and it coincides with ζ−1φζ on U(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then ¯fL has in U(L) two fixed points: the sink ζ−1(Q) and the saddle ζ−1(P), both being hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The unstable separatrice of the saddle ζ−1(P) lies in L (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Now project the dynamics onto the 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Denote by N(0, 0, 0, 1) the North Pole of the sphere S3 = {x = (x1, x2, x3, x4) : ||x|| = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' For every point x ∈ (S3 \\ {N}) there is the unique line passing through N and x in R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' This line intersects Quasi-energy function for Pixton diffeomorphisms 7 R Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 6 The phase portrait of the diffeomorphism ¯fL R3 in exactly one point ϑ+(x) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The point ϑ+(x) is the stereographic projection of the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' One can easily check that ϑ+(x1, x2, x3, x4) = � x1 1 − x4 , x2 1 − x4 , x3 1 − x4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Thus, the stereographic projection ϑ+ : S3 \\ {N} → R3 is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' N x ��(x) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 7 The stereographic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' By construction ¯fL coincides with h in some neighborhood of the point O and in some neighborhood of the infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Therefore, it induces on S3 the Morse-Smale fL(x) = � ϑ−1 + ( ¯fL(ϑ+(x))), x ̸= N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' N, x = N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' It follows directly from the construction that the non-wandering set of fL consists of exactly four fixed hyperbolic points: two sinks ω = ϑ−1 + (ζ−1(Q)), S, one saddle σ = ϑ−1 + (ζ−1(P)) and one source N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' We say the constructed diffeomorphism to be model and it is of Pixton class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 8 Timur Medvedev, Olga Pochinka Proposition 1 ([2]) – Any diffeomorphism f ∈ P is topologically conjugate to some model diffeomor- phism fL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' – Two model diffeomorphisms fL, fL′ are topologically conjugate if and only if their knots L, L′ are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 3 Genus of Hopf knot In this section we introduce the notion of genus for a Hopf knot and use it to estimate the number of critical points of the quasi-energy function of the Pixton diffeomorphism defined by this knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let L be a Hopf knot and let ¯L = p−1(L) be its cover in R3\\O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' We say a closed orientable surface Σ ⊂ S2 × S1 to be a secant surface of the knot L if it intersects L in a unique point and there is an h-compressible 3-manifold QΣ ⊂ R3 (that is h(QΣ) ⊂ int QΣ) with the boundary ¯Σ such that Σ = p( ¯Σ) and the intersection ¯L∩ ¯Σ is the unique point ¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The minimally possible genus gL of the secant surface is called the genus of the knot L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The secant surface of L of genus gL is said to be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Lemma 1 If Σ is a minimal secant surface of the knot L then the surface ¯Σ \\ ¯y is non-compressible in R3 \\ (O ∪ ¯L), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' any simple closed curve c ⊂ int ( ¯Σ \\ ¯y) is contractible on ¯Σ\\ ¯y if it bounds a smoothly embedded 2-disk D ⊂ int (R3\\(O∪ ¯L)) such that D ∩ ( ¯Σ \\ ¯y) = ∂D = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Proof Let Σ be a minimal secant surface of L and let ¯y be the unique point of the intersection ¯L ∩ ¯Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Assume the opposite: the surface ¯Σ \\ ¯y is compressible in R3 \\ (O ∪ ¯L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then there is a non-contractible simple closed curve c ⊂ int ( ¯Σ \\ ¯y) and there is the smoothly embedded 2-disk D ⊂ int (R3 \\ (O ∪ ¯L)) such that D ∩ ( ¯Σ \\ ¯y) = ∂D = c (see, for example, [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then we have two possibilities: (int D) ∩ � � k∈Z hk( ¯Σ) � = ∅, (1) (int D) ∩ � � k∈Z hk( ¯Σ) � ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' (2) In case (1) two subcases are possible: (1a) D ⊂ QΣ, (1b) D ⊂ (R3 \\ int QΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' For case 1a) let N(D) ⊂ QΣ be a tubular neighborhood of the disk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then exactly one connected component of the set QΣ \\ int N(D) intersects ¯L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' According to (1) this component is h-compressible and its boundary intersects ¯L at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The projection of this boundary into S2 × S1 is, therefore, the secant surface of L of genus less than gL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' This contradicts the fact that the surface Σ is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' In case 1b) let N(D) ⊂ (R3 \\ int QΣ) be a tubular neighborhood of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then due to (1) the set QΣ ∪ N(D) is h-compressible and its boundary intersects ¯L at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The projection of this boundary into S2 ×S1 is, therefore, the secant surface of L of genus less than gL and we have the same contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Quasi-energy function for Pixton diffeomorphisms 9 In case (2) without loss of generality assume the intersection int D∩( � k∈Z hk( ¯Σ)) to be transversal and denote it by Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let γ be a curve from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' We say the curve γ to be innermost if it is the boundary of the disk Dγ ⊂ D such that int Dγ contains no curves of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Consider this innermost curve γ ⊂ f k(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' There are two subcases: a) γ is essential on f k(Σ) and b) γ is contractible on f k(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' In case a) the arguments of the case (1) apply for the body f k(QΣ) and the disk Dγ and we get the contradiction to the minimality of the surface Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' In case b) denote by dγ ⊂ f k(Σ) the 2-disk bounded by γ and denote by Bγ ⊂ (R3 \\ O) the 3-ball bounded by the 2-sphere Dγ ∪ dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Consider: b1) Bγ ⊂ f k(QΣ) and b2) Bγ ⊂ (R3 \\ int f k(QΣ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' For b1) let N(Bγ) ⊂ f k(QΣ) be a tubular neighborhood of Bγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then the set QΣ \\ int N(Bγ) is h-compressible because the curve γ lies in its interior and the boundary of QΣ \\ int N(Bγ) intersects ¯L at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The projection of this boundary into S2 ×S1 is, therefore, the secant surface of the knot L of genus gL for which the number of connected components of the set Γ is less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' We get the same result for b2) for the set QΣ ∪ N(Bγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Thus, iterating the process we come either to the case a) or to the case (1) and get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Lemma 2 For any diffeomorphism f ∈ P the following estimation holds ρf ⩾ 4 + 2gLf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' (3) Proof Since Proposition 1 is true and since the number ρf of the critical points of a quasi-energy function of f ∈ P is invariant, from now on we consider model Pixton diffeomorphismsfL with the Hopf knot L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Denote by ℓ the non-stable separatrice of the saddle σ lying in the basin of the sink S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let pS : W s S \\ S → S2 × S1 be the natural projection sending a point w ∈ (W s S\\S) to the point p(f kw(w)), f kw(w) ∈ VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Since the diffeomorphism fL coincides with the homothety h in some neigh- borhood VS of S, the natural projection pS is well defined and pS(ℓ) = L by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Consider an arbitrary Morse-Lyapunov function ϕ : S3 → R of the diffeomor- phism fL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' To be definite let ϕ(S) = 0, ϕ(σ) = 1 and ϕ(N) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' From the definition of the Morse-Lyapunov function it follows that ϕ|ℓ monotonically decreases in some neighborhood of the saddle σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Therefore, there is ε1 ∈ (0, 1) such that the interval (1 − ε1, 1) contains no critical values of ϕ and the connected component ¯Σ1 of the level set ϕ−1(1 − ε1) intersects the separatrice ℓ at the unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Denote this point by w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let ¯Q1 be the connected component of the set ϕ−1([0, 1 − ε1]) which contains the segment [w1, S] of the closure of the separatrice ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Since ϕ decreases along the trajectories of f, the values of ϕ on W s σ are greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Therefore, the manifold ¯Q1 lies in the manifold W s S diffeomorphic to R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let the function ϕ| ¯ Q1 have kq, q ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' , 3} critical points of index q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Due to [5, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content='1] on the manifold ¯Q1 there exists a self-indexing Morse function ψ (the value of the function in a critical point equals the index of this point) which has kq critical points of index q and which is constant on ∂ ¯Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Thus, the manifold ¯Q1 is the surface ˜Q1 of 10 Timur Medvedev, Olga Pochinka genus g1 = 1 + k1 − k0 with attached handles of indexes 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then the genus of any surface of the set ∂ ¯Q1 cannot be greater than g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' On the other hand, the number of critical points of ϕ| ¯ Q1 is not less than k0+k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' If k0 ⩾ 1 and g1 = 1 + k1 − k0 then one gets k0 + k1 = g1 + 2k0 − 1 ⩾ g1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Thus, ϕ| ¯ Q1 has at least g1 + 1 critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Denote by ¯Σ1 the connected component of ∂ ¯Q1 which intersects the separatrice ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then the surface ¯Σ1 divides the manifold W s S ∼= R3 into two parts, one of which Q1 being an h-compressible body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' This means that Σ1 = pS( ¯Σ1) is the secant surface of L and, therefore, g1 ⩾ gL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Analogously, there is ε2 ∈ (0, 1) for which the interval (1, 1 + ε2) contains no critical points of ϕ and the connected component ¯Q2 of the level set ϕ−1([0, 1+ε2)] contains cl(W u σ ) in its interior while the intersection ¯Q2 with W s σ is the unique 2-disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Due to construction the function ϕ| ¯ Q2 has at least g1 + 3 critical points and genus of the connected components of ∂ ¯Q2 is less or equals g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Denote by ¯Σ2 the connected component of ∂ ¯Q2 which intersects W s σ and denote by g2 its genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' The surface ¯Σ2 divides the manifold W u N ∼= R3 into two parts, one of which Q2 being an h−1-compressible body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Arguing as above one comes to conclusion that the number of critical points of ϕ|Q2 is at least g2 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Therefore, the total number of critical points of ϕ is greater or equal to g1 + 3 + g2 + 1 ⩾ 4 + 2g1 ⩾ 4 + 2gLf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 4 The generalized Mazur knot Ln In this section we show that the genus gLn of a generalized Mazur knot equals n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' At first we give a detailed description of construction of Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content='1 Construction of the generalized Mazur knot Ln Recall that h : R3 → R3 is the homothety defined by h(x1, x2, x3) = �x1 2 , x2 2 , x3 2 � and p : R3 \\ O → S2 × S1 is the natural projection defined by p(x1, x2, x3) = � x1 ||x||, x2 ||x||, log2(||x||) (mod 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Consider the annulus K = � (x1, x2, x3) ∈ R3 : 1 4 ≤ x2 1 + x2 2 + x2 3 ≤ 1 � bounded by the spheres S2 = � (x1, x2, x3) ∈ R3 : x2 1 + x2 2 + x2 3 = 1 � , h(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Quasi-energy function for Pixton diffeomorphisms 11 Pick on the circle S1 = � (x1, x2, x3) ∈ R3 : x2 1 + x2 2 = 1, x3 = 0 � pairwise distinct points α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' , α2n+1 numbered in counter-clockwise order (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let ai, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' , 2n} be the arc of the circle S1 bounded by αi, αi+1 whose inte- rior contains no points of {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' , α2n+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let B, Ai ⊂ int K, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' , 2n} be pairwise disjoint smooth arcs such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' the boundary points of B are α2n+1, h(α1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' the boundary points of A2j−1 are α2j−1, α2j and the boundary points of A2j are h(α2j), h(α2j+1) for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' , n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' the closed curves c2j−1 = cl(a2j−1 ∪ A2j−1), c2j = cl(h(a2j) ∪ A2j) bound the 2-disks d2j−1, d2j, the transversal intersection of these disks being the arc lj with the boundary points b2j−1 = d2j−1 ∩ A2j and b2j = d2j ∩ A2j−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' the arc cl(h(A1) ∪ A2 ∪ · · · ∪ h(A2n−1) ∪ A2n ∪ B) is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 8 Construction of the knot Ln Let ¯Ln = � k∈Z hk(B ∪ A1 ∪ · · · ∪ A2n), Ln = p(¯Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content='2 The genus of the knot Ln Lemma 3 The genus gLn of the knot Ln equals n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 12 Timur Medvedev, Olga Pochinka Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 9 A secant surface of Ln of genus n Proof Since there is a secant surface of Ln of genus n, we have gLn ⩽ n (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Now we show that gLn ⩾ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' To that end we prove that for Ln there exists a minimal secant surface Σ such that ¯Σ ⊂ K and ¯Ln ∩ ¯Σ = h(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Indeed, let Σ0 be some minimal secant surface of Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then there exists the connected component ¯Σ0 of p−1(Σ0) such that it intersects the curve ¯Ln at the point ¯y0 situated on ¯Ln between α1, h(α1) and that bounds the h-compressible body QΣ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Without loss of generality let ¯y0 = h(α1) (otherwise the desired surface is constructed by removing the tubular neighborhood of the arc [¯y0, h(α1)] ⊂ ¯Ln from QΣ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Denote by k+, k− ⩾ 0 the maximal integers for whichf k( ¯Σ0)∩ ¯Σ0 ̸= ∅, f −k( ¯Σ0)∩ ¯Σ0 ̸= ∅, k ⩾ 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' If k+ = k− = 0 then ¯Σ0 is the desired surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Other- wise we show the way to decrease by 1 the number k+ > 0 (for k− the arguments are the same) using isotopy of the secant surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Notice that ¯Σ0 ∩ f k+(c2j−1) = ∅, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Without loss of generality let the intersection Γ = n� j=1 f k+(d2j−1) ∩ ¯Σ0 be transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Let γ be a curve from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Then γ bounds the unique disk Dγ ⊂ f k+(d2j−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' There are two possibilities: 1) b2j−1 /∈ Dγ, 2) b2j−1 ∈ Dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' In case 1) we say the curve γ to be innermost if it bounds the disk Dγ ⊂ f k+(d2j−1) such that int Dγ contains no curves of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Consider this innermost curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Due to Lemma 1 the surface ¯Σ0 \\ ¯y0 is non- compressible in R3 \\ (O ∪ ¯Ln) and, therefore, there exists the disk dγ ⊂ ( ¯Σ0 \\ ¯y0) bounded by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Denote by Bγ ⊂ (R3 \\(O∪ ¯Ln)) the 3-ball bounded by the 2-sphere Dγ ∪ dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+ page_content=' Consider two subcases: 1a) Bγ ⊂ QΣ0 and 1b) Bγ ⊂ (R3 \\ int QΣ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
293
+ page_content=' In case 1a) let N(Bγ) ⊂ QΣ0 be a tubular neighborhood of the ball Bγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
294
+ page_content=' Then the set QΣ \\ int N(Bγ) is h-compressible because the curve γ lies in its interior and its boundary intersects ¯Ln at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
295
+ page_content=' The projection of this boundary Quasi-energy function for Pixton diffeomorphisms 13 to S2 × S1 is, therefore, a secant surface of Ln of the same genus as Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
296
+ page_content=' For it the number of the connected components of the set Γ is less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
297
+ page_content=' One gets the same result in case 1b) for the set QΣ0 ∪ N(Bγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
298
+ page_content=' If we continue this process then we get the secant surface of Ln of the same genus as Σ0 and for which the set Γ contains no curves of type 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
299
+ page_content=' Denote the resulting surface again by Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
300
+ page_content=' Now the set Γ consists only of the curves γ bounding the disk Dγ ⊂ b2j−1 which contains the point b2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
301
+ page_content=' Since (b2j−1 ⊔ c2j−1) ⊂ (R3 \\QΣ0), the number of these curves on the disk d2j−1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
302
+ page_content=' Since the surface ¯Σ0\\¯y0 is non-compressible in R3\\(O∪ ¯Ln), all these curves are pairwise homotopic on ¯Σ0 \\ ¯y0 and, therefore, they lie in the annulus κ ⊂ ( ¯Σ0 \\ ¯y0) bounded by the pair of these curves γ1, γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
303
+ page_content=' Denote by ˜κ ⊂ d2j−1 the annulus bounded by the same curves on the disk d2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
304
+ page_content=' Let ˜Σ0 = ¯Σ0 \\ κ ∪ ˜κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
305
+ page_content=' Due to construction the surface ˜Σ0 is of the same genus as the surface ¯Σ0 and it bounds an h-compressible body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
306
+ page_content=' Having removed a tubular neighborhood of the annulus ˜κ from this body we get a h-compressible body whose boundary does not intersect the disk d2j−1 and whose projection to S2 ×S1 is the secant surface of the knot Ln of the same genus as Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
307
+ page_content=' If we continue this process then we get a secant surface of Ln of the same genus as Σ0 and for which the set Γ is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
308
+ page_content=' Denote this surface again by Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
309
+ page_content=' Without loss of generality let the intersections of the surface ¯Σ0 with the spheres f k(S2) be transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
310
+ page_content=' Denote by F the set of the connected components of the intersection f k+(K) ∩ ¯Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
311
+ page_content=' Now we show the way to reduce by 1 the number of the components in F using isotopy of the secant surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
312
+ page_content=' Denote by Q the set obtained by removal from the annulus f k+(K) of the tubular neighborhoods of the disks d2j−1 as well as the tubular neighborhoods of the curves A2j, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
313
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
314
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
315
+ page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
316
+ page_content=' Then Q is homeomorphic to the direct product of the 2-sphere with 2n + 1 deleted points and the segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
317
+ page_content=' Since Q ∩ ¯Σ0 = f k+(K)∩ ¯Σ0 and since ¯Σ0 \\ ¯y0 is non-compressible in R3 \\(O∪ ¯Ln), each connected component of F ∈ F is non-compressible in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
318
+ page_content=' Due to [16, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
319
+ page_content='2] there exists a surface ˜F ⊂ f k+−1(S2) diffeomorphic to F for which ∂F = ∂ ˜F and the surface F ∪ ˜F bounds in Q the body ∆ diffeomorphic to the direct product F ×[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
320
+ page_content=' Then we replace the part F of ¯Σ0 with ˜F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
321
+ page_content=' If we continue the process we get the desired secant surface Σ ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
322
+ page_content=' Notice (see, for instance, [4, Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
323
+ page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
324
+ page_content='1]) that the fundamental group π1(K\\ ¯Ln) has 2n generators γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
325
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
326
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
327
+ page_content=' , γ2n, each of which γi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
328
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
329
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
330
+ page_content=' , 2n} being the generator of the punctured disk di \\bi (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
331
+ page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
332
+ page_content=' Since b2j−1 ∈ int QΣ and c2j−1 ∩ QΣ = ∅, there exists the connected component of ˜d2j−1 of the intersection d2j−1 ∩ QΣ which contains the point b2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
333
+ page_content=' This component is the 2-disk bounded by the curve ˜γ2j−1 ⊂ ( ¯Σ \\ h(α1)) with holes and the curves γ2j−1, ˜γ2j−1 are homotopic on the punctured disk d2j−1 \\ b2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
334
+ page_content=' In the same way one finds the curves ˜γ2j ⊂ ( ¯Σ \\ h(α1)) homotopic to the curves γ2j on the punctured disk d2j \\ b2j (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
335
+ page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
336
+ page_content=' Due to Lemma 1 the surface ¯Σ \\ h(α1) is non-compressible in K \\ ¯Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
337
+ page_content=' Then the curves ˜γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
338
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
339
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
340
+ page_content=' , ˜γ2n are pairwise non-homotopic to the generators on the surface ¯Σ \\ h(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
341
+ page_content=' Therefore, the genus of the surface ¯Σ cannot be less than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
342
+ page_content=' 14 Timur Medvedev, Olga Pochinka Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
343
+ page_content=' 10 Generators of the group π1(K \\ ¯Ln) 5 Construction of a quasi-energy function for a Pixton diffeomorphism with the Hopf knot Ln Let f be a Pixton diffeomorphism constructed for a generalized Mazur knot Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
344
+ page_content=' Then its non-wandering set Ωf consists of four points: two sinks ω, S, a source N and a saddle σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
345
+ page_content=' Then W u σ \\σ consists of two separatrices ℓω, ℓS respective closures of which contain the sinks ω, S, the separatrice ℓω being tame while ℓS being wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
346
+ page_content=' Let ¯Σ be the surface of genus n bounding the handle-body QΣ of the same genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
347
+ page_content=' Now we construct for f a Morse-Lyapunov function with 6 + 2n critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
348
+ page_content=' Our construction of a quasi-energy function is analogous to the construction of an energy function in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
349
+ page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
350
+ page_content=' Choose an energy function ϕp : Up → R in the neighborhood of each fixed point p of f so that ϕp(p) = dim W u p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
351
+ page_content=' Let Bω, BS be the 3-balls which are the level sets of respective functions ϕω, ϕS and such that BS ⊂ int QΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
352
+ page_content=' Choose a tubular neighborhood Tσ of the arc W u σ \\ (Bω ∪ QΣ) so that the handle-body Bω ∪ QΣ ∪ Tσ of genus n is f-compressible and its intersection with W s σ is the 2-disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
353
+ page_content=' Denote by P + the smoothing of this body by addition of a small exterior collar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
354
+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
355
+ page_content=' Due to [7, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
356
+ page_content='3] there exists an energy function ϕ : P + \\ int QΣ whose value on ∂P + is 4/3, whose value on ¯Σ is 2/3 and which has exactly two critical points ω, σ of respective Morse indexes 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
357
+ page_content=' The disks d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
358
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
359
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
360
+ page_content=' , d2n−1 cut the handle-body QΣ making the 3-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
361
+ page_content=' Denote by BΣ the smoothing of this ball by removal of the interior collar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
362
+ page_content=' The results of the classic Morse theory (see, for example, [13]) allow to extend the function ϕ to the set QΣ \\ int BΣ in such way that it has n critical points of Morse index 1, one point lying on each disk d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
363
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
364
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
365
+ page_content=' , d2n−1, while the value of ϕ on ∂BΣ is 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
366
+ page_content=' Due to [7, Lemma Quasi-energy function for Pixton diffeomorphisms 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
367
+ page_content='2] the function ϕ can be extended to the ball BΣ by an energy function with the unique critical point S of Morse index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
368
+ page_content=' Since f(QΣ) ⊂ int BΣ, the constructed function decreases along the trajectories of the diffeomorphism f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
369
+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
370
+ page_content=' It follows from the definition of the knot Ln that P − = S3 \\ int P + is the handle-body of genus n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
371
+ page_content=' Moreover, the disks d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
372
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
373
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
374
+ page_content=' , d2n cut P − making the 3-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
375
+ page_content=' Denote by B− smoothing of this ball by removal of the interior collar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
376
+ page_content=' The results of the classic Morse theory (see, for example, [13]) allow extension of the function ϕ to the set P − \\ int B− in such way that it has n critical points of Morse index 2, one point lying on each disk d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
377
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
378
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
379
+ page_content=' , d2n, while the value of ϕ on ∂B− is 5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
380
+ page_content=' According to [7, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
381
+ page_content='2] the function ϕ can be extended to the ball B− by an energy function with unique critical point N of Morse index 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
382
+ page_content=' Since f(B−) ⊂ int P −, the constructed function decreases along the trajectories of the diffeomorphism f and, therefore, it is the desired quasi-energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
383
+ page_content=' Conflict of interest The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
384
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436
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1
+ Prepared for submission to JCAP
2
+ Stochastic gravitational wave
3
+ background from the collisions of dark
4
+ matter halos
5
+ Qiming Yana,b Xin Rena,b Yaqi Zhaoa,b Emmanuel N. Saridakisc,a,d
6
+ aDeep Space Exploration Laboratory/School of Physical Sciences, University of Science and
7
+ Technology of China, Hefei, Anhui 230026, China
8
+ bCAS Key Laboratory for Researches in Galaxies and Cosmology/Department of Astronomy,
9
+ School of Astronomy and Space Science, University of Science and Technology of China,
10
+ Hefei, Anhui 230026, China
11
+ cNational Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece
12
+ dDepartamento de Matem´aticas, Universidad Cat´olica del Norte, Avda.
13
+ Angamos 0610,
14
+ Casilla 1280 Antofagasta, Chile
15
+ E-mail: asadoubi233@mail.ustc.edu.cn, rx76@mail.ustc.edu.cn,
16
+ zxmyg86400@mail.ustc.edu.cn, msaridak@noa.gr
17
+ Abstract.
18
+ We investigate for the first time the effect of the dark matter (DM) halos colli-
19
+ sions, namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung,
20
+ on the stochastic gravitational wave background. We first calculate the gravitational wave
21
+ signal of a single collision event, assuming point masses and linear perturbation theory. Then
22
+ we proceed to the calculation of the energy spectrum of the collective effect of all dark matter
23
+ collisions in the Universe. Concerning the DM halo collision rate we show that it is given
24
+ by the product of the number density of DM halos, which is calculated by the extended
25
+ Press-Schechter (EPS) theory, with the collision rate of a single DM halo, which is given by
26
+ simulation results, with a function of the linear growth rate of matter density through cos-
27
+ mological evolution. Hence, integrating over all mass and distance ranges, we finally extract
28
+ the spectrum of the stochastic gravitational wave background created by DM halos collisions.
29
+ As we show, the resulting contribution to the stochastic gravitational wave background is
30
+ of the order of hc ≈ 10−30 in the pulsar timing array (PTA) band of f ≈ 10−9Hz, much
31
+ smaller than other GW sources, such as super-massive black-hole mergers. However, in very
32
+ low frequency band, it is larger. With current observational sensitivity it cannot be detected,
33
+ nevertheless it may be accessible by PTA in the future, where techniques of distinguishing
34
+ signal overlap should be used in order to isolate it and use it for cosmological studies.
35
+ arXiv:2301.02414v1 [astro-ph.CO] 6 Jan 2023
36
+
37
+ Contents
38
+ 1
39
+ Introduction
40
+ 1
41
+ 2
42
+ Gravitational waves emitted during a single collision
43
+ 2
44
+ 3
45
+ Effect on the stochastic gravitational wave background
46
+ 6
47
+ 3.1
48
+ Energy spectrum of a single GW event
49
+ 7
50
+ 3.2
51
+ Number density of GW sources
52
+ 8
53
+ 3.3
54
+ The energy spectrum of the stochastic gravitational wave background
55
+ 10
56
+ 4
57
+ Conclusions
58
+ 11
59
+ 1
60
+ Introduction
61
+ Recently, the gravitational wave (GW) detecting technology has been developing rapidly.
62
+ In 2015, the detection of binary black holes merger GW150914 by the LIGO experimental
63
+ cooperation signaled the first detection of gravitational waves [1], while in 2017, the joint
64
+ detection of GW170817 [2] and GRB170817A [3] opened the new era of multi-messenger
65
+ astronomy [4]. In general, with the increasing amount of detected gravitational wave events
66
+ [5] one has improved statistics that allows to track the history of the universe [6, 7] and
67
+ impose bounds on various cosmological parameters [8, 9], as well as constrain various theories
68
+ of gravity [10–14]. Moreover, for different frequencies and types of gravitational wave sources,
69
+ various detection means have been designed and implemented. Besides ground-based laser
70
+ interferometers such as LIGO, Virgo and KAGRA, which probe high frequency bands (10 −
71
+ 104 Hz), space-based laser interferometers such as LISA [15, 16] for intermediate frequency
72
+ gravitational waves (10−4 − 1 Hz), and the pulsar timing array (PTA) [17–20] for lower
73
+ frequency bands (10−9 − 10−6 Hz), are also raised. These observational avenues allow us to
74
+ acquire rich information from GWs of different types and sources, among which stochastic
75
+ gravitational wave background is attracting increasing interest.
76
+ Stochastic gravitational wave background (GWB) is a type of random background signal
77
+ that exists in an analogous way to the cosmic microwave background.
78
+ The contribution
79
+ of GWB can be roughly divided into cosmological sources and astrophysical sources [21].
80
+ Astrophysical originated GWB contains all types of unresolved GW emitting events, including
81
+ binary black hole mergers [22–28]. These signals can provide information about astrophysical
82
+ source populations and processes over the history of the universe [29–32]. On the other hand,
83
+ cosmological originated GWB mainly involves primordial gravitational perturbations during
84
+ the inflation epoch [33–35], or perturbations arising from primordial black holes fluctuations
85
+ [36–39]. GW signals typically remain unaffected during their propagation, and thus they
86
+ can provide valuable information about the very early stages of the universe. For instance,
87
+ different inflationary models can lead to different predictions for the GWB spectrum [40–52],
88
+ and thus GWB can be used as a probe of this primordial universe epoch. Since GWB can
89
+ provide us with important astrophysical and cosmological probes, it is crucial to understand
90
+ its composition and properties [53–63].
91
+ On the other hand, according to observations, dark matter (DM) constitutes a significant
92
+ fraction of the energy density of the universe [64–66]. Its microphysical nature and possible
93
+ – 1 –
94
+
95
+ interactions remain unknown [67–70], nevertheless we do know unambiguously that DM
96
+ interacts gravitationally [71, 72].
97
+ Current theory predicts that the main part of DM is
98
+ concentrated in dark halos, which coincide in position with galaxy or galaxy clusters [73].
99
+ These galaxies and galaxy clusters, and thus dark halos too, are typically accelerating and
100
+ merging through their mutual attraction [74–76]. Such processes can in principle release GW
101
+ signal through gravitational bremsstrahlung [77–87].
102
+ In this work, we are interested in investigating for the first time the possible GW signals
103
+ that could be emitted through bremsstrahlung during dark halo merger and collisions, and
104
+ their contribution to the stochastic GWB. In particular, we will first consider a single event of
105
+ two DM halos collision, and we will calculate the emitted GW signal. Then, we will calculate
106
+ the energy spectrum contribution to the stochastic GWB, taking the DM halo collision rate
107
+ into consideration. The structure of the article is as follows. In Section 2 we analyze the
108
+ GW emitted during the collision of two galaxies or two galaxy clusters. In Section 3 we
109
+ integrate over redshift and DM halos parameters to extract the contribution to stochastic
110
+ GWB. Finally, in Section 4 we conclude and discuss our results.
111
+ 2
112
+ Gravitational waves emitted during a single collision
113
+ In this section, we aim at estimating the gravitational waves emitted during a single collision
114
+ event. In particular, we calculate the GW radiated by the collision of two DM halos, which
115
+ corresponds to the collision of two galaxies or two galaxy clusters.
116
+ According to observations, such a collision typically has a huge duration, which in turn
117
+ implies that the energy radiated through GWs per unit time is not very large, and thus we
118
+ can safely use linear perturbation theory in the involved calculations. Specifically, we use
119
+ [88]
120
+ gµν = ηµν + hµν,
121
+ |hµν| ≪ 1,
122
+ (2.1)
123
+ ¯hij(t, x) = 2G
124
+ rc4
125
+ d2Iij (tr)
126
+ dt2
127
+ ,
128
+ tr = t − r
129
+ c,
130
+ (2.2)
131
+ where G is the gravitational constant, c is the speed of light, and r is the distance from us
132
+ to the center of mass of the two galaxies or galaxy clusters. Moreover, Iij is the quadruple
133
+ moment
134
+ Iij(t) =
135
+
136
+ yiyjT 00(t, y)d3y =
137
+
138
+ yiyjρ(t, y)d3y,
139
+ (2.3)
140
+ where T µν is energy-momentum tensor, ρ is energy density, and yi is the spatial coordinate.
141
+ Since the goal of our calculation is to acquire an estimation of the order of the magnitude of
142
+ the resulting signal, we can consider these two DM halos as mass points, with mass Ma and
143
+ position y(a)(t) at time t. Hence, the density ρ can be written as
144
+ ρ(t, y) =
145
+
146
+ a
147
+ Maδ3(y − y(a)(t)),
148
+ (2.4)
149
+ while the quadruple moment Iij(t) becomes
150
+ Iij(t) =
151
+
152
+ yiyjρ(t, y)d3y =
153
+
154
+ a
155
+ Mayi
156
+ (a)(t)yj
157
+ (a)(t).
158
+ (2.5)
159
+ – 2 –
160
+
161
+ Finally, since the relative speed of two galaxies or galaxy clusters is much smaller than the
162
+ speed of light, we can use Newtonian mechanics to handle their dynamics.
163
+ For simplicity we write the equations in the center-of-mass frame of these two mass
164
+ points. By definition, we have
165
+ MArA + MBrB = 0,
166
+ (2.6)
167
+ where MA, MB are the masses of the mass points A and B, with rA, rB their position vectors.
168
+ From Newtonian mechanics we have
169
+ ¨rA = −
170
+ GMB
171
+ |rA − rB|2
172
+ rA
173
+ |rA|,
174
+ (2.7)
175
+ which using (2.6) gives
176
+ ¨rA = −µB
177
+ rA
178
+ |rA|3 ,
179
+ (2.8)
180
+ where we have defined µB ≡
181
+ GMB
182
+ (1+ MA
183
+ MB )2 . Additionally, we assume that the two points are ini-
184
+ tially at infinite distance, their relative speed is v∞ = vA∞ +vB∞, and the impact parameter
185
+ is b = bA + bB. From Newtonian mechanics we know that the trajectory of each point is a
186
+ hyperbola and the two points are moving in a plane (we set this plane as z = 0 plane, and
187
+ thus rA = (xA, yA, 0)), while the total energy of the system is positive. Additionally, the
188
+ mass center of these two DM halos will not follow a hyperbolic trajectory at all times, in
189
+ order to acquire a collision. In Fig. 1 we depict an illustrative representation of the initial
190
+ conditions of the collision.
191
+ Figure 1.
192
+ An illustrative representation of the initial conditions of the collision. The two galaxies
193
+ or galaxy clusters are considered as points with masses MA and MB, where bA and bB are the impact
194
+ parameters.
195
+ Let us start with the beginning of the collision, when the two DM halos start moving
196
+ towards each other. For point A we have
197
+ (xA + aAeA)2 − (yA)2 = a2
198
+ A,
199
+ (2.9)
200
+ – 3 –
201
+
202
+ Y
203
+ MB
204
+ UB8
205
+ bB
206
+ O center of mass
207
+ X
208
+ L= 8
209
+ b A
210
+ VA8
211
+ MAwhere
212
+ aA =
213
+ µB
214
+ (vA∞)2 ,
215
+ (2.10)
216
+ eA =
217
+
218
+ 1 + vA4∞b2
219
+ A
220
+ (µB)2 ,
221
+ (2.11)
222
+ a = aA + aB.
223
+ (2.12)
224
+ We proceed by defining λA through
225
+ eA sinh(λA) − λA = vA∞t
226
+ aA
227
+ ,
228
+ (2.13)
229
+ hence
230
+ r1
231
+ A = xA = aA [eA − cosh(λA)] ,
232
+ (2.14)
233
+ r2
234
+ A = yA = aA
235
+ ��
236
+ e2
237
+ A − 1 sinh(λA)
238
+
239
+ .
240
+ (2.15)
241
+ Note that t = 0 corresponds to the time when the two mass points have the shortest distance.
242
+ In order to obtain the GW amplitude hij, we proceed to the calculation of the quadrupole
243
+ moment Iij(t) and its second time derivative. We have
244
+ Iij = MAri
245
+ Arj
246
+ A + MBri
247
+ Brj
248
+ B,
249
+ (2.16)
250
+ d2Iij
251
+ dt2
252
+ = MA(¨ri
253
+ Arj
254
+ A + ri
255
+ A¨rj
256
+ A + 2 ˙ri
257
+ A ˙rj
258
+ A) + MB(¨ri
259
+ Brj
260
+ B + ri
261
+ B¨rj
262
+ B + 2 ˙ri
263
+ B ˙rj
264
+ B).
265
+ (2.17)
266
+ From (2.14), (2.15) we find
267
+ ˙xA = −
268
+ aA sinh(λA)
269
+
270
+ a3
271
+ A
272
+ µB [eA cosh(λA) − 1]
273
+ ,
274
+ (2.18)
275
+ ¨xA =
276
+ µB(cosh(λA) − e)
277
+ a2
278
+ A[eA cosh(λA) − 1]3 ,
279
+ (2.19)
280
+ ˙yA =
281
+ aA
282
+
283
+ e2
284
+ A − 1 cosh(λA)
285
+
286
+ a3
287
+ A
288
+ µB [eA cosh(λA) − 1]
289
+ ,
290
+ (2.20)
291
+ ¨yA = −
292
+
293
+ e2
294
+ A − 1µB sinh(λA)
295
+ a2
296
+ A[eA cosh(λA) − 1]3 ,
297
+ (2.21)
298
+ and thus inserting into (2.17) we extract all the second time derivatives of the quadrupole
299
+ – 4 –
300
+
301
+ moment Iij(t), namely
302
+ d2I11
303
+ dt2
304
+ = µBMA{7eA cosh(λA) + eA[cosh(3λA) − 4eA] − 4 cosh(2λA)}
305
+ 2aA[eA cosh(λA) − 1]3
306
+ +µAMB{7eB cosh(λB) + eB[cosh(3λB) − 4eB] − 4 cosh(2λB)}
307
+ 2aB[eB cosh(λB) − 1]3
308
+ ,
309
+ (2.22)
310
+ d2I12
311
+ dt2
312
+ = −
313
+
314
+ e2
315
+ A − 1µBMA sinh(λA){eA[cosh(2λA) + 3] − 4 cosh(λA)}
316
+ aA[eA cosh(λA) − 1]3
317
+
318
+
319
+ e2
320
+ B − 1µBMA sinh(λB){eB[cosh(2λB) + 3] − 4 cosh(λB)}
321
+ aB[eB cosh(λB) − 1]3
322
+ ,
323
+ (2.23)
324
+ d2I22
325
+ dt2
326
+ =
327
+
328
+ e2
329
+ A − 1
330
+
331
+ µBMA[3eA cosh(λA) + eA cosh(3λA) − 4 cosh(2λA)]
332
+ 2aA[eA cosh(λA) − 1]3
333
+ +
334
+
335
+ e2
336
+ B − 1
337
+
338
+ µAMB[3eB cosh(λB) + eB cosh(3λB) − 4 cosh(2λB)]
339
+ 2aB[eB cosh(λB) − 1]3
340
+ .
341
+ (2.24)
342
+ Figure 2.
343
+ The dimensionless components of the gravitational wave signal arising from a single
344
+ event of the collision of two DM halos, i.e., the collision of two galaxies or clusters of galaxies. The
345
+ left panel shows the ¯h11 component, the middle panel the ¯h12 component and the right panel the ¯h22
346
+ component. The time t = 0 corresponds to the shortest distance between the two DM halos, that is
347
+ the moment in which ¯h11 and ¯h22 reach their peaks and ¯h12 exhibits the largest variation. We have
348
+ imposed the typical values MA = MB = 109M⊙, vA∞ = vB∞ = 300km/s, bA = bB = 104ly, and we
349
+ have assumed that the distance from Earth is ∼ 109ly. Time t is measured in seconds.
350
+ We can now use (2.2) in order to obtain the GW signal in the time domain. As typical
351
+ values we set MA = MB = 109M⊙, namely the order of mass of a (dwarf) galaxy, where M⊙
352
+ is the mass of the Sun, and we use vA∞ = vB∞ = 300km/s, bA = bB = 104ly, which are
353
+ the typical values for galaxy collisions. Moreover, we assume that the collision happens at a
354
+ distance of 109ly from the Earth, which is roughly the distance of the source of GW150914.
355
+ Hence, we can estimate the magnitude of the GW signal. In Fig. 2 we present the obtained
356
+ dimensionless GW signal ¯hij, as a function of time t. Since t = 0 corresponds to the time of
357
+ shortest distance, the change rate of ¯hij is fastest at this time, as expected. As we observe,
358
+ the variation of ¯hij is of the order of 5 × 10−22 during the collision. However, this variation
359
+ – 5 –
360
+
361
+
362
+ -1.0×1017 -5.0×1016
363
+ 5.0 ×1016
364
+ 1.0×1017
365
+ -1. × 10-22
366
+ -3. × 10-22
367
+ -5. × 10-22h12
368
+ 2.×10
369
+
370
+ -1.0×1017 -5.0×1016
371
+ 5.0 × 1016
372
+ 6 1.0×1017
373
+ -1. × 10-22
374
+ -2. × 10-22h22
375
+ 1.2538 × 10-18
376
+ 1.2534 × 10-18
377
+ t
378
+ -1.0×1017 -5.0×1016
379
+ 5.0×1016
380
+ 1.0 ×1017corresponds to a large time scale (about 1015s), which implies that a single signal of this kind
381
+ of GW is extremely hard to be detected. Additionally, we can see that the evolution of ¯h12
382
+ is faster than that of ¯h11, ¯h22, which implies that ¯h12 will be dominant in relatively higher
383
+ frequency than that of ¯h11, ¯h22.
384
+ We proceed by taking the Fourier transformation of ¯hij, in order to investigate its
385
+ spectrum. In particular, we use
386
+ ˜¯hij(ω) =
387
+ � t=+∞
388
+ t=−∞
389
+ dt eiωt ¯hij(t) ,
390
+ (2.25)
391
+ where ω = 2πf, with f the frequency. In Fig. 3 we present the dependence of ˜¯hij(f) on f.
392
+ As we observe, all ˜¯hij(f) obey the power law in a very good approximation for a very wide
393
+ frequency range. Besides, as ˜¯h11, ˜¯h22 ∝ 1/f2, while ˜¯h12 ∝ 1/f, we can infer that ˜¯h11, ˜¯h22
394
+ will be dominant in the low frequency band while ˜¯h12 will be dominant in relatively high
395
+ frequencies.
396
+ Figure 3. The spectrum of the gravitational waves as a function of the frequency. The upper left panel
397
+ shows the ˜¯h11(f) component, the upper right panel the ˜¯h12(f) component and the right panel the ˜¯h22(f)
398
+ component. The blue dots represent the exact results at the time of shortest distance, while the red solid
399
+ curves are power-law fits, specifically ˜¯h11(f) ≈ 1.7 × 10−53(Hz/f)2, ˜¯h12(f) ≈ 1.2 × 10−38(Hz/f)2,
400
+ ˜¯h22(f) ≈ 2.0 × 10−53(Hz/f)2.
401
+ 3
402
+ Effect on the stochastic gravitational wave background
403
+ In this section, we calculate the contribution of the DM halos collisions to the stochastic
404
+ gravitational wave background. Specifically, we integrate the gravitational wave spectrum of
405
+ – 6 –
406
+
407
+ 10-23
408
+ 10~24
409
+ 10~25
410
+ 10~26
411
+ 10~27
412
+ 10-28
413
+ 5.×10-15 10-14
414
+ 5. × 10-1.4 10-13
415
+ 5.× 10-13h11(f)f/H
416
+ 21. × 10-22
417
+ 5. × 10-23
418
+ 1.×10-23
419
+ 5.× 10-24
420
+ 1. × 10-24
421
+ 5. × 10-25
422
+ 5. × 10-15 10-14
423
+ 5. × 10-1.4 10-13
424
+ 5.×10-13h12(f)f/H
425
+ 210~23
426
+ 10~24
427
+ 10~25
428
+ 10-26
429
+ 10~27
430
+ 10-28
431
+ 5.× 10-15 10-14
432
+ 5. × 10-1.4 10-13
433
+ 5.× 10-13h22(f)f/H
434
+ 2a single collision event over the number density of GW sources.
435
+ In principle, in order to compare a theoretical model with observations, one uses both
436
+ the fractional energy density spectrum Ωgw(f), as well as the characteristic strain amplitude
437
+ hc(f) [56]. They are related to the energy spectrum of GWB through the expression
438
+ π
439
+ 4 f2h2
440
+ c(f) = ρcΩgw(f) = dρgw(f)
441
+ d ln f ,
442
+ (3.1)
443
+ where f is the frequency of GW detected on Earth, and ρc ≡ 3c2H2
444
+ 0/8πG is the critical
445
+ energy density. The energy spectrum of the stochastic GWB, dρgw
446
+ d ln f , can be written as
447
+ dρgw(f)
448
+ d ln f
449
+ =
450
+ � ∞
451
+ 0
452
+ dz
453
+ 1
454
+ 1 + z
455
+
456
+ dξ dn
457
+ dzdξ
458
+ dE(ξ)gw
459
+ d ln fr
460
+ ����
461
+ fr=f(1+z)
462
+ ,
463
+ (3.2)
464
+ with z the redshift at the GW emission. Additionally, dE(ξ)gw
465
+ d ln(fr) is the energy spectrum of a
466
+ single GW event, which is calculated through the analysis of the previous subsection, and fr
467
+ is the GW frequency in the rest frame of GW sources, and thus fr = (1 + z)f.
468
+ We mention that we denote the parameters related to the number density of GW sources
469
+ collectively by ξ = {ξ1, . . . , ξm}, and therefore
470
+ dn
471
+ dξ1...dξmdzdξ1 . . . dξmdz ≡
472
+ dn
473
+ dξdzdξdz is the
474
+ number density of sources in the redshift interval [z, z + dz] and with source parameters
475
+ in the interval [ξ, ξ + dξ]. Hence, in the simple single event of two DM halos collision of
476
+ the previous section we have ξ = {M, x, v∞, b}, where M = MA + MB, x = MA/MB,
477
+ v∞ = vA∞ + vB∞ and b = bA + bB.
478
+ Let us now calculate the full distribution function
479
+ dn
480
+ dzdξ =
481
+ dn
482
+ dzdMdxdv∞db. As we have
483
+ checked numerically, the variance of b, v∞ has a minor effect on the final result, not affecting
484
+ the order of magnitude. Hence, it is a good approximation to omit the change of b, v∞, and
485
+ consider that ξ = {M, x}. Hence, we have
486
+ dρgw(f)
487
+ d ln f
488
+ =
489
+ � ∞
490
+ 0
491
+ dz
492
+ 1
493
+ 1 + z
494
+
495
+ dξ dn
496
+ dzdξ
497
+ dE(ξ)gw
498
+ d ln fr
499
+ ����
500
+ fr=f(1+z)
501
+
502
+ � 10
503
+ 0
504
+ dz
505
+ 1
506
+ 1 + z
507
+ � Mmax=1015M⊙
508
+ Mmin=109M⊙
509
+ dM
510
+ � xmax=105
511
+ xmin=1
512
+ dx
513
+ dn
514
+ dzdMdx
515
+ dE(ξ)gw
516
+ d ln, fr
517
+ �����
518
+ fr=f(1+z)
519
+ , (3.3)
520
+ where the varying range of M and x is taken from [89].
521
+ In the following subsections we will separately calculate the energy spectrum of a single
522
+ GW event dE(ξ)gw
523
+ d ln fr , and the number density of GW sources
524
+ dn
525
+ dzdMdx.
526
+ 3.1
527
+ Energy spectrum of a single GW event
528
+ The energy density of a single GW event can be calculated from the (traceless) second time
529
+ derivative of the quadrupole moment, namely [90]
530
+ dE(ξ)gw
531
+ d ln fr
532
+ ≈ fr
533
+ 2G
534
+ 5c5 (2πfr)2( ¨˜Qij(M, x; fr))( ¨˜Qij(M, x; fr)),
535
+ (3.4)
536
+ – 7 –
537
+
538
+ where Qij is the traceless quadrupole moment and ¨˜Qij is the Fourier transformation of the
539
+ second time derivative of Qij, which is related to Iij via
540
+ Q11 = 2
541
+ 3I11 − 1
542
+ 3I22,
543
+ (3.5)
544
+ Q22 = −1
545
+ 3I11 + 2
546
+ 3I22,
547
+ (3.6)
548
+ Q33 = −1
549
+ 3I11 − 1
550
+ 3I22,
551
+ (3.7)
552
+ Q21 = Q12 = I12,
553
+ (3.8)
554
+ while all other Qij are equal to zero. Now, from Newtonian mechanics Iij can be written as
555
+ ¨˜Iij(M, x; fr) = 4
556
+
557
+ x
558
+ (1 + x)3 +
559
+ 1/x
560
+ (1 + 1/x)3
561
+ � �
562
+ M
563
+ 2 × 1012M⊙
564
+ �2 ¨˜IG
565
+ ij (fr),
566
+ (3.9)
567
+ where x is the mass ratio of the two masses, and IG
568
+ ij is defined as Iij(M = 2×1012M⊙, x = 1).
569
+ Therefore, from the calculation of Section 2, we can extract the values of ¨˜IG
570
+ ij (fr) as
571
+ ¨˜IG
572
+ 11(fr) = 2.86 × 1021
573
+ �Hz
574
+ fr
575
+ �2
576
+ kg m2s−1,
577
+ (3.10)
578
+ ¨˜IG
579
+ 22(fr) = 5.72 × 1020
580
+ �Hz
581
+ fr
582
+ �2
583
+ kg m2s−1,
584
+ (3.11)
585
+ ¨˜IG
586
+ 12(fr) = ¨˜IG
587
+ 21(fr) = 1.29 × 1037
588
+ �Hz
589
+ fr
590
+
591
+ kg m2s−1.
592
+ (3.12)
593
+ Hence, inserting the above into (3.4) gives us the energy density of a single GW event.
594
+ 3.2
595
+ Number density of GW sources
596
+ Let us now calculate the number density of GW sources (per redshift, total mass and mass
597
+ ratio interval)
598
+ dn
599
+ dzdMdx. This number density is equal to the DM matter halos mergers rate,
600
+ which can be calculated by combining the extended Press-Schechter (EPS) theory [91] and
601
+ numerical simulations [89]:
602
+ dn
603
+ dzdMdx = nhalo(M, z)dω
604
+ dz
605
+
606
+ 1
607
+ nhalo
608
+ dnmerger
609
+ dωdx
610
+
611
+ ,
612
+ (3.13)
613
+ where nhalo(M, z) is the number density of dark matter halos (per redshift per mass inter-
614
+ val in the co-moving space), ω = ω(z) is a redshift-dependent function given below, and
615
+ (
616
+ 1
617
+ nhalo
618
+ dnmerger
619
+ dωdx
620
+ ) is the merger rate (at some ω) for a pair of DM halos with fixed total mass
621
+ M and mass ratio x. In the following we handle these terms separately.
622
+ We start with the definition of ω(z) [91]
623
+ ω(z) = 1.69
624
+ D(z),
625
+ (3.14)
626
+ where D(z) is the linear growth rate of matter density. D(z) can be written as
627
+ D(z) =
628
+ 1
629
+ g(z = 0)
630
+ � g(z)
631
+ 1 + z
632
+
633
+ ,
634
+ (3.15)
635
+ – 8 –
636
+
637
+ where a good approximation of g(z) is
638
+ g(z) ≈ 5
639
+ 2Ωm(z)
640
+
641
+ Ω4/7
642
+ m (z) − ΩΛ(z) + [1 + Ωm(z)/2] [1 + ΩΛ(z)/70]
643
+ �−1
644
+ ,
645
+ (3.16)
646
+ with ΩΛ(z), Ωm(z) the density parameters of dark energy and matter sectors given by
647
+ ΩΛ(z) = ΩΛ,0
648
+ E2(z);
649
+ Ωm(z) = Ωm,0(1 + z)3
650
+ E2(z)
651
+ ,
652
+ (3.17)
653
+ where the normalized Hubble function E(z) ≡ H(z)/H0 reads as
654
+ E(z) ≈
655
+
656
+ ΩΛ,0 + Ωm,0(1 + z)3�1/2 ,
657
+ (3.18)
658
+ with the value of the Hubble function at present time given as [64]
659
+ H0 ≈ 67.3 km s−1Mpc−1,
660
+ (3.19)
661
+ and with the values ΩΛ,0, Ωm,0 at present time taken as [64]
662
+ ΩΛ,0 ≈ 0.685,
663
+ (3.20)
664
+ Ωm,0 ≈ 0.317.
665
+ (3.21)
666
+ Note that in the above we consider that the underlying cosmology is ΛCDM concordance
667
+ scenario, i.e., the dark energy sector is the cosmological constant.
668
+ We continue by using the EPS theory in order to write the formula of the number
669
+ density of DM halos nhalo. We consider that the halos merge when the redshift is between z
670
+ and z +dz , and that the emitted GW signals are detected at Earth at present. In co-moving
671
+ space those halos are in the volume ∆V = 4πr2(z)d(r(z)). Now, the EPS theory provides
672
+ the number density of DM halos nEPS(M, z) at some redshift z and mass M. Therefore, we
673
+ have
674
+ nhalo = 4πr2(z)dr(z)
675
+ dz nEPS(M, z),
676
+ (3.22)
677
+ where the radius in the co-moving space r(z) is [91]
678
+ r(z) =
679
+ c
680
+ H0
681
+ � z
682
+ 0
683
+ dz
684
+
685
+ 1
686
+ E(z
687
+ ′),
688
+ (3.23)
689
+ while the formula of nEPS(M, z) is [91]
690
+ nEPS(M, z) =
691
+
692
+ 2
693
+ π
694
+ ¯ρ
695
+ M2
696
+ δc
697
+ σ exp
698
+
699
+ − δ2
700
+ c
701
+ 2σ2
702
+ � ����
703
+ d ln σ
704
+ d ln M
705
+ ���� .
706
+ (3.24)
707
+ In the above expression ¯ρ = ρcΩm,0 is the mean density of the matter component, δc = ω =
708
+ 1.69
709
+ D(z), while σ(M) is the variance of the matter density perturbation which can be estimated
710
+ as [91]
711
+ σ(M) ≈ σ8
712
+ � R
713
+ r8
714
+ �−β
715
+ ,
716
+ (3.25)
717
+ – 9 –
718
+
719
+ with M = 4π
720
+ 3 ¯ρR3 , σ8 ≈ 1, β ≈ 0.6 + 0.8(Ωm,0h), h = 0.673 , and r8 = 8 Mpc h−1, leading to
721
+ ����
722
+ d ln σ
723
+ d ln M
724
+ ���� = β
725
+ 3 .
726
+ (3.26)
727
+ Finally, the last term of (3.13), namely (
728
+ 1
729
+ nhalo
730
+ dnmerger
731
+ dωdx
732
+ ) (dimensionless since both ω, x
733
+ are dimensionless), can be found in [89] and it is given by
734
+
735
+ 1
736
+ nhalo
737
+ dnmerger
738
+ dωdx
739
+
740
+ = A
741
+
742
+ M
743
+ 1012M⊙
744
+ �α
745
+ xb exp [(˜x/x)γ] ,
746
+ (3.27)
747
+ where the best-fit parameters from simulations are A = 0.065, α = 0.15, b = −0.3, ˜x = 2.5,
748
+ γ = 0.5 [89].
749
+ In summary, inserting (3.14), (3.22) and (3.27) into (3.13), provides the value of the
750
+ number density of GW sources
751
+ dn
752
+ dzdMdx.
753
+ 3.3
754
+ The energy spectrum of the stochastic gravitational wave background
755
+ We have now all the ingredients needed in order to calculate the energy spectrum of the
756
+ stochastic gravitational wave background. This is given by (3.3), in which the energy spec-
757
+ trum of a single GW event dE(ξ)gw
758
+ d ln fr
759
+ was calculated in subsection 3.1, while the number density
760
+ of GW sources
761
+ dn
762
+ dzdMdx was calculated in subsection 3.2. Assembling everything, we finally
763
+ obtain the stochastic gravitational wave background resulting from DM halos collisions in
764
+ the universe, which is calculated numerically and it is shown in Fig. 4. Additionally, for com-
765
+ parison we also depict the corresponding result of of the contribution of the super-massive
766
+ black-hole mergers (SMBHM), which follows hc(f) ∝ f−2/3, as well as the current observa-
767
+ tional sensitivity [92].
768
+ 10
769
+ 16
770
+ 10
771
+ 14
772
+ 10
773
+ 12
774
+ 10
775
+ 10
776
+ 10
777
+ 8
778
+ 10
779
+ 6
780
+ 10
781
+ 4
782
+ 10
783
+ 2
784
+ Frequency f /Hz
785
+ 10
786
+ 34
787
+ 10
788
+ 30
789
+ 10
790
+ 26
791
+ 10
792
+ 22
793
+ 10
794
+ 18
795
+ 10
796
+ 14
797
+ 10
798
+ 10
799
+ Characteristic Strain hc
800
+ DM halos collisions
801
+ SMBHM
802
+ current observation sensitivity
803
+ Figure 4. The characteristic strain hc(f) as a function of the frequency of the stochastic gravita-
804
+ tional wave background created by DM halos, namely galaxies and galaxy clusters, collisions (blue-solid
805
+ curve). For comparison, with the green-dashed curve we depict the corresponding result of of the con-
806
+ tribution of the super-massive black-hole mergers (SMBHM), which follows hc(f) ∝ f −2/3. Finally,
807
+ with the red-dotted curve we present the current observational sensitivity [92].
808
+ – 10 –
809
+
810
+ As we can see, the contribution of GW radiated from the collisions of DM halos, namely
811
+ galaxies and galaxy clusters, is quite small comparing to other sources. In the pulsar timing
812
+ array (PTA) band, where f ≈ 10−9Hz, and where the current observational limit is hc ≈
813
+ 10−15 [93], we obtain an effect of the order of hc ≈ 10−30. Nevertheless, in very low frequency
814
+ band hc will be larger. In general, with current observational sensitivity the effect of the DM
815
+ halos collisions on the stochastic gravitational wave background cannot be detected, however
816
+ it will be accessible in the future, in which case one could use the advanced techniques of
817
+ distinguishing signal overlap in order to isolate it from other sources [94–97].
818
+ Note that
819
+ one could try to extend the analysis, by considering, instead of point masses, a group of
820
+ mass points with Navarro, Frenk & White (NFW) density profile [91] to simulate DM halo
821
+ collisions, nevertheless the results are expected to be at the same order of magnitude.
822
+ 4
823
+ Conclusions
824
+ In this work we investigated for the first time the effect of the dark matter halos collisions,
825
+ namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung, on
826
+ the stochastic gravitational wave background.
827
+ In order to achieve this goal, we first calculated the gravitational wave signal of a single
828
+ DM halo collision event. As an estimation of the order of magnitude, we handled the two
829
+ DM halos as mass points. Furthermore, since the strength of such GW signals is weak, we
830
+ adopted linear perturbation theory of General Relativity, namely we extracted the GW signal
831
+ using the second time derivative of the quadruple moment. Additionally, since the velocity
832
+ of DM halos is small, we applied non-relativistic Newtonian Mechanics. Hence, we extracted
833
+ the GW signal through bremsstrahlung from a single DM halo collision. As we showed, ¯hij is
834
+ of the order of 10−22, and it becomes maximum at the time of shortest distance as expected.
835
+ However, since such an event typically corresponds to durations of the order of 1015s, we
836
+ deduce that a single signal of this kind of GW is extremely hard to be detected.
837
+ As a next step we proceeded to the calculation of the energy spectrum of the collective
838
+ effect of all DM halos collisions in the Universe. This can arise by the energy spectrum of
839
+ a GW signal radiated by a single collision, multiplied by the DM halo collision rate, and
840
+ integrating over the whole Universe.
841
+ Firstly, knowing the signal of a single collision we
842
+ calculated its energy spectrum. Secondly, concerning the DM halo collision rate we showed
843
+ that it is given by the product of the number density of DM halos, which is calculated by the
844
+ EPS theory, with the collision rate of a single DM halo, which is given by simulation results,
845
+ with a function of the linear growth rate of matter density through cosmological evolution.
846
+ Hence, integrating over all mass and distance ranges, we finally extracted the spectrum of
847
+ the stochastic gravitational wave background created by DM halos collisions.
848
+ As we saw, the resulting contribution to the stochastic gravitational wave background
849
+ is of the order of hc ≈ 10−30 in the pulsar timing array (PTA) band of f ≈ 10−9Hz, much
850
+ smaller than other GW sources, such as super-massive black-hole mergers. However, in very
851
+ low frequency band, hc is larger. With current observational sensitivity it cannot be detected,
852
+ nevertheless it may be accessible by PTA in the future, where techniques of distinguishing
853
+ signal overlap should be used in order to isolate it and use it for cosmological studies.
854
+ In summary, with the current and future significant advance in gravitational-wave as-
855
+ tronomy, and in particular with the tremendous improvement on the sensitivity bounds that
856
+ Collaborations like Laser Interferometer Space Antenna (LISA), Einstein Telescope (ET),
857
+ Cosmic Explorer (CE), etc will bring, it is both interesting and necessary to investigate all
858
+ – 11 –
859
+
860
+ possibles contributions to the stochastic gravitational wave background. And the gravita-
861
+ tional bremsstrahlung during galaxies and galaxy clusters collisions is one of them.
862
+ Acknowledgments
863
+ We are grateful to Yifu Cai, Jiewen Chen, Zihan Zhou and Jiarui Li for helpful discussions.
864
+ This work is supported in part by the National Key R&D Program of China (2021YFC2203100),
865
+ by the NSFC (11961131007, 11653002), by the Fundamental Research Funds for Central Uni-
866
+ versities, by the CSC Innovation Talent Funds, by the CAS project for young scientists in
867
+ basic research (YSBR-006), by the USTC Fellowship for International Cooperation, and by
868
+ the USTC Research Funds of the Double First-Class Initiative. ENS acknowledges partici-
869
+ pation in the COST Association Action CA18108 “Quantum Gravity Phenomenology in the
870
+ Multimessenger Approach (QG-MM)”. All numerics were operated on the computer clusters
871
+ LINDA & JUDY in the particle cosmology group at USTC.
872
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873
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914
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915
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916
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926
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928
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929
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1
+ UserSimCRS: A User Simulation Toolkit for Evaluating
2
+ Conversational Recommender Systems
3
+ Jafar Afzali
4
+ University of Stavanger
5
+ j.afzali@stud.uis.no
6
+ Aleksander Mark Drzewiecki
7
+ University of Stavanger
8
+ am.drzewiecki@stud.uis.no
9
+ Krisztian Balog
10
+ University of Stavanger
11
+ Stavanger, Norway
12
+ krisztian.balog@uis.no
13
+ Shuo Zhang
14
+ Bloomberg
15
+ London, United Kingdom
16
+ szhang611@bloomberg.net
17
+ ABSTRACT
18
+ We present an extensible user simulation toolkit to facilitate auto-
19
+ matic evaluation of conversational recommender systems. It builds
20
+ on an established agenda-based approach and extends it with sev-
21
+ eral novel elements, including user satisfaction prediction, persona
22
+ and context modeling, and conditional natural language generation.
23
+ We showcase the toolkit with a pre-existing movie recommender
24
+ system and demonstrate its ability to simulate dialogues that mimic
25
+ real conversations, while requiring only a handful of manually
26
+ annotated dialogues as training data.
27
+ CCS CONCEPTS
28
+ • Information systems → Recommender systems.
29
+ KEYWORDS
30
+ Conversational recommender systems; user simulation
31
+ ACM Reference Format:
32
+ Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, and Shuo Zhang.
33
+ 2023. UserSimCRS: A User Simulation Toolkit for Evaluating Conversational
34
+ Recommender Systems. In Proceedings of the Sixteenth ACM International
35
+ Conference on Web Search and Data Mining (WSDM ’23), February 27-March
36
+ 3, 2023, Singapore, Singapore. ACM, New York, NY, USA, 4 pages. https:
37
+ //doi.org/10.1145/3539597.3573029
38
+ 1
39
+ INTRODUCTION
40
+ Conversational recommender systems (CRSs) elicit user preferences
41
+ via multi-turn real-time interactions using natural language [6, 9].
42
+ There has been a great deal of progress in recent years on various
43
+ aspects, including question-based user preference elicitation [5,
44
+ 10, 29], multi-turn conversational recommendation strategies [12],
45
+ and natural language understanding and generation [13, 27]. A
46
+ major challenges that remains, however, is evaluation [6]. Due to
47
+ the dynamic nature of interactions, measuring performance on
48
+ Permission to make digital or hard copies of all or part of this work for personal or
49
+ classroom use is granted without fee provided that copies are not made or distributed
50
+ for profit or commercial advantage and that copies bear this notice and the full citation
51
+ on the first page. Copyrights for components of this work owned by others than ACM
52
+ must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
53
+ to post on servers or to redistribute to lists, requires prior specific permission and/or a
54
+ fee. Request permissions from permissions@acm.org.
55
+ WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
56
+ © 2023 Association for Computing Machinery.
57
+ ACM ISBN 978-1-4503-9407-9/23/02...$15.00
58
+ https://doi.org/10.1145/3539597.3573029
59
+ the conversation level is not possible using offline test collections.
60
+ While online evaluation with users of a live service is an option,
61
+ it is expensive and does not scale. A promising solution to these
62
+ issues is user simulation [1, 6]. The idea there is to build a simulated
63
+ user that mimics how a real human would respond in a given
64
+ dialogue situation [19, 25]. Simulation thus offers a repeatable and
65
+ reproducible means of evaluation. (We note that it is not meant to
66
+ replace, but rather to complement human evaluation.)
67
+ There is indeed an emerging focus in recent research on using
68
+ simulation for evaluating conversational information access sys-
69
+ tems in general [1, 3, 17, 20, 23] and conversational recommenders
70
+ in particular [25, 26]. The current work aims to contribute to the de-
71
+ velopment of novel CRSs by recognizing the need for better tooling
72
+ for user simulation. In particular, we provide an extensible open-
73
+ source toolkit that is designed specifically for evaluation. Our work
74
+ is unique in at least three regards. First, it focuses on the task of
75
+ conversational recommendation and hence place a strong emphasis
76
+ on both the recommendation-specific conversation flow and on the
77
+ human-likeness of the generated user utterances. Second, it centers
78
+ around evaluation as opposed to other uses of simulation (most
79
+ commonly, synthetic data generation for reinforcement learning).
80
+ Third, it is designed to work with existing CRSs, without needing ac-
81
+ cess to source code or knowledge of their inner workings. It merely
82
+ requires collecting and annotating a small sample of dialogues.
83
+ Building on an established agenda-based simulator [25], we intro-
84
+ duce novel components, motivated by recent research [17, 23, 26],
85
+ for modeling user satisfaction, persona and context, and condi-
86
+ tional natural language generation. Given its modular design, the
87
+ toolkit can also be easily extended with other modeling options or
88
+ additional components. The toolkit is comprised of two Python li-
89
+ braries, which are made publicly available on GitHub: DialogueKit1
90
+ is a collection of generic and reusable dialogue components, and
91
+ UserSimCRS2 is an extensible user simulator built on top.
92
+ 2
93
+ RELATED WORK
94
+ While there are several efforts on simulation toolkits for recom-
95
+ mender systems [8, 11, 14, 16, 21], our work differs from those in
96
+ two major ways. First, we focus on the task of conversational recom-
97
+ mendations and hence place a strong emphasis on natural language
98
+ understanding and generation. Thus, unlike others that operate in
99
+ 1https://github.com/iai-group/DialogueKit
100
+ 2https://github.com/iai-group/UserSimCRS
101
+ arXiv:2301.05544v1 [cs.IR] 13 Jan 2023
102
+
103
+ WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
104
+ Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, & Shuo Zhang
105
+ Figure 1: Conceptual overview of the user simulator. The parts in blue follow [25], while the yellow ones are novel additions.
106
+ the “intent space,” we operate in the “language space.” Second, our
107
+ objective is system evaluation, as opposed to training end-to-end
108
+ recommender systems using reinforcement learning (RL).
109
+ Our toolkit implements an agenda-based simulator [18], building
110
+ on and extending the approach in [25]. Alternatively, model-based
111
+ simulation could also be employed as it has been done recently for
112
+ task-based dialogue systems. Shi et al. [22] demonstrate how to
113
+ build model-based user simulators that rely on a simple Seq2seq di-
114
+ alogue system with copy and attention mechanisms, to facilitate RL-
115
+ based dialogue system training. ConvLab-2 [28] is an open-source
116
+ toolkit that enables researchers to build task-oriented dialogue sys-
117
+ tems, where user simulators are provided to support end-to-end
118
+ evaluation. These simulators can be assembled by equipping a neu-
119
+ ral network-based user policy with NLU and NLG components.
120
+ Tseng et al. [24] propose a learning framework for developing dia-
121
+ logue systems that perform joint optimization with an LSTM-based
122
+ user simulator, which consists of a dialogue manager, an NLG model,
123
+ and a dialogue context encoder. The dialogue systems and user sim-
124
+ ulator models are pre-trained using supervised learning and then
125
+ fine-tuned using reinforcement learning based on the generated
126
+ dialogues. Importantly, such model-based approaches can also be
127
+ incorporated into our framework in the future.
128
+ 3
129
+ CONCEPTUAL OVERVIEW
130
+ The goal of user simulation is to mimic how real users would re-
131
+ spond in given dialogue situation [19, 25]. Conceptually, our user
132
+ simulator follows the architecture of a typical task-based dialogue
133
+ system, which consists of natural language understanding, response
134
+ generation, and natural language generation components. Addi-
135
+ tionally, there is a dedicated user modeling component; see Fig. 1.
136
+ We opt for a modular design, as opposed to an end-to-end trainable
137
+ system, in order to have complete control over how responses are
138
+ generated and to allow for flexible extensions. Our work builds on
139
+ and extends the approach proposed in [25] as detailed below.
140
+ Natural language understanding (NLU) is responsible for
141
+ obtaining a structured representation of text utterances. Conven-
142
+ tionally, it entails intent classification and entity recognition. Addi-
143
+ tionally, motivated by recent research [17, 23], we also include a
144
+ classifier for user satisfaction prediction.3
145
+ Response generation is currently based on agenda-based simu-
146
+ lation [18], however, it could be replaced with other approaches in
147
+ the future. Following [25], response generation is based on an in-
148
+ teraction model, which is responsible for initializing the agenda and
149
+ 3User satisfaction prediction is only used in the training stage to annotate dialogues.
150
+ updating it. Updates to the agenda can be summarized as follows:
151
+ if the agent responds in an expected manner, the interaction model
152
+ pulls the next action off the agenda; otherwise, it either repeats the
153
+ same action as the previous turn or samples a new action.
154
+ User modeling consists of three sub-components. The preference
155
+ model captures users’ likes and dislikes. Following [25], it is modeled
156
+ as a personal knowledge graph [2], where nodes correspond to items
157
+ and attributes. Novel to our work is the modeling of persona, which
158
+ can capture user-specific traits, e.g., patience or cooperativeness,
159
+ and context, which can characterize the situation of the user, e.g.,
160
+ temporal (time of the day and weekday vs. weekend), relational
161
+ (alone vs. group setting), or conversational (user satisfaction). We
162
+ focus on contextual aspects as these represent a so far unexplored
163
+ area of user modeling [9] and there is evidence suggesting that
164
+ language usage depends on persona and context [15, 23].
165
+ Natural language generation (NLG) is currently template-
166
+ based, that is, given the output of the response generation mod-
167
+ ule, a fitting textual response is chosen and may be instantiated
168
+ with preferences. Additionally, we extend the NLG such that it can
169
+ be conditioned on context. For example, user responses might be
170
+ shorter/longer depending on the time of the day or users could use
171
+ a stronger language when getting dissatisfied with the system.
172
+ 4
173
+ SOFTWARE ARCHITECTURE
174
+ The toolkit is written in Python and is based on a modular archi-
175
+ tecture to support additional components, different models, and
176
+ custom features to be added in the future. There are two main li-
177
+ braries that are stacked on each other: DialogueKit provides basic
178
+ dialogue management functionalities, while UserSimCRS contains
179
+ simulation-specific models and logic. See Fig. 2 for an overview of
180
+ the main packages and their dependencies. Both libraries are made
181
+ available in the Python Package Index (PyPI).
182
+ 4.1
183
+ DialogueKit
184
+ DialogueKit models dialogue participants (users and agents), do-
185
+ mains (which define the types of slots for a particular application),
186
+ utterances, and annotations as base concepts. Utterances may be
187
+ annotated with intents and slot-value pairs. DialogueKit currently
188
+ supports two models for annotation, a cosine classifier for intents
189
+ and a minimal pipeline DIET classifier [4] for slot-value pairs.4 A
190
+ dialogue connector is included to orchestrate and store the conver-
191
+ sation between participants (human-human, human-machine, or
192
+ 4The DIET classifier can be used for intent detection as well.
193
+
194
+ User Simulator
195
+ Natural Language
196
+ Response Generation
197
+ Understanding (NLU)
198
+ Agenda-based simulator
199
+ User satisfaction prediction
200
+ [Future simulators]
201
+ Interaction model
202
+ Conversational
203
+ Agent
204
+
205
+ Natural Language
206
+ User Modeling
207
+ Generation (NLG)
208
+ <.
209
+ Context model
210
+ Conditional NLG
211
+ Preference model
212
+ PersonaUserSimCRS: A User Simulation Toolkit for Evaluating Conversational Recommender Systems
213
+ WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
214
+ DialogueKit
215
+ UserSimCRS
216
+ Dialogue connector
217
+ Platforms
218
+ Utilities
219
+ Evaluator
220
+ NLU
221
+ Entity extractor
222
+ Intent classifier
223
+ Satisfaction classifier
224
+ NLG
225
+ Template-based NLG
226
+ Core components
227
+ Annotation
228
+ Intent
229
+ Dialogue
230
+ Utterance
231
+ Domain
232
+
233
+ Agenda-based
234
+ simulator
235
+ Interaction model
236
+ User modeling
237
+ Preference model
238
+ Context model
239
+ Persona
240
+ Items
241
+ Conditional NLG
242
+ Participant
243
+ Agent
244
+ User
245
+ Item
246
+ Ratings
247
+ Figure 2: Overview of the main packages (in yellow) with
248
+ some of the core modules highlighted (in white). Arrows in-
249
+ dicate intra-library dependencies (in blue) and inter-library
250
+ dependencies (in black).
251
+ machine-machine). Furthermore, the evaluation component pro-
252
+ vides functionality required to evaluate a set of conversations with
253
+ respect to standard metrics (such as AvgTurns and AvgSuccess).
254
+ 4.2
255
+ UserSimCRS
256
+ The UserSimCRS library implements the simulation-specific compo-
257
+ nents in Fig. 1, specifically, response generation and user modeling.
258
+ During a conversation, any time the user is asked to provide pref-
259
+ erences, the preference model is consulted. Context is modeled in
260
+ a generic way such that it can capture, among others, temporal,
261
+ relational, and conversational factors. The generation of user utter-
262
+ ances may be conditioned on the user’s context and persona. Next,
263
+ we elaborate on how to use UserSimCRS for system evaluation.
264
+ Note that the library may also be used for training agents, but that
265
+ is outside the focus of the current paper.
266
+ 5
267
+ SYSTEM EVALUATION USING SIMULATION
268
+ This section discusses how to employ simulation for evaluating an
269
+ existing CRS and illustrates this with a case study.
270
+ 5.1
271
+ Methodology
272
+ The main objective of simulation-based evaluation in this work
273
+ is to establish a relative comparison between two systems. These
274
+ may be different variants of the same CRS or two different systems.
275
+ Importantly, the user simulator needs to target the differences that
276
+ we care about. For the sake of illustration, assume that there is
277
+ a baseline conversational movie recommender that understands
278
+ movie genres and an improved version that also recognizes plot
279
+ keywords. Having a user simulator that asks only for genres but not
280
+ for plot keywords will not capture the differences between these
281
+ two systems. Therefore, as a general principle, the user simulator
282
+ needs to be co-developed with the CRS and customized to mimic
283
+ the targeted user behavior.
284
+ 5.2
285
+ Setting up Simulation
286
+ A unique feature of our toolkit is that it allows for the evaluation
287
+ of any existing CRS by treating it as a “black box.” That is, it does
288
+ not require access to the source code or assume knowledge of its
289
+ inner workings—it merely relies on observable behavior. Setting up
290
+ an existing CRS with our simulator involves the following steps:
291
+ (1) Prepare domain and item collection: A config file with domain-
292
+ specific slot names must be prepared for the preference model.
293
+ Additionally, a file containing the item collection is required.
294
+ (2) Provide preference data: Preference data is consumed in the
295
+ form of item ratings (user ID, item ID, and rating triples).
296
+ (3) Dialogue sample: A small sample of dialogues with the CRS
297
+ needs to be collected. The sample size depends on the complex-
298
+ ity of the system, in terms of action space and language variety,
299
+ but is generally in the order of 5-50 dialogues.
300
+ (4) Define interaction model: A config file containing the space
301
+ of user and agent intents (i.e., possible actions), as well as the set
302
+ of expected agent responses for each user intent, is required for
303
+ the interaction model. The baseline (CRSv1) interaction model
304
+ shipped with the UserSimCRS library offers a starting point,
305
+ which may be further tailored according to the behavior and
306
+ capabilities of the given CRS.
307
+ (5) Annotate sample: The sample of dialogues must contain utter-
308
+ ance-level annotations in terms of intents and entities, as this
309
+ is required to train the NLU and NLG components. Note that
310
+ the slots used for annotation should be the same as the ones
311
+ defined in the domain file (cf. Step 1) and intents should follow
312
+ the ones defined in the interaction model (cf. Step 4.).
313
+ (6) Define user model/population: Simulation is seeded with a
314
+ user population that needs to be characterized, for example,
315
+ in terms of the different contexts (e.g., weekday vs. weekend,
316
+ alone vs. group setting) and personas (e.g., patient and impa-
317
+ tient users). Further, the number of users to be generated is to
318
+ be specified. Each user will have their own preference model,
319
+ which may be instantiated by grounding it in actual preferences
320
+ (i.e., the ratings dataset given in Step 2).
321
+ (7) Train simulator: The NLU, NLG, and response generation
322
+ components of the simulator are trained using the annotated
323
+ dialogue sample.
324
+ (8) Run simulation: Running the simulator will generate a set of
325
+ simulated conversations for each user with the CRS and save
326
+ those to files.
327
+ (9) Perform evaluation: Evaluation takes the set of simulated
328
+ dialogues generated in the previous step as input, and measures
329
+ the performance of the CRS in terms of the metrics implemented
330
+ in DialogueKit.
331
+ 5.3
332
+ Case Study
333
+ To see our user simulator in action, we conducted a case study with
334
+ IAI MovieBot [7],5 which is an open-source conversational movie
335
+ recommender system. This required creating a connector agent in
336
+ DialogueKit, which can talk to IAI MovieBot via a RESTful API.
337
+ We followed the steps listed above to prepare the user simulator.
338
+ This included collecting a sample of 8 dialogues, configuring the
339
+ domain (with title, genre, and keyword as slots), and annotating
340
+ user and system utterances using intents (according to our CRSv1
341
+ interaction model) and slot-value pairs. As it can be seen from the
342
+ sample dialogue in Fig. 3, the simulator could successfully complete
343
+ dialogues with the CRS, mimicking the behavior of users observed
344
+ in the training data it was exposed to.
345
+ 5https://github.com/iai-group/MovieBot
346
+
347
+ WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
348
+ Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, & Shuo Zhang
349
+ Figure 3: Sample dialogue between IAI MovieBot (Left, in
350
+ green) and the user simulator (Right, in blue).
351
+ 6
352
+ CONCLUSION AND FUTURE DIRECTIONS
353
+ We have presented a user simulation toolkit, organized into two
354
+ Python libraries around general dialogue management and specific
355
+ user simulation functionality, to facilitate research on both conver-
356
+ sational recommender systems and simulation-based evaluation.
357
+ The toolkit is shipped with solid baseline models for each of the
358
+ components, a detailed set of instructions, and a working exam-
359
+ ple with an existing CRS, in order to make it easy for researchers
360
+ and developers to start conducting simulation-based experiments.
361
+ Future work is concerned with extending the components with ad-
362
+ ditional modeling options, including alternatives to agenda-based
363
+ simulation. We also plan to evaluate additional existing CRSs to
364
+ ensure that our framework generalizes to diverse systems.
365
+ Acknowledgment. We thank Nolwenn Bernard for her extensive
366
+ contributions to the toolkit, made after the submission of this paper.
367
+ REFERENCES
368
+ [1] Krisztian Balog. 2021. Conversational AI from an Information Retrieval Perspec-
369
+ tive: Remaining Challenges and a Case for User Simulation. In Proc. of DESIRES
370
+ ’21. 80–90.
371
+ [2] Krisztian Balog and Tom Kenter. 2019. Personal Knowledge Graphs: A Research
372
+ Agenda. In Proc. of ICTIR ’19. 217–220.
373
+ [3] Krisztian Balog, David Maxwell, Paul Thomas, and Shuo Zhang. 2021. Report on
374
+ the 1st Simulation for Information Retrieval Workshop (Sim4IR 2021) at SIGIR
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+ 2021. SIGIR Forum 55, 2, Article 10 (dec 2021).
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+ [4] Tanja Bunk, Daksh Varshneya, Vladimir Vlasov, and Alan Nichol. 2020.
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+ DIET:
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+ Lightweight
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+ Language
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+ Understanding
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+ for
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+ Dialogue
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+ arXiv:2004.09936 [cs.CL]
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+ [5] Konstantina Christakopoulou, Filip Radlinski, and Katja Hofmann. 2016. Towards
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+ Conversational Recommender Systems. In Proc. of KDD ’16. 815–824.
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+ [6] Chongming Gao, Wenqiang Lei, Xiangnan He, Maarten de Rijke, and Tat-Seng
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+ Chua. 2021. Advances and Challenges in Conversational Recommender Systems:
389
+ A Survey. AI Open 2 (2021), 100–126.
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+ [7] Javeria Habib, Shuo Zhang, and Krisztian Balog. 2020. IAI MovieBot: A Conver-
391
+ sational Movie Recommender System. In Proc. of CIKM ’20. 3405–3408.
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+ [8] Eugene Ie, Chih wei Hsu, Martin Mladenov, Vihan Jain, Sanmit Narvekar, Jing
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+ Wang, Rui Wu, and Craig Boutilier. 2019. RecSim: A Configurable Simulation
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+ Platform for Recommender Systems. arXiv:1909.04847 [cs.LG]
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+ [9] Dietmar Jannach, Ahtsham Manzoor, Wanling Cai, and Li Chen. 2021. A Survey
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+ on Conversational Recommender Systems. ACM Comput. Surv. 54, 5 (2021).
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+ [10] Ivica Kostric, Krisztian Balog, and Filip Radlinski. 2021. Soliciting User Preferences
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+ in Conversational Recommender Systems via Usage-Related Questions. In Proc.
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+ of RecSys ’21. 724–729.
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+ [11] Karl Krauth, Sarah Dean, Alex Zhao, Wenshuo Guo, Mihaela Curmei, Benjamin
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+ Recht, and Michael I. Jordan. 2020. Do Offline Metrics Predict Online Performance
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+ in Recommender Systems? arXiv:2011.07931 [cs.LG]
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+ [12] Wenqiang Lei, Xiangnan He, Yisong Miao, Qingyun Wu, Richang Hong, Min-Yen
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+ Kan, and Tat-Seng Chua. 2020. Estimation-Action-Reflection: Towards Deep
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+ Interaction Between Conversational and Recommender Systems. In Proc. of
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+ WSDM ’20. 304–312.
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+ [13] Raymond Li, Samira Kahou, Hannes Schulz, Vincent Michalski, Laurent Charlin,
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+ and Chris Pal. 2018. Towards Deep Conversational Recommendations. In Proc. of
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+ NIPS ’18. 9748–9758.
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+ [14] Martin Mladenov, Chih-Wei Hsu, Vihan Jain, Eugene Ie, Christopher Colby,
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+ Nicolas Mayoraz, Hubert Pham, Dustin Tran, Ivan Vendrov, and Craig Boutilier.
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+ 2021. RecSim NG: Toward Principled Uncertainty Modeling for Recommender
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+ Ecosystems. arXiv:2103.08057 [cs.LG]
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+ [15] Namkee Park, Kyungeun Jang, Seonggyeol Cho, and Jinyoung Choi. 2021. Use of
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+ Offensive Language in Human-Artificial Intelligence Chatbot Interaction: The
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+ Effects of Ethical Ideology, Social Competence, and Perceived Humanlikeness.
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+ Comput. Hum. Behav. 121 (2021), 106795.
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+ [16] David Rohde, Stephen Bonner, Travis Dunlop, Flavian Vasile, and Alexan-
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+ dros Karatzoglou. 2018.
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+ RecoGym: A Reinforcement Learning Environ-
421
+ ment for the problem of Product Recommendation in Online Advertising.
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+ arXiv:1808.00720 [cs.IR]
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+ [17] Alexandre Salle, Shervin Malmasi, Oleg Rokhlenko, and Eugene Agichtein. 2021.
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+ Studying the Effectiveness of Conversational Search Refinement Through User
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+ Simulation. In Proc. of ECIR ’21. 587–602.
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+ [18] Jost Schatzmann, Blaise Thomson, Karl Weilhammer, Hui Ye, and Steve Young.
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+ 2007. Agenda-Based User Simulation for Bootstrapping a POMDP Dialogue
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+ System. In Proc. of NAACL ’07. 149–152.
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+ [19] Jost Schatzmann, Karl Weilhammer, Matt Stuttle, and Steve Young. 2006. A
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+ Survey of Statistical User Simulation Techniques for Reinforcement-Learning of
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+ Dialogue Management Strategies. Knowl. Eng. Rev. 21, 2 (June 2006), 97–126.
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+ [20] Ivan Sekulić, Mohammad Aliannejadi, and Fabio Crestani. 2022. Evaluating
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+ Mixed-Initiative Conversational Search Systems via User Simulation. In Proc. of
434
+ WSDM ’22. 888–896.
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+ [21] Bichen Shi, Makbule Gulcin Ozsoy, Neil Hurley, Barry Smyth, Elias Z. Tragos,
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+ James Geraci, and Aonghus Lawlor. 2019. PyRecGym: A Reinforcement Learning
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+ Gym for Recommender Systems. In Proc. of RecSys ’19. 491–495.
438
+ [22] Weiyan Shi, Kun Qian, Xuewei Wang, and Zhou Yu. 2019. How to Build User
439
+ Simulators to Train RL-based Dialog Systems. In Proc. of EMNLP-IJCNLP ’19.
440
+ 1990–2000.
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+ [23] Weiwei Sun, Shuo Zhang, Krisztian Balog, Zhaochun Ren, Pengjie Ren, Zhumin
442
+ Chen, and Maarten de Rijke. 2021. Simulating User Satisfaction for the Evaluation
443
+ of Task-Oriented Dialogue Systems. In Proc. of SIGIR ’21. 2499–2506.
444
+ [24] Bo-Hsiang Tseng, Yinpei Dai, Florian Kreyssig, and Bill Byrne. 2021. Transferable
445
+ Dialogue Systems and User Simulators. In Proc. of ACL ’21. 152–166.
446
+ [25] Shuo Zhang and Krisztian Balog. 2020. Evaluating Conversational Recommender
447
+ Systems via User Simulation. In Proc. of KDD ’20. 1512–1520.
448
+ [26] Shuo Zhang, Mu-Chun Wang, and Krisztian Balog. 2022. Analyzing and Simulat-
449
+ ing User Utterance Reformulation in Conversational Recommender Systems. In
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+ Proc. of SIGIR ’22. 133–143.
451
+ [27] Yongfeng Zhang, Xu Chen, Qingyao Ai, Liu Yang, and W. Bruce Croft. 2018.
452
+ Towards Conversational Search and Recommendation: System Ask, User Respond.
453
+ In Proc. of CIKM ’18. 177–186.
454
+ [28] Qi Zhu, Zheng Zhang, Yan Fang, Xiang Li, Ryuichi Takanobu, Jinchao Li, Baolin
455
+ Peng, Jianfeng Gao, Xiaoyan Zhu, and Minlie Huang. 2020. ConvLab-2: An
456
+ Open-Source Toolkit for Building, Evaluating, and Diagnosing Dialogue Systems.
457
+ In Proc. of ACL ’20. 142–149.
458
+ [29] Jie Zou, Yifan Chen, and Evangelos Kanoulas. 2020. Towards Question-Based
459
+ Recommender Systems. In Proc. of SIGIR ’20. 881–890.
460
+
461
+ Hi there. I am IAl MovieBot, your movie recommending buddy. I
462
+ can recommend you movies based on your preferences. I will ask
463
+ you a few questions and based on your answers, I will try to find a
464
+ movie for you.",
465
+ Hello.
466
+ DISCLOSE.NON-DISCLOSE
467
+ Do you have any specific genres in mind?
468
+ DISCLOSE.NON-DISCLOSE
469
+ How about Crime?
470
+ INQUIRE.ELICIT
471
+ There are almost 5300 crime films. Please answer a few
472
+ DISCLOSE
473
+ more questions to help me find a good movie.Can you give
474
+ me a few keywords? For example, 'organized crime' or
475
+ 'contact lens'",
476
+ Pixar animation
477
+ INQUIRE.ELICIT
478
+ Do you like: Cat City 8.4 96 min
479
+ DISCLOSE
480
+ Tell me something about it
481
+ REAVEAL
482
+ What would you like to know about Cat City?
483
+ INQUIRE.ITEMINFO
484
+
485
+ INQUIRE.MORE
486
+ Director name
487
+ Its directed by Béla Ternovszky.
488
+ INQUIRE.MORE
489
+ I like this recommendation
490
+ DISCLOSE.MORE
491
+ NOTE.ACCEPT
492
+ Please choose your next step.
493
+ I would like to quite now.
494
+ INQUIRE.NEXT
495
+ I hope you had a good experience. Bye.
496
+ COMPLETE
497
+ END
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+ page_content='UserSimCRS: A User Simulation Toolkit for Evaluating Conversational Recommender Systems Jafar Afzali University of Stavanger j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='no Aleksander Mark Drzewiecki University of Stavanger am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='drzewiecki@stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='no Krisztian Balog University of Stavanger Stavanger, Norway krisztian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='balog@uis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='no Shuo Zhang Bloomberg London, United Kingdom szhang611@bloomberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='net ABSTRACT We present an extensible user simulation toolkit to facilitate auto- matic evaluation of conversational recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' It builds on an established agenda-based approach and extends it with sev- eral novel elements, including user satisfaction prediction, persona and context modeling, and conditional natural language generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' We showcase the toolkit with a pre-existing movie recommender system and demonstrate its ability to simulate dialogues that mimic real conversations, while requiring only a handful of manually annotated dialogues as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' CCS CONCEPTS Information systems → Recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' KEYWORDS Conversational recommender systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' user simulation ACM Reference Format: Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, and Shuo Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' UserSimCRS: A User Simulation Toolkit for Evaluating Conversational Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' In Proceedings of the Sixteenth ACM International Conference on Web Search and Data Mining (WSDM ’23), February 27-March 3, 2023, Singapore, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' ACM, New York, NY, USA, 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='1145/3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='3573029 1 INTRODUCTION Conversational recommender systems (CRSs) elicit user preferences via multi-turn real-time interactions using natural language [6, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' There has been a great deal of progress in recent years on various aspects, including question-based user preference elicitation [5, 10, 29], multi-turn conversational recommendation strategies [12], and natural language understanding and generation [13, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' A major challenges that remains, however, is evaluation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Due to the dynamic nature of interactions, measuring performance on Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' WSDM ’23, February 27-March 3, 2023, Singapore, Singapore © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' ACM ISBN 978-1-4503-9407-9/23/02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='1145/3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='3573029 the conversation level is not possible using offline test collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' While online evaluation with users of a live service is an option, it is expensive and does not scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' A promising solution to these issues is user simulation [1, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The idea there is to build a simulated user that mimics how a real human would respond in a given dialogue situation [19, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Simulation thus offers a repeatable and reproducible means of evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' (We note that it is not meant to replace, but rather to complement human evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=') There is indeed an emerging focus in recent research on using simulation for evaluating conversational information access sys- tems in general [1, 3, 17, 20, 23] and conversational recommenders in particular [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The current work aims to contribute to the de- velopment of novel CRSs by recognizing the need for better tooling for user simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' In particular, we provide an extensible open- source toolkit that is designed specifically for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Our work is unique in at least three regards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' First, it focuses on the task of conversational recommendation and hence place a strong emphasis on both the recommendation-specific conversation flow and on the human-likeness of the generated user utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Second, it centers around evaluation as opposed to other uses of simulation (most commonly, synthetic data generation for reinforcement learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Third, it is designed to work with existing CRSs, without needing ac- cess to source code or knowledge of their inner workings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' It merely requires collecting and annotating a small sample of dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Building on an established agenda-based simulator [25], we intro- duce novel components, motivated by recent research [17, 23, 26], for modeling user satisfaction, persona and context, and condi- tional natural language generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Given its modular design, the toolkit can also be easily extended with other modeling options or additional components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The toolkit is comprised of two Python li- braries, which are made publicly available on GitHub: DialogueKit1 is a collection of generic and reusable dialogue components, and UserSimCRS2 is an extensible user simulator built on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 2 RELATED WORK While there are several efforts on simulation toolkits for recom- mender systems [8, 11, 14, 16, 21], our work differs from those in two major ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' First, we focus on the task of conversational recom- mendations and hence place a strong emphasis on natural language understanding and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Thus, unlike others that operate in 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='com/iai-group/DialogueKit 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='com/iai-group/UserSimCRS arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='05544v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='IR] 13 Jan 2023 WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, & Shuo Zhang Figure 1: Conceptual overview of the user simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The parts in blue follow [25], while the yellow ones are novel additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' the “intent space,” we operate in the “language space.” Second, our objective is system evaluation, as opposed to training end-to-end recommender systems using reinforcement learning (RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Our toolkit implements an agenda-based simulator [18], building on and extending the approach in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Alternatively, model-based simulation could also be employed as it has been done recently for task-based dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' [22] demonstrate how to build model-based user simulators that rely on a simple Seq2seq di- alogue system with copy and attention mechanisms, to facilitate RL- based dialogue system training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' ConvLab-2 [28] is an open-source toolkit that enables researchers to build task-oriented dialogue sys- tems, where user simulators are provided to support end-to-end evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' These simulators can be assembled by equipping a neu- ral network-based user policy with NLU and NLG components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Tseng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' [24] propose a learning framework for developing dia- logue systems that perform joint optimization with an LSTM-based user simulator, which consists of a dialogue manager, an NLG model, and a dialogue context encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The dialogue systems and user sim- ulator models are pre-trained using supervised learning and then fine-tuned using reinforcement learning based on the generated dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Importantly, such model-based approaches can also be incorporated into our framework in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 3 CONCEPTUAL OVERVIEW The goal of user simulation is to mimic how real users would re- spond in given dialogue situation [19, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Conceptually, our user simulator follows the architecture of a typical task-based dialogue system, which consists of natural language understanding, response generation, and natural language generation components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Addi- tionally, there is a dedicated user modeling component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' We opt for a modular design, as opposed to an end-to-end trainable system, in order to have complete control over how responses are generated and to allow for flexible extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Our work builds on and extends the approach proposed in [25] as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Natural language understanding (NLU) is responsible for obtaining a structured representation of text utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Conven- tionally, it entails intent classification and entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Addi- tionally, motivated by recent research [17, 23], we also include a classifier for user satisfaction prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='3 Response generation is currently based on agenda-based simu- lation [18], however, it could be replaced with other approaches in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Following [25], response generation is based on an in- teraction model, which is responsible for initializing the agenda and 3User satisfaction prediction is only used in the training stage to annotate dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' updating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Updates to the agenda can be summarized as follows: if the agent responds in an expected manner, the interaction model pulls the next action off the agenda;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' otherwise, it either repeats the same action as the previous turn or samples a new action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' User modeling consists of three sub-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The preference model captures users’ likes and dislikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Following [25], it is modeled as a personal knowledge graph [2], where nodes correspond to items and attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Novel to our work is the modeling of persona, which can capture user-specific traits, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=', patience or cooperativeness, and context, which can characterize the situation of the user, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=', temporal (time of the day and weekday vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' weekend), relational (alone vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' group setting), or conversational (user satisfaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' We focus on contextual aspects as these represent a so far unexplored area of user modeling [9] and there is evidence suggesting that language usage depends on persona and context [15, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Natural language generation (NLG) is currently template- based, that is, given the output of the response generation mod- ule, a fitting textual response is chosen and may be instantiated with preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Additionally, we extend the NLG such that it can be conditioned on context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' For example, user responses might be shorter/longer depending on the time of the day or users could use a stronger language when getting dissatisfied with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 4 SOFTWARE ARCHITECTURE The toolkit is written in Python and is based on a modular archi- tecture to support additional components, different models, and custom features to be added in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' There are two main li- braries that are stacked on each other: DialogueKit provides basic dialogue management functionalities, while UserSimCRS contains simulation-specific models and logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 2 for an overview of the main packages and their dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Both libraries are made available in the Python Package Index (PyPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='1 DialogueKit DialogueKit models dialogue participants (users and agents), do- mains (which define the types of slots for a particular application), utterances, and annotations as base concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Utterances may be annotated with intents and slot-value pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' DialogueKit currently supports two models for annotation, a cosine classifier for intents and a minimal pipeline DIET classifier [4] for slot-value pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='4 A dialogue connector is included to orchestrate and store the conver- sation between participants (human-human, human-machine, or 4The DIET classifier can be used for intent detection as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' User Simulator Natural Language Response Generation Understanding (NLU) Agenda-based simulator User satisfaction prediction [Future simulators] Interaction model Conversational Agent 金 Natural Language User Modeling Generation (NLG) <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Context model Conditional NLG Preference model PersonaUserSimCRS: A User Simulation Toolkit for Evaluating Conversational Recommender Systems WSDM ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' February 27-March 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Singapore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='DialogueKit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='UserSimCRS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Dialogue connector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Platforms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Utilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Evaluator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='NLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Entity extractor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Intent classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Satisfaction classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='NLG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Template-based NLG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Core components ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Annotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Intent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Dialogue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Domain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Agenda-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='simulator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Interaction model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='User modeling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Preference model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Context model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Persona ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Items ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Conditional NLG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Participant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Agent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='User ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Item ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Ratings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='Figure 2: Overview of the main packages (in yellow) with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='some of the core modules highlighted (in white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Arrows in- dicate intra-library dependencies (in blue) and inter-library dependencies (in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' machine-machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Furthermore, the evaluation component pro- vides functionality required to evaluate a set of conversations with respect to standard metrics (such as AvgTurns and AvgSuccess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='2 UserSimCRS The UserSimCRS library implements the simulation-specific compo- nents in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 1, specifically, response generation and user modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' During a conversation, any time the user is asked to provide pref- erences, the preference model is consulted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Context is modeled in a generic way such that it can capture, among others, temporal, relational, and conversational factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The generation of user utter- ances may be conditioned on the user’s context and persona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Next, we elaborate on how to use UserSimCRS for system evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Note that the library may also be used for training agents, but that is outside the focus of the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 5 SYSTEM EVALUATION USING SIMULATION This section discusses how to employ simulation for evaluating an existing CRS and illustrates this with a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='1 Methodology The main objective of simulation-based evaluation in this work is to establish a relative comparison between two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' These may be different variants of the same CRS or two different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Importantly, the user simulator needs to target the differences that we care about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' For the sake of illustration, assume that there is a baseline conversational movie recommender that understands movie genres and an improved version that also recognizes plot keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Having a user simulator that asks only for genres but not for plot keywords will not capture the differences between these two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Therefore, as a general principle, the user simulator needs to be co-developed with the CRS and customized to mimic the targeted user behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='2 Setting up Simulation A unique feature of our toolkit is that it allows for the evaluation of any existing CRS by treating it as a “black box.” That is, it does not require access to the source code or assume knowledge of its inner workings—it merely relies on observable behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Setting up an existing CRS with our simulator involves the following steps: (1) Prepare domain and item collection: A config file with domain- specific slot names must be prepared for the preference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Additionally, a file containing the item collection is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' (2) Provide preference data: Preference data is consumed in the form of item ratings (user ID, item ID, and rating triples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' (3) Dialogue sample: A small sample of dialogues with the CRS needs to be collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The sample size depends on the complex- ity of the system, in terms of action space and language variety, but is generally in the order of 5-50 dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' (4) Define interaction model: A config file containing the space of user and agent intents (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=', possible actions), as well as the set of expected agent responses for each user intent, is required for the interaction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The baseline (CRSv1) interaction model shipped with the UserSimCRS library offers a starting point, which may be further tailored according to the behavior and capabilities of the given CRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' (5) Annotate sample: The sample of dialogues must contain utter- ance-level annotations in terms of intents and entities, as this is required to train the NLU and NLG components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Note that the slots used for annotation should be the same as the ones defined in the domain file (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Step 1) and intents should follow the ones defined in the interaction model (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' (6) Define user model/population: Simulation is seeded with a user population that needs to be characterized, for example, in terms of the different contexts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=', weekday vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' weekend, alone vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' group setting) and personas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=', patient and impa- tient users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Further, the number of users to be generated is to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Each user will have their own preference model, which may be instantiated by grounding it in actual preferences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=', the ratings dataset given in Step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' (7) Train simulator: The NLU, NLG, and response generation components of the simulator are trained using the annotated dialogue sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' (8) Run simulation: Running the simulator will generate a set of simulated conversations for each user with the CRS and save those to files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' (9) Perform evaluation: Evaluation takes the set of simulated dialogues generated in the previous step as input, and measures the performance of the CRS in terms of the metrics implemented in DialogueKit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='3 Case Study To see our user simulator in action, we conducted a case study with IAI MovieBot [7],5 which is an open-source conversational movie recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' This required creating a connector agent in DialogueKit, which can talk to IAI MovieBot via a RESTful API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' We followed the steps listed above to prepare the user simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' This included collecting a sample of 8 dialogues, configuring the domain (with title, genre, and keyword as slots), and annotating user and system utterances using intents (according to our CRSv1 interaction model) and slot-value pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' As it can be seen from the sample dialogue in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 3, the simulator could successfully complete dialogues with the CRS, mimicking the behavior of users observed in the training data it was exposed to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content='com/iai-group/MovieBot WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, & Shuo Zhang Figure 3: Sample dialogue between IAI MovieBot (Left, in green) and the user simulator (Right, in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 6 CONCLUSION AND FUTURE DIRECTIONS We have presented a user simulation toolkit, organized into two Python libraries around general dialogue management and specific user simulation functionality, to facilitate research on both conver- sational recommender systems and simulation-based evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' The toolkit is shipped with solid baseline models for each of the components, a detailed set of instructions, and a working exam- ple with an existing CRS, in order to make it easy for researchers and developers to start conducting simulation-based experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Future work is concerned with extending the components with ad- ditional modeling options, including alternatives to agenda-based simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' We also plan to evaluate additional existing CRSs to ensure that our framework generalizes to diverse systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' We thank Nolwenn Bernard for her extensive contributions to the toolkit, made after the submission of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+ page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
317
+ page_content=' Agenda-Based User Simulation for Bootstrapping a POMDP Dialogue System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
318
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
319
+ page_content=' of NAACL ’07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
320
+ page_content=' 149–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
321
+ page_content=' [19] Jost Schatzmann, Karl Weilhammer, Matt Stuttle, and Steve Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
322
+ page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
323
+ page_content=' A Survey of Statistical User Simulation Techniques for Reinforcement-Learning of Dialogue Management Strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
324
+ page_content=' Knowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
325
+ page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
326
+ page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
327
+ page_content=' 21, 2 (June 2006), 97–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
328
+ page_content=' [20] Ivan Sekulić, Mohammad Aliannejadi, and Fabio Crestani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
329
+ page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
330
+ page_content=' Evaluating Mixed-Initiative Conversational Search Systems via User Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
331
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
332
+ page_content=' of WSDM ’22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
333
+ page_content=' 888–896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
334
+ page_content=' [21] Bichen Shi, Makbule Gulcin Ozsoy, Neil Hurley, Barry Smyth, Elias Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
335
+ page_content=' Tragos, James Geraci, and Aonghus Lawlor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
336
+ page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
337
+ page_content=' PyRecGym: A Reinforcement Learning Gym for Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
338
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
339
+ page_content=' of RecSys ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
340
+ page_content=' 491–495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
341
+ page_content=' [22] Weiyan Shi, Kun Qian, Xuewei Wang, and Zhou Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
342
+ page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
343
+ page_content=' How to Build User Simulators to Train RL-based Dialog Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
344
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
345
+ page_content=' of EMNLP-IJCNLP ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
346
+ page_content=' 1990–2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
347
+ page_content=' [23] Weiwei Sun, Shuo Zhang, Krisztian Balog, Zhaochun Ren, Pengjie Ren, Zhumin Chen, and Maarten de Rijke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
348
+ page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
349
+ page_content=' Simulating User Satisfaction for the Evaluation of Task-Oriented Dialogue Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
350
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
351
+ page_content=' of SIGIR ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
352
+ page_content=' 2499–2506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
353
+ page_content=' [24] Bo-Hsiang Tseng, Yinpei Dai, Florian Kreyssig, and Bill Byrne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
354
+ page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
355
+ page_content=' Transferable Dialogue Systems and User Simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
356
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
357
+ page_content=' of ACL ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
358
+ page_content=' 152–166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
359
+ page_content=' [25] Shuo Zhang and Krisztian Balog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
360
+ page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
361
+ page_content=' Evaluating Conversational Recommender Systems via User Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
362
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
363
+ page_content=' of KDD ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
364
+ page_content=' 1512–1520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
365
+ page_content=' [26] Shuo Zhang, Mu-Chun Wang, and Krisztian Balog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
366
+ page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
367
+ page_content=' Analyzing and Simulat- ing User Utterance Reformulation in Conversational Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
368
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
369
+ page_content=' of SIGIR ’22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
370
+ page_content=' 133–143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
371
+ page_content=' [27] Yongfeng Zhang, Xu Chen, Qingyao Ai, Liu Yang, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
372
+ page_content=' Bruce Croft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
373
+ page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
374
+ page_content=' Towards Conversational Search and Recommendation: System Ask, User Respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
375
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
376
+ page_content=' of CIKM ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
377
+ page_content=' 177–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
378
+ page_content=' [28] Qi Zhu, Zheng Zhang, Yan Fang, Xiang Li, Ryuichi Takanobu, Jinchao Li, Baolin Peng, Jianfeng Gao, Xiaoyan Zhu, and Minlie Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
379
+ page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
380
+ page_content=' ConvLab-2: An Open-Source Toolkit for Building, Evaluating, and Diagnosing Dialogue Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
381
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
382
+ page_content=' of ACL ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
383
+ page_content=' 142–149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
384
+ page_content=' [29] Jie Zou, Yifan Chen, and Evangelos Kanoulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
385
+ page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
386
+ page_content=' Towards Question-Based Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
387
+ page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
388
+ page_content=' of SIGIR ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
389
+ page_content=' 881–890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
390
+ page_content=' Hi there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
391
+ page_content=' I am IAl MovieBot, your movie recommending buddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
392
+ page_content=' I can recommend you movies based on your preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
393
+ page_content=' I will ask you a few questions and based on your answers, I will try to find a movie for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
394
+ page_content=' ", Hello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
395
+ page_content=' DISCLOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
396
+ page_content='NON-DISCLOSE Do you have any specific genres in mind?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
397
+ page_content=' DISCLOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
398
+ page_content='NON-DISCLOSE How about Crime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
399
+ page_content=' INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
400
+ page_content='ELICIT There are almost 5300 crime films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
401
+ page_content=' Please answer a few DISCLOSE more questions to help me find a good movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
402
+ page_content='Can you give me a few keywords?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
403
+ page_content=' For example, \'organized crime\' or \'contact lens\'", Pixar animation INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
404
+ page_content='ELICIT Do you like: Cat City 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
405
+ page_content='4 96 min DISCLOSE Tell me something about it REAVEAL What would you like to know about Cat City?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
406
+ page_content=' INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
407
+ page_content='ITEMINFO 藍 INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
408
+ page_content='MORE Director name Its directed by Béla Ternovszky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
409
+ page_content=' INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
410
+ page_content='MORE I like this recommendation DISCLOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
411
+ page_content='MORE NOTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
412
+ page_content='ACCEPT Please choose your next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
413
+ page_content=' I would like to quite now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
414
+ page_content=' INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
415
+ page_content='NEXT I hope you had a good experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
416
+ page_content=' Bye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
417
+ page_content=' COMPLETE END' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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1
+ A Practical Runtime Security Policy
2
+ Transformation Framework for Software Defined
3
+ Networks
4
+ Yunfei Menga, Changbo Keb, Zhiqiu Huangc, Guohua Shenc, ChunQiang Liua,
5
+ Xiaojie Fenga
6
+ aCollege of Information Engineering, Qingdao Binhai University, Qingdao 266555, China
7
+ bSchool of Computer Science and Technology, Nanjing University of Posts and
8
+ Telecommunications, Nanjing 210023, China
9
+ cCollege of Computer Science and Technology, Nanjing University of Aeronautics and
10
+ Astronautics, Nanjing 211106, China
11
+ Abstract
12
+ Software-defined networking (SDN) has been widely utilized to enforce the se-
13
+ curity of traditional networks, thereby promoting the process of transforming
14
+ traditional networks into SDN networks. However, SDN-based security enforce-
15
+ ment mechanisms rely heavily on the security policies containing the underlying
16
+ information of data plane, such as MAC address, IP address or switch ports.
17
+ These security policies need to be specifically developed by the network opera-
18
+ tors, and loaded into the control plane by manual inputting. With increasing the
19
+ scale of underlying network, the current security policy management mechanism
20
+ will confront more and more challenges. The security policy transformation for
21
+ SDN networks is to research how to transform the high-level security policy
22
+ without containing the underlying information of data plane into the practical
23
+ flow entries used by the OpenFlow switches automatically, thereby implement-
24
+ ing the automation of security policy management. Based on this insight, a
25
+ practical runtime security policy transformation framework is proposed in this
26
+ paper. First of all, we specify the security policies used by SDN networks as a
27
+ system model of security policy (SPM). From the theoretical level, we establish
28
+ the system model for SDN network and propose a formal method to transform
29
+ SPM into the system model of flow entries automatically. From the practical
30
+ level, we propose a runtime security policy transformation framework to solve
31
+ the problem of how to find a connected path for each relationship of SPM in
32
+ the data plane, as well as how to generate the practical flow entries according
33
+ to the system model of flow entries. In order to validate the feasibility and ef-
34
+ fectiveness of the framework, we set up an experimental system and implement
35
+ the framework with POX controller and Mininet emulator. The experimental
36
+ results illustrate the framework can synchronously perceive the changes caused
37
+ by cutting down one edge or changing SPM, and keep the data plane holding
38
+ the security properties defined by SPM continuously at runtime.
39
+ Keywords:
40
+ SDN, security policy, model transformation, data plane.
41
+ Preprint submitted to Elsevier
42
+ January 11, 2023
43
+ arXiv:2301.03790v1 [cs.CR] 10 Jan 2023
44
+
45
+ 1. Introduction
46
+ Software-defined networking (SDN) is a novel networking technique or ar-
47
+ chitecture that changes the limitation of traditional network infrastructures by
48
+ breaking the vertical integration, decoupling the control logics from the underly-
49
+ ing forwarding devices, promoting the centralization of control and introducing
50
+ the abilities to program the network directly[1]. In SDN networks, the control
51
+ logics of network, such as routing, traffic engineering or security policy devel-
52
+ oped in the application plane, are loaded into the control plane via the north-
53
+ bound interfaces (NBI) and transformed into a set of forward entries used by
54
+ the OpenFlow switches. After that, the control plane distributes the generated
55
+ flow entries to the associated switches in the data plane via the southbound
56
+ interfaces (SBI). Because of its programmable, centralized intelligent control
57
+ as well as global traffic view, SDN has been widely utilized to enhance the
58
+ security of tradition networks, thereby promoting the process of transforming
59
+ traditional networks into SDN networks. For instances, Garay et al.[2] proposed
60
+ a SDN-based network access control mechanism, flownac, which is a centralized
61
+ EAP (extensible authentication protocol) for IEEE 802.1x wireless local area
62
+ network (WLAN). Yakasai et al.[3] proposed a network access control mech-
63
+ anism, flowidentity. This mechanism integrates EAP security authentication
64
+ mechanism into the SDN controller. Hu et al.[4] proposed a dynamic firewall
65
+ mechanism, flowguard, based on SDN. Koerner et al.[5] proposed a device se-
66
+ curity authentication mechanism based on MAC address and SDN.
67
+ However, SDN-based security enforcement mechanisms rely heavily on the
68
+ security policies containing the underlying information of the data plane, such
69
+ as MAC address, IP address or switch ports. These security policies need to
70
+ be specifically developed by the network operators, and loaded into the control
71
+ plane by means of the manual inputting. With increasing the scale of underlying
72
+ network, the current security policy management mechanism will confront more
73
+ and more challenges. First of all, it is nearly impossible for any operator to
74
+ completely understand all the information of underlying network. In addition,
75
+ with the emergence of multi-controller SDN[6], network operators need to man-
76
+ age a variety of heterogeneous controllers at the same time. In this case, the
77
+ same security policy often needs to be developed and deployed for the different
78
+ types of controller, which inevitably increases the complexity and difficulty for
79
+ network management. Therefore, a novel security policy management mecha-
80
+ nism which can be completely transparent to the underlying information of data
81
+ plane is urgently needed for SDN networks. That is, it can permit the operators
82
+ only to define the high-level security policy without containing any underlying
83
+ information, then by means of the security policy transformation, the high-level
84
+ security policy can be automatically transformed into its corresponding flow
85
+ entries used by the OpenFlow switches in the data plane, thereby implementing
86
+ the automation of security policy management for SDN networks.
87
+ 2
88
+
89
+ Based on these insights, we have proposed a security policy model transfor-
90
+ mation and verification approach for SDN networks and published the approach
91
+ in our previous paper[7]. In that paper, we proposed a security policy transfor-
92
+ mation method to transform the high-level security policy model (SPM) without
93
+ containing the underlying information into its corresponding low-level security
94
+ policy model (LSPM) containing the underlying information.
95
+ To verify the
96
+ soundness of proposed security policy model transformation method, we further
97
+ proposed a security policy verification method and proved that the problem of
98
+ whether the data plane can satisfy the security properties defined by SPM is
99
+ equivalent to the problem of searching the connected paths related with SPM in
100
+ the data plane, that is, as long as each access control relationship Ri ∈ SPM can
101
+ be transformed into a corresponding connected path Pi in the data plane, next
102
+ transforms Pi into a set of flow entries used by the OpenFlow switches, then
103
+ the data plane must can hold the security properties defined by SPM. However,
104
+ that paper only proposed the method from the theoretical level, and did not
105
+ specifically implement this method. Moreover, it did not solve the problem of
106
+ how to find a connected path for each relationship of SPM in the data plane,
107
+ and how to transform LSPM into the practical flow entries used by the switches.
108
+ And based on the theoretical foundation of that paper, we propose a run-
109
+ time security policy transformation framework for SDN networks in this paper.
110
+ First of all, this paper further improves the system model of SDN networks and
111
+ solves the problem of how to transform SPM into the flow entries used by the
112
+ OpenFlow switches from the theoretical level. Moreover, this paper proposes
113
+ a runtime security policy transformation framework from the practical level,
114
+ thereby solving the problem of how to find a connected path Pi for each rela-
115
+ tionship Ri ∈ SPM in the data plane, as well as how to transform the system
116
+ model of flow entries into the practical flow entries used by the switches at run-
117
+ time. In addition, this paper further implements the proposed framework with
118
+ an experimental system. The experimental result illustrate the framework is
119
+ completely effective at runtime.
120
+ Hence, the contributions of this paper can be summarized as follows:
121
+ • We specify the security policies used by SDN networks, such as access
122
+ control policies or firewall policies, as a system model of security policy (SPM).
123
+ SPM is of a high-level system model without containing any underlying infor-
124
+ mation of data plane.
125
+ • From the theoretical level, we establish the system model for SDN network,
126
+ and propose a formal method to transform SPM into the system model of flow
127
+ entries automatically. The system model of flow entry is of a low-level system
128
+ model containing the underlying information of data plane.
129
+ • From the practical level, we propose a runtime security policy transforma-
130
+ tion framework which consists of the security policy module, topology discovery
131
+ module, runtime monitoring module, path generation module and flow entry
132
+ generation module. Leveraging these functional modules, the framework can
133
+ solve the problem of how to find a connected path for each relationship of SPM
134
+ in the data plane, how to transform the path into the system model of flow en-
135
+ tries, as well we how to generate the practical flow entries by using the system
136
+ 3
137
+
138
+ model of flow entries.
139
+ • In order to validate the feasibility and effectiveness of the framework, we
140
+ set up an experimental system and implement the framework by using POX con-
141
+ troller and Mininet emulator. The experimental result illustrate the framework
142
+ is completely effective at runtime.
143
+ The remainder of this paper is structured as follows. Section 2 discusses
144
+ some related works. Section 3 proposes the system model and elaborates on
145
+ how to transform SPM into the flow entries from the theoretical level. Sec-
146
+ tion 4 proposes the runtime security policy transformation framework from the
147
+ practical level and introduces its functional modules. Section 5 implements the
148
+ framework with an experimental system and elaborates on how to evaluate the
149
+ effectiveness and performance of the framework. Finally, Section 6 concludes
150
+ this paper and presents some future directions.
151
+ 2. Related Work
152
+ In this section, we discuss some research works related with the policy model
153
+ transformation and the security policy verification.
154
+ 2.1. Policy Model Transformation
155
+ According to the definitions of model-driven architecture (MDA), the model
156
+ transformation refers to the process of transforming the platform independent
157
+ model (PIM) to its corresponding platform specific model (PSM)[8]. As far as
158
+ the literatures we have read, the researches towards the policy model transforma-
159
+ tion can be roughly divided into three categories, they are the template-based
160
+ transformation, RBAC-oriented transformation as well as the transformation
161
+ based on the system model and mapping rules[9]. Due to the limitation of tem-
162
+ plate, the template-based model transformation has very limited transformation
163
+ capability. Generally, RBAC-oriented model transformation[10] is only suitable
164
+ for transforming RBAC (role-based access control) policies, and does not have
165
+ enough capability to describe the complex system, so that these two methods
166
+ are not suitable for SDN networks.
167
+ At present, the model transformation based on the system model and map-
168
+ ping rules has been widely used for transforming the policy models. The main
169
+ idea of this method can be summarized as follows: (1) System Model: it de-
170
+ fines the objects of system and the relationship between the system objects;
171
+ (2) Policy Model: it defines the policy object and the relationship between the
172
+ policy objects; (3) Mapping Rules: it establishes the mapping rules between the
173
+ upper-level policy objects and the lower-level system objects[11][12]. The trans-
174
+ formation based on the system model and mapping rules first establishes the
175
+ policy model and the system model which can describe the underlying system,
176
+ then establishes the mapping rules between the policy objects and the system
177
+ objects, then transforms the upper-level policy model into its corresponding
178
+ lower-level policy model by means of the established mapping rules. In partic-
179
+ ular, Davy et al.[13] proposed a policy model transformation method based on
180
+ 4
181
+
182
+ mapping rules, in which the policy model is defined as a tuple ( event, condi-
183
+ tion, behavior, subject, object ) and used the ontology to establish the mapping
184
+ rules between the different system layers. Luck et al.[14] proposed a method
185
+ to transform RBAC model defined in service layer into the policy model used
186
+ in the system layer. In this method, the system model is divided into three
187
+ layers: roles and object (RO), subject and resources (SR) and processes and
188
+ hosts (PH), and the mapping rules between the three layers have been estab-
189
+ lished. Based on the Luck’s research, Porto et al.[15] further decomposes the
190
+ PH layer into two sub layers, namely DAS (diagram abstract subsystem) layer
191
+ and PH layer. DAS layer is mainly used to describe the network topology in
192
+ the original PH layer, while PH layer is used to describe the specific network
193
+ information in DAS layer. In addition, the authors also proposed a policy ver-
194
+ ification framework, which can be used to verify the consistency problems in
195
+ the process of policy transformation. In addition, Lampson et al.[16] proposed
196
+ a network policy model transformation method for the distributed computing
197
+ environment. Maullo et al.[17] proposed a policy transformation system based
198
+ on the first-order predicate logics, which transforms the high-level policy model
199
+ into the low-level network configuration policy through the network topology
200
+ and other information. Nanxi et al.[18] proposed a SDN-oriented access control
201
+ policy transformation framework. In this paper, In this paper, we also propose a
202
+ security policy transformation framework based on the system model and map-
203
+ ping rules. We first establish the system model of security policy (SPM) and
204
+ data plane, then establish the transformation rules between the policy objects
205
+ of SPM and the objects of the data plane, thereby transforming SPM into the
206
+ system model of flow entries automatically.
207
+ 2.2. Security Policy Verification
208
+ To assure the information systems running securely, security mechanisms
209
+ of information system need to be validated whether it can satisfy the security
210
+ properties defined by the security policy. The traditional validation methods
211
+ based on the testing and simulation can only confirm the system can work prop-
212
+ erly under the different testing scenarios, but it is difficult to find some hidden
213
+ scenarios that occur with little probability. Formal verification methods have
214
+ been applied to overcome the shortcomings existed in the traditional valida-
215
+ tion methods. At present, the formal verification methods for validating the
216
+ security policy mainly include theorem proving and model checking[19]. The-
217
+ orem proving is unsuitable to validate the properties of complex systems due
218
+ to its lower efficiency. Model checking[20] can be used to validate whether the
219
+ system model can satisfy the expected dynamic behaviors and specific static
220
+ properties.
221
+ Model checking technique has been widely used for the security
222
+ policy verification. For instances, Al-Shaer et al.[21] proposed a static policy
223
+ inconsistency detection method for the firewall policies of network. Bandara et
224
+ al.[22] proposed a security policy verification framework based on event calcu-
225
+ lus (EC) and used the reasoning techniques for the policy conflict identification.
226
+ May et al.[23] verified the privacy policies by means of an asynchronous model
227
+ checker. Rubio-Loyola et al.[24] proposed a goal-oriented policy refinement and
228
+ 5
229
+
230
+ conflict detection method by means of the model checking technique and linear
231
+ temporal Logic (LTL). Graham et al.[25] proposed a policy conflict detection
232
+ method with the model checking and an extended decision table. Baliosan and
233
+ Serrat[26] proposed a specific finite automata based method for the policy con-
234
+ flict detection.
235
+ 3. Problem Formalization
236
+ The security policy transformation for SDN networks is to research how to
237
+ transform the high-level security policy without containing the underlying infor-
238
+ mation into the set of practical flow entries used by the OpenFlow switches in
239
+ the data plane automatically, thereby implementing the automation of security
240
+ policy management in SDN network. In the following of this section, we first
241
+ establish the system model for SDN network, then propose a formal method to
242
+ transform the security policy (SPM) into the system model of flow entries from
243
+ the theoretical level.
244
+ 3.1.
245
+ System Model
246
+ Definition 1. (Security Policy):
247
+ The high-level security policy is defined as
248
+ a finite set of access control relationships: SPM = { R0, R1,...,Rn | ∀ Ri = (
249
+ si, oj, a ) }, where si ∈ S represents the subject of the relationship, oj ∈ O
250
+ represents the object of the relationship, a represents the access authorization,
251
+ i.e., the subject can access the object.
252
+ Definition 2. (Host):
253
+ The host existed in the data plane is defined as a tuple:
254
+ hi=( ipi, swi, portm
255
+ i
256
+ ), where ipi represents the host’s IP address in the data
257
+ plane, swi represents the OpenFlow switch connected with the host, portm
258
+ swi
259
+ represents the port connected with the host in swi.
260
+ Definition 3. (OpenFlow Switch):
261
+ The OpenFlow switch existed the data
262
+ plane is defined as a finite set of flow entries: swi = { f0, f1, ..., fn }.
263
+ Definition 4. (Flow Entry):
264
+ The flow entry existed in the OpenFlow switch
265
+ is defined as a tuple: fi = ( ipsrc, ipdst, portin
266
+ swi =⇒ portout
267
+ swi ), where ipsrc
268
+ represents the traffic’s source IP address, ipdst represents the traffic’s destination
269
+ IP address, portin
270
+ swi =⇒ portout
271
+ swi represents the traffic input from portin
272
+ swi will be
273
+ outputted from portout
274
+ swi in the switch swi.
275
+ Definition 5. (Edge):
276
+ The edge existed in the data plane is defined as: ei
277
+ = portout
278
+ start �−→ portin
279
+ end, where portout
280
+ start represents the port connected with the
281
+ edge in the start switch swstart, portin
282
+ end represents the port connected with the
283
+ edge in the end switch swend, so that the direction of the edge is from swstart
284
+ to swend.
285
+ Definition 6. (Topology):
286
+ The topology of the data plane is defined as an
287
+ graph: G = ( SW, E ), where SW represents a finite set of OpenFlow switches,
288
+ E represents a finite set of edges.
289
+ 6
290
+
291
+ Definition 7. (Connected Path):
292
+ The connected path between the host hi
293
+ and the host hj in the topology G is defined as: Pi = hi �−→ hj = { hi, e0, e1,...,
294
+ en, hj | ∀ei ∈ E }, where the direction of the path is from hi to hj.
295
+ 3.2. Transforming SPM into the Flow Entries
296
+ Based on the established system model, we propose a formal method to
297
+ transform SPM into the system model of flow entries from the theoretical level.
298
+ The method can be described as Figure 1 and summarized as follows: First of all,
299
+ for ∀Ri ∈ SPM, the subject si ∈ Ri is transformed into a host h(si) in the data
300
+ plane, the object oj ∈ Ri is transformed into a host h(oi) in the data plane;
301
+ Next, the access authorization a ∈ Ri is transformed into a connected path
302
+ Pi between h(si) and h(oj); After that, the connected path Pi is transformed
303
+ into a set of flow entries used by the switches which are passed by Pi; Finally,
304
+ SPM is transformed into the set of flow entries ∆ when all relationships of
305
+ SPM have been transformed.
306
+ As shown in Figure 1, the relationship R1 ∈
307
+ SPM is transformed into the connected path P1 between h1 and h2, thus P1
308
+ is transformed into the flow entries deployed in sw1, so as to implement the
309
+ security policy transformation from the system model level. The soundness of
310
+ the formal method has been proven in our previous paper[7]. That is, if the
311
+ security properties defined by SPM is denoted as ϕ, the system model of data
312
+ plane is denoted as D, the flow entries generated by the method is denoted as ∆,
313
+ then the method can ensure the data plane D loaded with ∆ can synchronously
314
+ and continuously hold the security properties ϕ at runtime, i.e., D(∆) |= ϕ.
315
+ Specifically, the formal method to transform SPM into the system model of
316
+ flow entries is defined as follows.
317
+ • Transforming the Subject Given an access control relationship Ri ∈
318
+ SPM and the subject si ∈ Ri, then si is transformed into a corresponding
319
+ host existed in the data plane. The rule of transforming the subject of SPM is
320
+ formally defined as follows:
321
+ si ∈ Ri
322
+ h(si) = (ipsrc, swsrc, portin
323
+ src, )
324
+ (1)
325
+ where h(si) represents the host mapped with si in the data plane, ipsrc repre-
326
+ sents the IP address of h(si) in the data plane, swsrc represents the OpenFlow
327
+ switch connected with h(si) in the data plane, portin
328
+ src represents the port con-
329
+ nected with h(si) in swsrc.
330
+
331
+ Transforming the Object Given an access control relationship Ri ∈
332
+ SPM and the object oj ∈ Ri, then oj is transformed into a corresponding host
333
+ existed in the data plane. The rule of transforming the object of SPM is formally
334
+ defined as follows:
335
+ oj ∈ Ri
336
+ h(oj) = (ipdst, swdst, portout
337
+ dst, )
338
+ (2)
339
+ where h(oj) represents the host mapped with oj in the data plane, ipdst repre-
340
+ sents the IP address of h(oj) in the data plane, swdst represents the OpenFlow
341
+ 7
342
+
343
+ Figure 1:
344
+ The method of transforming SPM into the system model of flow entries.
345
+ 8
346
+
347
+ -
348
+ Si
349
+ R1
350
+ S
351
+ 0
352
+ Application PN
353
+ h3
354
+ P1
355
+ sw1
356
+ h1
357
+ h2
358
+ sw2switch connected with h(oj), portout
359
+ dst represents the port connected with h(oj)
360
+ in swdst.
361
+ • Transforming the Authorization Given an access control relationship
362
+ Ri ∈ SPM and the access authorization a ∈ Ri, if there existing a connected
363
+ path Pi between h(si) and h(oj) in the topology, then a is transformed into the
364
+ connected Pi. The rule of transforming the authorization is formally defined as
365
+ follows:
366
+ a ∈ Ri
367
+ h(si) �−→ h(oj) ⊂ G
368
+ (3)
369
+ where h(si) �−→ h(oj) represents a directional connected path from h(si) to
370
+ h(oj) in the topology.
371
+ • Transforming the Path The connected path h(si) �−→ h(oj) is trans-
372
+ formed into a set of flow entries deployed in the OpenFlow switches which are
373
+ passed by the path. The rule of transforming the connected path is formally
374
+ defined as follows:
375
+ h(si) �−→ h(oj)
376
+ dst
377
+
378
+ k=src
379
+ fi(k)
380
+ (4)
381
+ where fi(k) =( ipsrc, ipdst, portin
382
+ k =⇒ portout
383
+ k
384
+ ) represents a flow entry deployed
385
+ in the switch swk which is passed by h(si) �−→ h(oj). Leveraging the definitions
386
+ of the system model, we can proof that the rule of transforming the connected
387
+ path is sound.
388
+ proof :
389
+ h(si) �−→ h(oj)
390
+ = { h(si), e0, e1, ..., en, h(oj) }
391
+ = {(ipsrc, swsrc, portin
392
+ src), (portout
393
+ src �−→ portin
394
+ 0 )
395
+ , ..., (portout
396
+ n
397
+ �−→ portin
398
+ dst), (ipdst, swdst, portout
399
+ dst)}
400
+ = {(portin
401
+ src =⇒ portout
402
+ src), (portin
403
+ 0 =⇒ portout
404
+ 0
405
+ ),
406
+ , ..., (portin
407
+ n =⇒ portout
408
+ n ), (portin
409
+ dst =⇒ portout
410
+ dst)
411
+ (ipsrc, ipdst), (swsrc, swdst)}
412
+ = {
413
+ dst
414
+
415
+ k=src
416
+ (portin
417
+ k =⇒ portout
418
+ k )} � (ipsrc, ipdst)
419
+ =
420
+ dst
421
+
422
+ k=src
423
+ {(portin
424
+ k =⇒ portout
425
+ k ) � (ipsrc, ipdst)}
426
+ =
427
+ dst
428
+
429
+ k=src
430
+ fi(k).
431
+ Therefore, ∀Ri ∈ SPM, the subject si ∈ Ri, the object oj ∈ Ri and the
432
+ autherization a ∈ Ri, if ∃Pi = h(si) �−→ h(oj) in the topology, then Ri can
433
+ be transformed into a corresponding set of flow entries
434
+ dst
435
+
436
+ k=src
437
+ fi(k) by using the
438
+ system model step by step, it can be formally defined as follows:
439
+ Ri
440
+ dst
441
+
442
+ k=src
443
+ fi(k)
444
+ (5)
445
+ 9
446
+
447
+
448
+ Transforming SPM SPM is transformed into a corresponding set of
449
+ flow entries ∆ by using the equation (5). If || SPM || = m, then the rule of
450
+ transforming SPM is formally defined as follows:
451
+ ∆ =
452
+ m
453
+
454
+ i=1
455
+ {
456
+ dst
457
+
458
+ k=src
459
+ fi(k)}
460
+ (6)
461
+ 4. The Security Policy Transformation Framework
462
+ The problem of how to transform SPM into the corresponding flow entries
463
+ used by the OpenFlow switches has been solved from the theoretical level in
464
+ Section 3. However, we cannot solve the problem of how to find a connected
465
+ path in the data plane for each relationship Ri ∈ SPM. In addition, by means
466
+ of the security policy transformation method, SPM can be transformed into the
467
+ corresponding flow entries, but the flow entry transformed from SPM is of the
468
+ system model, i.e. it is only the formal description of the real flow entry and
469
+ cannot be used by the real OpenFlow switch directly, so that we need to further
470
+ solve the problem of how to generate the practical flow entries by using the
471
+ system model of flow entries. Based on this insight, a runtime security policy
472
+ transformation framework for SDN networks is proposed from the practical level
473
+ in this section. By means of the framework, we can solve the problem of how
474
+ to find the connected path for each relationship defined by SPM, as well as
475
+ the problem of how to generate the practical flow entries based on the system
476
+ model at runtime. As shown in Figure 2, this framework consists of 5 functional
477
+ modules, i.e., the security policy module, the topology discovery module, the
478
+ runtime monitoring module, the path generation module and the flow entry
479
+ generation module.
480
+ 4.1. Overview of the Framework
481
+ • Security policy module is deployed in the application plane and responsible
482
+ for maintaining the security policy (SPM). Each access control relationship Ri ∈
483
+ SPM is designed as a 3-tuple: ( si, oj, fixed ) based on the system model, where
484
+ si is the subject; oj is the object; fixed={ 0, 1 } is a tag bit, fixed=1 represents
485
+ the relationship has been updated by user, fixed=0 represents it is unchanged.
486
+ SPM is stored as a text document and can be updated by the user at runtime.
487
+ • Topology discovery module is deployed in the control plane and responsible
488
+ for creating a dynamic real-time topology of the entire data plane by capturing
489
+ the link events transmitted from the OpenFlow switches. Based on the system
490
+ model, each edge in the topology is designed as a tuple: ei = ( swsrc, portsrc,
491
+ swdst, portdst, using, c ), where using={ True, False } is a tag bit and can
492
+ be changed by the real-time link events at runtime, using=True represents the
493
+ edge can be used now, using=False represents the edge is interrupted now. For
494
+ the convenient of researching, the cost of each edge is set to 1, i.e., c =1. The
495
+ generated topology is stored as a text document and can also be updated by
496
+ the real-time link events at runtime.
497
+ 10
498
+
499
+ Figure 2:
500
+ The security policy transformation framework for SDN networks.
501
+ • Runtime monitoring module is deployed in the control plane and respon-
502
+ sible for monitoring all the traffics in the data plane by capturing the real-time
503
+ packet-in events transmitted from the OpenFlow switches. When a new packet-
504
+ in event arrives in the controller, the module first invokes the path generation
505
+ module to transform the latest security policy (SPM) into a set of connected
506
+ paths in the data plane, then invokes the flow entry generation module to trans-
507
+ form all the connected paths into their corresponding flow entries deployed in
508
+ the OpenFlow switches which are passed by these paths. As SPM and the topol-
509
+ ogy of data plane will be evolved with the runtime environment, this module is
510
+ designed to be triggered by the real-time packet-in events continuously, so that,
511
+ when SPM is changed (i.e., occurring fixed=1) or the topology is changed (i.e.,
512
+ occurring using=False) at runtime, the module will first delete all the current
513
+ flow entries deployed in the OpenFlow switches, then update all the flow tables
514
+ by using the latest generated flow entries, so as to ensure the security properties
515
+ defined by SPM can be synchronously and continuously hold in the data plane
516
+ at runtime.
517
+ • Path generation module is deployed in the control plane and invoked by the
518
+ runtime monitoring module at runtime. The module is responsible for trans-
519
+ forming each Ri ∈ SPM input from the runtime monitoring module into a
520
+ corresponding path Pi in the data plane by using the latest topology file and
521
+ the path searching algorithm. Specifically, it first transforms the subject si ∈ Ri
522
+ and the object oj ∈ Ri into the hosts h(si) and h(oj) in the data plane respec-
523
+ 11
524
+
525
+ Security Policy
526
+ Update
527
+ SPM
528
+ Module
529
+ Application Plane
530
+ SPM
531
+ Path
532
+ Topology
533
+ -Topology
534
+ Generation
535
+ Module
536
+ Control Plane
537
+ Update
538
+ Invoke
539
+ Path
540
+ Topology
541
+ Flow Entry
542
+ Runtime
543
+ Discovery
544
+ Generation
545
+ Path
546
+ Monitoring
547
+ Module
548
+ Module
549
+ Module
550
+ Link Events
551
+ Flow Entries
552
+ Packet-In Events
553
+ OpenFlow Switches
554
+ Data Planetively, then finds a shortest connected path Pi between h(si) and h(oj) by using
555
+ the path searching algorithm, finally all the connected paths transformed from
556
+ SPM are returned to the runtime monitoring module.
557
+ • Flow entry generation module is also deployed in the control plane and in-
558
+ voked by the runtime monitoring module at runtime. The module is responsible
559
+ for transforming the connected path into a set of flow entries deployed in the
560
+ OpenFlow switches which are passed by the path, then utilizing the instructions
561
+ provided by the controller to generate the practical flow entries and distributing
562
+ these flow entries to the corresponding OpenFlow switches at runtime.
563
+ 4.2. Runtime Monitoring Algorithm
564
+ The runtime monitoring algorithm deployed in the runtime monitoring mod-
565
+ ule plays the role of coordinator in the framework, and can be described as Al-
566
+ gorithm 1 in pseudo code. First of all, the algorithm creates two dynamic lists
567
+ S and T by reading the latest SPM and Topo file respectively. If there existing
568
+ an access control relationship has been changed by user (Ri.fixed=1) or a edge
569
+ has been shut down in the topology (ei.using=False) at runtime, it will clear
570
+ all the current flow entries deployed in the OpenFlow switches for ready of the
571
+ updating. In the following, for each access control relationship Ri ∈ S, it maps
572
+ the subject si ∈ Ri and the object oj ∈ Ri with the switches swsrc and swdst
573
+ in the data plane, then transforms Ri into a corresponding connected path Pi
574
+ by invoking the path searching algorithm djk-route(swsrc, swdst, N). Based
575
+ on the transformation rules, Pi can be further transformed into a set of flow
576
+ entries. When all the relationships in the List S having been transformed, SPM
577
+ has been transformed into a corresponding set of flow entries ∆, the algorithm
578
+ invokes the flow entry generation module to update the data plane by using
579
+ ∆. Since the algorithm is designed to be triggered by the packet-in events at
580
+ runtime, so that it ensures the framework can perceive any changes in time
581
+ when the security policy or the topology of data plane has been evolved with
582
+ the environment, and then update the data plane synchronously at runtime.
583
+ 4.3. Path Searching Algorithm
584
+ Another important algorithm in the framework is the path searching algo-
585
+ rithm deployed in the path generation module. The algorithm is improved from
586
+ the classic Dijkstra algorithm and can be described as the Algorithm 2 in pseudo
587
+ code. First of all, the algorithm creates a dynamic matrix djk[N][N] by using
588
+ the sum of Openflow switches N and the topology file. In the following, it cal-
589
+ culates a shortest connected path between swsrc and swdst in the data plane
590
+ by using djk[N][N] and the created stacks. After multi-round calculating, the
591
+ shortest path Pi between swsrc and swdst is found and returned to the runtime
592
+ monitoring module. As the cost of each edge has been set to 1 and not consider-
593
+ ing of the quality of services (QoS) of edges, so that the shortest path Pi found
594
+ by the algorithm is generated by calculating the minimum number of hops in
595
+ the topology. Moreover, since the matrix djk[N][N] is dynamically created by
596
+ the topology file, so that the searched shortest path will be evolved with the
597
+ changing of the topology at runtime.
598
+ 12
599
+
600
+ 5. Implementation and Evaluations
601
+ In order to validate the feasibility and effectiveness of the framework pro-
602
+ posed in Section 4, we set up an experimental system and implement the frame-
603
+ work with POX controller[27] and Mininet emulator[28]. First of all, we im-
604
+ plement a virtual SDN network by using the Mininet emulator. As shown in
605
+ Figure 3, the topology of the network consists of 6 hosts (h1 ∼ h6) and 11 Open-
606
+ Flow switches (sw1 ∼ sw11). We further implement the security policy module,
607
+ topology discovery module, runtime monitoring module, path generation mod-
608
+ ule and flow entry generation module with Python 3.6.1 and integrate these
609
+ modules with the core of POX controller. The experimental system consists of
610
+ a Lenovo workstation with Windows OS, Intel-i7 32Cores 2.60GHz CPU, 32GB
611
+ RAM and a Raspberry platform with Linux OS, ARM-v7 CPU and 945MB
612
+ RAM. The POX controller and the functional modules are deployed in Lenovo
613
+ workstation, Mininet emulator is deployed in Raspberry platform, and Lenovo
614
+ workstation is connected with Raspberry platform using coaxial cable directly.
615
+ Table 1: The high-level security policy (SPM)
616
+ R1
617
+ ( 1, 5, 1 )
618
+ R2
619
+ ( 5, 1, 1 )
620
+ R3
621
+ ( 2, 4, 1 )
622
+ R4
623
+ ( 4, 2, 1 )
624
+ 5.1. Effectiveness Evaluation
625
+ The security policy used for validating the effectiveness of the framework is
626
+ shown in Table I. Since any effective interaction is bidirectional in SDN networks,
627
+ i.e., the subject’s host and the object’s host must be ensured they can access
628
+ each other in the data plane, so that we design the security policy as 4 access
629
+ control relationships (R1 ∼ R4) to ensure h1 (1) and h5 (5) can access each
630
+ other, h2 (2) and h4 (4) can access each other, and all the relationships of
631
+ SPM are set as fixed=1, i.e., having been updated by user. In the following,
632
+ the effectiveness evaluations towards the framework will be carried out under 4
633
+ different scenarios at runtime, they are the effectiveness after loading the flow
634
+ entries, the effectiveness after cutting down the path and the effectiveness after
635
+ changing SPM.
636
+ 5.1.1. Effectiveness after Loading the Flow Entries
637
+ The purpose of this experiment is to validate whether the data plane after
638
+ loading the generated flow entries can hold the security properties defined by
639
+ SPM. First of all, the subjects and objects of SPM shown in Table I, i,e., the
640
+ 1, 2, 4 and 5, are transformed into their corresponding hosts in the data plane
641
+ by using security policy transformation. Specifically, the 1 is transformed into
642
+ 13
643
+
644
+ Figure 3:
645
+ The topology of the virtual SDN network.
646
+ h1=( 10.0.0.1, sw1, 1 ), the 2 is transformed into h2=( 10.0.0.2, sw2, 1 ), the 4
647
+ is transformed into h4=( 10.0.0.4, sw9, 1 ) and the 5 is transformed into h5=(
648
+ 10.0.0.5, sw10, 1 ) respectively. In the following, the path searching algorithm,
649
+ i.e., Algorithm 2, searches the shortest path between the subject’s host and the
650
+ object’s host for each Ri ∈ SPM based on the latest topology generated from
651
+ the topology file.
652
+ After that, the relationships shown in Table I have been transformed into 4
653
+ corresponding shortest connected paths (P1 ∼ P4) in the data plane. As shown
654
+ in Figure 4, the R1 is transformed into P1={ h1, e1, e2, e3, h5 }, the R2 is
655
+ transformed into P2={ h5, e3, e2, e1, h1 }, the R3 is transformed into P3={ h2,
656
+ e4, e5, h4 } and the R4 is transformed into P4={ h4, e5, e4, h2 }, where the
657
+ P1 and P2 are depicted with the blue lines, the P3 and P4 are depicted with
658
+ the orange lines. In the following, the flow entry generation module transforms
659
+ each path into a set of flow entries deployed in the switches passed by the path.
660
+ Specifically, P1 and P2 are transformed into the flow entries deployed in the
661
+ switches { sw1, sw5, sw8, sw10 }, while P3 and P4 are transformed into the flow
662
+ entries deployed in the switches { sw2, sw6, sw9 }.
663
+ After that, we execute the pingall instruction in the Mininet CLI and observe
664
+ the reachability of the entire data plane. As shown in Figure 5, h1 and h5 can
665
+ access each other, h2 and h4 can access each other either, so that the data plane
666
+ after loading the generated flow entries has been proven that it can hold all the
667
+ security properties defined by SPM.
668
+ 14
669
+
670
+ Figure 4:
671
+ The shortest paths searched by Algorithm 2.
672
+ Figure 5:
673
+ The result of executing the pingall instruction.
674
+ 15
675
+
676
+ 1
677
+ e
678
+ es
679
+ e3pi@raspberrypi:/mininet/custom
680
+
681
+ X
682
+ s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11
683
+ *** Adding links:
684
+ (hl, sl) (h2, s2) (h3, s7) (h4, s9) (h5, s10) (h6, sl1) (sl, s2) (sl, s3) (sl, s
685
+ 5) (s2, s4) (s2, s6) (s3, s4) (s3, s7) (s4, s5) (s4, s7) (s4, s8) (s5, s6) (s5,
686
+ s8) (s5. s9) (s6, s9) (s7. s8) (s7. s10) (s8. s9) (s8. s10) (s8. s11) (s9, s11)
687
+ (s10, sl1)
688
+ *** Configuring hosts
689
+ hl h2 h3 h4 h5 h6
690
+ *** starting controller
691
+ c0
692
+ *** starting ll switches
693
+ s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11
694
+ *** Starting CLI:
695
+ mininet> pingall
696
+ *** Ping: testing ping reachability
697
+ hl
698
+ -> X X X h5 X
699
+ h2
700
+ -> x x h4 x x
701
+ 13
702
+ XXXXX<-
703
+ h4 -> x h2 X X
704
+ n5 -> hl x x x X
705
+ X X X X X <- 9u
706
+ *** Results: 86% dropped (4/30 received)
707
+ mininet>
708
+ mininet>5.1.2. Effectiveness after Cutting down the Path
709
+ The purpose of this experiment is to validate whether the security properties
710
+ defined by SPM can be continuously hold in the data plane after the connected
711
+ path between the subject’s host and the object’s host is shut down at runtime.
712
+ If the framework can synchronously perceive this change from the topology
713
+ and automatically find another new connected path to keep the data plane
714
+ holding the security properties at runtime, then the framework will be proven
715
+ to be effective under this scenario. First of all, we let h1 and h5 can access
716
+ each other by loading the corresponding flow entries into the data plane, and
717
+ the shortest path between h1 and h5 is initialed with { h1, e1, e2, e3, h5 }.
718
+ In the following, we make a continuous TCP traffic sent from h1 to h5 by
719
+ using the iperf instruction in the Mininet CLI, we set the duration time of the
720
+ experiment equals 60 seconds and record the throughput of the traffics in h5.
721
+ When the time reaches 23 seconds, we shut down the edge between sw8 and
722
+ sw10 existed in the path P1 by using the instruction in the Mininet CLI and let
723
+ the experiment going on. When the experiment is finished, we read the data
724
+ recorded in h5 and plot them in Figure 6. As shown in Figure 6, the throughput
725
+ of the traffic sent from h1 to h5 is sharply declined after the edge is shut down at
726
+ 23 seconds, and completely becomes zero from 25 seconds to 37 seconds. After
727
+ 38 seconds, the traffic quickly returns to normal until the end of the experiment.
728
+ The experimental result has illustrated that the framework can synchronously
729
+ perceive the change caused by cutting down one edge between sw8 and sw10,
730
+ and automatically find another new shortest connected path between h1 and
731
+ h5, i.e., { h1, e6, e7, e8, h5 }, so as to make the traffic returning to normal
732
+ quickly and keep the data plane holding the security properties defined by SPM
733
+ at runtime. The new shortest connected path searched by Algorithm 2 after
734
+ cutting down the using path is shown in Figure 7.
735
+ 5.1.3. Effectiveness after Changing SPM
736
+ The purpose of this experiment is to validate whether the security properties
737
+ defined by SPM can be continuously hold in the data plane after SPM is changed
738
+ by the user at runtime. If the framework can synchronously perceive this change
739
+ from SPM and automatically update the flow entries deployed in the switches
740
+ to keep the data plane holding the evolved security properties at runtime, then
741
+ the framework will be proven to be effective under this scenario. First of all,
742
+ we let h1 and h5 can access each other by loading a corresponding flow entries
743
+ into the data plane, then make a continuous TCP traffic sent from h1 to h5 by
744
+ using the iperf instruction in the Mininet CLI, we set the duration time of the
745
+ experiment equals 60 seconds and record all the throughput data of the traffic
746
+ in h5.
747
+ In the following, we first validate the effectiveness of the framework under
748
+ the scenario of changing SPM from h1 and h5 can access each other to h2 and h5
749
+ can access each other at runtime, and the experimental result under this scenario
750
+ is plotted in Figure 8. As shown in Figure 8, the throughput of the traffic sent
751
+ from h1 to h5, which is depicted with the red line, is quickly declined when we
752
+ load the new SPM into the controller at 29 seconds, and completely becomes
753
+ 16
754
+
755
+ Figure 6:
756
+ The blue line represents the throughput of the traffic sent from h1 to h5. The
757
+ dotted line represents we shut down the connected path at 23 seconds.
758
+ Figure 7:
759
+ The new connected path searched by Algorithm 2 at runtime.
760
+ 17
761
+
762
+ Throughput (Gbits/sec)
763
+ the traffic from h1 to h5
764
+ -shut down the path at runtime
765
+ 20
766
+ 60
767
+ Time (sec)e6
768
+ eA
769
+ 1
770
+ e7
771
+ es
772
+ es
773
+ eszero after 31 seconds. From 31 seconds until to the end of the experiment, h5
774
+ can only receive the traffic sent from h2 which is depicted with the blue line.
775
+ The experimental result illustrates the framework can synchronously perceive
776
+ this change and keep the data plane holding the evolved security properties after
777
+ changing SPM from h1 and h5 can access each other to h2 and h5 can access
778
+ each other at runtime.
779
+ Figure 8:
780
+ The red line represents the throughput of the traffic sent from h1 to h5, the blue
781
+ line represents the throughput of the traffic sent from h2 to h5, the dotted line represents we
782
+ change SPM at 29 seconds.
783
+ After that, we further validate the effectiveness of the framework under the
784
+ scenario of adding a new relationship into SPM, i.e., adding h2 and h5 can access
785
+ each other, at runtime. The experimental result under this scenario is plotted
786
+ in Figure 9. As shown in Figure 9, the throughput of the traffic sent from h1 to
787
+ h5, which is depicted with the red line, still keeps normal before we load the new
788
+ SPM into the controller at 43 seconds. From 44 seconds until to the end of the
789
+ experiment, h5 can receive the continuous traffic sent from h2 which is depicted
790
+ with the blue line, and can also receive the traffic sent from h1 at the same time.
791
+ Due to the crowding of the traffic sent from h2, the throughput of the traffic
792
+ from h1 is declined from 3.71GB/s to 1.99 GB/s. The throughput of the traffic
793
+ sent from h2 is still kept between 1.6GB/s and 2.2GB/s after 44 seconds. The
794
+ experimental result illustrates the framework can synchronously perceive this
795
+ change and keep the data plane holding the evolved security properties after
796
+ adding a new relationship into SPM at runtime.
797
+ 18
798
+
799
+ Throughput (Gbits/sec)
800
+ the traffic from h1 to h5
801
+ the traffic from h2 to h5
802
+ -load the new SPM at runtime
803
+ Time (sec)Figure 9:
804
+ The red line represents the throughput of the traffic sent from h1 to h5, the blue
805
+ line represents the throughput of the traffic sent from h2 to h5, the dotted line represents we
806
+ change SPM at 43 seconds.
807
+ 5.2. Performance Evaluation
808
+ As the critical algorithm used for implementing the security policy trans-
809
+ formation, the performance of the path searching algorithm, i.e., Algorithm 2,
810
+ needs to be further evaluated. First of all, the sum of access control relation-
811
+ ships of the security policy (SPM) is denoted as M, and the sum of OpenFlow
812
+ switches in the topology is denoted as N in this performance evaluation. Then
813
+ by leveraging the Python programming, the execution time of Algorithm 2 have
814
+ been recorded in milliseconds (ms) for calculating the shortest paths under set-
815
+ ting the different value of M and N.
816
+ The experimental result is plotted in
817
+ Figure 10. As shown in Figure 10, with gradually amplifying the value of M
818
+ from 2 to 10, and the value of N from 11 to 400 respectively, the execution time
819
+ of Algorithm 2 shows an obvious exponential upward trend. Moreover, accord-
820
+ ing to the description of Algorithm 2, the time complexity for calculating only
821
+ one shortest path will reach O(N 2), because the Algorithm 2 needs to create
822
+ a dynamic matrix djk[N][N] and further calculates the while loop, so that the
823
+ time complexity for transforming all the access control relationships defined by
824
+ SPM into their corresponding shortest paths will reach O(M × N 2).
825
+ 6. Conclusion
826
+ In this paper, we propose a practical runtime security policy transforma-
827
+ tion framework for SDN networks. First of all, we specify the security policies
828
+ used by SDN networks, such as access control policies or firewall policies, as a
829
+ 19
830
+
831
+ Throughput (Gbits/sec)
832
+ the traffic from h1 to h5
833
+ 一the traffic from h2 to h5
834
+ -load the new SPM at runtime
835
+ Time (sec)Figure 10:
836
+ The execution time of Algorithm 2 recorded in milliseconds (ms) for calculating
837
+ the shortest paths under setting the different value of M and N, where M represents the sum
838
+ of access control relationships of SPM, N represents the sum of OpenFlow switches in the
839
+ topology.
840
+ system model of security policy (SPM). SPM is of a high-level system model
841
+ without containing any underlying information of data plane. From the theoret-
842
+ ical level, we establish the system model for SDN network and propose a formal
843
+ method to transform SPM into the corresponding flow entries automatically.
844
+ The flow entry transformed from SPM is of a low-level system model containing
845
+ the underlying information of data plane. From the practical level, we propose
846
+ a runtime security policy transformation framework which consists of the se-
847
+ curity policy module, topology discovery module, runtime monitoring module,
848
+ path generation module, as well as flow entry generation module. Leveraging
849
+ these functional modules, the framework can solve the problem of how to find
850
+ a connected path for each relationship defined by SPM in the data plane, how
851
+ to transform the path into the system model of flow entries, as well as how to
852
+ generate the practical flow entries by using the system model of flow entries.
853
+ In order to validate the feasibility and effectiveness of the framework, we set
854
+ up an experimental system and implement the framework by using POX con-
855
+ troller and Mininet emulator. The experimental result illustrate the framework
856
+ is completely effective at runtime.
857
+ However, there still exists some problems needed to be further researched
858
+ in the future. The current path searching algorithm, i.e., Algorithm 2, used
859
+ by the framework is improved from the classic Dijkstra algorithm and finds
860
+ the shortest path by calculating the minimum number of hops in the topology.
861
+ However, in the real SDN networks, the problem of searching a connected path
862
+ between the two hosts need to consider the matters of quality of service (QoS),
863
+ 20
864
+
865
+ 350
866
+ when M =2
867
+ when M =4
868
+ M
869
+ 300-
870
+ when M =6
871
+ when M =8
872
+ when M =10
873
+ 250
874
+ (sw)
875
+ e 200
876
+ Performance
877
+ W
878
+ 150
879
+ P
880
+ 100 -
881
+ 50 -
882
+ +0
883
+ 0
884
+ 50
885
+ 100
886
+ 150
887
+ 200
888
+ 250
889
+ 300
890
+ 350
891
+ 400
892
+ Sum of OpenFlow Switches (N)load balance, as well as some specific requirements about the traffic engineering
893
+ at runtime, so that the framework needs to be further improved by employing
894
+ some novel path searching algorithms based on multi-object optimization[29] or
895
+ reinforcement learning[30] methods.
896
+ Acknowledgment
897
+ This paper has been sponsored and supported by National Key Research
898
+ and Development Program of China (Grant No.2018YFB0803400), Doctoral
899
+ Foundation of Qingdao Binhai University (Grant No. BS2022A10), partially
900
+ supported by Key Program of National Natural Science Foundation of China
901
+ (Grant No.61932013).
902
+ References
903
+ References
904
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+
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1
+ 𝑘-Means SubClustering: A Differentially Private Algorithm
2
+ with Improved Clustering Quality
3
+ Devvrat Joshi1,*,†, Janvi Thakkar1,*,†
4
+ 1Indian Institute of Technology Gandhinagar, India
5
+ Abstract
6
+ In today’s data-driven world, the sensitivity of information has been a significant concern. With this data and additional
7
+ information on the person’s background, one can easily infer an individual’s private data. Many differentially private iterative
8
+ algorithms have been proposed in interactive settings to protect an individual’s privacy from these inference attacks. The
9
+ existing approaches adapt the method to compute differentially private(DP) centroids by iterative Llyod’s algorithm and
10
+ perturbing the centroid with various DP mechanisms. These DP mechanisms do not guarantee convergence of differentially
11
+ private iterative algorithms and degrade the quality of the cluster. Thus, in this work, we further extend the previous work on
12
+ ‘Differentially Private 𝑘-Means Clustering With Convergence Guarantee’ by taking it as our baseline. The novelty of our
13
+ approach is to sub-cluster the clusters and then select the centroid which has a higher probability of moving in the direction
14
+ of the future centroid. At every Lloyd’s step, the centroids are injected with the noise using the exponential DP mechanism.
15
+ The results of the experiments indicate that our approach outperforms the current state-of-the-art method, i.e., the baseline
16
+ algorithm, in terms of clustering quality while maintaining the same differential privacy requirements. The clustering quality
17
+ significantly improved by 4.13 and 2.83 times than baseline for the Wine and Breast_Cancer dataset, respectively.
18
+ Keywords
19
+ differential privacy, 𝑘-means clustering, convergence guarantee
20
+ 1. Introduction
21
+ Achieving extraordinary results is dependent on the data
22
+ on which the machine learning models are trained. Data
23
+ curators have a responsibility to provide datasets such
24
+ that the privacy of data is not compromised. However,
25
+ attackers use other public datasets to perform inference
26
+ and adversarial attacks to get information about an indi-
27
+ vidual in the dataset. Differential privacy is a potential
28
+ technique for giving customers a mathematical guarantee
29
+ of the privacy of their data[1]. There are two fundamen-
30
+ tal settings in which differential privacy is used on data:
31
+ in interactive setting data curator holds the data and re-
32
+ turns the response based on the queries requested by
33
+ third parties; while in non-interactive setting the curator
34
+ sanitized the data before publishing[2].
35
+ Iterative clustering algorithms provide important in-
36
+ sights about the dataset, which helps in a large number of
37
+ applications. They are prone to privacy threats because
38
+ they can reveal information about an individual with ad-
39
+ ditional knowledge. Existing approaches obtain the set
40
+ of centroids using Lloyd’s K-means algorithm, then per-
41
+ turb them with a differentially private mechanism to add
42
+ CIKM-PAS’22: PRIVACY ALGORITHMS IN SYSTEMS (PAS) Workshop,
43
+ Conference on Information and Knowledge Management, October 21,
44
+ 2022, CIKM-PAS
45
+ *Corresponding author.
46
+ †These authors contributed equally.
47
+ � devvrat.joshi@iitgn.ac.in (D. Joshi); janvi.thakkar@iitgn.ac.in
48
+ (J. Thakkar)
49
+ © 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License
50
+ Attribution 4.0 International (CC BY 4.0).
51
+ CEUR
52
+ Workshop
53
+ Proceedings
54
+ http://ceur-ws.org
55
+ ISSN 1613-0073
56
+ CEUR Workshop Proceedings (CEUR-WS.org)
57
+ privacy [3]. In contrast to Lloyd’s K-means algorithm,
58
+ which guarantees convergence, these algorithms do not
59
+ provide any convergence guarantee. Getting differen-
60
+ tially private centroids might not help in getting quality
61
+ inferences because of this non-convergence. We studied
62
+ an existing approach that provides this guarantee and
63
+ converges in twice the number of iterations to Lloyd’s al-
64
+ gorithm while maintaining the same differential privacy
65
+ requirements as existing works [4] [5]. Their algorithm
66
+ perturbs the centroids in a random direction from the
67
+ center of the cluster. However, this lowers the quality of
68
+ clustering, which is necessary for making inferences.
69
+ In this work, we propose a variant of the existing ap-
70
+ proach, which provides better clustering quality while
71
+ using the same privacy budget. We used the intuition
72
+ of Lloyd’s algorithm that the next centroid will move
73
+ in the direction where there is a higher number of data
74
+ points. Finally, we give the mathematical proof that our
75
+ approach at any instance gives better clustering quality
76
+ than the existing approaches. We have tested our ap-
77
+ proach on breat_cancer, wine, iris, and digits datasets.
78
+ We were able to get a significant improvement from the
79
+ previous approach in terms of clustering quality.
80
+ Interactive setting implies that the dataset is not dis-
81
+ closed to the user, however, the data curator returns the
82
+ response of each query received from the user by manip-
83
+ ulating it using DP strategy.
84
+ Our main contribution includes:
85
+ 1. We proposed SubClustering approach which has
86
+ better clustering quality than the baseline (which
87
+ is the current SOTA in terms of clustering qual-
88
+ arXiv:2301.02896v1 [cs.LG] 7 Jan 2023
89
+
90
+ ity). For the Wine and Breast_cancer dataset, the
91
+ clustering quality improved by 4.13 and 2.83 times
92
+ respectively.
93
+ 2. In addition to improving the clustering quality,
94
+ our algorithm used same privacy budget as that
95
+ of the existing work.
96
+ 2. Related Work
97
+ The concept of differential privacy has inspired a plethora
98
+ of studies, particularly in the area of differentially private
99
+ k-means clustering [6][7][8] in an interactive setting. The
100
+ important mechanisms of DP in the literature include:
101
+ the Laplace mechanisms (LapDP) [9], the exponential
102
+ mechanisms (ExpDP) [10], and the sample and aggregate
103
+ framework [11]. To achieve differential privacy, many im-
104
+ plementations included infusing Laplace noise into each
105
+ iteration of Lloyd’s algorithm. The proportion of noise
106
+ added was based on a fixed privacy budget. Some of the
107
+ strategies for allocating privacy budget included splitting
108
+ the overall privacy budget uniformly to each iteration
109
+ [12]. However, this requires us to calculate the number of
110
+ iterations for the convergence, prior to the execution of
111
+ algorithm, thus increasing the computational cost. Fur-
112
+ ther, researchers overcome this weakness by allocating
113
+ theoretically guaranteed optimal allocation method [6],
114
+ but the major assumption taken in this approach was
115
+ that every cluster has the same size, which does not align
116
+ with the real-world datasets. In another work, Mohan
117
+ et al. [8] proposed GUPT, which uses Lloyd’s algorithm
118
+ for local clustering of each bucket where the items were
119
+ uniformly sampled to different buckets. The final result
120
+ was the mean of locally sampled points in each bucket
121
+ with added Laplace noise. But, the clustering quality of
122
+ GUPT was unsatisfying because a large amount of noise
123
+ was added in the aggregation stage.
124
+ Based on the study of past literature on differentially
125
+ private k-means clustering, Zhigang et al. [3] concluded
126
+ that convergence of an iterative algorithm is important to
127
+ the clustering quality. To solve this, they introduced the
128
+ concept of the convergent zone and orientation controller.
129
+ With the help of a convergent zone and orientation con-
130
+ troller, they further create a sampling zone for selecting
131
+ a potential centroid for the 𝑖𝑡ℎ iteration. The approach
132
+ iteratively adds noise with an exponential mechanism
133
+ (ExpDP) by using prior and future knowledge of the po-
134
+ tential centroid at every step of Lloyd’s algorithm. The ap-
135
+ proach maintains the same DP requirements as existing
136
+ literature, with guaranteed convergence and improve-
137
+ ment in clustering quality. However, their algorithm
138
+ perturbs the centroids in a random direction from the
139
+ center of the cluster, degrading the quality of clustering.
140
+ Thus, in this work, we further build upon the approach
141
+ and significantly improve the clustering quality with the
142
+ same epsilon privacy.
143
+ 3. Preliminaries
144
+ The definitions used in this work are briefly discussed
145
+ in this section. The following is a formal definition of
146
+ Differential Privacy:
147
+ Definition 1 (𝜖-DP [9]). A randomised mechanism T
148
+ is 𝜖- differentially private if for all neighbouring datasets
149
+ 𝑋 and 𝑋′ and for an arbitrary answer 𝑠 ∈ 𝑅𝑎𝑛𝑔𝑒(𝑇), T
150
+ satisfies
151
+ 𝑃𝑟[𝑇(𝑋) = 𝑠] ≤ 𝑒𝑥𝑝(𝜖) · 𝑃𝑟[𝑇(𝑋′) = 𝑠],
152
+ where 𝜖 is the privacy budget.
153
+ Here, 𝑋 and 𝑋′ differ by only one item. Smaller val-
154
+ ues of 𝜖 imply a better privacy guarantee. It is because
155
+ the difference between the two neighboring datasets is
156
+ reflected by the privacy budget. In this work, we use the
157
+ ExpDP and LapDP. In exponential DP for non-numeric
158
+ computation, they introduce the concept of scoring func-
159
+ tion 𝑞(𝑋, 𝑥), which represents the effectiveness of the
160
+ pair (𝑋, 𝑥). Here 𝑋 is the dataset and 𝑥 is the response
161
+ to the 𝑞(𝑋, 𝑥) on X.
162
+ The formal definition of Exponential DP mechanism
163
+ is defined as follow:
164
+ Definition
165
+ 2
166
+ (Exponential
167
+ Mechanism
168
+ [10]).
169
+ Given a scoring function of a dataset 𝑋, 𝑞(𝑋, 𝑥),
170
+ which reflects the quality of query respond x.
171
+ The
172
+ exponential mechanism T provides 𝜖-differential privacy,
173
+ if 𝑇(𝑋) = {𝑃𝑟[𝑥] ∝ 𝑒𝑥𝑝( 𝜖·𝑞(𝑋,𝑥)
174
+ 2Δ𝑞
175
+ )}, where ∆𝑞 is the
176
+ sensitivity of scoring function q(X,x), 𝜖 is the privacy
177
+ budget.
178
+ Definition 3 (Convergent & Sampling Zones[3]).
179
+ A region whose points satisfies the condition: { Node S:
180
+ ‖𝑆 − 𝑆𝑖
181
+ (𝑡)‖ < ‖𝑆𝑖
182
+ (𝑡−1) − 𝑆𝑖
183
+ (𝑡)‖} is the convergent zone.
184
+ 𝑆𝑖
185
+ (𝑡) is defined as the mean of 𝐶𝑖
186
+ (𝑡). A sub-region inside
187
+ convergent zone is defined as a sampling zone.
188
+ Definition 4 (Orientation Controller[3]). 𝑋𝑖
189
+ (𝑡) is
190
+ a direction from the center of the convergent zone to a point
191
+ on its circumference. This is the direction along which the
192
+ center of the sampling zone will be sampled, defined as the
193
+ orientation controller.
194
+ 4. Approach
195
+ In this section, we explain our proposed approach and
196
+ the baseline approach.
197
+ 4.1. Overview - KMeans Guarantee
198
+ (Baseline)
199
+ We took "Differentially Private K-Means Clustering with
200
+ Convergence Guarantee" [3] as our baseline and im-
201
+ proved the clustering quality by further building on it.
202
+
203
+ Figure 1: Overview of KMeans Guarantee Approach
204
+ The key concept of the algorithm is to use ExpDP to in-
205
+ troduce bounded noise into centroids at each iteration of
206
+ Lloyd’s algorithm. The technique is designed in a way
207
+ that it ensures the new centroid is different from the cen-
208
+ troid of Lloyd’s algorithm while maintaining constraint
209
+ given in Lemma 1. The constraint guarantees that the
210
+ perturbed centroid will eventually converge with the
211
+ centroid of Lloyd’s algorithm.
212
+ Their algorithm has four main steps to update the
213
+ centroids at each Lloyd step t [3]. The overview of their
214
+ approach can be seen in (Figure : 1).
215
+ 1. Let the differentially private centroid at iteration
216
+ 𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1). Using this centroid,
217
+ run one iteration of Lloyd’s algorithm to get the
218
+ current Lloyd’s centroid 𝑆𝑖
219
+ (𝑡) for each cluster 𝑖.
220
+ 2. Using 𝑆𝑖
221
+ (𝑡) and 𝑆𝑖
222
+ (𝑡−1), generate a conver-
223
+ gent zone for each cluster 𝑖 as described in
224
+ 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3.
225
+ 3. Generate a sampling zone in the convergence zone
226
+ and an orientation controller 𝑋𝑖
227
+ (𝑡) for each cluster
228
+ i as defined in 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3 𝑎𝑛𝑑 4 respectively.
229
+ 4. Sample a differentially private 𝑆𝑖ˆ (𝑡) with ExpDP
230
+ in the sampling zone generated in step 3.
231
+ The definition for the convergent zone (for convergence
232
+ guarantee) and sampling zone (for centroid updating) is
233
+ defined in Definition 3.
234
+ 4.2. Overview - SubCluster Guarantee
235
+ We build upon the KMeans Guarantee algorithm to
236
+ achieve better clustering quality. Our idea differs from
237
+ the baseline in terms of creating a sampling zone. For
238
+ each cluster, we execute Lloyd’s algorithm over its con-
239
+ vergent zone to generate its sub-clustering. Further, we
240
+ assign each sub-cluster with a probability linearly pro-
241
+ portional to the number of points it contains. Finally, we
242
+ sample the sub-cluster based on the assigned probability
243
+ and define it as the sampling zone of the convergent zone.
244
+ Drawing analogy from the KMeans Guarantee algorithm,
245
+ our orientation controller is this sub-clustering and sam-
246
+ pling technique. Intuitively, our algorithm ensures that
247
+ Algorithm 1: Differentially Private 𝑘−Means
248
+ SubClustering Algorithm
249
+ Input: X = {𝑥1, 𝑥2, ...., 𝑥𝑁}: Dataset with N
250
+ data points
251
+ k: number of clusters
252
+ 𝜖𝑒𝑥𝑝: ExpDP privacy budget
253
+ 𝜖𝑙𝑎𝑝: Laplacian privacy budget for the converged
254
+ centroids.
255
+ 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾: number of sub-clusters per cluster
256
+ Output: S: Final clustering centroids
257
+ 1 Select 𝑘 centroids S(0) = (𝑆(0)
258
+ 1 , 𝑆(0)
259
+ 2 , ..., 𝑆(0)
260
+ 𝑘 )
261
+ uniformly from X.
262
+ 2 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝐹𝑜𝑟𝐿𝑙𝑜𝑦𝑑 = number of iterations to
263
+ run the algorithm.
264
+ 3 for iters i in 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝐹𝑜𝑟𝐿𝑙𝑜𝑦𝑑 do
265
+ 4
266
+ for each Cluster i at Iteration t do
267
+ 5
268
+ 𝐶(𝑡)
269
+ 𝑖
270
+ ← assign each 𝑥𝑗 to its closest
271
+ centroid 𝑆𝑖
272
+ 𝑡−1;
273
+ 6
274
+ 𝑆𝑖
275
+ 𝑡 ← centroid of 𝐶𝑖
276
+ 𝑡;
277
+ 7
278
+ 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖
279
+ (𝑡) ← List of data
280
+ points inside the spherical region having
281
+ 𝑆𝑖
282
+ 𝑡 and 𝑆𝑖
283
+ 𝑡−1 as the endpoints of its
284
+ radius.
285
+ 8
286
+ 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖
287
+ (𝑡) ← run Algorithm 2
288
+ using 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖
289
+ (𝑡) ,
290
+ 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾;
291
+ 9
292
+ 𝑆𝑖ˆ (𝑡) ← sample from 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖
293
+ (𝑡)
294
+ using ExpDP with 𝑞 and 𝜖𝑒𝑥𝑝;
295
+ 10
296
+ 𝑆𝑖
297
+ (𝑡) ← 𝑆𝑖ˆ (𝑡)
298
+ 11 Publish: 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖
299
+ (𝑡), 𝑞, 𝜖𝑒𝑥𝑝, 𝑆𝑖
300
+ (𝑡)
301
+ 12 S ← add laplace noise with 𝜖𝑙𝑎𝑝 to S(𝑡);
302
+ Algorithm 2: SubClusterSamplingAlgorithm
303
+ Input: ConvergentZone: Convergent Zone
304
+ internalK: Subclustering K
305
+ Output: 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑡
306
+ 𝑖
307
+ 1 S(𝑡): Mean of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖
308
+ (𝑡)
309
+ 2 ConvergentZoneClusters ← Cluster
310
+ ConvergentZone using Lloyd’s algorithm and
311
+ 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾
312
+ 3 ConvergentZoneProbability ← Assign
313
+ probabilities to the
314
+ ConvergentZoneClusters proportional to
315
+ the number of points inside each cluster.
316
+ 4 SamplingZonei
317
+ (t) ← Sample a cluster from
318
+ the ConvergentZoneClusters using
319
+ ConvergentZoneProbability
320
+ 5 Return: SamplingZonei
321
+ (t);
322
+
323
+ Cluster i at Iteration t (
324
+ 415
325
+ sampling zone
326
+ ()人
327
+ α(t+1)
328
+ convergence zone
329
+ orientation
330
+ controllerFigure 2: Overview of SubCluster Guarantee Approach
331
+ the sampling zone lies towards the region containing a
332
+ higher number of data points in an expected case. With
333
+ this, we guarantee that our differentially private centroid
334
+ moves in the direction where the number of data points
335
+ is higher, incorporating the intuition of Lloyd’s algorithm
336
+ without compromising on the 𝜖-differential privacy. The
337
+ probability of a differentially private centroid at 𝑖 − 1𝑡ℎ
338
+ iteration to move in the direction of a more populated re-
339
+ gion at the 𝑖𝑡ℎ step of Lloyd’s algorithm is also high. Thus,
340
+ we introduce the concept of sub-clustering in the conver-
341
+ gent zone and consequently sample one sub-cluster as
342
+ our sampling zone.
343
+ We sample the centroid from the sampling zone using
344
+ the ExpDP mechanism. Finally, we inject Laplace noise
345
+ in the centroids of the clustering when our algorithm
346
+ converges. It is because the differentially private cen-
347
+ troids obtained are a subset of one of the local minima
348
+ at which Lloyd’s algorithm converges. The overview of
349
+ the proposed approach can be seen in (Figure : 2). We
350
+ show that a randomized iterative algorithm satisfies an
351
+ invariant (given in the claim of Lemma 1) and always
352
+ converges (Proof: refer Lemma 1). Finally, we show
353
+ that the SubCluster algorithm is a randomized iterative
354
+ algorithm that satisfies the invariant(given in Lemma 1)
355
+ (Proof: Refer Lemma 2).
356
+ We have four main steps to update the centroids at
357
+ each Lloyd step t.
358
+ 1. Let the differentially private centroid at iteration
359
+ 𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1). Using this centroid,
360
+ run one iteration of Lloyd’s algorithm to get the
361
+ current Lloyd’s centroid 𝑆𝑖
362
+ (𝑡) for each cluster 𝑖.
363
+ 2. Using 𝑆𝑖
364
+ (𝑡) and 𝑆𝑖
365
+ (𝑡−1), generate a conver-
366
+ gent zone for each cluster 𝑖 as described in
367
+ 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3.
368
+ 3. SubCluster the convergence zone and sample one
369
+ of the sub-cluster as our sampling zone based on
370
+ the probability assigned to each sub-cluster. The
371
+ probability assignment is directly proportional to
372
+ the number of points in each sub-cluster.
373
+ 4. Sample a differentially private 𝑆𝑖ˆ (𝑡) with EXpDP
374
+ in the sampling zone generated in step 3.
375
+ Our approach surpasses the baseline approach in terms
376
+ of clustering quality while maintaining the same DP re-
377
+ quirements as that of the KMeans Guarantee approach,
378
+ which is evident from the results obtained (Figure :
379
+ 3). The better clustering quality is a result of our sub-
380
+ clustering strategy to perturb centroid with a higher prob-
381
+ ability than the baseline approach towards the direction
382
+ of the actual centroid generated by Lloyd’s algorithm.
383
+ The pseudo-code of our approach is shown in the Algo-
384
+ rithm 1 and Algorithm 2.
385
+ Lemma 1: [3] A randomised iterative algorithm
386
+ 𝜏 is convergent if, in 𝐶(𝑡)
387
+ 𝑖 (Cluster i at iteration t),
388
+ 𝑆𝑖ˆ (𝑡)(sampled centroid using 𝜏), 𝑆𝑖
389
+ (𝑡−1)(centroid before
390
+ recentering) and 𝑆𝑖
391
+ (𝑡)(centroid of 𝐶(𝑡)
392
+ 𝑖 ) satisfies the in-
393
+ variant, ||𝑆𝑖ˆ (𝑡)−𝑆𝑖
394
+ (𝑡)|| < ||𝑆𝑖
395
+ (𝑡)−𝑆𝑖
396
+ (𝑡−1)|| in Euclidean
397
+ distance, ∀𝑡, 𝑖.
398
+ We reproduce this lemma from our baseline approach
399
+ [3]. Lemma1 and Lemma 2 together provides the com-
400
+ pleteness and proof for the convergence of our approach.
401
+ If the distance between the sampled centroid 𝑆ˆ(𝑡)
402
+ 𝑖
403
+ from
404
+ the 𝐶(𝑡)
405
+ 𝑖
406
+ and the new centroid 𝑆(𝑡)
407
+ 𝑖
408
+ is less than the dis-
409
+ tance between the new 𝑆(𝑡)
410
+ 𝑖
411
+ and the old centroid 𝑆(𝑡−1)
412
+ 𝑖
413
+ ,
414
+ then the random iterative algorithm will always converge.
415
+ Intuitively, the loss of 𝐶(𝑡)
416
+ 𝑖
417
+ is minimum if the mean of
418
+ 𝐶(𝑡)
419
+ 𝑖
420
+ is taken as centroid. But, if we slightly shift from
421
+ the mean of 𝐶(𝑡)
422
+ 𝑖 , then the loss will increase. However, if
423
+ we can ensure that any sampled point from 𝐶(𝑡)
424
+ 𝑖
425
+ fulfills
426
+ the condition: ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖
427
+ (𝑡)|| < ||𝑆𝑖
428
+ (𝑡) − 𝑆𝑖
429
+ (𝑡−1)||, it
430
+ will lead to a lesser loss than 𝐽𝑆(𝑡−1)
431
+ 𝑖
432
+ , thus, resulting into
433
+ convergence of the randomised iterative algorithm. For
434
+ the mathematical proof, refer [3].
435
+ Lemma 2: Differentially Private 𝑘−Means SubClus-
436
+ tering approach (SubClustering) is a randomised iterative
437
+ algorithm that satisfies the invariant ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖
438
+ (𝑡)|| <
439
+ ||𝑆𝑖
440
+ (𝑡) − 𝑆𝑖
441
+ (𝑡−1)||.
442
+ Proof: SubClustering is an iterative algorithm that
443
+ samples a set of centroids for each iteration with Ex-
444
+ pDP mechanism, thus, making it a randomised itera-
445
+ tive algorithm. It subclusters the points lying inside
446
+ 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡)
447
+ 𝑖 . After subclustering, it samples
448
+ one subcluster (sampling zone) with the assigned proba-
449
+ bilities (linearly proportional to the number of data points
450
+ in subcluster). Finally, it samples a datapoint from the
451
+ sampled subcluster with ExpDP and call it as the cen-
452
+
453
+ Cluster i at iteration t (C(t)
454
+ subclusters
455
+ S;(t-1) > S;(t-1)
456
+ S,(t+7)
457
+ Srt+s
458
+ convergent zone
459
+ ≤.(t)
460
+ sampling zoneFigure 3: Above figures plots the graph between costGap and epsilon budget for two approaches, the baseline as KmeansGuar-
461
+ antee and our approach SubClusterGuarantee. The algorithm was tested on four dataset, Digits (top-left), Wine (top-right),
462
+ Breast Cancer (bottom-left), and Iris (bottom-right) datasets.
463
+ troid of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡)
464
+ 𝑖 . Thus, our sampling zone
465
+ always lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡)
466
+ 𝑖 . Therefore, the
467
+ sampled point lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡)
468
+ 𝑖
469
+ and it sat-
470
+ isfies the invariant ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖
471
+ (𝑡)|| < ||𝑆𝑖
472
+ (𝑡) − 𝑆𝑖
473
+ (𝑡−1)||.
474
+ 5. Experimental Setup
475
+ 5.1. Dataset Used
476
+ We used following four datasets to test our work Sub-
477
+ Cluster Guarantee upon the baseline:
478
+ 1. Iris [13] dataset comprises total of 150 datapoints
479
+ with four features and three classes.
480
+ 2. Wine[13] dataset comprises total of 178 data-
481
+ points with 13 features and three classes.
482
+ 3. Breast Cancer[13] dataset comprises total of
483
+ 569 datapoints with 30 features and two classes.
484
+ 4. Digits[13] dataset comprises of 1797 datapoints
485
+ with 64 dimensions and 10 classes.
486
+ 5.2. Metric for Clustering Quality
487
+ To evaluate the clustering quality, we used the following
488
+ equation to calculate the normalised difference between
489
+ the differentially private algorithms (here, SubCluster
490
+ Guarantee approach) (𝐶𝑜𝑠𝑡𝐷𝑃 ) and Lloyd’s algorithm
491
+ (𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑):
492
+ 𝐶𝑜𝑠𝑡𝐺𝑎𝑝 = |𝐶𝑜𝑠𝑡𝐷𝑃 − 𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑|
493
+ 𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑
494
+ (1)
495
+ The smaller CostGap [3] represents the better quality of
496
+ clustering. In the experiments, we compare the clustering
497
+ quality of SubCluster Guarantee with KMeans Guarantee.
498
+ 6. Results and Discussion
499
+ We tested our algorithm on four datasets. All the datasets
500
+ have different dimensions ranging from 4 to 64 dimen-
501
+ sions and training sets ranging from 150 to 1800. As
502
+ defined in metric smaller gap represents the better clus-
503
+ tering quality. From the (Figure : 3) we can observe
504
+ that, cost gap for all the dataset is smaller or equal to
505
+ the baseline. Thus, it is evident that our algorithm has
506
+ better clustering quality than the existing work for all the
507
+ datasets experimented. We varied internalK (parameter
508
+ for number of sub-clusters) from 2 to 5.
509
+ Each experiment was conducted 30 times in the case
510
+ of the Iris, Wine, and Breast cancer dataset and 10 times
511
+ for digits dataset due to computational constraints. Fi-
512
+ nally, for each dataset, we took the average of all the
513
+ experiments as our final result for plotting the graphs.
514
+
515
+ KmeansGuaranteevsSubClusterGuaranteeDataset:Digits
516
+ KmeansGuaranteevsSubClusterGuaranteeDataset:Wine
517
+ 0.200
518
+ 0.08
519
+ KmeansGuarantee
520
+ KmeansGuarantee
521
+ 0.175
522
+ SubClusterGuarantee
523
+ 0.07
524
+ SubClusterGuarantee
525
+ 0.150
526
+ 0.06
527
+ 0.125
528
+ 0.05
529
+ costGap
530
+ 0.04
531
+ 0.075
532
+ E00
533
+ 0.050
534
+ 0.02
535
+ 0.025
536
+ 0.01
537
+ 0.000
538
+ 0.05
539
+ 0.10
540
+ 0.15
541
+ 0.20
542
+ 0.25
543
+ 0.30
544
+ 0.05
545
+ 0.10
546
+ 0.15
547
+ 0.20
548
+ 0.25
549
+ 0.30
550
+ epsilon
551
+ epsilon
552
+ KmeansGuaranteevsSubClusterGuaranteeDataset:BreastCancer
553
+ KmeansGuaranteevsSubClusterGuaranteeDataset:Iris
554
+ 0.035
555
+ 14
556
+ KmeansGuarantee
557
+ 12
558
+ SubClusterGuarantee
559
+ 0.030
560
+ 10
561
+ 0.025
562
+ KmeansGuarantee
563
+ 8
564
+ SubClusterGuarantee
565
+ 6
566
+ 0.015
567
+ 4
568
+ 0.010
569
+ 2
570
+ 0.005
571
+ 0.05
572
+ 0.10
573
+ 0.15
574
+ 0.20
575
+ 0.25
576
+ 0.30
577
+ 0.05
578
+ 0.10
579
+ 0.15
580
+ 0.20
581
+ 0.25
582
+ 0.30
583
+ epsilon
584
+ epsilonFigure 4: Above figures plots the graph between costGap and epsilon budget for different internalK in SubClusterGuarantee
585
+ Algorithm. The algorithm was tested for internalK=2,3,4,5 for all the four datasets, Digits (top-left), Wine (top-right), Breast
586
+ Cancer (bottom-left), and Iris (bottom-right). Please note: K and internalK are the same parameter
587
+ Comparing the SubCluster Guarantee (proposed ap-
588
+ proach) and K-means Guarantee approach (baseline) by
589
+ taking an average of all the cost gaps for varied epsilon,
590
+ and finally taking the ratio between K-means and Sub-
591
+ Cluster approach:
592
+ 1. In case of Iris dataset, the cost gap is 1.1 times
593
+ smaller than baseline algorithm.
594
+ 2. In case of Wine dataset, the cost gap is 4.13 times
595
+ smaller than baseline algorithm.
596
+ 3. In case of Breast_Cancer dataset, the cost gap
597
+ is 2.83 times smaller than baseline algorithm.
598
+ 4. In case of Digits dataset, the cost gap is almost
599
+ same as that of baseline algorithm.
600
+ 6.1. Detailed Analysis
601
+ 1. Iris: Iris dataset has four dimensions and a very
602
+ small training set of 150 data points. Our al-
603
+ gorithm achieves better clustering quality than
604
+ the baseline algorithm for smaller epsilon values.
605
+ Since the number of data points is less in Iris, the
606
+ impact of sub-clustering reduces, resulting in its
607
+ performance similar to that of the baseline ap-
608
+ proach. From (Figure : 4), we can observe that
609
+ changing the value of intenalK has a small impact
610
+ on the costGap due to a small number of points
611
+ in each sub-cluster. This is because there is a pos-
612
+ sibility that a sub-cluster has no data point when
613
+ internalK is increased causing zero probability
614
+ sub-cluster regions.
615
+ 2. Wine: The wine dataset has 13 dimensions and
616
+ 178 data points in the training set. Our algorithm
617
+ performs significantly better than the baseline, as
618
+ observed in (Figure : 3). It is because the baseline
619
+ algorithm is constrained to choose a theta in any
620
+ abrupt direction ranging from [−𝜋/2, 𝜋/2] as
621
+ shown in (Figure : 1). In contrast, our algorithm
622
+ shifts the centroids in the direction where the fu-
623
+ ture centroid of Lloyd’s algorithm is more likely
624
+ to move (in the expected case). From (Figure : 4),
625
+ it is evident that internalK=4 for the wine dataset
626
+ performs better than the rest of the internalK val-
627
+ ues. Here, the number of dimensions is more than
628
+ Iris. Therefore, the spatial arrangement will be in
629
+ an n-sphere which allows better sub-clustering.
630
+ 3. Breast_Cancer: Breast_Cancer dataset has 569
631
+ data points in its training set and 30 dimensions.
632
+ Our algorithm performs exceptionally better than
633
+ the baseline, with internalK equal to 4. From
634
+ (Figure : 3), we can observe that there is no
635
+ monotonous trend for the costGap. Trends are
636
+ visible in other datasets due to the larger num-
637
+ ber of classification classes, whereas this dataset
638
+ has only two classes. Thus, adding Laplace noise
639
+ does not have a relation to the clustering quality.
640
+ Increasing the internalK improves the clustering
641
+
642
+ VaryinginternalKforSubClusteringDataset:Digits
643
+ VaryinginternalKforSubClusteringDataset:Wine
644
+ K=2
645
+ 0.0035
646
+ K=2
647
+ 0.20
648
+ K=3
649
+ K=3
650
+ K=4
651
+ K=4
652
+ 0.0030
653
+ K=5
654
+ K=5
655
+ 0.15
656
+ 0.0025
657
+ 0.10
658
+ 0.0020
659
+ 0.05
660
+ 0.0015
661
+ 0.00
662
+ 0.0010
663
+ 0.05
664
+ 0.10
665
+ 0.15
666
+ 0.20
667
+ 0.25
668
+ 0.30
669
+ 0.05
670
+ 0.10
671
+ 0.15
672
+ 0.20
673
+ 0.25
674
+ 0.30
675
+ epsilon
676
+ epsilon
677
+ VaryinginternalkforSubClusteringDataset:BreastCancer
678
+ VaryinginternalKforSubClusteringDataset:Iris
679
+ K=2
680
+ 10
681
+ K=2
682
+ 0.025
683
+ K=3
684
+ K=3
685
+ K=4
686
+ K=4
687
+ 0.020
688
+ 8
689
+ K=5
690
+ h
691
+ 0.010
692
+ 0.005
693
+ 0.05
694
+ 0.10
695
+ 0.15
696
+ 0.20
697
+ 0.25
698
+ 0.30
699
+ 0.05
700
+ 0.10
701
+ 0.15
702
+ 0.20
703
+ 0.25
704
+ 0.30
705
+ epsilon
706
+ epsilonquality, with internalK being 4 having the least
707
+ loss. It is because this dataset has a high number
708
+ of dimensions and a larger number of training
709
+ points than other datasets.
710
+ 4. Digits: It has 64 dimensions and 1797 data points
711
+ in the training dataset. Although it has a large
712
+ number of dimensions, our algorithm has a very
713
+ small improvement over the baseline algorithm as
714
+ seen in (Figure : 3). Because of the higher time
715
+ complexity of our algorithm, it is hard to tune
716
+ the internalK parameter. As the number of sam-
717
+ ples in a dataset increases, the internalK should
718
+ increase because a single cluster can contain a
719
+ large number of data points. But, due to limited
720
+ computational resources, we were not able to ex-
721
+ periment with it further. We took internalK to
722
+ be 5 for our experiments as it performed best in
723
+ the range [2, 5] as in the (Figure : 4). One of the
724
+ intriguing findings in the dataset’s results is that
725
+ the curves based on the internalK have a clearly
726
+ evident trend, which is a result of the large num-
727
+ ber of training data points.
728
+ Our proposed algorithm significantly improves over the
729
+ baseline in terms of clustering quality, especially for the
730
+ wine and breast cancer dataset. In addition our algorithm
731
+ maintains the same DP requirements as that of existing
732
+ works.
733
+ 7. Conclusion
734
+ This work presents a novel method for improving the
735
+ clustering quality of differentially private k-means al-
736
+ gorithms while ensuring convergence. The novelty of
737
+ our approach is the sub-clustering of the cluster to select
738
+ the differentially private centroid, which has a higher
739
+ probability of moving in the direction of the next cen-
740
+ troid. We proved that our work surpasses the current
741
+ state-of-the-art algorithms in terms of clustering quality.
742
+ Especially for the Wine and Breast_Cancer dataset, the
743
+ clustering quality was significantly improved by 4.13 and
744
+ 2.83 times than the baseline. In addition, we maintain
745
+ the same DP requirements as that of baseline and other
746
+ existing approaches.
747
+ 8. Future Work
748
+ • In this work, we proved our claim using empirical
749
+ results. We further plan to validate the results
750
+ by providing mathematical bounds for the con-
751
+ vergence degree and rate of the SubClustering
752
+ Lloyd’s algorithm. In terms of clustering qual-
753
+ ity, the proposed algorithm in this work is com-
754
+ pared with k-means guarantee clustering only;
755
+ to prove the effectiveness of our work, we plan
756
+ to experiment with other algorithms in the lit-
757
+ erature including, PrivGene [14], GUPT [8] and
758
+ DWork [7].
759
+ • The DP requirements in this work are the same
760
+ as that of past literature, but in the future, we
761
+ plan to explore ways to improve the current DP
762
+ guarantees while maintaining the same clustering
763
+ quality as in this work.
764
+ • We used Exponential and Laplace mechanisms
765
+ of DP in the proposed approach; we further plan
766
+ to explore the third mechanisms, i.e., sample and
767
+ aggregate framework, by integrating it with the
768
+ current algorithm.
769
+ • In our algorithm, the number of data points inside
770
+ a cluster is variable. Thus we plan to choose an
771
+ internalK, custom to the size of the cluster to
772
+ improve the clustering quality.
773
+ Acknowledgement
774
+ We would like to thank Prof. Anirban Dasgupta
775
+ (IIT Gandhinagar) for his continuous support and
776
+ guidance throughout the research.
777
+ References
778
+ [1] C. Dwork, Differential privacy: A survey
779
+ of results, in: International conference on
780
+ theory and applications of models of com-
781
+ putation, Springer, 2008, pp. 1–19.
782
+ [2] A. Narayanan, Data privacy: The non-
783
+ interactive setting, The University of Texas
784
+ at Austin, 2009.
785
+ [3] Z. Lu, H. Shen,
786
+ Differentially private k-
787
+ means clustering with convergence guar-
788
+ antee, IEEE Transactions on Dependable
789
+ and Secure Computing (2020).
790
+ [4] D. Su, J. Cao, N. Li, E. Bertino, H. Jin, Dif-
791
+ ferentially private k-means clustering, in:
792
+ Proceedings of the sixth ACM conference
793
+ on data and application security and pri-
794
+ vacy, 2016, pp. 26–37.
795
+ [5] J. Lei, Differentially private m-estimators,
796
+ Advances in Neural Information Processing
797
+ Systems 24 (2011).
798
+ [6] D. Su, J. Cao, N. Li, E. Bertino, M. Lyu, H. Jin,
799
+ Differentially private k-means clustering
800
+ and a hybrid approach to private optimiza-
801
+ tion, ACM Transactions on Privacy and
802
+ Security (TOPS) 20 (2017) 1–33.
803
+ [7] C. Dwork, A firm foundation for private
804
+ data analysis, Communications of the ACM
805
+ 54 (2011) 86–95.
806
+
807
+ [8] P. Mohan, A. Thakurta, E. Shi, D. Song,
808
+ D. Culler, Gupt: privacy preserving data
809
+ analysis made easy, in: Proceedings of the
810
+ 2012 ACM SIGMOD International Confer-
811
+ ence on Management of Data, 2012, pp. 349–
812
+ 360.
813
+ [9] C. Dwork, F. McSherry, K. Nissim, A. Smith,
814
+ Calibrating noise to sensitivity in private
815
+ data analysis, in: Theory of cryptography
816
+ conference, Springer, 2006, pp. 265–284.
817
+ [10] F. McSherry, K. Talwar, Mechanism design
818
+ via differential privacy, in: 48th Annual
819
+ IEEE Symposium on Foundations of Com-
820
+ puter Science (FOCS’07), IEEE, 2007, pp. 94–
821
+ 103.
822
+ [11] K. Nissim, S. Raskhodnikova, A. Smith,
823
+ Smooth sensitivity and sampling in private
824
+ data analysis, in: Proceedings of the thirty-
825
+ ninth annual ACM symposium on Theory
826
+ of computing, 2007, pp. 75–84.
827
+ [12] A. Blum, C. Dwork, F. McSherry, K. Nis-
828
+ sim, Practical privacy: the sulq framework,
829
+ in: Proceedings of the twenty-fourth ACM
830
+ SIGMOD-SIGACT-SIGART symposium on
831
+ Principles of database systems, 2005, pp.
832
+ 128–138.
833
+ [13] A. Asuncion, Uci machine learning reposi-
834
+ tory, university of california, irvine, school
835
+ of information and computer sciences,
836
+ http://www. ics. uci. edu/˜ mlearn/MLRepos-
837
+ itory. html (2007).
838
+ [14] J. Zhang, X. Xiao, Y. Yang, Z. Zhang,
839
+ M. Winslett, Privgene: differentially pri-
840
+ vate model fitting using genetic algorithms,
841
+ in: Proceedings of the 2013 ACM SIGMOD
842
+ International Conference on Management
843
+ of Data, 2013, pp. 665–676.
844
+
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1
+ filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf,len=432
2
+ page_content='𝑘-Means SubClustering: A Differentially Private Algorithm with Improved Clustering Quality Devvrat Joshi1,*,†, Janvi Thakkar1,*,† 1Indian Institute of Technology Gandhinagar, India Abstract In today’s data-driven world, the sensitivity of information has been a significant concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
3
+ page_content=' With this data and additional information on the person’s background, one can easily infer an individual’s private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
4
+ page_content=' Many differentially private iterative algorithms have been proposed in interactive settings to protect an individual’s privacy from these inference attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
5
+ page_content=' The existing approaches adapt the method to compute differentially private(DP) centroids by iterative Llyod’s algorithm and perturbing the centroid with various DP mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
6
+ page_content=' These DP mechanisms do not guarantee convergence of differentially private iterative algorithms and degrade the quality of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
7
+ page_content=' Thus, in this work, we further extend the previous work on ‘Differentially Private 𝑘-Means Clustering With Convergence Guarantee’ by taking it as our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
8
+ page_content=' The novelty of our approach is to sub-cluster the clusters and then select the centroid which has a higher probability of moving in the direction of the future centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
9
+ page_content=' At every Lloyd’s step, the centroids are injected with the noise using the exponential DP mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
10
+ page_content=' The results of the experiments indicate that our approach outperforms the current state-of-the-art method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
11
+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
12
+ page_content=', the baseline algorithm, in terms of clustering quality while maintaining the same differential privacy requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
13
+ page_content=' The clustering quality significantly improved by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
14
+ page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
15
+ page_content='83 times than baseline for the Wine and Breast_Cancer dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
16
+ page_content=' Keywords differential privacy, 𝑘-means clustering, convergence guarantee 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
17
+ page_content=' Introduction Achieving extraordinary results is dependent on the data on which the machine learning models are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
18
+ page_content=' Data curators have a responsibility to provide datasets such that the privacy of data is not compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
19
+ page_content=' However, attackers use other public datasets to perform inference and adversarial attacks to get information about an indi- vidual in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
20
+ page_content=' Differential privacy is a potential technique for giving customers a mathematical guarantee of the privacy of their data[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
21
+ page_content=' There are two fundamen- tal settings in which differential privacy is used on data: in interactive setting data curator holds the data and re- turns the response based on the queries requested by third parties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
22
+ page_content=' while in non-interactive setting the curator sanitized the data before publishing[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
23
+ page_content=' Iterative clustering algorithms provide important in- sights about the dataset, which helps in a large number of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
24
+ page_content=' They are prone to privacy threats because they can reveal information about an individual with ad- ditional knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
25
+ page_content=' Existing approaches obtain the set of centroids using Lloyd’s K-means algorithm, then per- turb them with a differentially private mechanism to add CIKM-PAS’22: PRIVACY ALGORITHMS IN SYSTEMS (PAS) Workshop, Conference on Information and Knowledge Management, October 21, 2022, CIKM-PAS Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
26
+ page_content=' †These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
27
+ page_content=' � devvrat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
28
+ page_content='joshi@iitgn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
29
+ page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
30
+ page_content='in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
31
+ page_content=' Joshi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
32
+ page_content=' janvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
33
+ page_content='thakkar@iitgn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
34
+ page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
35
+ page_content='in (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
36
+ page_content=' Thakkar) © 2022 Copyright for this paper by its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
37
+ page_content=' Use permitted under Creative Commons License Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
38
+ page_content='0 International (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
39
+ page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
40
+ page_content=' CEUR Workshop Proceedings http://ceur-ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
41
+ page_content='org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
42
+ page_content='org) privacy [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
43
+ page_content=' In contrast to Lloyd’s K-means algorithm, which guarantees convergence, these algorithms do not provide any convergence guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
44
+ page_content=' Getting differen- tially private centroids might not help in getting quality inferences because of this non-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
45
+ page_content=' We studied an existing approach that provides this guarantee and converges in twice the number of iterations to Lloyd’s al- gorithm while maintaining the same differential privacy requirements as existing works [4] [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
46
+ page_content=' Their algorithm perturbs the centroids in a random direction from the center of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
47
+ page_content=' However, this lowers the quality of clustering, which is necessary for making inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
48
+ page_content=' In this work, we propose a variant of the existing ap- proach, which provides better clustering quality while using the same privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
49
+ page_content=' We used the intuition of Lloyd’s algorithm that the next centroid will move in the direction where there is a higher number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
50
+ page_content=' Finally, we give the mathematical proof that our approach at any instance gives better clustering quality than the existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
51
+ page_content=' We have tested our ap- proach on breat_cancer, wine, iris, and digits datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
52
+ page_content=' We were able to get a significant improvement from the previous approach in terms of clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
53
+ page_content=' Interactive setting implies that the dataset is not dis- closed to the user, however, the data curator returns the response of each query received from the user by manip- ulating it using DP strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
54
+ page_content=' Our main contribution includes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
55
+ page_content=' We proposed SubClustering approach which has better clustering quality than the baseline (which is the current SOTA in terms of clustering qual- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
56
+ page_content='02896v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
57
+ page_content='LG] 7 Jan 2023 ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
58
+ page_content=' For the Wine and Breast_cancer dataset, the clustering quality improved by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
59
+ page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
60
+ page_content='83 times respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
61
+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
62
+ page_content=' In addition to improving the clustering quality, our algorithm used same privacy budget as that of the existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
63
+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
64
+ page_content=' Related Work The concept of differential privacy has inspired a plethora of studies, particularly in the area of differentially private k-means clustering [6][7][8] in an interactive setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The important mechanisms of DP in the literature include: the Laplace mechanisms (LapDP) [9], the exponential mechanisms (ExpDP) [10], and the sample and aggregate framework [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' To achieve differential privacy, many im- plementations included infusing Laplace noise into each iteration of Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The proportion of noise added was based on a fixed privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Some of the strategies for allocating privacy budget included splitting the overall privacy budget uniformly to each iteration [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' However, this requires us to calculate the number of iterations for the convergence, prior to the execution of algorithm, thus increasing the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Fur- ther, researchers overcome this weakness by allocating theoretically guaranteed optimal allocation method [6], but the major assumption taken in this approach was that every cluster has the same size, which does not align with the real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' In another work, Mohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' [8] proposed GUPT, which uses Lloyd’s algorithm for local clustering of each bucket where the items were uniformly sampled to different buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The final result was the mean of locally sampled points in each bucket with added Laplace noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' But, the clustering quality of GUPT was unsatisfying because a large amount of noise was added in the aggregation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Based on the study of past literature on differentially private k-means clustering, Zhigang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' [3] concluded that convergence of an iterative algorithm is important to the clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' To solve this, they introduced the concept of the convergent zone and orientation controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' With the help of a convergent zone and orientation con- troller, they further create a sampling zone for selecting a potential centroid for the 𝑖𝑡ℎ iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The approach iteratively adds noise with an exponential mechanism (ExpDP) by using prior and future knowledge of the po- tential centroid at every step of Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The ap- proach maintains the same DP requirements as existing literature, with guaranteed convergence and improve- ment in clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' However, their algorithm perturbs the centroids in a random direction from the center of the cluster, degrading the quality of clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Thus, in this work, we further build upon the approach and significantly improve the clustering quality with the same epsilon privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Preliminaries The definitions used in this work are briefly discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The following is a formal definition of Differential Privacy: Definition 1 (𝜖-DP [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' A randomised mechanism T is 𝜖- differentially private if for all neighbouring datasets 𝑋 and 𝑋′ and for an arbitrary answer 𝑠 ∈ 𝑅𝑎𝑛𝑔𝑒(𝑇), T satisfies 𝑃𝑟[𝑇(𝑋) = 𝑠] ≤ 𝑒𝑥𝑝(𝜖) · 𝑃𝑟[𝑇(𝑋′) = 𝑠], where 𝜖 is the privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Here, 𝑋 and 𝑋′ differ by only one item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Smaller val- ues of 𝜖 imply a better privacy guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' It is because the difference between the two neighboring datasets is reflected by the privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' In this work, we use the ExpDP and LapDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' In exponential DP for non-numeric computation, they introduce the concept of scoring func- tion 𝑞(𝑋, 𝑥), which represents the effectiveness of the pair (𝑋, 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Here 𝑋 is the dataset and 𝑥 is the response to the 𝑞(𝑋, 𝑥) on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The formal definition of Exponential DP mechanism is defined as follow: Definition 2 (Exponential Mechanism [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Given a scoring function of a dataset 𝑋, 𝑞(𝑋, 𝑥), which reflects the quality of query respond x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The exponential mechanism T provides 𝜖-differential privacy, if 𝑇(𝑋) = {𝑃𝑟[𝑥] ∝ 𝑒𝑥𝑝( 𝜖·𝑞(𝑋,𝑥) 2Δ𝑞 )}, where ∆𝑞 is the sensitivity of scoring function q(X,x), 𝜖 is the privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Definition 3 (Convergent & Sampling Zones[3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' A region whose points satisfies the condition: { Node S: ‖𝑆 − 𝑆𝑖 (𝑡)‖ < ‖𝑆𝑖 (𝑡−1) − 𝑆𝑖 (𝑡)‖} is the convergent zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 𝑆𝑖 (𝑡) is defined as the mean of 𝐶𝑖 (𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' A sub-region inside convergent zone is defined as a sampling zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Definition 4 (Orientation Controller[3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 𝑋𝑖 (𝑡) is a direction from the center of the convergent zone to a point on its circumference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' This is the direction along which the center of the sampling zone will be sampled, defined as the orientation controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Approach In this section, we explain our proposed approach and the baseline approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Overview - KMeans Guarantee (Baseline) We took "Differentially Private K-Means Clustering with Convergence Guarantee" [3] as our baseline and im- proved the clustering quality by further building on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Figure 1: Overview of KMeans Guarantee Approach The key concept of the algorithm is to use ExpDP to in- troduce bounded noise into centroids at each iteration of Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The technique is designed in a way that it ensures the new centroid is different from the cen- troid of Lloyd’s algorithm while maintaining constraint given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The constraint guarantees that the perturbed centroid will eventually converge with the centroid of Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Their algorithm has four main steps to update the centroids at each Lloyd step t [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The overview of their approach can be seen in (Figure : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Let the differentially private centroid at iteration 𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Using this centroid, run one iteration of Lloyd’s algorithm to get the current Lloyd’s centroid 𝑆𝑖 (𝑡) for each cluster 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Using 𝑆𝑖 (𝑡) and 𝑆𝑖 (𝑡−1), generate a conver- gent zone for each cluster 𝑖 as described in 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Generate a sampling zone in the convergence zone and an orientation controller 𝑋𝑖 (𝑡) for each cluster i as defined in 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3 𝑎𝑛𝑑 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Sample a differentially private 𝑆𝑖ˆ (𝑡) with ExpDP in the sampling zone generated in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The definition for the convergent zone (for convergence guarantee) and sampling zone (for centroid updating) is defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Overview - SubCluster Guarantee We build upon the KMeans Guarantee algorithm to achieve better clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Our idea differs from the baseline in terms of creating a sampling zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' For each cluster, we execute Lloyd’s algorithm over its con- vergent zone to generate its sub-clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Further, we assign each sub-cluster with a probability linearly pro- portional to the number of points it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Finally, we sample the sub-cluster based on the assigned probability and define it as the sampling zone of the convergent zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Drawing analogy from the KMeans Guarantee algorithm, our orientation controller is this sub-clustering and sam- pling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Intuitively, our algorithm ensures that Algorithm 1: Differentially Private 𝑘−Means SubClustering Algorithm Input: X = {𝑥1, 𝑥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='., 𝑥𝑁}: Dataset with N data points k: number of clusters 𝜖𝑒𝑥𝑝: ExpDP privacy budget 𝜖𝑙𝑎𝑝: Laplacian privacy budget for the converged centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾: number of sub-clusters per cluster Output: S: Final clustering centroids 1 Select 𝑘 centroids S(0) = (𝑆(0) 1 , 𝑆(0) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=', 𝑆(0) 𝑘 ) uniformly from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 2 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝐹𝑜𝑟𝐿𝑙𝑜𝑦𝑑 = number of iterations to run the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 3 for iters i in 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝐹𝑜𝑟𝐿𝑙𝑜𝑦𝑑 do 4 for each Cluster i at Iteration t do 5 𝐶(𝑡) 𝑖 ← assign each 𝑥𝑗 to its closest centroid 𝑆𝑖 𝑡−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 6 𝑆𝑖 𝑡 ← centroid of 𝐶𝑖 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 7 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖 (𝑡) ← List of data points inside the spherical region having 𝑆𝑖 𝑡 and 𝑆𝑖 𝑡−1 as the endpoints of its radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 8 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖 (𝑡) ← run Algorithm 2 using 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖 (𝑡) , 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 9 𝑆𝑖ˆ (𝑡) ← sample from 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖 (𝑡) using ExpDP with 𝑞 and 𝜖𝑒𝑥𝑝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 10 𝑆𝑖 (𝑡) ← 𝑆𝑖ˆ (𝑡) 11 Publish: 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖 (𝑡), 𝑞, 𝜖𝑒𝑥𝑝, 𝑆𝑖 (𝑡) 12 S ← add laplace noise with 𝜖𝑙𝑎𝑝 to S(𝑡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Algorithm 2: SubClusterSamplingAlgorithm Input: ConvergentZone: Convergent Zone internalK: Subclustering K Output: 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑡 𝑖 1 S(𝑡): Mean of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖 (𝑡) 2 ConvergentZoneClusters ← Cluster ConvergentZone using Lloyd’s algorithm and 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾 3 ConvergentZoneProbability ← Assign probabilities to the ConvergentZoneClusters proportional to the number of points inside each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 4 SamplingZonei (t) ← Sample a cluster from the ConvergentZoneClusters using ConvergentZoneProbability 5 Return: SamplingZonei (t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Cluster i at Iteration t ( 415 sampling zone ()人 α(t+1) convergence zone orientation controllerFigure 2: Overview of SubCluster Guarantee Approach the sampling zone lies towards the region containing a higher number of data points in an expected case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' With this, we guarantee that our differentially private centroid moves in the direction where the number of data points is higher, incorporating the intuition of Lloyd’s algorithm without compromising on the 𝜖-differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The probability of a differentially private centroid at 𝑖 − 1𝑡ℎ iteration to move in the direction of a more populated re- gion at the 𝑖𝑡ℎ step of Lloyd’s algorithm is also high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Thus, we introduce the concept of sub-clustering in the conver- gent zone and consequently sample one sub-cluster as our sampling zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' We sample the centroid from the sampling zone using the ExpDP mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Finally, we inject Laplace noise in the centroids of the clustering when our algorithm converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' It is because the differentially private cen- troids obtained are a subset of one of the local minima at which Lloyd’s algorithm converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The overview of the proposed approach can be seen in (Figure : 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' We show that a randomized iterative algorithm satisfies an invariant (given in the claim of Lemma 1) and always converges (Proof: refer Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Finally, we show that the SubCluster algorithm is a randomized iterative algorithm that satisfies the invariant(given in Lemma 1) (Proof: Refer Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' We have four main steps to update the centroids at each Lloyd step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Let the differentially private centroid at iteration 𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Using this centroid, run one iteration of Lloyd’s algorithm to get the current Lloyd’s centroid 𝑆𝑖 (𝑡) for each cluster 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Using 𝑆𝑖 (𝑡) and 𝑆𝑖 (𝑡−1), generate a conver- gent zone for each cluster 𝑖 as described in 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' SubCluster the convergence zone and sample one of the sub-cluster as our sampling zone based on the probability assigned to each sub-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The probability assignment is directly proportional to the number of points in each sub-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Sample a differentially private 𝑆𝑖ˆ (𝑡) with EXpDP in the sampling zone generated in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Our approach surpasses the baseline approach in terms of clustering quality while maintaining the same DP re- quirements as that of the KMeans Guarantee approach, which is evident from the results obtained (Figure : 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The better clustering quality is a result of our sub- clustering strategy to perturb centroid with a higher prob- ability than the baseline approach towards the direction of the actual centroid generated by Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The pseudo-code of our approach is shown in the Algo- rithm 1 and Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Lemma 1: [3] A randomised iterative algorithm 𝜏 is convergent if, in 𝐶(𝑡) 𝑖 (Cluster i at iteration t), 𝑆𝑖ˆ (𝑡)(sampled centroid using 𝜏), 𝑆𝑖 (𝑡−1)(centroid before recentering) and 𝑆𝑖 (𝑡)(centroid of 𝐶(𝑡) 𝑖 ) satisfies the in- variant, ||𝑆𝑖ˆ (𝑡)−𝑆𝑖 (𝑡)|| < ||𝑆𝑖 (𝑡)−𝑆𝑖 (𝑡−1)|| in Euclidean distance, ∀𝑡, 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' We reproduce this lemma from our baseline approach [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Lemma1 and Lemma 2 together provides the com- pleteness and proof for the convergence of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' If the distance between the sampled centroid 𝑆ˆ(𝑡) 𝑖 from the 𝐶(𝑡) 𝑖 and the new centroid 𝑆(𝑡) 𝑖 is less than the dis- tance between the new 𝑆(𝑡) 𝑖 and the old centroid 𝑆(𝑡−1) 𝑖 , then the random iterative algorithm will always converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Intuitively, the loss of 𝐶(𝑡) 𝑖 is minimum if the mean of 𝐶(𝑡) 𝑖 is taken as centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' But, if we slightly shift from the mean of 𝐶(𝑡) 𝑖 , then the loss will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' However, if we can ensure that any sampled point from 𝐶(𝑡) 𝑖 fulfills the condition: ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖 (𝑡)|| < ||𝑆𝑖 (𝑡) − 𝑆𝑖 (𝑡−1)||, it will lead to a lesser loss than 𝐽𝑆(𝑡−1) 𝑖 , thus, resulting into convergence of the randomised iterative algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' For the mathematical proof, refer [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Lemma 2: Differentially Private 𝑘−Means SubClus- tering approach (SubClustering) is a randomised iterative algorithm that satisfies the invariant ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖 (𝑡)|| < ||𝑆𝑖 (𝑡) − 𝑆𝑖 (𝑡−1)||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Proof: SubClustering is an iterative algorithm that samples a set of centroids for each iteration with Ex- pDP mechanism, thus, making it a randomised itera- tive algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' It subclusters the points lying inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' After subclustering, it samples one subcluster (sampling zone) with the assigned proba- bilities (linearly proportional to the number of data points in subcluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Finally, it samples a datapoint from the sampled subcluster with ExpDP and call it as the cen- Cluster i at iteration t (C(t) subclusters S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='(t-1) > S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='(t-1) S,(t+7) Srt+s convergent zone ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' (t) sampling zoneFigure 3: Above figures plots the graph between costGap and epsilon budget for two approaches, the baseline as KmeansGuar- antee and our approach SubClusterGuarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The algorithm was tested on four dataset, Digits (top-left), Wine (top-right), Breast Cancer (bottom-left), and Iris (bottom-right) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' troid of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Thus, our sampling zone always lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Therefore, the sampled point lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡) 𝑖 and it sat- isfies the invariant ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖 (𝑡)|| < ||𝑆𝑖 (𝑡) − 𝑆𝑖 (𝑡−1)||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Experimental Setup 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Dataset Used We used following four datasets to test our work Sub- Cluster Guarantee upon the baseline: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Iris [13] dataset comprises total of 150 datapoints with four features and three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Wine[13] dataset comprises total of 178 data- points with 13 features and three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Breast Cancer[13] dataset comprises total of 569 datapoints with 30 features and two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Digits[13] dataset comprises of 1797 datapoints with 64 dimensions and 10 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Metric for Clustering Quality To evaluate the clustering quality, we used the following equation to calculate the normalised difference between the differentially private algorithms (here, SubCluster Guarantee approach) (𝐶𝑜𝑠𝑡𝐷𝑃 ) and Lloyd’s algorithm (𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑): 𝐶𝑜𝑠𝑡𝐺𝑎𝑝 = |𝐶𝑜𝑠𝑡𝐷𝑃 − 𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑| 𝐶𝑜𝑠𝑡𝐿𝑙𝑜𝑦𝑑 (1) The smaller CostGap [3] represents the better quality of clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' In the experiments, we compare the clustering quality of SubCluster Guarantee with KMeans Guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Results and Discussion We tested our algorithm on four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' All the datasets have different dimensions ranging from 4 to 64 dimen- sions and training sets ranging from 150 to 1800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' As defined in metric smaller gap represents the better clus- tering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' From the (Figure : 3) we can observe that, cost gap for all the dataset is smaller or equal to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Thus, it is evident that our algorithm has better clustering quality than the existing work for all the datasets experimented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' We varied internalK (parameter for number of sub-clusters) from 2 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Each experiment was conducted 30 times in the case of the Iris, Wine, and Breast cancer dataset and 10 times for digits dataset due to computational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Fi- nally, for each dataset, we took the average of all the experiments as our final result for plotting the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' KmeansGuaranteevsSubClusterGuaranteeDataset:Digits KmeansGuaranteevsSubClusterGuaranteeDataset:Wine 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='08 KmeansGuarantee KmeansGuarantee 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='175 SubClusterGuarantee 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='07 SubClusterGuarantee 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='05 costGap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='075 E00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='30 epsilon epsilon KmeansGuaranteevsSubClusterGuaranteeDataset:BreastCancer KmeansGuaranteevsSubClusterGuaranteeDataset:Iris 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='035 14 KmeansGuarantee 12 SubClusterGuarantee 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='030 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='025 KmeansGuarantee 8 SubClusterGuarantee 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='015 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='010 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='30 epsilon epsilonFigure 4: Above figures plots the graph between costGap and epsilon budget for different internalK in SubClusterGuarantee Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' The algorithm was tested for internalK=2,3,4,5 for all the four datasets, Digits (top-left), Wine (top-right), Breast Cancer (bottom-left), and Iris (bottom-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Please note: K and internalK are the same parameter Comparing the SubCluster Guarantee (proposed ap- proach) and K-means Guarantee approach (baseline) by taking an average of all the cost gaps for varied epsilon, and finally taking the ratio between K-means and Sub- Cluster approach: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' In case of Iris dataset, the cost gap is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='1 times smaller than baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' In case of Wine dataset, the cost gap is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='13 times smaller than baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' In case of Breast_Cancer dataset, the cost gap is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='83 times smaller than baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' In case of Digits dataset, the cost gap is almost same as that of baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Detailed Analysis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Iris: Iris dataset has four dimensions and a very small training set of 150 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Our al- gorithm achieves better clustering quality than the baseline algorithm for smaller epsilon values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Since the number of data points is less in Iris, the impact of sub-clustering reduces, resulting in its performance similar to that of the baseline ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' From (Figure : 4), we can observe that changing the value of intenalK has a small impact on the costGap due to a small number of points in each sub-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' This is because there is a pos- sibility that a sub-cluster has no data point when internalK is increased causing zero probability sub-cluster regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Wine: The wine dataset has 13 dimensions and 178 data points in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Our algorithm performs significantly better than the baseline, as observed in (Figure : 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' It is because the baseline algorithm is constrained to choose a theta in any abrupt direction ranging from [−𝜋/2, 𝜋/2] as shown in (Figure : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' In contrast, our algorithm shifts the centroids in the direction where the fu- ture centroid of Lloyd’s algorithm is more likely to move (in the expected case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' From (Figure : 4), it is evident that internalK=4 for the wine dataset performs better than the rest of the internalK val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Here, the number of dimensions is more than Iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Therefore, the spatial arrangement will be in an n-sphere which allows better sub-clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Breast_Cancer: Breast_Cancer dataset has 569 data points in its training set and 30 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Our algorithm performs exceptionally better than the baseline, with internalK equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' From (Figure : 3), we can observe that there is no monotonous trend for the costGap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Trends are visible in other datasets due to the larger num- ber of classification classes, whereas this dataset has only two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Thus, adding Laplace noise does not have a relation to the clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Increasing the internalK improves the clustering VaryinginternalKforSubClusteringDataset:Digits VaryinginternalKforSubClusteringDataset:Wine K=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='0035 K=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='20 K=3 K=3 K=4 K=4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='0030 K=5 K=5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='30 epsilon epsilon VaryinginternalkforSubClusteringDataset:BreastCancer VaryinginternalKforSubClusteringDataset:Iris K=2 10 K=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='025 K=3 K=3 K=4 K=4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='020 8 K=5 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
331
+ page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
332
+ page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
333
+ page_content='30 epsilon epsilonquality, with internalK being 4 having the least loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
334
+ page_content=' It is because this dataset has a high number of dimensions and a larger number of training points than other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
335
+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
336
+ page_content=' Digits: It has 64 dimensions and 1797 data points in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
337
+ page_content=' Although it has a large number of dimensions, our algorithm has a very small improvement over the baseline algorithm as seen in (Figure : 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
338
+ page_content=' Because of the higher time complexity of our algorithm, it is hard to tune the internalK parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
339
+ page_content=' As the number of sam- ples in a dataset increases, the internalK should increase because a single cluster can contain a large number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
340
+ page_content=' But, due to limited computational resources, we were not able to ex- periment with it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
341
+ page_content=' We took internalK to be 5 for our experiments as it performed best in the range [2, 5] as in the (Figure : 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
342
+ page_content=' One of the intriguing findings in the dataset’s results is that the curves based on the internalK have a clearly evident trend, which is a result of the large num- ber of training data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
343
+ page_content=' Our proposed algorithm significantly improves over the baseline in terms of clustering quality, especially for the wine and breast cancer dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
344
+ page_content=' In addition our algorithm maintains the same DP requirements as that of existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
345
+ page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+ page_content=' Conclusion This work presents a novel method for improving the clustering quality of differentially private k-means al- gorithms while ensuring convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
347
+ page_content=' The novelty of our approach is the sub-clustering of the cluster to select the differentially private centroid, which has a higher probability of moving in the direction of the next cen- troid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
348
+ page_content=' We proved that our work surpasses the current state-of-the-art algorithms in terms of clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
349
+ page_content=' Especially for the Wine and Breast_Cancer dataset, the clustering quality was significantly improved by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
350
+ page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
351
+ page_content='83 times than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
352
+ page_content=' In addition, we maintain the same DP requirements as that of baseline and other existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
353
+ page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
354
+ page_content=' Future Work In this work, we proved our claim using empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
355
+ page_content=' We further plan to validate the results by providing mathematical bounds for the con- vergence degree and rate of the SubClustering Lloyd’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
356
+ page_content=' In terms of clustering qual- ity, the proposed algorithm in this work is com- pared with k-means guarantee clustering only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
357
+ page_content=' to prove the effectiveness of our work, we plan to experiment with other algorithms in the lit- erature including, PrivGene [14], GUPT [8] and DWork [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
358
+ page_content=' The DP requirements in this work are the same as that of past literature, but in the future, we plan to explore ways to improve the current DP guarantees while maintaining the same clustering quality as in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
359
+ page_content=' We used Exponential and Laplace mechanisms of DP in the proposed approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
360
+ page_content=' we further plan to explore the third mechanisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
361
+ page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
362
+ page_content=', sample and aggregate framework, by integrating it with the current algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
363
+ page_content=' In our algorithm, the number of data points inside a cluster is variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
364
+ page_content=' Thus we plan to choose an internalK, custom to the size of the cluster to improve the clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
365
+ page_content=' Acknowledgement We would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
366
+ page_content=' Anirban Dasgupta (IIT Gandhinagar) for his continuous support and guidance throughout the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
367
+ page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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1
+
2
+
3
+
4
+
5
+
6
+ Manuscript Template
7
+ Page 1 of 22
8
+
9
+ FRONT MATTER
10
+
11
+ Title
12
+ • Controlling Electromagnetic Surface Waves with Conformal Transformation Optics
13
+
14
+ Authors
15
+ Xiaoyu Zhao,1† Hong Deng,1† Xiaoke Gao,1† Xikui Ma,1 Tianyu Dong1*
16
+ 1School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China.
17
+
18
+ †These authors contributed equally to this work.
19
+ *To whom correspondence should be addressed; E-mail: tydong@mail.xjtu.edu.cn.
20
+
21
+ Abstract
22
+ The application of transformation optics to the development of intriguing electromagnetic
23
+ devices can produce weakly anisotropic or isotropic media with the assistance of quasi-
24
+ conformal and/or conformal mapping, as opposed to the strongly anisotropic media
25
+ produced by general mappings; however, it is typically limited to two-dimensional
26
+ applications. By addressing the conformal mapping between two manifolds embedded in
27
+ three-dimensional space, we demonstrate that electromagnetic surface waves can be
28
+ controlled without introducing singularity and anisotropy into the device parameters.
29
+ Using fruitful surface conformal parameterization methods, a near-perfect conformal
30
+ mapping between smooth manifolds with arbitrary boundaries can be obtained.
31
+ Illustrations of cloaking and illusions, including surface Luneburg and Eaton lenses and
32
+ black holes for surface waves, are provided. Our work brings the manipulation of surface
33
+ waves at microwave and optical wavelengths one step closer.
34
+
35
+ Teaser
36
+ Waves can be controlled at will on arbitrary open surfaces without holes, showing
37
+ fascinating applications such as invisible bumps for surface waves, reproducing
38
+ scatterings of one bump on other smooth surfaces, and controlling light beams on surfaces
39
+ to focus, to bend and/or to be absorbed akin to black holes without visible scatterings.
40
+
41
+ MAIN TEXT
42
+
43
+ Introduction
44
+ Since its inception in the design of electromagnetic cloaks (1, 2), transformation optics
45
+ (TO) has proven to be a powerful tool for understanding and customizing the physics in
46
+ acoustics (3), optics (4), mechanics (5), thermodynamics (6, 7), etc. Following the
47
+ groundbreaking work of cloaking, a number of other electromagnetic devices have been
48
+ reported within the theoretical framework of TO, such as electromagnetic concentrators
49
+ (8, 9), field rotators (10), optical lenses (11, 12) and optical illusion devices (13, 14). In
50
+ practice, however, traditional TO often yields significant anisotropy in a designed medium
51
+ (15). Thus, metamaterials are often used to infer spatial changes from coordinate
52
+ transformation geometry, which is based on the mathematical equivalence between
53
+ geometry and material (16).
54
+
55
+ Manuscript Template
56
+ Page 2 of 22
57
+
58
+ To reduce the anisotropy of the functional medium induced by TO, various approaches
59
+ have been developed. By constructing mapping in non-Euclidean space, for instance, it is
60
+ possible to remove singular points formed by traditional TO (17), hence minimizing
61
+ anisotropy in part. But for wavelengths comparable to the size of the transform region,
62
+ non-Euclidean TO may perform even worse (18); thus, several research projects focus on
63
+ conformal or quasi-conformal mappings to achieve isotropy (19). In ℝ2, the concept of a
64
+ carpet cloak that resembles a flat ground plane is successfully realized with an isotropic
65
+ medium produced by minimizing the Modified-Liao functional under sliding boundary
66
+ conditions (20), or equivalently by constructing the quasi-conformal mapping via solving
67
+ inverse Laplace's equations (21). Although the concept of carpet cloak has been extended
68
+ to ℝ3 by the extrusion or revolution of a two-dimensional refractive index profile to
69
+ control the reflection of free-space waves, it is only applicable to surfaces with
70
+ translational or rotational symmetry (22).
71
+ Previous research has focused largely on controlling propagating waves by TO, whereas
72
+ less attention has been attached to the manipulation of surface waves (12, 23, 24). Perfect
73
+ surface wave cloaks have been proposed by equating the optical path length of a ray
74
+ traversing a flat plane with a homogeneous refractive index to the optical path on a curved
75
+ surface with an angle-dependent refractive index for two orthogonal paths (25, 26), which
76
+ have been experimentally validated (27). Although an electrically large object may be
77
+ hidden by such a cloak with an inhomogeneous isotropic medium, this approach is limited
78
+ to rotationally symmetric surfaces. By linking the governing eikonal equations on a virtual
79
+ flat plane and on a curved surface by transformation optics, the projection mapping yields
80
+ surface wave cloaks for non-rotationally symmetric geometries but with high anisotropy
81
+ (14, 28). Considerable effort has been devoted to reducing such anisotropy by employing
82
+ efficient numerical conformal algorithms such as boundary first flattening (29), yet only
83
+ non-rotationally symmetric surfaces with circular boundary are investigated (30).
84
+ In this work, we show how to manipulate surface waves on smooth manifolds embedded
85
+ in ℝ3 within the framework of conformal TO, requiring an effective isotropic material
86
+ under the regime of geometrical optics. Fig. 1 illustrates a conformal surface mapping
87
+ between two smooth manifolds in ℝ2 and ℝ3, i.e., 𝑓: ℳ′ → ℳ. The curved manifold ℳ
88
+ shown in Fig. 1A has been 𝑢𝑣-parameterized and the mesh grid can be regarded as the
89
+ mapping result of the Cartesian coordinate system {𝑥′, 𝑦′} in Fig. 1B. When the mapping
90
+ is conformal or quasi-conformal, the face element d𝑆 remains right-angled, indicating that
91
+ elements are just scaled up with little distortion. From the local coordinate systems on d𝑆
92
+ and d𝑆′ (Fig. S5), one can derive the Jacobian matrix 𝐉 of mapping 𝑓 with two singular
93
+ values 𝜎J1 = 𝜎J2 = 𝜎J that state equal scaling in two orthogonal directions (31).
94
+ Consequently, an isotropic cloaking medium distribution 𝑛 = 1/√det(𝐉) = 1/𝜎J may be
95
+ obtained based on the conformal TO (19), representing the ratio of line element d𝑙′ in
96
+ virtual space to the scaled element d𝑙 in physical space for compensating optical path
97
+ length (2). As a result, light propagating on curved ℳ behaves as propagating on flat ℳ′.
98
+ In practice, it is more convenient to describe mesh vertices in ℝ3 in a Cartesian coordinate
99
+ system {𝑥, 𝑦, 𝑧} and the Jacobian derived from the local coordinate system forms an
100
+ asymmetric rank-two matrix 𝐉3×2. In addition, the possible quasi-conformal mappings can
101
+ be measured by the conformality, i.e., the ratio 𝑄 = max(𝜎J1/𝜎J2, 𝜎J2/𝜎J1). A unity ratio
102
+ 𝑄 allows an effective cloaking medium expressed as 𝑛cloak = 1/√𝜎J1𝜎J2 for every face
103
+ element (20).
104
+
105
+ Manuscript Template
106
+ Page 3 of 22
107
+
108
+
109
+ Results
110
+ Having obtained a conformal mapping between the manifolds ℳ ∈ ℝ3 and ℳ′ ∈ ℝ2, we
111
+ first design an isotropic surface wave cloak under the perspective of conformal TO and
112
+ compare its performance with the traditional surface wave cloak with anisotropic medium
113
+ (14). Simulations were conducted on a double-camelback bump with an elliptical base
114
+ profile embedded in ℝ3, as shown in Fig. 2. In comparison with the scattering when the
115
+ surface has no index profile (Fig. S1A), one can observe that the surface wave cloaking is
116
+ successfully achieved by two distinct approaches: one induced by the projection mapping
117
+ proposed in (14) (Fig. 2A) and the other originated from the proposed quasi-conformal
118
+ mapping (Fig. 2B). The corresponding material characteristics for the two types of cloaks
119
+ are displayed in Fig. 2C, indicating that the former is strongly anisotropic while the latter
120
+ is almost isotropic. In addition, the isotropic refractive index 𝑛c,double (the subscript "c"
121
+ denotes the cloak, and "double" denotes the double-camelback bump) ranges from 0.83 to
122
+ 1, which decreases as the bump height rises because a longer geometrical distance need to
123
+ be compensated by a smaller refractive index in order to attain equal optical path length.
124
+ The proposed scheme based on conformal TO has achieved near-perfect surface wave
125
+ cloaking while eliminating the anisotropy in the transformation medium that the
126
+ traditional scheme presents. The distribution of 𝑛c,double in Fig. 2C outlines an
127
+ asymmetric geometric profile, manifesting that the effectiveness of this scheme is
128
+ independent from any symmetry. Such an achievement demands mappings with high
129
+ conformality rather than those bringing large distortion such as the projection mapping
130
+ (14). The numerical method we adopt here (29) can obtain a quasi-conformal mapping
131
+ with 𝑄 < 1.03, as shown in Fig. S1B, which is sufficient for designing an effective
132
+ isotropic cloaking medium distribution.
133
+ As the antithesis of cloaking, optical illusion devices can reproduce the scattering
134
+ characteristics of a specific object on other objects through a transformation medium (13,
135
+ 14). Fig. 3A depicts the surface electromagnetic wave scattered by a single-camelback
136
+ bump ℳ filled with homogeneous material. Traditionally, if one wants to reproduce its
137
+ scattering on a plane region ℳ′, the quasi-conformal mapping for designing the illusion
138
+ device is 𝑓′: ℳ → ℳ′ with a Jacobian matrix 𝚲2×3. Fig. 3B shows the accurately
139
+ recurring scattering characteristics on plane region ℳ′ filled with 𝑛i,plane = 1/√𝜎Λ1𝜎Λ2
140
+ (the subscript "i" denotes the illusion, and "plane" denotes the plane region), where 𝜎Λ1
141
+ and 𝜎Λ2 are singular values of 𝚲2×3. Furthermore, Fig. 3C illustrates that the double-
142
+ camelback bump filled with a carefully designed isotropic medium distribution can
143
+ reproduce the same scattering pattern as shown in Fig. 3A. Such an illusion is realized by
144
+ cascading two conformal mappings described in Fig. S3, i.e., 𝑓1 from ℝ3 (virtual space) to
145
+ ℝ2 (intermediate space), and 𝑓2 from ℝ2 to the ℝ3 (physical space). Thus, the illusion
146
+ medium for the double-camelback bump reads 𝑛i,double = 𝑛i,plane ∙ 𝑛c,double. Fig. 3D
147
+ displays the profiles of 𝑛i,plane (for Fig. 3B) and 𝑛i,double (for Fig. 3C), respectively,
148
+ which range from 1 to 1.25 (𝑛i,plane) and from 0.85 to 1.21 (𝑛i,double).
149
+ The scattering pattern of the single-camelback bump (Fig. 3A) has been successfully
150
+ reproduced on the plane region (Fig. 3B) and on the double-camelback bump (Fig. 3C),
151
+ which demonstrates that the proposed scheme is a general solution to illusion design on
152
+ smooth two-dimensional manifolds. The cascading method to construct mappings between
153
+ manifolds embedded in ℝ3 can even tackle surfaces with different base profiles, since a
154
+
155
+ Manuscript Template
156
+ Page 4 of 22
157
+
158
+ conformal mapping between simply-connected regions in ℝ2 exists according to the
159
+ Riemann mapping theorem (32). Moreover, the quasi-conformal ratios 𝑄 of the two
160
+ mappings for the double-camelback and single-camelback bump are smaller than 1.03
161
+ (Fig. S1B) and 1.012 (Fig. S2C), respectively, implicating that the cascaded mapping
162
+ meets the requirement for high conformality. The range of 𝑛i,single (1 to 1.25) is the
163
+ inverse of that of the cloaking refractive index 𝑛c,single (0.8 to 1) shown in Fig. S2B,
164
+ because the illusion can be regarded as the inverse design of the cloaking such that the
165
+ Jacobian matrices of their corresponding mappings are the Moore–Penrose pseudo-inverse
166
+ of each other (31).
167
+ Now that the wave behavior on the curved manifold can be manipulated flexibly, it is
168
+ natural to consider designing various complicated devices on it, such as surface wave
169
+ Luneburg lens, Eaton lens and black hole for surface waves (12, 23, 33, 34). Traditional
170
+ designs are usually based on spherical or circular profiles with a constant radius. While for
171
+ an elliptical profile without a constant radius, we adopt the distance from the point on the
172
+ ellipse to the center, also the coordinate origin, as the generalized radius, i.e., 𝑅(𝜃) =
173
+ √(𝑎 cos 𝜃)2 + (𝑏 sin 𝜃)2 (35–37). Thus, the refractive index of the considered Luneburg
174
+ lens can be expressed as
175
+
176
+ 𝑛L(𝑟, 𝜃) = √2 − (𝑟/𝑅(𝜃))2,
177
+ (1)
178
+ where 𝑟 = √𝑥2 + 𝑦2 and 𝜃 = arctan(𝑦/𝑥). Similar to the traditional circular Luneburg
179
+ lens, such a distribution retains 𝑛L = 1 on the boundary and 𝑛L = √2 at the center 𝑟 = 0
180
+ (38). Next, the medium distribution for a Luneburg lens on the double-camelback bump
181
+ can be expressed as 𝑛Luneburg = 𝑛c,double ⋅ 𝑛L. As illustrated in Fig. 4A, two Gaussian
182
+ beams with a free-space wavelength 𝜆G = 50 mm are incident along the 𝑥-direction at the
183
+ position ±0.8𝑏 on the 𝑦-direction and reflected by the Luneburg lens to interfere at the
184
+ focus point. The focal distance reads 20𝜆G that is identical to the unit circular Luneburg
185
+ lens. For the Eaton lens, the refractive index 𝑛E reads as
186
+
187
+ 𝑛E(𝑟, 𝜃) = √2𝑅(𝜃)/𝑟 − 1,
188
+ (2)
189
+ which can approach infinity when 𝑟 = 0, leaving a singular point to be cared for. Fig. 4B
190
+ describes that a Gaussian beam going along the 𝑥-direction bends to the inverse 𝑥-
191
+ direction after passing through the Eaton lens on the double-camelback bump. The
192
+ proposed surface wave Luneburg and Eaton lenses may be deployed in optical imaging,
193
+ signal acquisition and novel designs for surface wave microwave antennas. Another
194
+ functional device that can rotate beam propagation is the peripheral of the two-layer
195
+ optical black hole, where light is compelled to travel in a spiral path into the absorbing
196
+ medium at the core. The piece-wise refractive index distribution function 𝑛B can be
197
+ expressed as
198
+
199
+ 𝑛B(𝑟, 𝜃) = {
200
+ 1,
201
+ 𝑟 > 𝑅(𝜃)
202
+ 𝑅(𝜃)/𝑟,
203
+ 𝑟𝑐 ⋅ 𝑅(𝜃) < 𝑟 < 𝑅(𝜃)
204
+ 1/𝑟𝑐 + i𝛾,
205
+ 𝑟 < 𝑟𝑐 ⋅ 𝑅(𝜃)
206
+ ,
207
+ (3)
208
+ where 𝑟𝑐 = 0.4 is the scaling factor of the internal ellipse core compared with the base
209
+ profile and 𝛾 = 0.1 is the loss factor. The refractive index distribution 𝑛Blackhole =
210
+ 𝑛c,double ⋅ 𝑛B on the double-camelback bump is depicted in Fig. 4D. The real part of
211
+ material parameters is matched on the inner boundary, and the imaginary part for
212
+ absorbing energy ranging from 0.083 to 0.097 only exists in the core. The same Gaussian
213
+
214
+ Manuscript Template
215
+ Page 5 of 22
216
+
217
+ beam that was used for the Eaton lens is employed, and the result in Fig. 4C shows that
218
+ the beam bends around 90∘ before it reaches the inner boundary and is absorbed by the
219
+ lossy core without reflection, showing potential application in interference reduction and
220
+ energy harvesting for electronic devices. Note that, the overall sizes of the simulation
221
+ models are larger than ten times the operating wavelength, demonstrating that the
222
+ proposed scheme is capable of managing surface wave behaviors on electrically large
223
+ objects. Moreover, the excellent performance of these functional surface wave devices
224
+ demonstrates that, based on the proposed scheme, a variety of novel devices may be
225
+ realized on smooth curved manifolds, which may facilitate the development of
226
+ miniaturized and integrated photonic devices.
227
+
228
+ Discussion
229
+ Our theory and method are based on geometrical optics. It requires small curvature and
230
+ little variation in wavelength (see (7) and (8) in Materials and Methods), which can be
231
+ expressed as
232
+
233
+ 𝑤 = |∇𝜆| = |∇(𝜆0/𝑛)| = 𝜆0|∇𝑛|/𝑛2 ≪ 1,
234
+ (4)
235
+
236
+ 𝜌 = |𝑅𝑖𝑗|𝜆2 = |𝐾𝑔𝑖𝑗|(𝜆0/𝑛)2 = det(𝑔𝑖𝑗) 𝐾2𝜆0
237
+ 2/𝑛2 = 𝐾2𝜆0
238
+ 2/𝑛6 ≪ 1,
239
+ (5)
240
+ where 𝑅𝑖𝑗 is the Ricci curvature tensor, 𝐾 is the Gaussian curvature, and 𝑔𝑖𝑗 is the metric
241
+ tensor. Both the wavelength index 𝑤 and the curvature index 𝜌 are inversely proportional
242
+ to powers of the refractive index 𝑛. In order to prevent 𝑤 and 𝜌 from increasing
243
+ drastically, a height lower than half of the base radius is favorable, and thereby the optical
244
+ path length can be compensated with a near-unity refractive index. On this basis,
245
+ requirements (4) and (5) demand shorter wavelength 𝜆0 and smoother geometric structure
246
+ to ease the changing rate |∇𝑛| and the Gaussian curvature 𝐾. As a negative example, a
247
+ hemisphere surface wave cloak is reviewed and results are displayed in Fig. S4, whose
248
+ refractive index 𝑛c,sphere is between 0.5 and 1 and the maximum of quasi-conformal ratio
249
+ 𝑄 is smaller than 1.012. The visible scattering appearing in Fig. S4C implies the failure of
250
+ geometrical optics because of the high curvature index 𝜌 > 20 residing in the right-angle
251
+ connection between the hemisphere and the plane, as is depicted in Fig. S4F, and the
252
+ average curvature index 𝜌̅ = 1.57 is also larger than 1. The non-smooth connection causes
253
+ the phase distortion in the backward scattering, and the maximum of the forward
254
+ scattering |𝐸𝑧 − 𝐸b𝑧|max = 0.75 V/m implies a phase difference arcsin (0.75) = 48.6∘
255
+ resulted from the reconstruction of wave fronts. In comparison, Fig. S1C and Fig. S2D
256
+ display the average curvature index 𝜌̅ = 0.54 for double-camelback bump and 𝜌̅ = 0.39
257
+ for single-camelback bump, respectively, both satisfying the requirement (5) and leaving
258
+ near-zero 𝜌 on smooth boundaries. One may notice that the wavelength index 𝑤 for the
259
+ cloaks shown in Fig. S1D, Fig. S2E and Fig. S4E is smaller than unity everywhere
260
+ because it is related to lower powers of 𝜆0 and 𝑛; thus, it is much easier to meet the
261
+ requirement of (4) compared to (5). These selected curvature and wavelength
262
+ characteristics that validate the approximation of geometrical optics are indispensable for
263
+ the excellent performance of electromagnetic devices.
264
+
265
+ The isotropic case that determines the expression of requirements (4) and (5) is based on
266
+ the conformal or quasi-conformal mappings between two-dimensional manifolds.
267
+ Benefiting from the rapid development in conformal parameterization, a series of mapping
268
+ methods can be employed to design surface wave carpet cloak (29, 39, 40). The boundary
269
+ first flattening (BFF) method (29) adopted in our study can establish near-perfect
270
+ conformal mappings not only between smooth manifolds but also surfaces with cuspidal
271
+
272
+ Manuscript Template
273
+ Page 6 of 22
274
+
275
+ points, such as sharp corners and cone singularities, offering exhilarating promise for
276
+ wave manipulation on more complicated surfaces. In addition, there are algorithms aimed
277
+ at constructing quasi-conformal mappings between high-genus manifolds (41, 42), which
278
+ can be used to deal with phase regulation on surfaces with holes. One noteworthy idea is
279
+ to map a high-genus surface to a zero-genus plane region by transforming holes to slits
280
+ (43, 44) that implies the possibility for the scheme conducted in simply-connected regions
281
+ to manipulate wave behaviors on multiply-connected surfaces. By reasonably utilizing
282
+ advanced algorithms for a variety of particular cases, our method has the potential to be a
283
+ universal scheme for controlling surface electromagnetic waves on an arbitrary two-
284
+ dimensional manifold.
285
+
286
+ In summary, we have proposed a general method to manipulate electromagnetic waves on
287
+ smooth two-dimensional manifolds without rotational symmetry by means of a certain
288
+ isotropic refractive index distribution derived from the quasi-conformal mapping. The
289
+ relationship between medium and mappings is induced from the wave equation on the
290
+ manifold under the geometrical optics approximation. Numerical quasi-conformal
291
+ algorithms are introduced to construct mappings between manifolds, and consequent
292
+ functional mediums are validated by cloaking surfaces and generating illusions on plane
293
+ regions. By cascading mappings between ℝ2 and ℝ3 to obtain a mapping between ℝ3, we
294
+ succeed in reproducing the scattering of a surface on another surface. In addition,
295
+ functional devices such as surface Luneburg lenses, surface Eaton lenses, and black holes
296
+ for surface waves are designed based on carpet cloaks. Finally, the indices required by
297
+ geometrical optics are reviewed to demonstrate the validity of the approximation on
298
+ simulation models. Our method paves the way for the regulation of surface
299
+ electromagnetic waves on any two-dimensional manifold, and can be utilized to control
300
+ surface waves in other fields, such as acoustics, mechanics, and thermodynamics.
301
+
302
+ Materials and Methods
303
+ Conformal transformation optics for surface waves
304
+ Wave equation on curved manifold. The concept of transformation medium stems from
305
+ the equivalence between geometry and media. Within the Einstein summation convention,
306
+ the Maxwell's wave equation for the electric field ∇ℳ × ∇ℳ × 𝐄 − 𝜇0𝜀0𝜕𝑡
307
+ 2𝐄 = 0 in free
308
+ space can be expressed as (16)
309
+
310
+ 𝛻𝑗𝛻𝑗𝐸𝑖 − 𝑅𝑖𝑗𝐸𝑗 − 𝑐0
311
+ −2𝜕𝑡
312
+ 2𝐸𝑖 = 0,
313
+ (6)
314
+ where 𝑐0 = 1/√𝜇0𝜀0 is the light velocity in free space; 𝑅𝑖𝑗 is the Ricci tensor of the
315
+ considered geometry ℳ. Supposing that the electromagnetic waves are confined nearby a
316
+ curved surface ℳ embedded in ℝ3 as surface waves, its local plane wave solution reads
317
+ as 𝐸𝑖 = ℰi𝑒i𝜑 with constant complex amplitudes ℰi, where the phase reads as 𝜑 = k ⋅ r −
318
+ 𝜔𝑡 with the wave vector 𝐤 = ∇ℳ𝜑 and angular frequency 𝜔 = −𝜕𝑡𝜑. For surface waves,
319
+ the wave vector k lies in the tangent space of the curved surface ℳ, i.e., 𝐤 ∈ 𝒯(ℳ).
320
+ Thus, (6) can be simplified and approximated in the regime of geometrical optics where
321
+ the wavelength 𝜆 = 2𝜋/𝑘 varies slowly with distance, i.e.,
322
+
323
+ |∇ℳ𝜆| ≪ 1.
324
+ (7)
325
+ In addition, the effective curvature of the curved surface should be small enough
326
+ compared to the wavelength so that the assumption of locally plane waves is valid, i.e.,
327
+
328
+ |𝑅𝑖𝑗|𝜆2 ≪ 1.
329
+ (8)
330
+
331
+ Manuscript Template
332
+ Page 7 of 22
333
+
334
+ As a result, inserting 𝐸𝑖 = ℰi𝑒i𝜑 into (6) and considering that the (spatial and temporal)
335
+ derivatives of ℰi vanish, one can obtain the dispersion relation for the surface wave
336
+ propagating on ℳ, which reads as
337
+
338
+ 𝑘2 = 𝑘𝑗𝑘𝑗 = 𝑔𝑖𝑗𝑘𝑖𝑘𝑗 = 𝜔2/𝑐0
339
+ 2.
340
+ (9)
341
+ Here, 𝑔𝑖𝑗 is the induced metric tensor for the curved surface ℳ, which can be determined
342
+ from the transformation Jacobian matrix from the manifold ℳ′ in ℝ2 to ℳ (31),.
343
+
344
+ Wave equation on a flat plane. Alternatively, if ℳ is flat (i.e., 𝑅𝑖𝑗 = 0) and filled with
345
+ anisotropic medium denoted by relative permeability tensor 𝜇𝑖𝑗, (6) becomes
346
+
347
+ ∇ × ∇ × 𝐄 − 𝜇0𝜀0𝝁 ⋅ 𝜕𝑡
348
+ 2𝐄 = 0.
349
+ (10)
350
+ Suppose that the electromagnetic waves are confined nearby ℳ and the electric field 𝐄 is
351
+ perpendicularly polarized. In a Cartesian coordinate system, if ℳ can be placed into 𝑥𝑦
352
+ plane, we focus on the case that the electric field vector 𝐄 lies in the normal space of the
353
+ flat plane ℳ, i.e., 𝐄 ∈ 𝒩(ℳ), and the global wave solution may read as 𝐸𝑧 = ℰ𝑧𝑒i𝜑.
354
+ Thus, the phase 𝜑 is independent of 𝑧 and the wave vector just lies on the plane as 𝐤 =
355
+ (𝑘𝑥, 𝑘𝑦, 0), because a flat plane is coincident with its tangent space. Since the flat
356
+ manifold ℳ has a zero-curvature tensor, the condition (7) holds naturally. Once the other
357
+ condition (8) that wavelength varies slowly is satisfied, one may disregard the derivatives
358
+ of complex amplitude after inserting 𝐸𝑧 = ℰ𝑧𝑒i𝜑 into (10) and obtain the dispersion
359
+ relation for the surface wave propagating on ℳ, which reads as (𝜇𝑥𝑥𝑘𝑥2 + 2𝜇𝑥𝑦𝑘𝑥𝑘𝑦 +
360
+ 𝜇𝑦𝑦𝑘𝑦2)/det(𝝁) = 𝜔2/𝑐0
361
+ 2. By excluding consideration of the particular polarization, the
362
+ dispersion equation can be recast within the Einstein summation convention as
363
+
364
+ 1
365
+ det(𝝁) 𝜇𝑖𝑗𝑘𝑖𝑘𝑗 =
366
+ 𝜔2
367
+ 𝑐0
368
+ 2.
369
+ (11)
370
+
371
+ Transformation medium and geometry. For electromagnetic waves that behave
372
+ identically on two manifolds, one can obtain the equivalence between geometry and
373
+ material properties by comparing (9) and (11), which yields
374
+
375
+ 𝜇𝑖𝑗
376
+ det(𝝁) = 𝑔𝑖𝑗.
377
+ (12)
378
+ The relative permeability tensor 𝜇𝑖𝑗 actually creates an illusion on the flat plane because a
379
+ spatial point filled with medium 𝝁 is equivalent to be with a metric 𝒈 = det(𝝁)𝝁−1. If the
380
+ local Cartesian coordinate system at this point is aligned along the orthogonal
381
+ eigenvectors of 𝝁, the real and symmetric permeability tensor will reduce to
382
+ diag(𝜇𝑥, 𝜇𝑦, 𝜇𝑧) so that the square of the line element on 𝑥 direction is d𝑠2 = 𝑔𝑥𝑥d𝑥2 =
383
+ 𝜇𝑦𝜇𝑧d𝑥2, which is also the square of optical path length in curved free space. In
384
+ comparison to d𝑠2 = 𝑛𝑥2d𝑥2 on the flat manifold, one can derive 𝑛𝑥2 = 𝜇𝑦𝜇𝑧 and similar
385
+ results on 𝑦 and 𝑧 directions. Consequently, the relationship between the relative
386
+ permeability tensor 𝝁 and the refractive index tensor 𝒏 may be expressed as 𝒏2 =
387
+ det(𝝁)𝝁−1 and one may further obtain
388
+
389
+ 𝒏illustion
390
+ 2
391
+ = 𝒈.
392
+ (13)
393
+ by referring to (12).
394
+
395
+
396
+ Manuscript Template
397
+ Page 8 of 22
398
+
399
+ Surface transformation and TO medium. The metric tensor in equation (13) is induced
400
+ from the mapping 𝑓: ℳ′ → ℳ and can be constructed by the Jacobian matrix 𝐉3×2 as 𝒈 =
401
+ 𝐉T𝐉 (31). Nevertheless, we prefer to associate 𝒏illusion with the Jacobian matrix 𝚲2×3 that
402
+ represents the transformation from ℝ3 (virtual space) to ℝ2 (physical space). Actually, the
403
+ asymmetric Jacobian matrices 𝐉3×2 and 𝚲2×3 can be denoted as the Moore–Penrose
404
+ pseudo-inverse of each other (31), i.e., 𝐉 = 𝚲†, where the superscript '†' denotes pseudo-
405
+ inverse. Thus, one can rewrite the equivalence (13) as
406
+
407
+ 𝒏illustion
408
+ 2
409
+ = 𝒈 = 𝐉T𝐉 = (𝚲𝚲T)−1.
410
+ (14)
411
+ Similar relationship can be obtained for cloaking medium 𝒏cloak and corresponding
412
+ Jacobian matrix 𝐉3×2 from ℝ2 (virtual space) to ℝ3 (physical space) as
413
+
414
+ 𝒏cloak
415
+ 2
416
+ = (𝐉T𝐉)−1.
417
+ (15)
418
+ For the mapping between ℝ3 (Fig. S3), which is formed by cascading two transformations
419
+ between ℝ3 and ℝ2, the consequent medium for the illusion can be recast as the
420
+ combination of the cloaking and illusion refractive index tensors, i.e.,
421
+
422
+ 𝒏illustion
423
+ 2
424
+ = (𝚲1𝚲1
425
+ T)−1 ⋅ (𝐉2
426
+ T𝐉2)−1.
427
+ (16)
428
+ where 𝚲1 and 𝐉2 are Jacobian matrices for mappings 𝑓1 and 𝑓2, as illustrated in Fig. S3,
429
+ respectively. In particular, when the mappings are conformal, the refractive index
430
+ becomes isotropic, and the corresponding Jacobian matrix has two identical singular
431
+ values. By taking the determinants of (14) and (15), the refractive indices can be denoted
432
+ by singular values of Jacobian matrices as 𝑛cloak = 1/𝜎J and 𝑛illusion = 1/𝜎Λ.
433
+
434
+ Discrete conformal mapping and transformation medium
435
+ Review on discrete conformal mapping. It has been demonstrated that an isotropic
436
+ refractive index distribution can be achieved by solving equations for equal optical path
437
+ length only on rotationally-symmetric surfaces (25). As to the non-rotationally symmetric
438
+ cloak, high anisotropy is introduced by the projection mapping that distorts the coordinate
439
+ grid (14). However, numerical algorithms for surface parameterization provide possible
440
+ conformal mappings for arbitrary surfaces. For example, the angle-based flattening (ABF)
441
+ method (45, 46) has been proposed to construct conformal parameterization by
442
+ minimizing a punishing functional to decrease angular distortion while its nonlinearity
443
+ reduces computational efficiency. Also, the so-called least-squares method (LSCM) (47)
444
+ and spectral method (SCP) (48) have been introduced to attain higher efficiency,
445
+ benefiting from their linearity. Their disadvantages are free target boundaries and non-
446
+ bijectivity, whereas we expect a one-to-one mapping that includes every point on physical
447
+ and virtual space with controlled boundaries. Further research, like disk conformal
448
+ mapping (DCM) (40), has been reported as a linear and bijective conformal mapping
449
+ method but with a fixed disk boundary. Not until boundary first flattening (BFF) (29)
450
+ enabled editing boundary as demand were the drawbacks totally eliminated. To deal with
451
+ a certain electromagnetic circumstance, one could choose an appropriate algorithm among
452
+ the preceding techniques (49, 50).
453
+
454
+ Triangulation and Jacobian matrices. Supposing that the conformal mapping reads
455
+ 𝑓1: ℳ2 → ℳ1 (or 𝑓2: ℳ1 → ℳ2) between manifolds ℳ1 ⊂ ℝ3 and ℳ2 ⊂ ℝ2, as shown in
456
+ Fig. S5A, one can find that a simplex 𝒮1 on meshed ℳ1 and its counterpart on meshed ℳ2
457
+ are a pair of similar triangles, which allows 𝒮1 and 𝒮2 to share a same barycentric
458
+ coordinate system. This local coordinate system, as shown in Fig. S5B, can represent any
459
+
460
+ Manuscript Template
461
+ Page 9 of 22
462
+
463
+ point inside the simplex as the linear combination of three vertices and helps quickly
464
+ induce the Jacobian matrix of numerical mappings based on triangular mesh
465
+ parameterization. For example, the location of the point 𝐪(𝑥′, 𝑦′) on 𝒮2 can be expressed
466
+ as 𝑥′ = ∑
467
+ 𝜆𝑖𝑥𝑖
468
+
469
+ 3
470
+ 𝑖=1
471
+ and 𝑦′ = ∑
472
+ 𝜆𝑖𝑦𝑖
473
+
474
+ 3
475
+ 𝑖=1
476
+ with 𝜆1 + 𝜆2 + 𝜆3 = 1, i.e., a linear combination of
477
+ vertices 𝐪1(𝑥1′,𝑦1′), 𝐪2(𝑥2′, 𝑦2′) and 𝐪3(𝑥3′,𝑦3′). For the triangulation mesh, we can
478
+ obtain the barycentric coordinates, which read as
479
+
480
+ 𝜆1 = [(𝑦2
481
+ ′ − 𝑦3
482
+ ′)(𝑥′ − 𝑥3
483
+ ′) + (𝑥3
484
+ ′ − 𝑥2
485
+ ′)(𝑦′ − 𝑦3
486
+ ′)]/det (𝐌),
487
+ (17)
488
+
489
+ 𝜆2 = [(𝑦3
490
+ ′ − 𝑦1
491
+ ′)(𝑥′ − 𝑥3
492
+ ′) + (𝑥1
493
+ ′ − 𝑥3
494
+ ′)(𝑦′ − 𝑦3
495
+ ′)]/det (𝐌),
496
+ (18)
497
+
498
+ 𝜆3 = [(𝑦1
499
+ ′ − 𝑦2
500
+ ′)(𝑥′ − 𝑥2
501
+ ′) + (𝑥2
502
+ ′ − 𝑥1
503
+ ′)(𝑦′ − 𝑦2
504
+ ′)]/det (𝐌),
505
+ (19)
506
+ where det(𝐌) = |(𝐪1 − 𝐪3) × (𝐪2 − 𝐪3)|, with 𝐪𝑖(𝑥𝑖′, 𝑦𝑖′) being the 𝑖-th vertices (𝑖 =
507
+ 1, 2, 3). Here, (17), (18) and (19) show that the barycentric coordinate system (𝜆1, 𝜆2, 𝜆3)
508
+ can be expressed by the Cartesian coordinate system (𝑥′, 𝑦′). Regarding the point
509
+ 𝐩(𝑥, 𝑦, 𝑧) on 𝒮1 ⊂ ℝ3, mapped from the point 𝐪 in ℝ2, we have 𝑥 = ∑
510
+ 𝜆𝑖𝑥𝑖
511
+ 3
512
+ 𝑖=1
513
+ , 𝑦 =
514
+
515
+ 𝜆𝑖𝑦𝑖
516
+ 3
517
+ 𝑖=1
518
+ and 𝑧 = ∑
519
+ 𝜆𝑖𝑧𝑖
520
+ 3
521
+ 𝑖=1
522
+ as the linear combination of 𝐩1(𝑥1, 𝑦1, 𝑧1), 𝐩2(𝑥2, 𝑦2, 𝑧2) and
523
+ 𝐩3(𝑥3, 𝑦3, 𝑧3), since 𝒮1 and 𝒮2 share the same barycentric coordinates 𝜆𝑖. As a result, the
524
+ Jacobian matrix 𝐉3×2 of the mapping from 𝒮2 ⊂ ℝ2 to 𝒮1 ⊂ ℝ3 can be derived according
525
+ to the derivatives of (𝜆1, 𝜆2, 𝜆3) with respect to (𝑥′, 𝑦′), which reads as
526
+
527
+ 𝐉3×2 = (
528
+ 𝜕𝑥′𝑥
529
+ 𝜕𝑦′𝑥
530
+ 𝜕𝑥′𝑦
531
+ 𝜕𝑦′𝑦
532
+ 𝜕𝑥′𝑧
533
+ 𝜕𝑦′𝑧
534
+ ) =
535
+ 1
536
+ det(𝐌) (
537
+ 𝑥1
538
+ 𝑥2
539
+ 𝑥3
540
+ 𝑦1
541
+ 𝑦2
542
+ 𝑦3
543
+ 𝑧1
544
+ 𝑧2
545
+ 𝑧3
546
+ ) (
547
+ 𝑦2
548
+ ′ − 𝑦3′
549
+ 𝑥3
550
+ ′ − 𝑥2
551
+
552
+ 𝑦3
553
+ ′ − 𝑦1
554
+
555
+ 𝑥1
556
+ ′ − 𝑥3
557
+
558
+ 𝑦1
559
+ ′ − 𝑦2
560
+
561
+ 𝑥2
562
+ ′ − 𝑥1
563
+
564
+ ).
565
+ (20)
566
+ In a similar manner, one can derive the Jacobian matrix 𝚲2×3 of the numerical mapping
567
+ from 𝒮1 to 𝒮2; alternatively, one may calculate the Moore–Penrose pseudoinverse of 𝐉3×2
568
+ as 𝚲2×3 (31). By calculating the Jacobian matrices 𝐉3×2 or 𝚲2×3 on each simplex, the
569
+ information of mapping 𝑓1 or 𝑓2 can be fully described.
570
+
571
+ Simulation methods
572
+ FEM simulation. The wave behavior of electromagnetic devices is simulated using the
573
+ finite element method. The geometric model is an optical thin-film waveguide whose
574
+ thickness is less than one fifth of the wavelength. On the outer surfaces of the waveguide,
575
+ the perfect electric conductor (PEC) boundary condition is applied to emulate the
576
+ propagation of the surface wave on a two-dimensional manifold. Thus, the propagation of
577
+ the plane wave or Gaussian beam is restricted within the optical thin film. To mimic an
578
+ open and non-reflecting infinite domain, perfectly matched layers (PMLs) are applied on
579
+ the boundary of the propagating plane. The designed medium is configured to the
580
+ waveguide as a fitting function interpolated from the discrete data set calculated on extra
581
+ dense meshes.
582
+
583
+ References
584
+ 1. J. B. Pendry, D. Schurig, D. R. Smith, Controlling electromagnetic fields. Science. 312,
585
+ 1780–1782 (2006).
586
+ 2. U. Leonhardt, Optical conformal mapping. Science. 312, 1777–1780 (2006).
587
+ 3. H. Chen, C. Chan, Acoustic cloaking in three dimensions using acoustic metamaterials.
588
+ Appl Phys Lett. 91, 183518 (2007).
589
+
590
+ Manuscript Template
591
+ Page 10 of 22
592
+
593
+ 4. X. Wang, H. Chen, H. Liu, L. Xu, C. Sheng, S. Zhu, Self-focusing and the talbot effect in
594
+ conformal transformation optics. Phys Rev Lett. 119, 033902 (2017).
595
+ 5. D. Genov, S. Zhang, X. Zhang, Mimicking celestial mechanics in metamaterials. Nat
596
+ Phys. 5, 687–692 (2009).
597
+ 6. S. Guenneau, C. Amra, D. Veynante, Transformation thermodynamics: cloaking and
598
+ concentrating heat flux. Opt Express. 20, 8207–8218 (2012).
599
+ 7. M. Raza, Y. Liu, E. Lee, Y. Ma, Transformation thermodynamics and heat cloaking: a
600
+ review. J Opt. 18, 044002 (2016).
601
+ 8. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, J. B. Pendry, Design of
602
+ electromagnetic cloaks and concentrators using form-invariant coordinate transformations
603
+ of Maxwell’s equations. Photon Nanostruct Fundam Applic. 6, 87–95 (2008).
604
+ 9. M. M. Sadeghi, S. Li, L. Xu, B. Hou, H. Chen, Transformation optics with Fabry-Pérot
605
+ resonances. Sci Rep. 5, 1–7 (2015).
606
+ 10. Y. Luo, H. Chen, J. Zhang, L. Ran, J. A. Kong, Design and analytical full-wave validation
607
+ of the invisibility cloaks, concentrators, and field rotators created with a general class of
608
+ transformations. Phys Rev B. 77, 125127 (2008).
609
+ 11. J. Perczel, U. Leonhardt, Partial transmutation of singularities in optical instruments. New
610
+ J Phys. 13, 1005–1008 (2008).
611
+ 12. Y. Liu, T. Zentgraf, G. Bartal, X. Zhang, Transformational Plasmon Optics. Nano Lett.
612
+ 10, 1991–1997 (2010).
613
+ 13. Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z.-Q. Zhang, C. Chan, Illusion optics: the optical
614
+ transformation of an object into another object. Phys Rev Lett. 102, 253902 (2009).
615
+ 14. T. McManus, J. Valiente Kroon, S. Horsley, Y. Hao, Illusions and cloaks for surface
616
+ Waves. Sci Rep. 4, 5977 (2014).
617
+ 15. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. Starr, D. R. Smith,
618
+ Metamaterial electromagnetic cloak at microwave frequencies. Science. 314, 977–980
619
+ (2006).
620
+ 16. U. Leonhardt, T. Philbin, Geometry and Light: The Science of Invisibility (Courier
621
+ Corporation, 2010).
622
+ 17. U. Leonhardt, T. Tyc, Broadband invisibility by non-Euclidean cloaking. Science. 323,
623
+ 110–112 (2009).
624
+ 18. T. Tyc, H. Chen, C. T. Chan, U. Leonhardt, Non-Euclidean cloaking for light Waves.
625
+ IEEE J Sel Top Quantum Electron. 16, 418–426 (2010).
626
+ 19. L. Xu, H. Chen, Conformal transformation optics. Nat. Photonics. 9, 15–23 (2015).
627
+ 20. J. Li, J. B. Pendry, Hiding under the carpet: a new strategy for cloaking. Phys Rev Lett.
628
+ 101, 203901 (2008).
629
+
630
+ Manuscript Template
631
+ Page 11 of 22
632
+
633
+ 21. C. Zheng, X. Zhou, J. Hu, G. Hu, Design method for quasi-isotropic transformation
634
+ materials based on inverse Laplace’s equation with sliding boundaries. Opt Express. 18,
635
+ 6089–6096 (2010).
636
+ 22. N. Landy, N. Kundtz, D. Smith, Designing three-dimensional transformation optical
637
+ media using quasiconformal coordinate transformations. Phys Rev Lett. 105, 193902
638
+ (2010).
639
+ 23. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, X. Zhang, Plasmonic Luneburg and
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+ Eaton lenses. Nat. Nanotechnol. 6, 151–155 (2011).
641
+ 24. A. Vakil, N. Engheta, Transformation optics using graphene. Science. 332, 1291–4
642
+ (2011).
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+ 25. R. C. Mitchell Thomas, T. M. McManus, O. Quevedo-Teruel, S. A. R. Horsley, Y. Hao,
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+ Perfect surface wave cloaks. Phys Rev Lett. 111, 213901 (2013).
645
+ 26. R. Mitchell Thomas, O. Quevedo Teruel, J. Sambles, A. Hibbins, Omnidirectional surface
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+ wave cloak using an isotropic homogeneous dielectric coating. Sci Rep. 6, 30984 (2016).
647
+ 27. L. La Spada, T. McManus, A. Dyke, S. Haq, L. Zhang, Q. Cheng, Y. Hao, Surface wave
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+ cloak from graded refractive index nanocomposites. Sci Rep. 6, 29363 (2016).
649
+ 28. T. M. McManus, L. L. Spada, Y. Hao, Isotropic and anisotropic surface wave cloaking
650
+ techniques. J Opt. 18, 044005 (2016).
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+ 29. R. Sawhney, K. Crane, Boundary first flattening. ACM Trans Graph. 37 (2017),
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+ doi:10.1145/3132705.
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+ 30. L. Xu, H. Ge, J. Li, R. He, H. Chen, Conformal landscape of a two-dimensional gradient
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+ refractive-index profile for geometrical optics. Phys Rev Appl. 13 (2020),
655
+ doi:10.1103/PhysRevApplied.13.054007.
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+ 31. K. Hormann, B. Levy, A. Sheffer, Mesh parameterization: theory and practice. ACM
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+ SIGGRAPH Int Conf Comput Graph Interact Techn. 2 (2008),
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+ doi:10.1145/1508044.1508091.
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+ 32. M. Floater, K. Hormann, Advances in multiresolution for geometric modelling (Springer
660
+ Berlin Heidelberg, New York, ed. 1, 2005).
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+ 33. E. Narimanov, A. Kildishev, Optical black hole: broadband omnidirectional light
662
+ absorber. Appl Phys Lett. 95, 041106 (2009).
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+ 34. J. A. Dockrey, M. J. Lockyear, S. J. Berry, S. A. R. Horsley, J. R. Sambles, A. P. Hibbins,
664
+ Thin metamaterial Luneburg lens for surface waves. Phys Rev B. 87, 125137 (2013).
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+ 35. J. A. Grzesik, Focusing properties of a three-parameter class of oblate, Luneburg-like
666
+ inhomogeneous lenses. J Electromagn. 19, 1005–1019 (2005).
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+ 36. A. Demetriadou, Y. Hao, Slim Luneburg lens for antenna applications. Opt Express. 19,
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+ 19925–34 (2011).
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+
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+ Manuscript Template
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+ Page 12 of 22
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+
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+ 37. J. Gómez Correa, V. Coello, A. Garza Rivera, P. Puente, S. Chávez-Cerda, Three-
674
+ dimensional ray tracing in spherical and elliptical generalized Luneburg lenses for
675
+ application in the human eye lens. Appl Opt. 55, 2002 (2016).
676
+ 38. R. Luneburg, E. Wolf, M. Herzberger, Mathematical theory of optics (Univ. California,
677
+ 1964).
678
+ 39. B. Springborn, P. Schröder, U. Pinkall, "Conformal equivalence of triangle meshes" in
679
+ ACM SIGGRAPH 2008 papers (Association for Computing Machinery, 2008), pp. 1–11.
680
+ 40. P. T. Choi, L. M. Lui, Fast disk conformal parameterization of simply-connected open
681
+ surfaces. J Sci Comput. 65, 1065–1090 (2015).
682
+ 41. M. Zhang, Y. Li, W. Zeng, X. Gu, Canonical conformal mapping for high genus surfaces
683
+ with boundaries. Comput Graph. 36, 417–426 (2012).
684
+ 42. G. Choi, Efficient conformal parameterization of multiply-connected surfaces using quasi-
685
+ conformal theory. J Sci Comput. 87 (2021), doi:10.1007/s10915-021-01479-y.
686
+ 43. X. Yin, J. Dai, S.-T. Yau, X. Gu, "Slit map: conformal parameterization for multiply
687
+ connected surfaces" in Geometric modeling and processing conference (2008), pp. 410–
688
+ 422.
689
+ 44. A. W. K. Sangawi, A. Murid, K. W. Lee, Circular slit maps of multiply connected regions
690
+ with application to brain image processing. Bull Malays. Math Sci Soc. 44 (2020),
691
+ doi:10.1007/s40840-020-00942-7.
692
+ 45. A. Sheffer, E. de Sturler, Parameterization of faceted surfaces for meshing using angle-
693
+ based flattening. Eng Comput. 17, 326–337 (2001).
694
+ 46. A. Sheffer, B. Lévy, M. Mogilnitsky, A. Bogomyakov, ABF++: fast and robust angle
695
+ based flattening. ACM Trans Graph. 24, 311–330 (2005).
696
+ 47. B. Lévy, S. Petitjean, N. Ray, J. Maillot, Least squares conformal maps for automatic
697
+ texture atlas generation. ACM Trans Graph. 21, 362–371 (2002).
698
+ 48. P. Mullen, Y. Tong, P. Alliez, M. Desbrun, Spectral conformal parameterization. Comp
699
+ Graph Forum. 27 (2008), doi:10.1111/j.1467-8659.2008.01289.x.
700
+ 49. M. Botsch, L. Kobbelt, M. Pauly, P. Alliez, B. Lévy, Polygon Mesh Processing (A. K.
701
+ Peters, Ltd., Natick, MA, 2010).
702
+ 50. X. Gu, F. Luo, S.-T. Yau, Computational Conformal Geometry Behind Modern
703
+ Technologies. Am Math Soc. 67, 1 (2020).
704
+
705
+ Acknowledgments
706
+ Funding: This work was supported by the National Natural Science Foundation of China
707
+ (NSFC) under grant no. 51977165.
708
+
709
+ Author contributions:
710
+
711
+ Conceptualization: X.Z., X.M., T.D.
712
+
713
+ Manuscript Template
714
+ Page 13 of 22
715
+
716
+
717
+ Methodology: X.Z., H.D., X.G., T.D.
718
+
719
+ Investigation: X.Z., H.D., X.G.
720
+
721
+ Visualization: X.Z., H.D., X.G.
722
+
723
+ Supervision: X.M., T.D.
724
+
725
+ Writing—original draft: All authors.
726
+
727
+ Writing—review & editing: All authors.
728
+
729
+ Competing interests: The authors declare that they have no competing interests.
730
+
731
+ Data and materials availability: All data needed to evaluate the conclusions in the paper
732
+ are present in the paper and/or the Supplementary Materials. Raw data and corresponding
733
+ simulation data are available upon request.
734
+
735
+ Figures and Tables
736
+
737
+ Fig. 1. The conformal mapping between manifolds. (A) A light beam crossing a curved
738
+ two-dimensional manifold ℳ embedded in ℝ3. (B) A light beam crossing a flat
739
+ two-dimensional manifold ℳ′ in ℝ2. The manifold ℳ is 𝑢𝑣-parameterized and
740
+ both manifolds are plotted with coordinate grid. One can obtain the manifold ℳ in
741
+ (A) from ℳ′ in (B) through a certain analytic or numerical mapping 𝑓: ℳ′ → ℳ.
742
+
743
+
744
+ A
745
+ McR3
746
+ u
747
+ V
748
+ f: M'-M
749
+ B
750
+ M' c R?
751
+ XManuscript Template
752
+ Page 14 of 22
753
+
754
+
755
+ Fig. 2. The field and medium distribution for cloaks. Normalized electric field
756
+ distribution of surface electromagnetic wave cloaks achieved by (A) anisotropic
757
+ relative permeability and (B) isotropic refractive index. (C) Components of
758
+ anisotropic relative permeability, 𝜇𝑥𝑥, 𝜇𝑦𝑦 and 𝜇𝑥𝑦, applied in (A) and isotropic
759
+ refractive index 𝑛c,double applied in (B). The excitation is a 𝑧-polarized plane wave
760
+ with a magnitude of |𝐸𝑧| = 1 V/m; and the wavelength in free-space is 𝜆0 =
761
+ 20 mm. The bump with a height of 1.25𝜆0 is located in the center of the square
762
+ waveguide with a width of 12𝜆0. For the elliptical boundary, the semi-minor and
763
+ semi-major axis length are 𝑎 = 3.75𝜆0 and 𝑏 = 5𝜆0, respectively, along with 𝑥-
764
+ and 𝑦-axes.
765
+
766
+
767
+
768
+ A
769
+ 12^o
770
+ Ez
771
+ B
772
+ c
773
+ μxx
774
+ Py
775
+ μxy
776
+ Nc.double
777
+ 1
778
+ 0.63
779
+ 0.63
780
+ 1
781
+ 0.19
782
+ 0.19
783
+ 0.83Manuscript Template
784
+ Page 15 of 22
785
+
786
+
787
+ Fig. 3. The field and medium distribution for illusions. Normalized electric field
788
+ distribution of surface electromagnetic wave scattering. (A) Scattering on the
789
+ single-camelback bump when filled with homogeneous medium. (B) Illusion of the
790
+ single-camelback bump appearing on the plane. (C) Illusion of the single-
791
+ camelback bump appearing on the double-camelback bump. (D) Isotropic
792
+ refractive indices: 𝑛i,plane for the elliptic region in (B) and 𝑛i,double for the double-
793
+ camelback bump in (C). The elliptical base profiles in (A), (B) and (C) are the
794
+ same.
795
+
796
+ A
797
+ 15^o
798
+ B
799
+ c
800
+ D
801
+ ni,plane
802
+ ni.double
803
+ 1.25
804
+ 0.85
805
+ 1.21Manuscript Template
806
+ Page 16 of 22
807
+
808
+
809
+
810
+ Fig. 4. The field and medium distribution for devices. Normalized electric field
811
+ distribution on surface electromagnetic wave devices. (A) Luneburg lens; (B)
812
+ Eaton lens; and (C) Black hole. Gaussian beam is applied to demonstrate their
813
+ functions. (D) Isotropic refractive indices; 𝑛Luneburg for Luneburg lens in (A),
814
+ decimal logarithm of 𝑛Eaton for Eaton lens in (B), real and imaginary part of
815
+ 𝑛Blackhole for black hole in (C).
816
+
817
+
818
+
819
+
820
+ 12Ao
821
+ A
822
+ focus
823
+ [EI2
824
+ B
825
+ c
826
+ D
827
+ nLuneburg
828
+ log1o(nEaton)
829
+ real(nglaothole)
830
+ imag(nglaochole)
831
+ 1.37
832
+ 0
833
+ 8
834
+ 2.44
835
+ 0.083
836
+ 0.097Manuscript Template
837
+ Page 17 of 22
838
+
839
+ Supplementary Materials for
840
+
841
+
842
+ Controlling Electromagnetic Surface Waves with Conformal
843
+ Transformation Optics
844
+
845
+ Xiaoyu Zhao et al.
846
+
847
+ *Corresponding author. Email: tydong@mail.xjtu.edu.cn.
848
+
849
+
850
+
851
+
852
+
853
+ This PDF file includes:
854
+
855
+ Figs. S1 to S5
856
+
857
+
858
+
859
+ Manuscript Template
860
+ Page 18 of 22
861
+
862
+
863
+ Fig. S1. (A) Normalized electric field distribution of surface electromagnetic wave scattering on
864
+ double-camelback bump when filled with homogeneous medium. (B) Quasi-conformal ratio 𝑸 of
865
+ the mapping applied to design the cloak shown in Fig. 2A. (C) Curvature index 𝝆. (D)
866
+ Wavelength index 𝒘.
867
+
868
+
869
+
870
+ A
871
+ Ez
872
+ B
873
+ C
874
+ D
875
+ p = 0.54
876
+ Q
877
+ d
878
+ W
879
+ 1.028
880
+ 0
881
+ 10.2
882
+ 0
883
+ 0.24Manuscript Template
884
+ Page 19 of 22
885
+
886
+
887
+ Fig. S2. (A) Normalized electric field distribution of the surface electromagnetic wave cloak on
888
+ single-camelback bump achieved by (B) isotropic refractive index 𝒏𝐜,𝐬𝐢𝐧𝐠𝐥𝐞. (C) Quasi-conformal
889
+ ratio 𝑸 of the mapping applied to design the cloak shown in (A). (D) Curvature index 𝝆. (E)
890
+ Wavelength index 𝒘.
891
+
892
+
893
+
894
+ A
895
+ 12^o
896
+ B
897
+ C
898
+ D
899
+ E
900
+ nc.single
901
+ p=0.39
902
+ Q
903
+ d
904
+ W
905
+ 0.8
906
+ 1.011
907
+ 0
908
+ 14.5
909
+ 0
910
+ 0.21Manuscript Template
911
+ Page 20 of 22
912
+
913
+
914
+ Fig. S3. A quasi-conformal mapping between two manifolds embedded in ℝ𝟑 constructed by
915
+ cascading two mappings between ℝ𝟑 and ℝ𝟐. (A) A single-camelback manifold 𝓜𝟏 embedded in
916
+ ℝ𝟑. (B) The plane region 𝓜𝟐 in ℝ𝟐 mapped from 𝓜𝟏 through mapping 𝒇𝟏. (C) The double-
917
+ camelback manifold 𝓜𝟑 embedded in ℝ𝟑 mapped from 𝓜𝟐 through mapping 𝒇𝟐.
918
+
919
+
920
+
921
+ A
922
+ M1 CR3
923
+ fi : M1 → M2
924
+ B
925
+ M2 C IR2
926
+ fz : M2 → M3
927
+ c
928
+ M3 CR3Manuscript Template
929
+ Page 21 of 22
930
+
931
+
932
+ Fig. S4. (A) Normalized electric field 𝑬𝒛, (B) background field 𝑬𝐛𝒛 and (C) scattering field 𝑬𝒛 −
933
+ 𝑬𝐛𝒛 of the hemisphere surface wave cloak achieved by (D) isotropic refractive index 𝒏𝐜,𝐬𝐩𝐡𝐞𝐫𝐞. (E)
934
+ Quasi-conformal ratio 𝑸 of the mapping applied to design the cloak shown in (A). (F) Curvature
935
+ index 𝝆. (G) Wavelength index 𝒘. The radius of the hemisphere is 𝟓𝝀𝟎.
936
+
937
+
938
+
939
+ A
940
+ Ez
941
+ 1
942
+ 15^o
943
+ B
944
+ Epz
945
+ Ez- Ebz
946
+ c
947
+ 0.75
948
+ 0.75
949
+ D
950
+ E
951
+ F
952
+ G
953
+ p = 1.57
954
+ nc.sphere
955
+ W
956
+ 0.5
957
+ 1.012
958
+ 0
959
+ 35
960
+ 0
961
+ 0.2Manuscript Template
962
+ Page 22 of 22
963
+
964
+
965
+ Fig. S5. (A) Simplices 𝓢𝟏 and 𝓢𝟐 as triangle elements in the mesh of double-camelback manifold
966
+ 𝓜𝟏 embedded in ℝ𝟑 and the region 𝓜𝟐 in ℝ𝟐, related by quasi-conformal mappings 𝒇𝟏 and 𝒇𝟐.
967
+ (B) The same barycentric coordinates on simplices 𝓢𝟏 and 𝓢𝟐.
968
+
969
+
970
+ A
971
+ B
972
+ Pi(X1,Y1,Z1)
973
+ Si
974
+ MICR3
975
+ P=入1P1+22P2+2P3
976
+ p(x,y,z)
977
+ Si
978
+ P3(X3,Y3,Z3)
979
+ P2(X2,Y2,Z2)
980
+ f2:Mi→M2
981
+ f:M2-→M1
982
+ Barycentric
983
+ Coordinates
984
+ =A,(x,y-x(x.y,z)
985
+ q1(xi,yi)
986
+ S2
987
+ M2CR2
988
+ q=191+22+3
989
+ q(x,y)
990
+ S2
991
+ q3(x3,y3)
992
+ q2 (x2,y2)
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1
+ arXiv:2301.00590v1 [astro-ph.IM] 2 Jan 2023
2
+ End-to-end simulations of a near-infrared pyramid sensor on
3
+ Keck II
4
+ C. Planteta, G. Agapitoa, C. Giordanoa, S. Espositoa, P. Wizinowichb, and C. Bondc
5
+ aINAF - Osservatorio di Arcetri, 50125 Firenze, Italy
6
+ bW. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743, USA
7
+ cInstitute for Astronomy, University of Hawaii, 640 N. Aohoku Place, Hilo, HI 96720
8
+ ABSTRACT
9
+ The future upgrade of Keck II telescope’s adaptive optics system will include a pyramid wavefront sensor working
10
+ in the near-infrared (J and H band). It will benefit from the recently developed avalanche photodiode arrays,
11
+ specifically the SAPHIRA (Selex) array, which provides a low noise (ă 1 e- at high frame rates). The system will
12
+ either work with a natural guide star (NGS) in a single conjugated adaptive optics system, or in a laser guide
13
+ star (LGS) mode. In this case, the pyramid would be used as a low-order sensor only. We report on a study
14
+ of the pyramid sensor’s performance via end-to-end simulations, applied to Keck’s specific case. We present the
15
+ expected Strehl ratio with optimized configurations in NGS mode, and the expected residual on low orders in
16
+ LGS mode. In the latter case, we also compare the pyramid to LIFT, a focal-plane sensor, demonstrating the
17
+ ability of LIFT to provide a gain of about 2 magnitudes for low-order sensing.
18
+ Keywords: Adaptive optics, Wavefront sensing, Infrared, Keck, Pyramid, LIFT
19
+ 1. INTRODUCTION
20
+ The future upgrade of Keck II telescope’s Adaptive Optics (AO) system1 will include a pyramid wavefront sensor2
21
+ working in the near-infrared (J and H band).3 The main goal of this upgrade is to perform direct imaging and
22
+ slit spectroscopy of exoplanets around M dwarfs. The flux from these stars is very faint at optical wavelengths,
23
+ but sufficient in the near-infrared to use as NGSs in a single conjugated AO system, given the adequate detector
24
+ technology. The recently developed avalanche photodiode arrays, such as the SAPHIRA (Selex), provide a low
25
+ noise (ă 1 e- at high frame rates) and are thus suitable for this application.4 In addition to the NGS mode, the
26
+ system will also provide a LGS mode. In this case, the pyramid would be used as a low-order sensor only. We
27
+ report on a study of the pyramid sensor’s performance via end-to-end simulations made with PASSATA.5 After
28
+ a quick summary of the simulation parameters (section 2), we present the expected Strehl ratio in NGS mode
29
+ (section 3), and the expected residual on low orders in LGS mode (section 4). In the latter case, the pyramid
30
+ will not benefit from a hardware rebinning of pixels, and thus will not be in a fully optimized configuration. For
31
+ this reason, we also compare the pyramid to LIFT,6 a focal-plane sensor, that could provide a better low-order
32
+ estimation at low flux.
33
+ 2. SIMULATIONS PARAMETERS
34
+ We list in Table 1 the simulation parameters used for the different cases of this study. The chosen values for
35
+ the parameters that are optimized (modulation, frequency, control gain. . . ) are given in each specific case, and
36
+ we only state here the explored ranges of values. The wavefront modes (turbulent Karhunen-Lo`eve and Zernike)
37
+ are considered to be perfectly reproduced by the deformable mirror (DM). In all cases, the correction is made
38
+ with an integrator command, and the delay depends on the frequency, with the following rules (taken from ERIS
39
+ simulations experience7):
40
+ • f ą 666 Hz : 3 frames delay
41
+ Further author information:
42
+ C.P.: E-mail: plantet@arcetri.astro.it
43
+
44
+ • 333 Hz ă f ď 666 Hz : 2 frames delay
45
+ • f ď 333 Hz : 1 frame delay
46
+ Table 1. Simulation parameters.
47
+ Parameter
48
+ NGS mode
49
+ LGS mode
50
+ 20ˆ20
51
+ 32ˆ32
52
+ 20ˆ20
53
+ 32ˆ32
54
+ Sensing band
55
+ 1.5 µm - 1.8 µm (H band)
56
+ Pupil mask
57
+ Keck primary on 512 pixels
58
+ Keck primary on 256 pixels
59
+ Mode basis
60
+ 250 KL modes
61
+ 245 KL modes + 5 first Zernike
62
+ Total transmission (including QE)
63
+ 0.3
64
+ Sky background in H
65
+ 14 mag/arcsec2
66
+ Seeing
67
+ 0.63”
68
+ Layers’ altitudes (km)
69
+ 0, 500, 1000, 2000, 4000, 8000, 16000
70
+ C2
71
+ n profile (normalized in energy)
72
+ 0.517, 0.119, 0.063, 0.061, 0.105, 0.081, 0.054
73
+ Mean wind speed
74
+ 9.5 m/s
75
+ Zenith angle
76
+ 300
77
+ Subaperture size
78
+ 0.5625 m
79
+ 0.35 m
80
+ 0.5625 m
81
+ 0.35 m
82
+ APD gain
83
+ 30
84
+ Excess noise factor
85
+ 1.4
86
+ Read-out noise
87
+ 0.1 or 1 e´
88
+ 1 e´
89
+ 0.8 e´
90
+ 1 e´
91
+ Dark current
92
+ 0 or 100 e´/s
93
+ 20 e´/s
94
+ 100 e´/s
95
+ 20 e´/s
96
+ Frequency range
97
+ 300-1000 Hz
98
+ 200-1000 Hz
99
+ Control gain range
100
+ 0.1-0.6
101
+ LIFT: 0.1-0.6
102
+ LIFT: 0.1-0.6
103
+ Pyramid: 0.15-10
104
+ Pyramid:0.25-5
105
+ Pyramid modulation radius range
106
+ 1-3 λ{D
107
+ 1-2 λ{D
108
+ 0-2 λ{D
109
+ FoV
110
+ 1”
111
+ Additional HO residual (non corrected)
112
+ 60 nm
113
+ 0 nm
114
+ 60 nm
115
+ To these parameters, we add the following precisions for the LGS mode:
116
+ • High-order loop parameters:
117
+ – Sensor: SH 20ˆ20 with quad-cells estimating 250 modes.
118
+ – LGS = high flux point source at finite distance.
119
+ – Tip/tilt filtered and replaced by a residual jitter of 106 mas rms + turbulent tip/tilt.
120
+ – Control gain: 0.3.
121
+ • Focus loop (only in 32ˆ32 case):
122
+ – Correction frequency: 10 Hz.
123
+ – Input: focus residual from high-order control + sinusoid of period 5 seconds and amplitude 100 nm
124
+ (80 nm rms).
125
+ – Control gain range: 0.1-1 for LIFT, 0.1-4 for the pyramid.
126
+ Finally, for consistency with the error budget used in a previous study,3 we add a constant error of 165 nm
127
+ rms to the residual in NGS mode, representing miscellaneous errors from undetermined sources.
128
+
129
+ 3. NGS MODE
130
+ In this section, we study the performance of the pyramid, in terms of Strehl ratio, for different pupil samplings.
131
+ We first considered a pupil sampling of 20ˆ20 subapertures, in agreement with the current DM’s number of
132
+ actuators. However, the DM should be upgraded to a MEMS 32ˆ32. We thus study in a second step the impact
133
+ of increasing the pupil sampling to 32ˆ32, or to 40ˆ40 for robustness reasons.
134
+ 3.1 Pyramid 20ˆ20
135
+ We present here the results of the simulations with a pyramid 20ˆ20 in NGS mode (Fig. 3.1). The parameters,
136
+ listed in Table 2, are optimized in the ranges described previously to get the highest Strehl ratio. This optimiza-
137
+ tion is simply made by running simulations going through the whole set of parameters and selecting the best
138
+ one.
139
+ As we lacked information on the detector’s noise, we considered two cases: low noise (no dark current, read-
140
+ out noise = 0.1 e´) and high noise (dark current = 100 e´/s, read-out noise = 1 e´). The difference between
141
+ those two cases is not very significant (0.5 magnitude at faint end).
142
+ Table 2. Optimized parameters (high noise/low noise) for the pyramid 20ˆ20 in NGS mode.
143
+ Magnitude
144
+ 8
145
+ 10
146
+ 12
147
+ 13
148
+ 14
149
+ 15
150
+ Frequency (Hz)
151
+ 1000/1000
152
+ 1000/1000
153
+ 1000/600
154
+ 1000/500
155
+ 600/300
156
+ 600/300
157
+ Number of modes
158
+ 250/250
159
+ 250/250
160
+ 170/152
161
+ 135/104
162
+ 65/54
163
+ 44/14
164
+ Gain
165
+ 0.3/0.3
166
+ 0.2/0.2
167
+ 0.15/0.25
168
+ 0.15/0.3
169
+ 0.3/0.55
170
+ 0.3/0.6
171
+ Modulation radius (λ{D)
172
+ 1.5
173
+ 1.5
174
+ 1.5
175
+ 1.5
176
+ 1.5
177
+ 2
178
+ 8
179
+ 9
180
+ 10
181
+ 11
182
+ 12
183
+ 13
184
+ 14
185
+ 15
186
+ H-magnitude
187
+ 0.0
188
+ 0.1
189
+ 0.2
190
+ 0.3
191
+ 0.4
192
+ 0.5
193
+ 0.6
194
+ 0.7
195
+ 0.8
196
+ 0.9
197
+ 1.0
198
+ SR
199
+ λ = 1265nm
200
+ λ = 1659nm
201
+ λ = 2145nm
202
+ λ = 2200nm
203
+ λ = 3805nm
204
+ λ = 4781nm
205
+ (a) Low noise
206
+ 8
207
+ 9
208
+ 10
209
+ 11
210
+ 12
211
+ 13
212
+ 14
213
+ 15
214
+ H-magnitude
215
+ 0.0
216
+ 0.1
217
+ 0.2
218
+ 0.3
219
+ 0.4
220
+ 0.5
221
+ 0.6
222
+ 0.7
223
+ 0.8
224
+ 0.9
225
+ 1.0
226
+ SR
227
+ λ = 1265nm
228
+ λ = 1659nm
229
+ λ = 2145nm
230
+ λ = 2200nm
231
+ λ = 3805nm
232
+ λ = 4781nm
233
+ (b) High noise
234
+ 8
235
+ 9
236
+ 10
237
+ 11
238
+ 12
239
+ 13
240
+ 14
241
+ 15
242
+ H-Magnitude
243
+ 0.0
244
+ 0.1
245
+ 0.2
246
+ 0.3
247
+ 0.4
248
+ 0.5
249
+ 0.6
250
+ 0.7
251
+ 0.8
252
+ 0.9
253
+ 1.0
254
+ SR(@K)
255
+ Dark = 0 -- RON = 0.1
256
+ Dark = 100 -- RON = 1
257
+ (c) Comparison
258
+ Figure 1. Strehl ratio as a function of H magnitude with a pyramid 20ˆ20 in NGS mode. Left: Low noise case. Middle:
259
+ High noise case. Right: Comparison of low and high noise cases in K band.
260
+ Overall, what we observe on the optimization of the parameters when we are going towards lower fluxes is:
261
+ • Decrease in frequency: to collect more flux and reduce the noise error.
262
+ • Increase the gain: we cannot remove the background in pyramid images, it is thus taken into account in
263
+ the normalization when computing the slopes. In the end, the slopes are proportional to the ratio star
264
+ flux/background, which decreases with respect to the magnitude. A higher gain is needed to compensate
265
+ that effect. The increase in gain is also needed at lower frequencies, as the correction is done less often.
266
+ • Increase in modulation: at low flux, the noise error makes the pyramid work in non-linear regime. The
267
+ modulation reduces the non-linearity error, at the price of a lower sensitivity (hence greater noise error).
268
+ A trade-off is made between those two errors to reach the lowest overall error. At high flux, using a high
269
+ modulation lowers the non-linearity error.
270
+
271
+ • Decrease the number of modes: estimating less modes improves the noise propagation behavior at low
272
+ orders.
273
+ These results are consistent with the ones presented in an earlier study,3 with a difference of only a few
274
+ percents of Strehl ratio in K band.
275
+ 3.2 Impact of a finer pupil sampling
276
+ Having a finer pupil sampling allows us to estimate more modes at high flux, but lowers the signal-to-noise ratio
277
+ (SNR) at low flux. We consider here only the impact at low flux, as it corresponds to more practical cases and
278
+ is more critical for the system design.
279
+ We simulated two different pupil samplings: 32ˆ32 and 40ˆ40 (in that case, only the subaperture size from
280
+ Table 1 is changed). The first one matches the MEMS mirror sampling, while the second would help calibrate
281
+ misregistration errors and thus gain in robustness.
282
+ The performance and optimized parameters at magnitude 14 are given in Table 3, for a dark current of 20
283
+ e´/s and a read-out noise of 1 e´. The performance for the high noise case of the pyramid 20ˆ20 is recalled for
284
+ reference. It should be noted that the dark current does not have a significant impact here, the results can thus
285
+ be fairly compared.
286
+ Table 3. Optimized parameters and Strehl ratios for the pyramid 32ˆ32 and 40ˆ40 in NGS mode.
287
+ Frequency (Hz)
288
+ Number of modes
289
+ Gain
290
+ Modulation radius (λ{D)
291
+ Strehl ratio (K)
292
+ 20ˆ20
293
+ 600
294
+ 65
295
+ 0.3
296
+ 1.5
297
+ 29.8%
298
+ 32ˆ32
299
+ 200
300
+ 65
301
+ 0.75
302
+ 1.5
303
+ 26.7%
304
+ 40ˆ40
305
+ 200
306
+ 65
307
+ 0.75
308
+ 1.5
309
+ 25.1%
310
+ The finer pupil sampling does not have a strong impact on performance: the loss of Strehl in Ks is 3% for
311
+ the 32ˆ32 and 5% for the 40ˆ40. Hence, it seems a reasonable choice to go towards a 40ˆ40 sampling, making
312
+ the system more reliable without a significant loss of performance at low flux.
313
+ 4. LGS MODE
314
+ The goal of this section is to assess the achievable residual on tip/tilt and focus in LGS mode, for a NGS on
315
+ axis or at 15” off axis. We compare the pyramid to LIFT, in order to evaluate the gain of having a focal-plane
316
+ sensor for this low-order estimation. Indeed, as we cannot do a hardware rebin of pixels on the camera, the
317
+ pyramid would still utilise a fine sampling and would thus have poorer noise propagation properties for low-order
318
+ estimation than with a coarse sampling.
319
+ As in the previous section, we first considered a pyramid with 20ˆ20, and then checked the impact of a finer
320
+ sampling. For LIFT, the only design parameter that will have an impact on the performance is the pixel scale.
321
+ We consider here a pixel of 15 or 30 mas, corresponding respectively to a Nyquist and a Nyquist/2 sampling in
322
+ H band.
323
+ 4.1 Pyramid 20ˆ20
324
+ In this part, we only evaluate the residual on tip/tilt, as it is the most important feature of the low-order sensor.
325
+ The focus estimation will be included in the next section. For practical reasons, the number of reconstructed
326
+ modes for the pyramid is either 2 (lowest noise error) or 250 (lowest aliasing error).
327
+ We list in Tables 4 to 6 the optimized parameters for LIFT and the pyramid in each case, as well as the
328
+ residual on tip/tilt. The residuals obtained with LIFT and the pyramid are compared in Fig. 4.1. We find that
329
+ LIFT provides a gain of up to 2 magnitudes over the pyramid, either on axis or off axis.
330
+ The behavior of the optimized parameters for the pyramid is as described in section 3.1. In particular, we
331
+ can notice an increase in modulation at high flux when going off-axis: this is due to the increase in amplitude
332
+
333
+ of high-order modes, for which the linearity must be improved. The flux is sufficiently high in that case to use
334
+ a strong modulation without a significant impact on noise error.
335
+ As concerns LIFT’s sampling, the pixel of 30 mas benefits from better noise propagation properties (better
336
+ SNR/pixel), but does not provide any significant improvement of the performance. On the contrary, it is less
337
+ efficient off axis, or at high flux in general. Indeed, the signal from high orders, normally far from the spot center,
338
+ gets more easily mixed with the low orders signal, which is within the spot center. This aliasing error is visible
339
+ at high flux, where the noise error is negligible, and gets higher when going off axis, where the Strehl ratio is
340
+ lower. The overall aliasing + noise error is in the end always better with the 15 mas pixel for the considered
341
+ magnitudes.
342
+ Table 4. Optimized parameters (on axis/off axis) for the pyramid 20ˆ20 in LGS mode.
343
+ Magnitude
344
+ 10
345
+ 12
346
+ 13
347
+ 14
348
+ 15
349
+ 16
350
+ Frequency (Hz)
351
+ 1000/1000
352
+ 1000/1000
353
+ 1000/1000
354
+ 1000/1000
355
+ 200/200
356
+ 200/200
357
+ Number of modes
358
+ 250/250
359
+ 250/250
360
+ 2/250
361
+ 2/2
362
+ 2/2
363
+ 2/2
364
+ Gain
365
+ 0.5/0.25
366
+ 0.5/0.65
367
+ 0.75/0.65
368
+ 1/1
369
+ 3.5/3.5
370
+ 5/5
371
+ Modulation radius (λ{D)
372
+ 0/2
373
+ 0/0
374
+ 0/0
375
+ 0/0
376
+ 0/0
377
+ 0/1
378
+ TT residual (nm rms)
379
+ 21.4/48
380
+ 32.3/59.4
381
+ 44.6/73.3
382
+ 57.8/89.4
383
+ 88.6/123
384
+ 153.7/195.6
385
+ Table 5. Optimized parameters (on axis/off axis) for LIFT with a 15 mas pixel in LGS mode.
386
+ Magnitude
387
+ 10
388
+ 12
389
+ 13
390
+ 14
391
+ 15
392
+ 16
393
+ Frequency (Hz)
394
+ 1000/1000
395
+ 1000/1000
396
+ 1000/333
397
+ 333/333
398
+ 333/200
399
+ 200/200
400
+ Gain
401
+ 0.3/0.3
402
+ 0.3/0.3
403
+ 0.2/0.5
404
+ 0.4/0.5
405
+ 0.4/0.4
406
+ 0.4/0.3
407
+ TT residual (nm rms)
408
+ 18.9/42.3
409
+ 23.2/45.1
410
+ 27.3/48.5
411
+ 33.1/56.4
412
+ 42.9/67.5
413
+ 63.5/93.2
414
+ Table 6. Optimized parameters (on axis/off axis) for LIFT with a 30 mas pixel in LGS mode.
415
+ Magnitude
416
+ 10
417
+ 12
418
+ 13
419
+ 14
420
+ 15
421
+ 16
422
+ Frequency (Hz)
423
+ 1000/1000
424
+ 1000/1000
425
+ 1000/1000
426
+ 333/1000
427
+ 333/500
428
+ 200/333
429
+ Gain
430
+ 0.4/0.3
431
+ 0.3/0.2
432
+ 0.2/0.2
433
+ 0.5/0.2
434
+ 0.4/0.2
435
+ 0.4/0.3
436
+ TT residual (nm rms)
437
+ 25.7/62.5
438
+ 28.5/63.7
439
+ 31.5/66.5
440
+ 37.6/68.3
441
+ 45.6/81.1
442
+ 64.5/101.2
443
+ (a) On axis
444
+ (b) 15” off axis
445
+ Figure 2. Comparison of the tip/tilt residual obtained with LIFT or the pyramid 20ˆ20 in LGS mode. Left: NGS on
446
+ axis. Right: NGS at 15” off axis.
447
+
448
+ 4.2 Finer pupil sampling
449
+ We now check the impact of having a finer-sampled pyramid on the performance at magnitude 14. We also verify
450
+ that we have a correct estimation of focus on both sensors, and we study the possibility of having 10 mas pixels
451
+ on LIFT (for design simplicity reasons).
452
+ The parameters and results are given in Table 7, for an optimization on tip/tilt correction only (the focus is
453
+ then added with the same loop parameters).
454
+ On axis, there is a clear advantage using LIFT for tip/tilt estimation, with a factor 2 in rms residual. The
455
+ estimation of focus does not affect the estimation of tip/tilt, whatever the sensor, and LIFT has a slight advantage
456
+ on this mode as well (factor 1.4).
457
+ Off axis, we still have a better estimation of tip/tilt with LIFT, but with less difference (factor 1.4 at best).
458
+ LIFT’s performance is actually weakly dependent on the frequency: at 1000 Hz, the tip/tilt residual is increased
459
+ by approximately 5 nm for the 15 mas and 10 mas pixels. The focus correction is similar in all cases, and the
460
+ tip/tilt estimation is always affected. This might be the result of the sensors getting less and less linear when
461
+ lowering the Strehl ratio (aliasing error discussed in the previous section). The effect seems stronger on LIFT at
462
+ fine samplings (15 and 10 mas), but for these cases, as well as the pyramid, it is equivalent to adding an error of
463
+ approximately 30-35 nm rms. For LIFT with 30 mas pixels, this error is lower, around 20 nm rms. This might
464
+ be due to the fact that the tip/tilt estimation alone was already affected by non-linear effects.
465
+ Table 7. Optimized parameters and residuals (on axis/off axis) for the pyramid 32ˆ32 and LIFT in LGS mode, with the
466
+ estimation of focus.
467
+ LIFT 30 mas
468
+ LIFT 15 mas
469
+ LIFT 10 mas
470
+ Pyramid
471
+ Frequency (Hz)
472
+ 333/1000
473
+ 333/200
474
+ 333/333
475
+ 333/333
476
+ Modulation radius (λ{D)
477
+ 0/0
478
+ Gain
479
+ 0.4/0.1
480
+ 0.4/0.5
481
+ 0.4/0.3
482
+ 4/4
483
+ TT residual (nm rms)
484
+ 39.3/80.5
485
+ 34.3/67.5
486
+ 34.7/69.8
487
+ 68.9/95.9
488
+ TT residual (with focus)
489
+ 36.2/82.7
490
+ 33.6/75.9
491
+ 34.8/77.7
492
+ 69.3/100
493
+ Gain on focus
494
+ 0.6/0.4
495
+ 0.5/0.4
496
+ 0.7/0.3
497
+ 2/1.5
498
+ Focus residual (nm rms)
499
+ 37/52.9
500
+ 38.4/53.1
501
+ 35.9/53.2
502
+ 50.5/50.8
503
+ 5. CONCLUSION
504
+ We have studied the performance of a near-infrared pyramid for the next generation AO of Keck II, which will
505
+ include a classical AO mode (NGS mode) and a LGS mode. In NGS mode, the pyramid will provide a Strehl
506
+ ratio in K band of
507
+ 37% at magnitude H = 14 and
508
+ 80% at high flux (20ˆ20 configuration). The latter can
509
+ be increased with a finer pupil sampling (32ˆ32, or 40ˆ40) and a higher degree of correction (i. e. 32ˆ32
510
+ DM), without degrading significantly the performance at low flux. The 40ˆ40 sampling would also provide more
511
+ robustness to errors such as misregistration. In LGS mode, the pyramid would not benefit from a hardware
512
+ rebin of pixels, and a focal plane sensor would be preferable to estimate low orders. We have demonstrated that
513
+ through a comparison with LIFT, which provides a gain of 2 magnitudes on tip/tilt up to 15” off axis and a
514
+ similar performance on focus. It was also shown that LIFT gives best results with images sampled at Nyquist
515
+ (15 mas pixels). In future works, we will explore more off-axis distances and seeing conditions to confirm the
516
+ advantage of using LIFT. We will also study the impact of the atmosphere dispersion on both sensors.
517
+ ACKNOWLEDGMENTS
518
+ This work was partly funded by INAF (Research Grant DD 27). The Keck II pyramid wavefront sensor is funded
519
+ by the National Science Foundation under Grant No. AST-1611623.
520
+
521
+ REFERENCES
522
+ [1] Wizinowich, P., Le Mignant, D., Bouchez, A. H., Campbell, R. D., Chin, J. C., Contos, A. R., van Dam,
523
+ M. A., Hartman, S. K., Johansson, E. M., Lafon, R. E., et al., “The WM Keck Observatory laser guide star
524
+ adaptive optics system: overview,” Publications of the Astronomical Society of the Pacific 118(840), 297
525
+ (2006).
526
+ [2] Ragazzoni, R., “Pupil plane wavefront sensing with an oscillating prism,” Journal of modern optics 43(2),
527
+ 289–293 (1996).
528
+ [3] Wizinowich, P., Chun, M., Mawet, D., Agapito, G., Dekany, R., Esposito, S., Fusco, T., Guyon, O., Hall, D.,
529
+ Plantet, C., and Rigaut, F., “Near-infrared wavefront sensing,” Proc.SPIE 9909, 9909 – 9909 – 13 (2016).
530
+ [4] Feautrier, P., Gach, J.-L., and Wizinowich, P., “State of the art IR cameras for wavefront sensing using
531
+ e-APD MCT arrays,” in [AO4ELT4 Proceedings], (2015).
532
+ [5] G. Agapito, A. Puglisi, S. E., “Passata: object oriented numerical simulation software for adaptive optics,”
533
+ Proc.SPIE 9909, 9909 – 9909 – 9 (2016).
534
+ [6] Meimon, S., Fusco, T., and Mugnier, L. M., “LIFT: a focal-plane wavefront sensor for real-time low-order
535
+ sensing on faint sources,” Optics letters 35(18), 3036–3038 (2010).
536
+ [7] Quir´os-Pacheco, F., Agapito, G., Riccardi, A., Esposito, S., Louarn, M. L., and Marchetti, E., “Performance
537
+ simulation of the eris pyramid wavefront sensor module in the vlt adaptive optics facility,” Proc.SPIE 8447,
538
+ 8447 – 8447 – 12 (2012).
539
+
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+ page_content=', Kamuela, HI 96743, USA cInstitute for Astronomy, University of Hawaii, 640 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
14
+ page_content=' Aohoku Place, Hilo, HI 96720 ABSTRACT The future upgrade of Keck II telescope’s adaptive optics system will include a pyramid wavefront sensor working in the near-infrared (J and H band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
15
+ page_content=' It will benefit from the recently developed avalanche photodiode arrays, specifically the SAPHIRA (Selex) array, which provides a low noise (ă 1 e- at high frame rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
16
+ page_content=' The system will either work with a natural guide star (NGS) in a single conjugated adaptive optics system, or in a laser guide star (LGS) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' In this case, the pyramid would be used as a low-order sensor only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We report on a study of the pyramid sensor’s performance via end-to-end simulations, applied to Keck’s specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We present the expected Strehl ratio with optimized configurations in NGS mode, and the expected residual on low orders in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' In the latter case, we also compare the pyramid to LIFT, a focal-plane sensor, demonstrating the ability of LIFT to provide a gain of about 2 magnitudes for low-order sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Keywords: Adaptive optics, Wavefront sensing, Infrared, Keck, Pyramid, LIFT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
22
+ page_content=' INTRODUCTION The future upgrade of Keck II telescope’s Adaptive Optics (AO) system1 will include a pyramid wavefront sensor2 working in the near-infrared (J and H band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
23
+ page_content='3 The main goal of this upgrade is to perform direct imaging and slit spectroscopy of exoplanets around M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The flux from these stars is very faint at optical wavelengths, but sufficient in the near-infrared to use as NGSs in a single conjugated AO system, given the adequate detector technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
25
+ page_content=' The recently developed avalanche photodiode arrays, such as the SAPHIRA (Selex), provide a low noise (ă 1 e- at high frame rates) and are thus suitable for this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
26
+ page_content='4 In addition to the NGS mode, the system will also provide a LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
27
+ page_content=' In this case, the pyramid would be used as a low-order sensor only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
28
+ page_content=' We report on a study of the pyramid sensor’s performance via end-to-end simulations made with PASSATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
29
+ page_content='5 After a quick summary of the simulation parameters (section 2), we present the expected Strehl ratio in NGS mode (section 3), and the expected residual on low orders in LGS mode (section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
30
+ page_content=' In the latter case, the pyramid will not benefit from a hardware rebinning of pixels, and thus will not be in a fully optimized configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
31
+ page_content=' For this reason, we also compare the pyramid to LIFT,6 a focal-plane sensor, that could provide a better low-order estimation at low flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' SIMULATIONS PARAMETERS We list in Table 1 the simulation parameters used for the different cases of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The chosen values for the parameters that are optimized (modulation, frequency, control gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' ) are given in each specific case, and we only state here the explored ranges of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The wavefront modes (turbulent Karhunen-Lo`eve and Zernike) are considered to be perfectly reproduced by the deformable mirror (DM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' In all cases, the correction is made with an integrator command, and the delay depends on the frequency, with the following rules (taken from ERIS simulations experience7): f ą 666 Hz : 3 frames delay Further author information: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' : E-mail: plantet@arcetri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='it 333 Hz ă f ď 666 Hz : 2 frames delay f ď 333 Hz : 1 frame delay Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Parameter NGS mode LGS mode 20ˆ20 32ˆ32 20ˆ20 32ˆ32 Sensing band 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 µm - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='8 µm (H band) Pupil mask Keck primary on 512 pixels Keck primary on 256 pixels Mode basis 250 KL modes 245 KL modes + 5 first Zernike Total transmission (including QE) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3 Sky background in H 14 mag/arcsec2 Seeing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='63” Layers’ altitudes (km) 0, 500, 1000, 2000, 4000, 8000, 16000 C2 n profile (normalized in energy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='517, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='119, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
52
+ page_content='063, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
53
+ page_content='061, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
54
+ page_content='105, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
55
+ page_content='081, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='054 Mean wind speed 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 m/s Zenith angle 300 Subaperture size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5625 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='35 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5625 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='35 m APD gain 30 Excess noise factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='4 Read-out noise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1 or 1 e´ 1 e´ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='8 e´ 1 e´ Dark current 0 or 100 e´/s 20 e´/s 100 e´/s 20 e´/s Frequency range 300-1000 Hz 200-1000 Hz Control gain range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='6 LIFT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='6 LIFT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='6 Pyramid: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='15-10 Pyramid:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='25-5 Pyramid modulation radius range 1-3 λ{D 1-2 λ{D 0-2 λ{D FoV 1” Additional HO residual (non corrected) 60 nm 0 nm 60 nm To these parameters, we add the following precisions for the LGS mode: High-order loop parameters: – Sensor: SH 20ˆ20 with quad-cells estimating 250 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' – LGS = high flux point source at finite distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' – Tip/tilt filtered and replaced by a residual jitter of 106 mas rms + turbulent tip/tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' – Control gain: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Focus loop (only in 32ˆ32 case): – Correction frequency: 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' – Input: focus residual from high-order control + sinusoid of period 5 seconds and amplitude 100 nm (80 nm rms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' – Control gain range: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1-1 for LIFT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1-4 for the pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Finally, for consistency with the error budget used in a previous study,3 we add a constant error of 165 nm rms to the residual in NGS mode, representing miscellaneous errors from undetermined sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' NGS MODE In this section, we study the performance of the pyramid, in terms of Strehl ratio, for different pupil samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We first considered a pupil sampling of 20ˆ20 subapertures, in agreement with the current DM’s number of actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' However, the DM should be upgraded to a MEMS 32ˆ32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We thus study in a second step the impact of increasing the pupil sampling to 32ˆ32, or to 40ˆ40 for robustness reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1 Pyramid 20ˆ20 We present here the results of the simulations with a pyramid 20ˆ20 in NGS mode (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The parameters, listed in Table 2, are optimized in the ranges described previously to get the highest Strehl ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' This optimiza- tion is simply made by running simulations going through the whole set of parameters and selecting the best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' As we lacked information on the detector’s noise, we considered two cases: low noise (no dark current, read- out noise = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1 e´) and high noise (dark current = 100 e´/s, read-out noise = 1 e´).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The difference between those two cases is not very significant (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 magnitude at faint end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Optimized parameters (high noise/low noise) for the pyramid 20ˆ20 in NGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Magnitude 8 10 12 13 14 15 Frequency (Hz) 1000/1000 1000/1000 1000/600 1000/500 600/300 600/300 Number of modes 250/250 250/250 170/152 135/104 65/54 44/14 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='15/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='15/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='6 Modulation radius (λ{D) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 2 8 9 10 11 12 13 14 15 H-magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='0 SR λ = 1265nm λ = 1659nm λ = 2145nm λ = 2200nm λ = 3805nm λ = 4781nm (a) Low noise 8 9 10 11 12 13 14 15 H-magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='0 SR λ = 1265nm λ = 1659nm λ = 2145nm λ = 2200nm λ = 3805nm λ = 4781nm (b) High noise 8 9 10 11 12 13 14 15 H-Magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='0 SR(@K) Dark = 0 -- RON = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1 Dark = 100 -- RON = 1 (c) Comparison Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Strehl ratio as a function of H magnitude with a pyramid 20ˆ20 in NGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Left: Low noise case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Middle: High noise case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Right: Comparison of low and high noise cases in K band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Overall, what we observe on the optimization of the parameters when we are going towards lower fluxes is: Decrease in frequency: to collect more flux and reduce the noise error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Increase the gain: we cannot remove the background in pyramid images, it is thus taken into account in the normalization when computing the slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' In the end, the slopes are proportional to the ratio star flux/background, which decreases with respect to the magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' A higher gain is needed to compensate that effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The increase in gain is also needed at lower frequencies, as the correction is done less often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Increase in modulation: at low flux, the noise error makes the pyramid work in non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The modulation reduces the non-linearity error, at the price of a lower sensitivity (hence greater noise error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' A trade-off is made between those two errors to reach the lowest overall error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' At high flux, using a high modulation lowers the non-linearity error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Decrease the number of modes: estimating less modes improves the noise propagation behavior at low orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' These results are consistent with the ones presented in an earlier study,3 with a difference of only a few percents of Strehl ratio in K band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='2 Impact of a finer pupil sampling Having a finer pupil sampling allows us to estimate more modes at high flux, but lowers the signal-to-noise ratio (SNR) at low flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We consider here only the impact at low flux, as it corresponds to more practical cases and is more critical for the system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We simulated two different pupil samplings: 32ˆ32 and 40ˆ40 (in that case, only the subaperture size from Table 1 is changed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The first one matches the MEMS mirror sampling, while the second would help calibrate misregistration errors and thus gain in robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The performance and optimized parameters at magnitude 14 are given in Table 3, for a dark current of 20 e´/s and a read-out noise of 1 e´.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The performance for the high noise case of the pyramid 20ˆ20 is recalled for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' It should be noted that the dark current does not have a significant impact here, the results can thus be fairly compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Optimized parameters and Strehl ratios for the pyramid 32ˆ32 and 40ˆ40 in NGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Frequency (Hz) Number of modes Gain Modulation radius (λ{D) Strehl ratio (K) 20ˆ20 600 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='8% 32ˆ32 200 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='7% 40ˆ40 200 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1% The finer pupil sampling does not have a strong impact on performance: the loss of Strehl in Ks is 3% for the 32ˆ32 and 5% for the 40ˆ40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Hence, it seems a reasonable choice to go towards a 40ˆ40 sampling, making the system more reliable without a significant loss of performance at low flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' LGS MODE The goal of this section is to assess the achievable residual on tip/tilt and focus in LGS mode, for a NGS on axis or at 15” off axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We compare the pyramid to LIFT, in order to evaluate the gain of having a focal-plane sensor for this low-order estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Indeed, as we cannot do a hardware rebin of pixels on the camera, the pyramid would still utilise a fine sampling and would thus have poorer noise propagation properties for low-order estimation than with a coarse sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' As in the previous section, we first considered a pyramid with 20ˆ20, and then checked the impact of a finer sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' For LIFT, the only design parameter that will have an impact on the performance is the pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We consider here a pixel of 15 or 30 mas, corresponding respectively to a Nyquist and a Nyquist/2 sampling in H band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1 Pyramid 20ˆ20 In this part, we only evaluate the residual on tip/tilt, as it is the most important feature of the low-order sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The focus estimation will be included in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' For practical reasons, the number of reconstructed modes for the pyramid is either 2 (lowest noise error) or 250 (lowest aliasing error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We list in Tables 4 to 6 the optimized parameters for LIFT and the pyramid in each case, as well as the residual on tip/tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The residuals obtained with LIFT and the pyramid are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We find that LIFT provides a gain of up to 2 magnitudes over the pyramid, either on axis or off axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The behavior of the optimized parameters for the pyramid is as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' In particular, we can notice an increase in modulation at high flux when going off-axis: this is due to the increase in amplitude of high-order modes, for which the linearity must be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The flux is sufficiently high in that case to use a strong modulation without a significant impact on noise error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' As concerns LIFT’s sampling, the pixel of 30 mas benefits from better noise propagation properties (better SNR/pixel), but does not provide any significant improvement of the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' On the contrary, it is less efficient off axis, or at high flux in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Indeed, the signal from high orders, normally far from the spot center, gets more easily mixed with the low orders signal, which is within the spot center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' This aliasing error is visible at high flux, where the noise error is negligible, and gets higher when going off axis, where the Strehl ratio is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The overall aliasing + noise error is in the end always better with the 15 mas pixel for the considered magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Optimized parameters (on axis/off axis) for the pyramid 20ˆ20 in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Magnitude 10 12 13 14 15 16 Frequency (Hz) 1000/1000 1000/1000 1000/1000 1000/1000 200/200 200/200 Number of modes 250/250 250/250 2/250 2/2 2/2 2/2 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='75/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='6/123 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='6 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Optimized parameters (on axis/off axis) for LIFT with a 15 mas pixel in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Magnitude 10 12 13 14 15 16 Frequency (Hz) 1000/1000 1000/1000 1000/333 333/333 333/200 200/200 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3 TT residual (nm rms) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1/56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='2 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Optimized parameters (on axis/off axis) for LIFT with a 30 mas pixel in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
261
+ page_content=' Magnitude 10 12 13 14 15 16 Frequency (Hz) 1000/1000 1000/1000 1000/1000 333/1000 333/500 200/333 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3 TT residual (nm rms) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='2 (a) On axis (b) 15” off axis Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Comparison of the tip/tilt residual obtained with LIFT or the pyramid 20ˆ20 in LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Left: NGS on axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Right: NGS at 15” off axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='2 Finer pupil sampling We now check the impact of having a finer-sampled pyramid on the performance at magnitude 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We also verify that we have a correct estimation of focus on both sensors, and we study the possibility of having 10 mas pixels on LIFT (for design simplicity reasons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The parameters and results are given in Table 7, for an optimization on tip/tilt correction only (the focus is then added with the same loop parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' On axis, there is a clear advantage using LIFT for tip/tilt estimation, with a factor 2 in rms residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The estimation of focus does not affect the estimation of tip/tilt, whatever the sensor, and LIFT has a slight advantage on this mode as well (factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Off axis, we still have a better estimation of tip/tilt with LIFT, but with less difference (factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='4 at best).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' LIFT’s performance is actually weakly dependent on the frequency: at 1000 Hz, the tip/tilt residual is increased by approximately 5 nm for the 15 mas and 10 mas pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The focus correction is similar in all cases, and the tip/tilt estimation is always affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' This might be the result of the sensors getting less and less linear when lowering the Strehl ratio (aliasing error discussed in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The effect seems stronger on LIFT at fine samplings (15 and 10 mas), but for these cases, as well as the pyramid, it is equivalent to adding an error of approximately 30-35 nm rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' For LIFT with 30 mas pixels, this error is lower, around 20 nm rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' This might be due to the fact that the tip/tilt estimation alone was already affected by non-linear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' Optimized parameters and residuals (on axis/off axis) for the pyramid 32ˆ32 and LIFT in LGS mode, with the estimation of focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' LIFT 30 mas LIFT 15 mas LIFT 10 mas Pyramid Frequency (Hz) 333/1000 333/200 333/333 333/333 Modulation radius (λ{D) 0/0 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3 4/4 TT residual (nm rms) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='9 TT residual (with focus) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3/100 Gain on focus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='3 2/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5 Focus residual (nm rms) 37/52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='9 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='4/53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='9/53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='5/50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' CONCLUSION We have studied the performance of a near-infrared pyramid for the next generation AO of Keck II, which will include a classical AO mode (NGS mode) and a LGS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' In NGS mode, the pyramid will provide a Strehl ratio in K band of 37% at magnitude H = 14 and 80% at high flux (20ˆ20 configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The latter can be increased with a finer pupil sampling (32ˆ32, or 40ˆ40) and a higher degree of correction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' 32ˆ32 DM), without degrading significantly the performance at low flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' The 40ˆ40 sampling would also provide more robustness to errors such as misregistration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' In LGS mode, the pyramid would not benefit from a hardware rebin of pixels, and a focal plane sensor would be preferable to estimate low orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' We have demonstrated that through a comparison with LIFT, which provides a gain of 2 magnitudes on tip/tilt up to 15” off axis and a similar performance on focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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+ page_content=' It was also shown that LIFT gives best results with images sampled at Nyquist (15 mas pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
351
+ page_content=' In future works, we will explore more off-axis distances and seeing conditions to confirm the advantage of using LIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
352
+ page_content=' We will also study the impact of the atmosphere dispersion on both sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
353
+ page_content=' ACKNOWLEDGMENTS This work was partly funded by INAF (Research Grant DD 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
354
+ page_content=' The Keck II pyramid wavefront sensor is funded by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
355
+ page_content=' AST-1611623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
356
+ page_content=' REFERENCES [1] Wizinowich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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