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+arXiv:2301.01690v1  [cs.LO]  4 Jan 2023
+Proofs as stateful programs: A first-order logic with abstract Hoare
+triples, and an interpretation into an imperative language
+Thomas Powell
+January 5, 2023
+Abstract
+We introduce an extension of first-order logic that comes equipped with additional predicates for
+reasoning about an abstract state. Sequents in the logic comprise a main formula together with pre-
+and postconditions in the style of Hoare logic, and the axioms and rules of the logic ensure that the
+assertions about the state compose in the correct way. The main result of the paper is a realizability
+interpretation of our logic that extracts programs into a mixed functional/imperative language.
+All
+programs expressible in this language act on the state in a sequential manner, and we make this intuition
+precise by interpreting them in a semantic metatheory using the state monad. Our basic framework is
+very general, and our intention is that it can be instantiated and extended in a variety of different ways.
+We outline in detail one such extension: A monadic version of Heyting arithmetic with a wellfounded
+while rule, and conclude by outlining several other directions for future work.
+1
+Introduction
+The Curry-Howard correspondence lies at the heart of theoretical computer science.
+Over the years, a
+multitude of different techniques for extracting programs from proofs have been developed, the majority
+of which translate formal proof systems into lambda calculi. As such, programs extracted from proofs are
+typically conceived as pure functional programs.
+Everyday programmers, on the other hand, often think and write in an imperative paradigm, in terms of
+instructions that change some underlying global state. This is reinforced by the fact that many of the most
+popular programming languages, including C and Python, lean towards this style. Imperative programs are
+nevertheless highly complex from a mathematical perspective, and while systems such as Hoare logic [15] or
+separation logic [27] have been designed to reason about them, the formal extraction of imperative programs
+from proofs has received comparatively little attention.
+In this paper, we propose a new idea in this direction, developing a formal system SL that enriches
+ordinary first-order logic with Hoare triples for reasoning about an abstract global state. Sequents will have
+the form Γ ⊢ {α · A · β}, where A is a formula and α, β assertions about the state, and proofs in the logic
+will include both ordinary introduction and elimination rules for predicate logic, together with special rules
+for reasoning about the state. We then construct a stateful realizability interpretation (based on Kreisel’s
+modified realizability [17]) that relates formulas in SL to terms in a mixed functional/imperative language
+ST. Our main result is a soundness theorem, which confirms that whenever a formula is provable in SL, we
+can extract a corresponding stateful realizing term in ST. While our initial soundness theorem focuses on
+pure predicate logic, we subsequently show that it can be extended to arithmetic, where in particular we are
+then able to extract programs that contain both recursion and controlled while loops.
+We are not the first to adapt traditional methods to extract imperative programs: A major achievement
+in this direction, for example, is the monograph [22], which sets up a variant of intuitionistic Hoare logic
+alongside a realizability translation into a standard imperative language. Other relevant examples include
+[3, 8, 10, 13, 19, 32]. However, these and almost all other prior work in this direction tend to focus on
+formal verification, with an eye towards using proof interpretations as a method for the synthesis of correct-
+by-construction software. In concrete terms, this means that the formal systems tend to be quite detailed
+1
+
+and oriented towards program analysis, while the starting point is typically a program for which we want to
+construct a verification proof, rather than a proof from which we hope to extract a potentially unfamiliar
+program.
+Our approach, on the other hand, is much more abstract, with an emphasis on potential applications in
+logic and proof theory. Our basic system SL makes almost no assumptions about the structure of the state
+and what we are allowed to do with it. Rather, we focus on producing a general framework for reasoning
+about ‘stateful formulas’, which can then be instantiated with additional axioms to model concrete scenarios.
+The simplicity and generality of our framework is its most important feature, and we consider this work to
+be a first step towards a number of potentially interesting applications. For this reason, we include not only
+an extension of our system to a monadic theory of arithmetic, but conclude by sketching out some additional
+ways in which we conjecture that our logic and interpretation could be used and expanded, including the
+computational semantics of proofs and probabilistic logic.
+We take ideas from three main sources.
+The first is a case study of Berger et al.
+[6], in which a
+realizability interpretation is used to extract a version of in-place quicksort, and where the imperative nature
+of the extracted program is presented in a semantic way using the state monad.
+While their program
+behaves imperatively “by-chance”, terms extracted from our logic are forced to be imperative, and thus
+our framework offers one potential solution to their open problem of designing a proof calculus which only
+yields imperative programs. Indeed, an implementation of the insert sort algorithm is formally extracted
+in Section 6 below. Our second source of inspiration is the thesis of Birolo [7], where a general monadic
+realizability interpretation is defined and then used to give an alternative, semantic presentation of learning-
+based interactive realizability [2, 4]. However, our work goes beyond this in that it also involves a monadic
+extension of the target logic, whereas Birolo’s applies to standard first-order logic. Finally, a number of
+ideas are taken from the author’s previous work [24] on extracting stateful programs using the Dialectica
+interpretation. While there the state is used in a very specific and restricted way, unlike our more general
+presentation here, we use an analogous call-by-value monadic translation on terms.
+It is important to stress that we do not claim that our work represents an optimal or complete method
+for extracting imperative programs from proofs, nor do we claim that it is superior to alternative methods,
+including the aforementioned works in the direction of verification, or, for instance, techniques based on
+Krivine’s classical realizability [18], which could be viewed as imperative in nature. We simply offer what
+we consider to be a new and interesting perspective that emphasises abstraction and simplicity, and propose
+that our framework could prove valuable in a number of different contexts.
+Overview of the paper
+The main technical work that follows involves the design of three different systems, a realizability interpre-
+tation that connects them, and an instantiation of this framework in the setting of first-order arithmetic,
+namely:
+• A novel extension SL of predicate logic with abstract Hoare triples, which can be extended with
+additional axioms for characterising the state (Section 2).
+• A standard calculus ST for lambda terms with imperative commands, which can again be extended
+with additional constants for interacting with the state (Section 3).
+• A metalanguage Sω into which both SL and ST can be embedded (Section 4), which is used to formulate
+the realizability relation and prove its soundness (Section 5).
+• An instantiation of SL as a theory of arithmetic, with programs extracted into an extension of ST with
+recursion and while loops (Section 6).
+Concrete examples are given, and potential applications surveyed in Section 7.
+2
+
+2
+The system SL: First-order logic with state
+We begin by introducing our target theory SL from which stateful programs will be extracted. This is an
+extension of ordinary first-order logic in the sense that the latter can always be embedded into SL (we will
+make this precise in Proposition 2.1 below). Ultimately, we are interested not so much in SL on its own, but
+in theories of the form SL + ∆H + ∆S, where ∆H and ∆S are collections of (respectively non-computational
+and computational) axioms that together characterise the state. Several concrete examples will be given to
+illustrate this, and in Section 6 we present a variant of SL that represents a theory of first-order arithmetic
+with state.
+2.1
+Intuitionistic first-order logic
+Before defining SL, we give a standard presentation of first-order intuitionistic predicate logic PL, which
+serves as an opportunity to fix our basic style of formal reasoning. The language of PL consists of the
+logical constants ∧, ∨, ⇒, ∀, ∃, ⊤, ⊥, variables x, y, z, . . ., along with function symbols f, g, h, . . . and predicate
+symbols P, Q, R, . . ., each with a fixed arity. We assume the existence of at least one constant c. Terms are
+built from variables and function symbols as usual, and formulas are built from prime formulas P(t1, . . . , tn),
+⊤ and ⊥ using the logical constants. We use the usual abbreviation ¬A :≡ A ⇒ ⊥. We work in a sequent
+style natural deduction calculus, where sequents have the form Γ ⊢I A for some context Γ and formula A,
+and a context is a set of labelled assumptions of the form Au1
+1 , . . . , Aun
+n
+for pairwise distinct labels ui. The
+axioms and rules of PL are as in Figure 1.
+Figure 1: Axioms and rules of PL
+Propositional logic
+Γ ⊢I A
+if Au ∈ Γ for some u
+Γ ⊢I ⊤
+Γ ⊢I A
+Γ ⊢I B
+∧I
+Γ ⊢I A ∧ B
+Γ ⊢I A ∧ B
+∧EL
+Γ ⊢I A
+Γ ⊢I A ∧ B
+∧ER
+Γ ⊢I B
+Γ ⊢I A
+∨IL
+Γ ⊢I A ∨ B
+Γ ⊢I B
+∨IR
+Γ ⊢I A ∨ B
+Γ ⊢I A ∨ B
+Γ, Au ⊢I C
+Γ, Bv ⊢I C
+∨E
+Γ ⊢I C
+Γ, Au ⊢I B
+⇒I
+Γ ⊢I A ⇒ B
+Γ ⊢I A ⇒ B
+Γ ⊢I A
+⇒E
+Γ ⊢I B
+Γ ⊢I ⊥
+⊥E
+Γ ⊢I A
+Quantifier rules
+Γ ⊢I A[y/x]
+∀I
+Γ ⊢I ∀xA
+Γ ⊢I ∀xA
+∀E
+Γ ⊢I A[t/x]
+Γ ⊢I A[t/x]
+∃I
+Γ ⊢I ∃xA
+Γ ⊢I ∃xA
+Γ, A[y/x]u ⊢I C
+∃E
+Γ ⊢I C
+for ∀I, y ≡ x or y not free in A, and y not free in Γ
+for ∃E, y ≡ x or y not free in A, and y not free in C or Γ.
+3
+
+2.2
+Stateful first-order logic
+We now define our new logical system SL, which is an extension of ordinary first-order logic with new state
+propositions. To be more precise, we extend the language of PL with a ternary operation {−·−·−}, together
+with special state predicate symbols p, q, r, . . ., which also have a fixed arity. Terms of SL are the same as
+those of PL. On the other hand, there are two kinds of formulas in SL: state formulas and main formulas.
+A state formula is defined using state predicate symbols and propositional connectives as follows:
+• ⊤ and ⊥ are state formulas,
+• if p a state predicate symbol of arity n and t1, . . . , tn are terms, then p(t1, . . . , tn) is a state formula,
+• if α, β are state formulas, so are α ∧ β, α ∨ β, α ⇒ β.
+A main formula (or just formula) of SL is now defined as:
+• ⊤ and ⊥ are formulas,
+• if P is an ordinary predicate symbol of arity n and t1, . . . , tn are terms, then P(t1, . . . , tn) is a formula,
+• if A, B are formulas, so are A ∧ B, A ∨ B and ∃xA,
+• if A, B are formulas and α, β state formulas, then A ⇒ {α · B · β} and ∀x {α · A · β} are formulas.
+The notions of free and bound variables, along with substitution α[t/x] and A[t/x] can be easily defined for
+both state and main formulas.
+Analogous to the construction of formulas, our basic proof system uses the auxiliary notion of a state
+proof in order to define a main proof. A state sequent has the form Γ ⊢H α where α is a state formula and
+Γ a set of labelled state formulas. A proof of Γ ⊢H α in SL is built from the axioms and rules of classical
+propositional logic i.e. the propositional axioms and rules as set out in Figure 1 plus the law of excluded
+middle Γ ⊢H α ∨ ¬α, together with a set ∆H of as yet unspecified state axioms of the form Γ ⊢H α.
+A main sequent of SL has the form Γ ⊢S {α · A · β}, where A is a formula and α, β state formulas, and
+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
+Figure 2, together with a set ∆S of additional axioms.
+We now make precise what we mean when we characterise SL as an extension of standard first-order
+logic. The following is provable with an easy induction over derivations in PL:
+Proposition 2.1. For any formula A of PL and state formula α, define the main formula Aα of SL by
+• Qα := Q for Q atomic,
+• (A ∧ B)α := Aα ∧ Bα, (A ∨ B)α := Aα ∨ Bα and (∃x A)α := ∃x Aα,
+• (A ⇒ B)α := Aα ⇒ {α · Bα · α} and (∀x A)α := ∀x {α · Aα · α}.
+Then whenever Γ ⊢I A is provable in PL, we have that Γα, ∆ ⊢S {α · Aα · α} is provable in SL, where ∆ is
+arbitrary and Γα := (A1)u1
+α , . . . , (An)un
+α
+for Γ := Au1
+1 , . . . , Aun
+n .
+2.3
+The intuition behind SL
+The intended semantic meaning of Γ ⊢H α is that α can be inferred from the assumptions Γ for any fixed state.
+More specifically, if we imagine a semantic variant [α](π) of each state formula where now the dependency
+on an underlying state π is made explicit, the semantics of Γ ⊢H α is just
+[Γ](π) ⇒ [α](π)
+4
+
+Figure 2: Axioms and rules of SL
+Propositional axioms and rules
+Γ ⊢S {α · A · α}
+if Au ∈ Γ for some u
+Γ ⊢S {α · ⊤ · α}
+Γ ⊢S {α · A · β}
+Γ ⊢S {β · B · γ}
+∧SI
+Γ ⊢S {α · A ∧ B · γ}
+Γ ⊢S {α · A ∧ B · β}
+∧SEL
+Γ ⊢S {α · A · β}
+Γ ⊢S {α · A ∧ B · β}
+∧SER
+Γ ⊢S {α · B · β}
+Γ ⊢S {α · A · β}
+∨SIL
+Γ ⊢S {α · A ∨ B · β}
+Γ ⊢S {α · B · β}
+∨SIR
+Γ ⊢S {α · A ∨ B · β}
+Γ ⊢S {α · A ∨ B · β}
+Γ, Au ⊢S {β · C · γ}
+Γ, Bv ⊢S {β · C · γ}
+∨SE
+Γ ⊢S {α · C · γ}
+Γ, Au ⊢S {α · B · β}
+⇒SI
+Γ ⊢S {γ · A ⇒ {α · B · β} · γ}
+Γ ⊢S {α · A ⇒ {γ · B · δ} · β}
+Γ ⊢S {β · A · γ}
+⇒SE
+Γ ⊢S {α · B · δ}
+Γ ⊢S {α · ⊥ · β}
+⊥SE
+Γ ⊢S {α · A · γ}
+Quantifier rules
+Γ ⊢S {α[y/x] · A[y/x] · β[y/x]}
+∀SI
+Γ ⊢S {γ · ∀x {α · A · β} · γ}
+Γ ⊢S {α · ∀x {β · A · γ} · β[t/x]}
+∀SE
+Γ ⊢S {α · A[t/x] · γ[t/x]}
+Γ ⊢S {α · A[t/x] · β}
+∃SI
+Γ ⊢S {α · ∃xA · β}
+Γ ⊢S {α · ∃xA · β}
+Γ, A[y/x]u ⊢S {β · C · γ}
+∃SE
+Γ ⊢S {α · C · γ}
+for ∀SI, y ≡ x or y not free in A, α, β, and y not free in Γ
+for ∃SE, y ≡ x or y not free in A, and y not free in C, α, β, γ or Γ.
+Basic Hoare rules
+α ⊢H β
+Γ ⊢S {β · A · γ}
+γ ⊢H δ
+cons
+Γ ⊢S {α · A · δ}
+⊢H α ∨ β
+Γ ⊢S {α ∧ γ �� A · δ}
+Γ ⊢S {β ∧ γ · A · δ}
+cond
+Γ ⊢S {γ · A · δ}
+Additional axioms
+state axioms ∆H of the form Γ ⊢H α
+main axioms ∆S of the form Γ ⊢S {α · A · β}
+5
+
+On the other hand, the intended meaning of Γ ⊢S {α · A · β} is that from assumptions Γ, if α holds with
+respect to some initial state, then we can infer that A is true and β holds with respect to some modified
+state, or more precisely:
+[Γ] ⇒ (∃π [α](π) ⇒ ([A] ∧ ∃π′ [β](π′)))
+(1)
+In particular, the computational interpretation of (1) above will be a program that takes some input state
+π satisfying [α](π) and returns a realizer-state pair ⟨x, π′⟩ such that x realizes A and [β](π′) holds.
+Our semantic interpretation [·] will be properly defined in Section 4. Crucially, in SL the state is implicit,
+and so there are no variables or terms of state type. The state will rather be made explicit in our metatheory
+Sω. The main axioms and rules of SL simply describe how this semantic interpretation propagates in a
+call-by-value manner through the usual axioms and rules of first-order logic. The state itself is brought into
+play through the Hoare rules along with the additional axioms ∆H and ∆S.
+The two Hoare rules of SL correspond to the consequence and conditional rules of traditional Hoare
+logic. The usual conditional rule falls out as a special case of ours since we assume Γ ⊢H α ∨ ¬α. Some of
+the other traditional Hoare rules are derivable: The empty statement axiom corresponds to our own axiom
+Γ ⊢S {α · ⊤ · α} while composition can be viewed as the special case of ∧SI for A = B = ⊤. In Section 6 we
+add a controlled while loop to our logic. But for now, we illustrate our logic with some very straightforward
+scenarios.
+Example 2.2 (Simple read-write). Consider a very simple state, which we imagine to contain just two memory
+locations (input and output), and on which we can perform the following three actions:
+1. Store any value from our domain of discourse in the input location.
+2. For the current value x in the input location, compute some y such that P(x, y) holds (where P is a
+fixed binary predicate symbol of the logic), and store it in the output location.
+3. Retrieve the computed value y from the state’s output location.
+We could formalise those three actions by including two unary state predicates stored and solved, where
+stored(x) denotes that x is currently stored in the input location, and solved(x) denotes that some y satisfying
+P(x, y) is stores in the output location. We would then add the following axioms to ∆S, which intuitively
+represent each of the above actions:
+1. Γ ⊢S {α · ⊤ · stored(x)} where α ranges over all state formulas,
+2. Γ ⊢S {stored(x) · ⊤ · solved(x)}
+3. Γ ⊢S {solved(x) · ∃y P(x, y) · ⊤}
+We can then, for example, derive the following in SL + ∆H + ∆S for ∆H = ∅, where α, β are any state
+formulas:
+⊢S {β · ∀x {α · ∃y P(x, y) · ⊤} · β}
+An example of such a derivation is, for example:
+⊢S {α · ⊤ · stored(x)}
+⊢S {stored(x) · ⊤ · solved(x)}
+∧SI
+⊢S {α · ⊤ ∧ ⊤ · solved(x)}
+∧SEL
+⊢S {α · ⊤ · solved(x)}
+⊢S {solved(x) · ∃y P(x, y) · ⊤}
+∧SI
+⊢S {α · ⊤ ∧ ∃y P(x, y) · ⊤}
+∧SEL
+⊢S {α · ∃y P(x, y) · ⊤}
+∀SI
+⊢S {β · ∀x {α · ∃y P(x, y) · ⊤} · β}
+We note that while state formulas and actions are used in the proof, if we set α = β = ⊤ then the components
+of the theorem itself are just formulas in ordinary first-order logic.
+6
+
+Example 2.3 (Fixed-length array sorting). Let us now consider our state as an array of length three, and
+elements in that array as having some order structure.
+We formalise this in SL by introducing 1, 2, 3
+as constants representing our three locations, along with two state predicates: a binary predicate ≤ for
+comparing elements at locations l and l′, and a nullary predicate sorted that declares that the state is sorted.
+These can be characterised by adding the following axiom schemes, but to ∆H rather than ∆S as they do
+not represent state actions:
+Γ ⊢H 1 ≤ 2 ∧ 2 ≤ 3 ⇒ sorted
+Γ ⊢H l ≤ l′ ∨ l′ ≤ l
+where l, l′ range over {1, 2, 3}
+We then allow a single action on our array, namely the swapping of a pair of elements in the list. Suppose
+that α is a state formula of the form
+α :≡ l1 ≤ l′
+1 ∧ . . . ∧ ln ≤ l′
+n
+(2)
+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
+l and l′ are swapped, so that if e.g. α = 3 ≤ 2 ∧ 1 ≤ 2 ∧ 1 ≤ 3 then
+α[2 ↔ 3] = 2 ≤ 3 ∧ 1 ≤ 3 ∧ 1 ≤ 2
+We axiomatise the swapping of the values in locations of some arbitrary pair l, l′ ∈ {1, 2, 3} by adding to ∆S
+all instances of
+Γ ⊢S {α · ⊤ · α[l ↔ l′]}
+where α ranges over state formulas of the form (2). The statement that all arrays of length three can be
+sorted is then formulated as
+⊢S {⊤ · ��� · sorted}
+Let us now give a proof of this statement in SL + ∆H + ∆S. First, let α := 1 ≤ 2 ∧ 1 ≤ 3, and define D1 as
+⊢S {2 ≤ 3 ∧ α · ⊤ · 2 ≤ 3 ∧ α}
+cons
+⊢S {2 ≤ 3 ∧ α · ⊤ · sorted}
+2↔3
+⊢S {3 ≤ 2 ∧ α · ⊤ · 2 ≤ 3 ∧ 1 ≤ 3 ∧ 1 ≤ 2}
+cons
+⊢S {3 ≤ 2 ∧ α · ⊤ · sorted}
+cond[2≤3∨3≤2]
+⊢S {α · ⊤ · sorted}
+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,
+and for the final instance of cond we use ⊢H 2 ≤ 3 ∨ 3 ≤ 2. Now let D2 be defined by
+1↔2
+⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · 1 ≤ 2 ∧ 1 ≤ 3}
+D1
+⊢S {1 ≤ 2 ∧ 1 ≤ 3 · ⊤ · sorted}
+∧SI
+⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ ∧ ⊤ · sorted}
+∧SEL
+⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · sorted}
+Then we have D3:
+D2
+⊢S {2 ≤ 1 ∧ 2 ≤ 3 · ⊤ · sorted}
+{1 ≤ 2 ∧ 2 ≤ 3 · ⊤ · 1 ≤ 2 ∧ 2 ≤ 3}
+cons
+⊢S {1 ≤ 2 ∧ 2 ≤ 3 · ⊤ · sorted}
+cond[2≤1∨1≤2]
+⊢S {2 ≤ 3 · ⊤ · sorted}
+where here cond uses ⊢H 2 ≤ 1 ∨ 1 ≤ 2, and finally
+⊢S {2 ≤ 3 · ⊤ · 2 ≤ 3}
+2↔3
+⊢S {3 ≤ 2 · ⊤ · 2 ≤ 3}
+cond[2≤3∨3≤2]
+⊢S {⊤ · ⊤ · 2 ≤ 3}
+D3
+⊢S {2 ≤ 3 · ⊤ · sorted}
+∧SI
+⊢S {⊤ · ⊤ ∧ ⊤ · sorted}
+∧SEL
+⊢S {⊤ · ⊤ · sorted}
+7
+
+In contrast to Example 2.2 above, this is an example of a purely imperative proof that involves no proposi-
+tional formulas other than ⊤. As we will see in Example 5.6, the proof corresponds to a purely imperative
+program.
+3
+The system ST: A simple functional/imperative term calculus
+We now define our calculus ST + ΛS whose terms will represent realizers for proofs in SL + ∆H + ∆S. This
+is a standard typed lambda calculus for mixed functional and imperative programs, and is defined to include
+basic terms together with additional constants in some set ΛS, where the latter are intuitively there to realize
+the axioms in ∆S. Semantics for the terms will be given via a monadic translation into the metalanguage
+defined in the next section. Types are defined by the grammmar
+X ::= D | C | X × X | X + X | X → X
+while basic terms are defined as
+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
+where f ranges over all function symbols of SL, c are constants in ΛS, and α ranges over state formulas of
+SL. Typing derivations of the form Γ ⊢ t : X are given below, where Γ is a set of typed variables. Note that
+the types of constants c ∈ ΛS are also left unspecified.
+Γ ⊢ f : Dn → D
+where f has arity n
+Γ ⊢ c : X
+Γ ⊢ x : X
+if x : X in Γ
+Γ ⊢ skip : C
+Γ ⊢ s : X
+Γ ⊢ t : Y
+Γ ⊢ s ◦ t : X × Y
+Γ ⊢ t : X × Y
+Γ ⊢ p0(t) : X
+Γ ⊢ t : X × Y
+Γ ⊢ p1(t) : Y
+Γ ⊢ t : X
+Γ ⊢ ι0(t) : X + Y
+Γ ⊢ t : Y
+Γ ⊢ ι1(t) : X + Y
+Γ ⊢ r : X + Y
+Γ ⊢ s : X → Z
+Γ ⊢ t : Y → Z
+Γ ⊢ elim r s t ⊢ Z
+Γ, x : X ⊢ t : Y
+Γ ⊢ λx.t : X → Y
+Γ ⊢ t : X → Y
+Γ ⊢ s : X
+Γ ⊢ ts : Y
+Γ ⊢ defaultX : X
+Γ ⊢ s : X
+Γ ⊢ t : X
+x : D ∈ Γ for all free variables of α
+Γ ⊢ if α then s else t : X
+The type C should be interpreted as a type of commands that act on the state but don’t return any
+values. It is be helpful to consider a derived operator for sequential composition:
+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
+Γ ⊢ s ∗ t : C.
+4
+A monadic embedding of SL and ST into a metatheory Sω
+We now give a semantic interpretation of both state formulas of SL + ∆H + ∆S and terms in ST + ΛS into
+a standard higher-order, many sorted logic Sω + ΛSω.
+4.1
+The system Sω
+This logic contains typed lambda terms along with equational axioms for reasoning about them, together
+with the usual axioms and rules of many-sorted predicate logic.
+Because most aspects of the logic are
+completely standard, and in any case it is purely a verifying system, we are less detailed in specifying it.
+Types are defined as follows:
+X ::= D | 1 | Bool | S | X × X | X → X
+8
+
+where D represents objects in the domain of SL (just as in ST), Bool a type of booleans, and states are
+now explicitly represented as objects of type S. Our metatheory is an equational calculus, with an equality
+symbol =X for all types. Typed terms include:
+• variables x, y, z, . . . for each type, where we denote state variables by π, π1, π2, . . .
+• a constant f : Dn → D for each n-ary function symbol of SL,
+• additional, as yet unspecified constant symbols c : X for interpreting objects in ΛS, along with axioms
+that characterise them,
+• a unit element () : 1 along with the axiom x = (),
+• boolean constants t and f, with the axiom x =Bool t ∨ x =Bool f,
+• pairing ⟨s, t⟩ and projection proj0(t), proj1(t) operators, with the usual axioms,
+• terms formed by lambda abstraction and application, with the rule (λx.t)s = t[s/x],
+• 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
+case t x y = y.
+We sometimes write xX instead of x : X, and we use abbreviations such as ⟨x, y, z⟩ for ⟨x, ⟨y, z⟩⟩. Atomic
+formulas of Sω include all ordinary predicate symbols P, Q, R, . . . of SL as atomic formulas, where an n-ary
+predicate P in SL takes arguments of type Dn in Sω, along with predicates p, q, r, . . . for each state predicate
+symbol of SL, but now, if p is an n-ary state predicate in SL, p takes arguments of type Dn × S in Sω.
+General formulas are built using the usual logical connectives, including quantifiers for all types. The axioms
+and rules of Sω include the axioms of rules of predicate logic (now in all finite types), axioms for the terms,
+along with the usual equality axioms (including full extensionality). Because Sω acts as a verifying theory,
+we freely use strong axioms (such as extensionality), without concerning ourselves with the minimal such
+system that works.
+4.2
+The embedding [·] on state formulas of SL
+The main purpose of our metalanguage is to allow us to reason semantically about SL and ST. To do this,
+we introduce an embedding of state formulas of SL and terms of ST into Sω. We use the same notation [·]
+for both, as there is no danger of ambiguity.
+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
+x �→ x : D and f(t1, . . . , tn) �→ f(t1 ◦ · · · ◦ tn) : D. Similarly, there is a natural interpretation of t into Sω,
+this time with f(t1, . . . , tn) �→ f(⟨t1, . . . , tn⟩). We use the same notation for t in each of the three systems,
+as there is no risk of ambiguity.
+Definition 4.2. For each state formula α of SL, we define a formula [α](π) of Sω, whose free variables are
+the same as those of α (but now typed with type D) with the potential addition of a single state variable π,
+as follows:
+• [⊤](π) := ⊤ and [⊥](π) := ⊥,
+• [p(t1, . . . , tn)](π) := p(t1, . . . , tn, π),
+• [α ∧ β](π) := [α](π) ∧ [β](π), and similarly for α ∨ β and α ⇒ β.
+The following Lemma is easily proven using induction over propositional derivations.
+Lemma 4.3. If Γ ⊢H α in SL then [α](π) is provable in Sω from the assumptions [Γ](π), where [Γ](π) :=
+[α1](π), . . . , [αn](π) for Γ := α1, . . . , αn. This extends to proofs in SL + ∆H provided that the embedding of
+any axiom in ∆H is provable in Sω + ΛSω.
+9
+
+We are now in a position to make the semantic meaning of main formulas of SL precise. While, technically
+speaking, this is not necessary in what follows, neither to formulate our realizability interpretation nor to
+prove our soundness theorem, for the sake of completeness we give the full definition.
+Definition 4.4. For each main formula A of , we define a formula [A] of Sω, whose free variables are the
+same as those of A (but now typed with type D), as follows:
+• [⊤] := ⊤ and [⊥] := ⊥,
+• [P(t1, . . . , tn)] := P(t1, . . . , tn),
+• [A ∧ B] := [A] ∧ [B], [A ∨ B] := [A] ∨ [B] and [∃x A] := ∃xD [A],
+• [A ⇒ {α · B · β}] := [A] ⇒ [{α · B · β}] and [∀x {α · A · β}] := ∀xD [{α · A · β}]
+where [{α · A · β}] := ∃πS [α](π) ⇒ [A] ∧ ∃π′ [β](π′).
+Similarly to Lemma 4.3, we can now prove the following by induction over derivations in SL. We omit
+the proof, because it is straightforward and in any case not necessary in what follows.
+Proposition 4.5. If Γ ⊢S {α · A · β} in SL then [{α · A · β}] is provable in Sω from the assumptions [Γ],
+where [Γ] := [A1], . . . , [An] for Γ := A1, . . . , An. This extends to proofs in SL + ∆H + ∆S provided that the
+embedding of any axiom in ∆H and ∆S is provable in Sω + ΛSω.
+4.3
+The embedding [·] on terms of ST
+Our translation on terms is a call-by-value monadic translation using the state monad S → X × S. We first
+define a translation on types of ST as follows:
+• [D] := D, [C] := 1 and [X × Y ] := [X] × [Y ],
+• [X + Y ] := Bool × [X] × [Y ]
+• [X → Y ] := [X] → S → [Y ] × S
+Lemma 4.6. For any type X of SL, the type [X] is inhabited, in the sense that we can define a canonical
+closed term 0X : [X].
+Proof. Induction on types, letting 0D := c for a constant symbol which is assumed to exist in SL. The only
+other nonstandard case is 0X→Y , which can be defined as λx, π . ⟨0Y , π⟩.
+Finally, before introducing our translation on terms, we need to add characteristic functions to Sω
+for all state formulas (analogous to the characteristic functions for quantifier-free formulas in [14]). For
+any state formula α[x1, . . . , xn] of SL, where x1, . . . , xn are the free variables of α, we introduce constants
+χα :Dn → S → X → X → X satisfying the axioms
+[x1, . . . , xn](π) ⇒ χα ⟨x1, . . . , xn⟩ π y z = y
+[¬α][x1, . . . , xn](π) ⇒ χα ⟨x1, . . . , xn⟩ π y z = z
+Definition 4.7. For each term Γ ⊢ t : X of ST we define a term [Γ] ⊢ [t] : S → [X] × S of Sω as follows,
+where [·] is defined on contexts as [x1 : X1, . . . , xn : Xn] := x1 : [X1], . . . , xn : [Xn]:
+• [x]π := ⟨x, π⟩,
+• [skip]π := ⟨(), π⟩,
+• [f]π := ⟨λxDn, π . ⟨fx, π⟩, π⟩,
+• [c]π is appropriately defined for each additional constant in ΛS,
+10
+
+• [s ◦ t]π := ⟨a, b, π2⟩ where ⟨a, π1⟩ := [s]π and ⟨b, π2⟩ := [t]π1,
+• [p0t]π := ⟨a, π1⟩ and [p1t]π := ⟨b, π1⟩ where ⟨a, b, π1⟩ := [t]π,
+• [ι0t]π := ⟨f, a, 0Y , π1⟩ and [ι1t] := ⟨t, 0X, b, π1⟩ for ⟨a, π1⟩ := [t]π,
+• [elim r s t]π := case e (faπ2) (gbπ3) for ⟨e, a, b, π1⟩ := [r]π, ⟨f, π2⟩ := [s]π1, ⟨g, π3⟩ := [t]π1,
+• [λx.t]π := ⟨λx[X].[t], π⟩,
+• [ts]π := faπ2 for ⟨f, π1⟩ := [t]π and ⟨a, π2⟩ := [s]π1,
+• [defaultX]π := ⟨0X, π⟩,
+• [if α[x1, . . . , xn] then s else t]π := χα ⟨x1, . . . , xn⟩ π ([s]π) ([t]π) where {x1, . . . , xn} are the free variables
+of α.
+The following lemmas will be useful when verifying our realizability interpretation in the next section.
+The first is by a simple induction on terms.
+Lemma 4.8. For any term t of SL, we have [t]π = ⟨t, π⟩ (cf. Definitions 4.1 and 4.7).
+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 :
+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
+[(λ∗v.t)s]π = [t][a/x, b/y]π1
+where ⟨a, b, π1⟩ := [s]π.
+Proof. By unwinding the definition of [·]. For any variable v : X × Y we have [p0v]π = ⟨proj0v, π⟩ and
+[p1v]π = ⟨proj1v, π⟩, and we also have [λx, y . t]π = ⟨λx, π.⟨λy.[t], π⟩, π⟩. We therefore calculate
+[(λx, y.t)(p0v)]π = (λx, π.⟨λy.[t], π⟩)(proj0v)π = ⟨λy.[t][proj0v/x], π⟩
+and thus
+[(λx, y.t)(p0v)(p1v)]π = (λy.[t][proj0v/x])(proj1v)π = [t][proj0v/x, proj1v/y]π
+Finally, we can see that if ⟨a, b, π1⟩ := [s]π then
+π = (λv.[(λx, y.t)(p0v)(p1v)])(⟨a, b⟩)π1
+= (λv.[t][proj0v/x, proj1v/y])(⟨a, b⟩)π1
+= [t][proj0v/x, proj1v/y][⟨a, b⟩/v]π1
+= [t][a/x, b/y]π1
+which completes the proof.
+5
+A realizability interpretation of SL into ST
+We now come to the main contribution of the paper, which is the definition of a realizability relation between
+terms of ST and formulas of SL, along with a soundness theorem that shows us how to extract realizers from
+proofs. Our metatheory Sω is used to define the realizability relation and prove the soundness theorem.
+Definition 5.1 (Types of realizers). To each main formula A of SL we assign a type τS(A) of ST as follows:
+• τS(⊤) = τS(⊥) = τS(P(t1, . . . , tn)) := C,
+• τS(A ∧ B) := τS(A) × τS(B),
+11
+
+• τS(A ∨ B) := τS(A) + τS(B),
+• τS(∃x A) := D × τS(A),
+• τS(A ⇒ {α · B · β}) := τS(A) → τS(B),
+• τS(∀x {α · A · β}) := D → τS(A).
+Definition 5.2 (Realizability relation). For each main formula A of SL we define a formula x sr A of Sω,
+whose free variables are contained in those of A (now typed with type D) together with a fresh variable
+x : [τS(A)], by induction on the structure of A as follows:
+• x sr Q := Q for Q = ⊤, ⊥ or P(t1, . . . , tn),
+• x sr A ∧ B := (proj0x sr A) ∧ (proj1x sr B),
+• x sr A ∨ B := (proj0x = f ⇒ proj0(proj1x) sr A) ∧ (proj0x = t ⇒ proj1(proj1x) sr B),
+• x sr ∃y A(y) := (proj1x sr A)[proj0x/y],
+• f sr (A ⇒ {α · B · β}) := ∀x[τS(A)] (x sr A ⇒ fx sr {α · B · β}),
+• f sr (∀x {α(x) · A(x) · β(x)}) := ∀xD (fx sr {α(x) · A(x) · β(x)}),
+where for x : S → [τS(A)] × S we define
+• x sr {α · A · β} := ∀πS ([α](π) ⇒ proj0(xπ) sr A ∧ [β](proj1(xπ))).
+The following substitution lemma is easily proven by induction on formulas of SL.
+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
+in s and on the right hand side we implicitly mean the natural interpretation of t in Sω (cf. Definition 4.1).
+Theorem 5.4 (Soundness). Suppose that
+Γ := Au1
+1 , . . . , Aun
+n ⊢S {α · A · β}
+is provable in SL. Then we can extract from the proof a term ∆, τS(Γ) ⊢ t : τS(A) of ST, where ∆ contains
+the free variables of Γ and {α · A · β} (typed with type D) and τS(Γ) := x1 : τS(A1), . . . , xn : τS(An) for fresh
+variables x1, . . . , xn, such that the formula
+[t] sr {α · A · β}
+is provable in Sω from the assumptions (x1 sr A1)u1, . . . , (xn sr An)un for xi : [τS(Ai)]. The theorem holds
+more generally for proofs in SL + ∆H + ∆S, now provably in Sω + ΛSω, if:
+• for any axiom Γ ⊢H α in ∆H, the corresponding axiom [Γ](π) ⇒ [α](π) is added to ΛSω,
+• for any axiom in ∆S there is a term t of ST+ΛS such that [t] realizes that axiom provably in Sω +ΛSω.
+Proof. Induction on the structure of derivations in SL.
+In all cases, we assume as global assumptions
+(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
+state variable π, then a sr A and [β](π1) hold for ⟨a, π1⟩ := [t]π.
+• For the axiom Γ ⊢S {α · A · α}, if Au ∈ Γ we define t := x for the corresponding variable x : τS(A).
+Then [x]π := ⟨x, π⟩ for x sr A and [α](π). For Γ ⊢S {α · ⊤ · α} we define t := skip and the verification
+is even simpler.
+12
+
+• (∧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
+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,
+therefore we have shown that [s ◦ t] sr {α · A ∧ B · γ}.
+• (∧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⟩ :=
+[t]π. But then [p0t] sr {α · A · β} and [p1t] sr {α · B · β}.
+• (∨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
+(b = f ⇒ a sr A) ∧ (b = t ⇒ 0τS(B) sr B)
+for b := f. Thus [ι0t] sr A ∨ B. By an entirely analogous argument we can show that [ι1t] sr A ∨ B
+whenever [t] sr B.
+• (∨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
+x sr A, and [t](y) sr {β · C · γ} assuming y sr B. We claim that
+[elim r (λx.s) (λy.t)] sr {α · C · γ}
+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.
+There are now two possibilities. If e = f then
+elim r (λx.s) (λy.t)tπ = faπ2
+for ⟨f, π2⟩ := [λx.s]π1 = ⟨λx.[s](x), π1⟩
+= (λx.[s](x))aπ1
+= [s](a)π1
+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⟩ :=
+[s](a)π1, which proves the main claim in the case e = f. An analogous argument works for the case
+e = t.
+• (⇒S I) If t(x) is such that [t](x) sr {α · B · β} whenever x sr A, then by definition we have
+λx.[t] sr A ⇒ {α · B · β}
+and therefore [λx.t] sr {γ · A ⇒ {α · B · β} · γ} for any γ.
+• (⇒S E) Assume that [s] sr {β · A · γ} and [t] sr {α · A ⇒ {γ · B · δ} · β}. If [α](π) holds then defining
+⟨f, π1⟩ := [t]π we have [β]π1 and
+x sr A ⇒ fx sr {γ · B · δ}
+Similarly, defining ⟨a, π2⟩ := [s]π1, it follows that [γ](π2) and a sr A. Finally, setting ⟨b, π3⟩ := faπ2 it
+follows that b sr B and [δ](π3), and we have therefore proven that [ts] sr {α · B · δ}.
+• (⊥SE) If [t] sr {α · ⊥ · β} then from [α](π) we can infer a sr ⊥ and [β](π2) for ⟨a, π1⟩ := [t]π. But
+a sr ⊥ = ⊥, and from ⊥ we can deduce anything, and in particular 0τS(A) sr A and [γ](π), from which
+it follows that [defaultτS(A)] sr {α · A · γ}.
+• (∀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
+{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
+deduce in Sω that
+∀xD [t](x) sr {α · A · β}
+and therefore λx.[t] sr ∀x {α · A · β}, and thus (just as for ⇒S I) we have
+[λx.t] sr {γ · ∀x {α · A · β} · γ}
+for any γ.
+13
+
+• (∀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}
+and [β][t/x](π1) for ⟨f, π⟩ := [s]π.
+Now, using Lemma 4.8 we have [st]π = ftπ1 for the natural
+interpretation of t in Sω, since we can prove in Sω that
+ft sr {β[t/x] · A[t/x] · γ[t/x]}
+it follows that a sr A[t/x] and [γ][t/x](π2) for ⟨a, π2⟩ := ftπ1, and therefore we have shown that
+[st] sr {α · A[t/x] · γ[t/x]}.
+• (∃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
+5.3 we therefore have (a sr A)[t/x], and therefore ⟨t, a⟩ sr ∃x A. Observing (using Lemma 4.8) that
+[t ◦ s]π = ⟨t, a, π1⟩, we have shown that [t ◦ s] sr {α · ∃x A · β}.
+• (∃SE) Suppose that s and t(x, z) are such that [s] sr {α · ∃x A · β} and
+z sr A[y/x] ⇒ [t](y, z) sr {β · C · γ}
+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
+z sr A[y/x] = (z sr A)[y/x] = ⟨y, z⟩ sr ∃x A we therefore have
+⟨y, z⟩ sr ∃x A ⇒ [t](y, z) sr {β · C · γ}
+Now, applying Lemma 4.9 to ∆, Γ, y : D, z : τS(A) ⊢ t : τS(C), we have
+[(λ∗v.t)s]π = [t](e, a)π1
+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
+[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
+have shown that [(λ∗v.t)s] sr {α · C · γ}.
+• (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
+in Sω (respectively Sω + ΛSω for the general version of the theorem) for any π : S. It is then easy to
+show that if [t] sr {β · A · γ} then we also have [t] sr {α · A · δ}.
+• (cond) Suppose that [s] sr {α ∧ γ · A · δ} and [t] sr {β ∧ γ · A · δ}. We claim that
+[if α then s else t] sr {γ · A · δ}
+To prove this, suppose that [γ](π) holds. Since ⊢H α ∨ β then [α](π) ∨ [β](π) is provable in Sω, and so
+we consider two cases. Let {x1, . . . , xn} be the free variables of α. If [α](π) holds, then
+[if α then s else t]π = χα ⟨x1, . . . , xn⟩ π ([s]π) ([t]π) = [s]π
+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
+[β](π) holds, then by an analogous argument we can show that a sr A and [δ](π1) for ⟨a, π1⟩ := [t]π =
+[if α then s else t]π, and we are done.
+The extension of the soundness theorem to SL + ∆H + ∆S is straightforward, as the soundness proof is
+modular and so any axioms along with their realizers can be added. The first condition is needed so that
+Lemma 4.3 (needed for the cons rule) continues to apply.
+For the free variable condition that the free variables of t are contained in those of Γ, {α · A · β} and
+τS(Γ), if this were not the case, we could simply ground those variables with a canonical constant c : D and
+we would still have ˜t sr {α · A · β} for the resulting term ˜t.
+14
+
+Corollary 5.5 (Program extraction). Suppose that the sentence
+⊢S {α · ∀x {β · ∃y P(x, y) · γ(x)} · β}
+is provable in SL + ∆S. Then we can extract a closed realizing term t : D → D × C in ST + ΛS such that
+defining g : D → S → D × S by gxπ := ⟨a, π2⟩ for ⟨f, π1⟩ := [t]π and ⟨a, (), π2⟩ := fxπ1, we have
+∀πS([α](π) ⇒ ∀xD (P(x, proj0(gxπ)) ∧ [γ](x)(proj1(gxπ))))
+provably in Sω + ΛSω.
+5.1
+Simplification and removal of unit types
+In presentations of modified realizability that use product types instead of type sequences, it is common to
+introduce the notion of a Harrop formula (a formula that does not contain disjunction or existential quantifi-
+cation in a positive position) and define realizability in a way that all Harrop formulas have unit realizability
+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
+as for us. We have avoided this simplification earlier on, as it would have added additional cases and bureau-
+cracy to our soundness theorem. However, we can compensate retroactively for this choice by introducing
+equivalences on types that eliminate unit types, namely the closure under contexts of
+1 × X ≃ 1 ≃ X × 1
+(1 → X) ≃ X
+(X → 1) ≃ 1
+along with corresponding equivalences on terms, also closed under contexts:
+t1×X ≃ proj1(t)X
+tX×1 ≃ proj0(t)X
+t1→X ≃ t()
+tX ≃ λx1.t
+tX→1 ≃ ()
+For example, in Corollary 5.5 we would then have
+[t]π : (D → S → D × 1 × S) × S ≃ (D → S → D × S) × S
+and
+gxπ ≃ fxπ1.
+For us, the equivalence relation ≃ will not play a formal role in the paper, but will be used to provide
+simplified descriptions of extracted programs.
+5.2
+Examples of program extraction
+We now continue the short illustrative examples we outlined in Section 2.3.
+Example 5.6 (Simple read-write). In Example 2.2 we considered a state where three actions were possible
+(writing to the state, performing a calculation, and reading the output from the state). We can formalise these
+three actions semantically in the metatheory Sω by including three constants in ΛSω, namely c1 : D → S → S,
+c2 : S → S and c3 : S → D, along with the characterising axioms:
+1. stored(x, c1xπ),
+2. stored(x, π) ⇒ solved(x, c2π),
+3. solved(x, π) ⇒ P(x, c3π).
+While we are able to use these constants to form terms in Sω such as λπ, π1, x . ⟨c1xπ, c2π1⟩, which could be
+viewed as non-sequential in the sense that we take two input states as arguments, we can force them to be
+applied in a sequential, call-by-value manner by adding three corresponding constants to our term calculus
+ST, namely including write : D → C, calc : C and read : D × C in ΛS, along with the embedding rules
+• [write]π := ⟨λx, π′ . ⟨(), c1xπ′⟩, π⟩ ≃ ⟨c1, π⟩,
+• [calc]π := ⟨(), c2π⟩ so that [calc] ≃ c2,
+15
+
+• [read]π := ⟨c3π, (), π⟩ ≃ ⟨c3π, π⟩.
+and then restricting out attention to terms of the form [t] for t ∈ ST + {write, calc, read}. We can then prove
+the following in Sω i.e. that all axioms in ∆S can be realised:
+• [write(x)] sr {α · ⊤ · stored(x)},
+• [calc] sr {stored(x) · ⊤ · solved(x)},
+• [read] sr {solved(x) · ∃y P(x, y) · ⊤}.
+and thus Theorem 5.4 applies to SL + ∆H + ∆S for ∆H = ∅. In particular, we have
+[t] sr {β · ∀x {α · ∃y P(x, y) · ⊤} · β}
+for t := λx . ((write(x) ∗ calc) ∗ read) where ∗ is sequential composition operator from Definition 3.1. A
+formal derivation of this term from the corresponding proof given in Example 2.2 is as follows:
+x : D ⊢ write(x) : C
+x : D ⊢ calc : C
+∧SI
+x : D ⊢ write(x) ◦ calc : C × C
+∧SEL
+x : D ⊢ write(x) ∗ calc : C
+x : D ⊢ read : D × C
+∧SI
+x : D ⊢ (write(x) ∗ calc) ◦ read : C × D × C
+∧SEL
+x : D ⊢ (write(x) ∗ calc) ∗ read : D × C
+∀SI
+⊢ λx . ((write(x) ∗ calc) ∗ read) : D → D × C
+Example 5.7 (Fixed-length array sorting). In Example 2.3 we considered a situation where we are allowed a
+single action on our state, namely to swap elements. Analogously to the previous example, we can formalise
+this in our semantic environment Sω by adding to ΛSω constants cl,l′ : S → S for each pair l, l′ ∈ {1, 2, 3}
+along with the axiom
+[α](π) ⇒ [α[l ↔ l′]](cl,l′π)
+ranging over state formulas α of the form (2) and locations l, l′ ∈ {1, 2, 3} of SL, together with axioms
+corresponding to those of ∆H i.e.
+(π) ⇒ sorted(π)
+and
+[l ≤ l′ ∨ l′ ≤ l](π)
+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
+swapl,l′ sr {α · ⊤ · α[l ↔ l′]}
+A derivation of a closed term t : C of ST + {swapl,l′} such that [t] sr {⊤ · ⊤ · sorted} is given below. In
+particular, we can prove in Sω that ∀πS sorted(proj1([t]π)), and so the term λπ . proj1([t]π) : S → S acts as
+a sorting program for arrays of length three. For an extracted term t corresponding to the proof given in
+Example 2.3, first we interpret D1 as
+⊢ skip : C
+cons
+⊢ skip : C
+2↔3
+⊢ swap2,3 : C
+cons
+⊢ swap2,3 : C
+cond[2≤3∨3≤2]
+⊢ t1 := if (2 ≤ 3) then (skip) else (swap2,3) : C
+and define t1 := if (2 ≤ 3) then (skip) else (swap2,3). Now D2 is interpreted as
+1↔2
+⊢ swap1,2 : C
+D1
+⊢ t1 : C
+∧SI
+⊢ swap1,2 ◦ t1 : C × C
+∧SEL
+⊢ t2 := swap1,2 ∗ t1 : C
+16
+
+where we define t2 := swap1,2 ∗ t1 : C. Continuing, D3 is interpreted as:
+D2
+⊢ t2 : C
+⊢ skip : C
+cons
+⊢ skip : C
+cond[2≤1∨1≤2]
+t3 := if (2 ≤ 1) then t2 else (skip) : C
+where t3 := if (2 ≤ 1) then t2 else (skip), and finally
+⊢ skip : C
+2↔3
+⊢ swap2,3 : C
+cond[2≤3∨3≤2]
+⊢ if (2 ≤ 3) then (skip) else (swap2,3) : C
+D3
+⊢ t3 : C
+∧SI
+⊢ (if (2 ≤ 3) then (skip) else (swap2,3)) ◦ t3 : C × C
+∧SEL
+⊢ t := (if (2 ≤ 3) then (skip) else (swap2,3)) ∗ t3 : C
+6
+An extension to arithmetic
+We now present an extension of our framework to a stateful version of first-order intuitionistic arithmetic.
+On the logic side, we will add not only a stateful induction rule, but also a Hoare-style while rule for iteration
+over the natural numbers. On the computational side, these will be interpreted by stateful recursion in all
+finite types, along with a controlled while loop. The addition of these constants will allow us to extract
+programs that are more interesting than those obtainable from proofs in pure predicate logic, and which
+can be clearly compared to well-known stateful algorithms. To exemplify this, we will present a formally
+synthesised version of insertion sort, and we stress that by further extending our framework with additional
+rules and terms, we would be able to extract an even richer variety of combined functional/stateful programs.
+6.1
+The system SA: First-order arithmetic with state
+Our system of stateful intuitionistic arithmetic SA builds on SL just as ordinary first-order Heyting arithmetic
+builds on first-order predicate logic. In both cases, we introduce a constant 0, a unary successor symbol succ,
+symbols for all primitive recursive functions, and our predicate symbols now include an equality relation =.
+In what follows we write x + 1 instead of succ(x). The axioms and rules of SA are, in turn, analogous to
+the additional axioms and rules we would require in ordinary first-order arithmetic: They include all axioms
+and rules of SL (based now on the language of SA), along with a collection of additional axioms and rules.
+These comprise not only basic axioms and rules for equality and the successor, and an induction rule (all now
+adapted to incorporate the state), but also a new while rule for stateful iteration, which now exploits our
+state and, as we will see, allows us to extract programs that contain while loops. These additional axioms
+and rules are outlined in Figure 6.1.
+Our formulation of stateful arithmetic follows the same basic idea as the construction of stateful predicate
+logic, incorporating standard rules but keeping track of an ambient state in a call-by-value manner, and
+adding new rules that explicitly correspond to stateful constructions. In particular, Proposition 2.1 clearly
+extends to SA, as the usual axioms and rules of arithmetic can be embedded into those of SA:
+Proposition 6.1. For any formula A of HA and state formula α, define the main formula Aα of SA as in
+Proposition 2.1. Then whenever Γ ⊢I A is provable in HA, we have that Γα, ∆ ⊢S {α · Aα · α} is provable in
+SA, where ∆ is arbitrary and Γ�� := (A1)u1
+α , . . . , (An)un
+α
+for Γ := Au1
+1 , . . . , Aun
+n .
+We can also derive a natural extensionality rule from our stateful equality rules, which assures us that
+whenever s = t in ordinary Heyting arithmetic, then we can replace s by t for stateful formulas:
+Proposition 6.2. Suppose that ⊢I s = t is provable in HA. Then from Γ ⊢S {α(s) · A(s) · β(s)} we can
+derive Γ ⊢S {α(t) · A(t) · β(t)} in SA.
+17
+
+Figure 3: Additional axioms and rules of SA
+Axioms and rules for equality
+Γ ⊢S {α · t = t · α}
+Γ ⊢S {α · s = t · β}
+Γ ⊢S {α · t = s · β}
+Γ ⊢S {α · r = s · β}
+Γ ⊢S {β · s = t · γ}
+Γ ⊢S {α · r = t · γ}
+Γ ⊢S {α · s = t · β}
+Γ ⊢S {β · A(s) · γ(s)}
+ext
+Γ ⊢S {α · A(t) · γ(t)}
+Axioms and rules for arithmetical function symbols
+Γ ⊢S {α · succ(t) ̸= 0 · α}
+{α · succ(s) = succ(t) · β}
+{α · s = t · β}
+Γ ⊢S {α · l = r · α}
+where l = r ranges across defining equations for prim. rec. functions
+Induction rule
+Γ ⊢S {α · A(0) · β(0)}
+Γ, A(x) ⊢S {β(x) · A(x + 1) · β(x + 1)}
+ind
+Γ ⊢S {γ · ∀x {α · A(x) · β(x)} · γ}
+While rule (over natural numbers)
+A1
+A2
+A3
+while
+Γ, A(x) ⊢S {α(x) · B · β}
+A1 := Γ, A(x + 1) ⊢S {γ(x + 1) ∧ α(x + 1) · A(x) · α(x)}
+A2 := Γ, A(x + 1) ⊢S {¬γ(x + 1) ∧ α(x + 1) · B · β}
+A3 := Γ, A(0) ⊢S {α(0) · B · β}
+for ind and while, x is not free in Γ, and for while it is not free in B or β
+Proof. By Proposition 6.1 for α := α(s) we have Γ ⊢S {α(s) · s = t · α(s)} and thus using the extensionality
+rule in SA we can derive
+Γ ⊢S {α(s) · s = t · α(s)}
+Γ ⊢S {α(s) · A(s) · β(s)}
+ext
+Γ ⊢S {α(s) · A(t) · β(t)}
+Since ⊢I t = s must also be provable in HA, another instance of Proposition 6.1 for α := α(t) along with the
+true axiom in SA gives us
+Γ ⊢S {α(t) · t = s · α(t)}
+Γ ⊢S {α(t) · ⊤ · α(t)}
+ext
+Γ ⊢S {α(t) · ⊤ · α(s)}
+Putting these together we obtain
+Γ ⊢S {α(t) · ⊤ · α(s)}
+Γ ⊢S {α(s) · A(t) · β(t)}
+∧SI
+Γ ⊢S {α(t) · ⊤ ∧ A(t) · β(t)}
+∧SEL
+Γ ⊢S {α(t) · A(t) · β(t)}
+which completes the derivation.
+18
+
+6.2
+An extended term calculus STN
+In order to give derivations in SA a computation interpretation, we need to extend our term calculus ST
+to include a recursor (for induction) and a controlled while loop (for the while rule). The remaining new
+axioms and rules of SA are dealt with in a straightforward manner.
+To be precise: the theory STN is defined to be the instance of ST for the case of arithmetic, with function
+symbols for zero, successor and all primitive recursive functions. Accordingly, we rename the base type D
+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]
+ranges over state formulas of SL with a specified free variable z. The typing rules for these new terms are
+Γ ⊢ s : X
+Γ ⊢ t : Nat → X → X
+Γ ⊢ rec s t : Nat → X
+for the recursor, while for the while loop we have
+Γ ⊢ r : Nat → X → X
+Γ ⊢ s : Nat → X → Y
+Γ ⊢ t : X → Y
+Γ ⊢ u : Nat
+Γ ⊢ whileu γ[z] r s t : X → Y
+under the additional variable condition that z /∈ Γ, but x : Nat ∈ Γ for all free variables of γ[z] outside of
+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
+condition. In order to give the appropriate semantics to our terms, we must add to our metatheory Sω
+axioms and rules for arithmetic in all finite types, including the ability to define functions of arbitrary type
+via recursion over the natural numbers, along the lines of E-HAω [30] (though as before the precise details
+are not important). We then define:
+• [rec s t]π := ⟨Rf, π1⟩ for ⟨f, π1⟩ := [t]π, where
+Rf0π := [s]π
+Rf(n + 1)π := gaπ′
+2 for ⟨a, π′
+1⟩ := Rfnπ′ and ⟨g, π′
+2⟩ := fnπ′
+1
+(3)
+• [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⟩ :=
+[u]π3, where
+Lf,g,h0yπ′ := hyπ′
+Lf,g,h(n + 1)yπ′ :=
+�
+Lf,g,hny′π2 for ⟨a, π′
+1⟩ := fnπ′ and ⟨y′, π′
+2⟩ := ayπ′
+1
+if [γ][n + 1](π′)
+byπ′
+1 for ⟨b, π′
+1⟩ := gnπ′
+if ¬[γ][n + 1](π′)
+(4)
+where in the case distinctions, we would technically speaking need to use the characteristic function
+χγ⟨x1, . . . , n, . . . , xk⟩ for γ, with n substituted for the special free variable z.
+6.3
+The soundness theorem for arithmetic
+We now need to show that the soundness proof for stateful predicate logic also holds in the extension to
+arithmetic.
+Theorem 6.3. The statement of Theorem 5.4 remains valid if we replace SL by SA and ST by STN.
+Proof. We need to extend the proof of Theorem 6.3 to show that the additional axioms and rules as in Figure
+6.1 can be realized by a term of the form [t] for t in STN.
+• For the non-extensionality equality and arithmetic axioms this is straightforward due to the fact that
+these are also true in Sω: For instance, given a realizer [s] sr {α · u = v · β} and [t] sr {β · v = w · γ},
+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,
+it follows that [p1(s ◦ t)] sr {α · u = w · γ}. The other axioms and rules are even simpler.
+19
+
+• (ext) Extensionality is similarly simple: If [s] sr {α · u = v · β} and [t] sr {β · A(u) · γ(u)}, then [α](π)
+implies that u = v and [β](π1) for ⟨. . . , π1⟩ := [s]π, and therefore a sr A(u) and [γ](u)(π2) for
+⟨a, π2⟩ := [t]π1. Now applying extensionality in Sω to the formula T (x) := a sr A(x) ∧ [γ](x)(π2),
+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
+[p2(s ◦ t)] sr {α · A(v) · γ(v)}.
+• (rec) Suppose that s and t(x, y) are such that [s] sr {α · A(0) · β(0)} and
+[t](x, y) sr {β(x) · A(x + 1) · β(x + 1)}
+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
+[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
+n : Nat we have
+Rfn sr {α · A(n) · β(n)}
+We prove this by induction: For the base case, we have Rf0 = [s] and the claim holds by assumption.
+For the induction step, let us assume that [α](π′) holds, and so by the induction hypothesis we have
+a sr A(n) and [β(n)](π′
+1) for ⟨a, π′
+1⟩ := Rfn. Since fnπ′
+1 = ⟨g, π′
+1⟩ for g := λy.[t](n, y), we have that
+Rf(n+1)π′ = [t](n, a)π′
+1, and since by the property of [t] we then have b sr A(n + 1) and [β(n+1)](π′
+2)
+for ⟨b, π′
+2⟩ := [t](n, a)π′
+1, we have shown that Rf(n + 1) sr {α · A(n + 1) · β(n + 1)}, which completes
+the induction.
+• (while) We suppose that
+1. [r](x, y) sr {γ(x + 1) ∧ α(x + 1) · A(x) · α(x)} assuming that y sr A(x + 1),
+2. [s](x, y) sr {¬γ(x + 1) ∧ α(x + 1) · B · β} assuming that y sr A(x + 1),
+3. [t](y) sr {α(0) · B · β} assuming that y sr A(0).
+Our aim is to show that
+[(whilex γ (λx′, y′.r) (λx′, y′.s) (λy′.t))y] sr {α(x) · B · β}
+for any x, y ∈ Nat with y sr A(x). We observe, unwinding the definition, that
+[(whilex γ (λx′, y′.r) (λx′, y′.s) (λy′.t))y]π = Lf,g,hxyπ
+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
+show by induction on n that if y sr A(n) then
+Lf,g,hny sr {α(n) · B · β}
+and then the result follows by setting n := x. The base case is straightforward since
+Lf,g,y0y = [t](y)
+and the claim follows by definition of [t]. For the induction step, suppose that y sr A(n + 1) and
+[α(n + 1)](π). There are two cases. If ¬[γ](n + 1)(π) we have
+Lf,g,h(n + 1)yπ = [s](n, y)π
+and the result holds by the property of [s]. On the other hand, if [γ](n + 1)(π) then
+Lf,g,h(n + 1)yπ = Lf,g,hny′π′
+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
+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
+so the result is proven for n + 1.
+This covers all the additional axioms and rules of SA.
+20
+
+6.4
+Worked example: Insertion sort
+We now illustrate our extended system by synthesising a list sorting program that, intuitively, forms an
+implementation of the insertion sort algorithm. Here our state will represent the structure that is to be
+sorted, and continuing the spirit of generality that we have adhered to throughout, we characterise this
+structure through a number of abstract axioms. Instantiating the state as, say, an array of natural numbers,
+would provide a model for our theory, but our sorting algorithm can be extracted on the more abstract level.
+Crucially, the proof involves both loop iteration and induction, and the corresponding program combines an
+imperative while loop with a functional recursor.
+We begin by axiomatising our state, just as in previous examples. An intuition here is that states represent
+an infinite array of elements a0, a1, . . . possessing some total order structure ≤, and we seek to extract a
+program that, for any input n, sorts the first n elements. We use this informal semantics throughout to
+indicate the intended meaning of our axioms, but stress that none of this plays a formal role in the proof or
+resulting computational interpretation.
+We introduce three state predicates to SA, with the intuition indicated in each case:
+• sort(N)
+Sorted: The first N + 1 elements of the array i.e. [a0, . . . , aN] are sorted
+• psort(n, N)
+Partially sorted with respect to an: if n < N then the list [a0, . . . , an−1, an+1, . . . , aN] is sorted and
+an ≤ an+1. For the base cases, if n = N then the list [a0, . . . , aN−1] is sorted, and if n > N then the
+list [a0, . . . , aN] is sorted.
+• comm(n)
+Comparison: true if an ≤ an−1, and always true if n = 0
+We formalise this intuition by adding the following state independent axioms to ∆H:
+1. Γ, sort(N) ⊢H psort(N + 1, N + 1)
+If the first N +1 elements are sorted, then they are also partially sorted with respect to the next element
+aN+1.
+2. Γ, ¬comm(n), psort(n, N) ⊢H sort(N)
+If [a0, . . . , an−1, an+1, . . . , aN] is sorted, an ≤ an+1, but also an−1 ≤ an, then the entire segment
+[a0, . . . , aN] must be sorted.
+3. Γ, psort(0, N) ⊢H sort(N)
+If [a1, . . . , aN] is sorted and a0 ≤ a1, then [a0, . . . , aN] is sorted.
+4. Γ ⊢H sort(0)
+The singleton array [a0] is defined to be sorted.
+We complete the axiomatisation by adding a single state-sensitive axiom to ∆S:
+5 Γ ⊢S {comm(n + 1) ∧ psort(n + 1, N) · ⊤ · psort(n, N)}
+If [a0, . . . , an, an+2, . . . , aN] is sorted and an+1 ≤ an+2, but an+1 ≤ an, then we can modify the state
+(i.e. swapping an and an+1 by setting ˜an := an+1 and ˜an+1 := an) so that [a0, . . . , an−1, ˜an+1, . . . , aN]
+is sorted and ˜an ≤ ˜an+1. The edge cases for n ≥ N are interpreted in a more straightforward way.
+In order to give a realizing term to this axiom, we representing element swapping semantically by adding a
+constant c : Nat → S → S to our metatheory Sω, which satisfies
+comm(n + 1, π) ∧ psort(n + 1, N, π) ⇒ psort(n, N, cnπ)
+21
+
+and a corresponding term swap : Nat → C to our term calculus, along with the embedding
+[swap]π := ⟨λn, π.⟨(), cnπ⟩, π⟩ ≃ ⟨c, π⟩
+so that we can prove
+[swap n] sr {comm(n + 1) ∧ psort(n + 1, N) · ⊤ · psort(n, N)}
+With this in place, we can now prove in SA that the first N elements of the state can be sorted, and extract
+a corresponding realizing term in STN.
+6.4.1
+Proof of ⊢S {γ · ∀N {α · ⊤ · sort(N)} · γ} in SA
+The core of our proof begins with an instance of the while rule parametrised by N, with Γ := ∅, A(n) := ⊤,
+α(n) := psort(n, N + 1), β := sort(N + 1) and γ(n) := comm(n):
+D1
+D2
+D3
+while
+⊤ ⊢S {psort(n, N + 1) · ⊤ · sort(N + 1)}
+∀SI
+⊤ ⊢S {psort(N + 1, N + 1) · ∀n {psort(n, N + 1) · ⊤ · sort(N + 1)} · psort(N + 1, N + 1)}
+∀SE
+⊤ ⊢S {psort(N + 1, N + 1) · ⊤ · sort(N + 1)}
+cons
+⊤ ⊢S {sort(N) · ⊤ · sort(N + 1)}
+where the final composition inference makes use of the first state independent axiom. Here D1 represents an
+instance of the state sensitive axiom
+⊤ ⊢S {comm(n + 1) ∧ psort(n + 1, N + 1) · ⊤ · psort(n, N + 1)}
+and D2 represents the derivation
+⊤ ⊢S {sort(N + 1) · ⊤ · sort(N + 1)}
+cons
+⊤ ⊢S {¬comm(n + 1) ∧ psort(n + 1, N + 1) · ⊤ · sort(N + 1)}
+where composition makes use of the second state independent axiom. Finally D3 is
+⊤ ⊢S {sort(N + 1) · ⊤ · sort(N + 1)}
+cons
+⊤ ⊢S {psort(0, N + 1) · ⊤ · sort(N + 1)}
+this time making use of the third state independent axiom. Finally we can prove that all lists can be sorted
+with an outer induction as follows:
+⊢S {α · ⊤ · α}
+cons
+⊢S {α · ⊤ · sort(0)}
+D
+⊤ ⊢S {sort(N) · ⊤ · sort(N + 1)}
+ind
+⊢S {γ · ∀N {α · ⊤ · sort(N)} · γ}
+where α is an arbitrary state predicate, the instance of cons uses the fourth state independent axiom, and
+D represents the derivation above.
+6.4.2
+Program extraction
+We now extract a program that corresponds to the above proof. First of all, we note that the three premises
+of our while rule are realised by swap n, skip and skip respectively, and so our derivation D corresponds to
+the following program:
+y : C ⊢ swap n : C
+y : C ⊢ skip : C
+y : C ⊢ skip : C
+while
+y : C ⊢ t(n)y : C
+∀SI
+y : C ⊢ λn.t(n)y : Nat → C
+∀SE
+y : C ⊢ (λn.t(n)y)(N + 1) : C
+cons
+y : C ⊢ (λn.t(n)y)(N + 1) : C
+22
+
+where
+t(n) := whilen comm[z] (λx, y.(swap x)) (λx, y.skip) (λy.skip)
+≃ whilen comm[z] (λx.(swap x)) (skip) (skip)
+Then our final induction generates the following program:
+⊢ skip : C
+y : C ⊢ (λn.t(n)y)(N + 1) : C
+ind
+⊢ rec (skip) (λx, y.((λn.t(n)y)(x + 1))) : Nat → C
+Thus our list sorting program is
+rec (skip) (λx, y.((λn.t(n)y)(x + 1)))
+≃ rec (skip) (λx.((λn.(whilen comm[z] (λx.(swap x)) (skip) (skip)()))(x + 1)))
+which is essentially an implementation of the insertion sort algorithm, with an outer recursion that sorts
+initial segments of the list in turn, and an inner loop that inserts new elements into the appropriate place in
+the current sorted list.
+7
+Directions for future work
+In this paper we have presented the central ideas behind a new method for extracting stateful programs
+from proofs, which include an extension of ordinary first-order logic with Hoare triples, a corresponding
+realizability interpretation, and a soundness theorem. We emphasise once again that our intention has been
+to offer an alternative approach to connecting proofs with stateful programs, one that seeks to complement
+rather than improve existing work by embracing simplicity and abstraction, and which might be well suited
+to a range of applications in proof theory or computability theory. In this spirit, we conclude with a very
+informal outline of a series interesting directions in which we anticipate that our framework could be applied.
+7.1
+Further extensions and program synthesis
+While our main results have been presented in the neutral setting of first-order predicate logic, it would be
+straightforward to extend SL to richer logics with more complex data structures and a imperative commands.
+Already, the addition of recursion and loops over natural numbers in Section 6 has allowed us to synthesise
+a standard in-place sorting algorithm using our abstract axiomatisation of an ordered state, in a similar
+spirit to [6]. However, further extensions are naturally possible, including the addition of general fixpoint
+operators and non-controlled while loops, which would then require a Sω to be replaced by a domain theoretic
+semantics that allows for partiality.
+Looking a step further ahead, by implementing all of this in a proof assistant, we would have at our
+disposal a new technique for synthesising correct-by-construction imperative programs. While we do not
+suggest that this pipeline would directly compete with existing techniques for verifying imperative programs,
+it could be well suited to synthesising and reasoning about programs in very specific domains, where we are
+interested in algorithms for which interactions with the state have a restricted form that could be suitably
+axiomatised within our logic. For example, a more detailed axiomatisation our state as an ordered array along
+the lines of Section 6.4, with a “swap” operation and a few other ways of interacting with the state, might
+give rise to an interesting theory of in-place sort algorithms. Stateful algorithms on other data structures,
+such as graphs, could presumably also be formalised within our framework.
+7.2
+Bar recursion and the semantics of extracted programs
+Two of the main starting points for this paper, the monadic realizability of Birolo [7] and the author’s own
+Dialectica interpretation with state [24], address the broader problem of trying to understand the operational
+semantics of programs extracted from proofs as stateful procedures (the origins and development of this
+general idea, from Hilbert’s epsilon calculus onwards, is brilliantly elucidated in Chapter 1 of Aschieri’s
+23
+
+thesis [2], who then sets out his own realizability interpretation based on learning). A number of case studies
+by the author and others [20, 21, 25, 26] have demonstrated that while terms extracted from nontrivial proofs
+can be extremely complex, they are often much easier to understand if one focuses on the way they interact
+with the mathematical environment. For example, in understanding a program extracted from a proof using
+Ramsey’s theorem for pairs [20], it could be illuminating to study the trace of the program as it queries a
+colouring at particular pairs, as this can lead to a simpler characterisation of the algorithm ultimately being
+implemented by the term.
+While the aforementioned analysis of programs has always been done in an informal way, our stateful
+realizability interpretation would in theory allow us to extract programs which store this trace formally in
+the state, where our abstract characterisation of state would allow us to implement it in whichever way is
+helpful in a given setting. For example, in the case of the Bolzano-Weierstrass theorem [21], our state might
+record information of the form xn ∈ I, collecting information about the location of sequence elements. For
+applications in algebra [26], one might instead store information about a particular maximal ideal.
+The aforementioned theorems are typically proven using some form of choice or comprehension, and that
+in itself leads to the interesting prospect of introducing both stateful recursors and while-loops that are
+computationally equivalent to variants of bar recursion [29]. In [23], several bar recursive programs that
+arise from giving a computational interpretation to arithmetical comprehension principles are formulated as
+simple while loops, and these could in principle be incorporated into our system with new controlled Hoare
+rules in the style of update recursion [5], that replace the conditions n < N and n ≥ N in the Ai above with
+e.g. n ∈ dom(f) and n /∈ dom(f), where f is some partial approximation to a comprehension function. An
+exploration of such while-loops from the perspective of higher-order computability theory might well be of
+interest in its own right.
+7.3
+A logic for probabilistic lambda calculi
+Probabilistic functional languages are a major topic of research at present. While work in this direction dates
+back to the late 1970s [16, 28] where it typically had a semantic flavour, a more recent theme [9, 11, 12] has
+been to study simple extensions of the lambda calculus with nondeterministic choice operators ⊕, where s⊕t
+evaluates nondeterministically (or probabilistically) to either s or t. While such calculi have been extensively
+studied, corresponding logics that map under some proof interpretation to probabilistic programs are far
+more rare (although there is some recent work in this direction e.g. [1]).
+We conjecture that our framework offers a bridge between logic and probabilistic computation through
+incorporating probabilistic disjunctions into our logic SL and taking states to be streams of outcomes of
+probabilistic events together with a current ‘counter’ that increases each time an event occurs. In a simple
+setting where only two outcomes are possible with equal probability, we can axiomatise this within SL by
+adding zero and successor functions (allowing us to create numerals n), along with a unary state predicate
+count(n). We can then model probabilistic events by adding the appropriate axioms to ∆S. Suppose, for
+example, we add two predicate constants H(x) and T (x) (for heads and tails), along with constants c1, c2, . . .
+representing coins. Then flipping a coin would be represented by the axiom schema
+Γ ⊢S {count(n) · H(ci) ∨ T (ci) · count(n + 1)}
+where n ranges over numerals and ci over coin constants, the counter indicating that a probabilistic event has
+occurred. The act of reading a probability from the state could be interpreted semantically by introducing
+a constant ω : S → Bool × S to Sω, with the axiom
+count(n, π) ⇒ (e = f ⇒ H(ci)) ∧ (e = t ⇒ T (ci)) ∧ count(n + 1, π1) for ⟨e, π1⟩ := ωπ
+(alternatively, we could simply define S := Nat × (Nat → Bool) for a type of Nat natural numbers, and
+define ω⟨n, a⟩ := ⟨a(n), ⟨n + 1, a⟩⟩ and count(n, ⟨m, a⟩) := m =Nat n).
+A probabilistic choice operator ⊕ can then be added to the language of ST, along with the typing rule
+Γ ⊢ s ⊕ t : X + Y for Γ ⊢ s : X and Γ ⊢ t : Y , and the interpretation
+[s ⊕ t]π := case e ([ι0s]π1) ([ι1t]π1) where ⟨e, π1⟩ := ωπ
+24
+
+In particular, defining flip := skip ⊕ skip : C + C we would have
+[flip] sr {count(n) · H(ci) ∨ T (ci) · count(n + 1)}
+although we stress that the operator ⊕ and would allow for much more complex probabilistic disjunctions,
+potentially involving additional computational content.
+Our soundness theorem, extended to these new probabilistic axioms and terms, would then facilitate the
+extraction of probabilistic programs from proofs. For instance, including a winner predicate W(x), two player
+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};
+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
+⊢S {count(n) · ∃x W(x) · count(n + 2)}
+expressing the fact that a winner can be determined after two flips. We can then extract a corresponding
+probabilistic term for realizing this statement, which would be isomorphic to the expected program that
+queries the state twice in order to determine the outcome of those flips, and returns either p1 or p2 as a
+realizer for ∃x W(x) depending on the content of the state.
+Of course, the details here need to be worked through carefully in order to properly substantiate the
+claim that our framework could be used to extract probabilistic programs in a natural and meaningful way.
+At the very least, it is likely that further additions to SL along with a more intricate state would be needed
+to incorporate more interesting probabilistic events, such as annotated disjunctions along the lines of [31].
+We leave such matters to future work.
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+

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+Mono-STAR: Mono-camera Scene-level Tracking and Reconstruction
+Haonan Chang1, Dhruv Metha Ramesh1, Shijie Geng1, Yuqiu Gan, Abdeslam Boularias1
+Abstract— We present Mono-STAR, the first real-time 3D
+reconstruction system that simultaneously supports semantic
+fusion, fast motion tracking, non-rigid object deformation, and
+topological change under a unified framework. The proposed
+system
+solves
+a
+new
+optimization
+problem
+incorporating
+optical-flow-based 2D constraints to deal with fast motion
+and a novel semantic-aware deformation graph (SAD-graph)
+for handling topology change. We test the proposed system
+under various challenging scenes and demonstrate that it
+significantly
+outperforms
+existing
+state-of-the-art
+methods.
+Supplementary material, including videos, can be found at
+https://github.com/changhaonan/Mono-STAR-demo.
+I. INTRODUCTION
+Real-time perception is a crucial component of modern
+robotic manipulation systems. Recently, You Demonstrate
+Only Once [1] has demonstrated that given the geometry
+model and 6D-pose trajectory of a manipulated object during
+an expert demonstration, a robot can quickly learn complex
+and contact-rich manipulation skills. Such progress shows
+the importance of geometric 3D reconstruction and tracking
+systems for robotic manipulation.
+However, a perception system that can perform both track-
+ing and reconstruction simultaneously is notoriously difficult
+to build because reconstruction and tracking inherently de-
+pend on each other. For example, tracking algorithms usually
+require geometry models, while dynamic scene reconstruc-
+tion relies on accurate tracking for producing those geometry
+models. Scene-level Tracking and Reconstruction (STAR) [2]
+refers to a category of perception systems that generate both
+the geometry and the pose of every visible object in a scene.
+This problem is related to the multiple-instance dynamic
+SLAM problem, where all movable objects in the scene are
+assumed to be rigid so that the problem can be decomposed
+into multiple dense-SLAM sub-problems. This approach was
+proposed in Co-Fusion [3] and MaskFusion [4], where a
+semantic neural network was employed first to decompose
+the scene into multiple objects and then deal with each
+object individually. This approach requires every object in
+the scene to be rigid or quasi-rigid. The same problem was
+investigated in MidFusion [5], where an octree was used to
+improve reconstruction and tracking. However, these systems
+are limited to scenes of rigid objects with slow motions.
+Instead of dealing with each object individually based
+on their semantic labels, one can also reconstruct all the
+objects in the scene as one large non-rigid object and segment
+them ulteriorly. This approach was however very challenging
+1 Authors are with the Department of Computer Science, Rutgers
+University, 08854 New Brunswick, USA. This work is supported by NSF
+awards 1734492, 1846043, and 2132972.
+to apply until the introduction of the first real-time non-
+rigid reconstruction DynamicFusion [6], where the non-
+rigid reconstruction problem was decomposed into two sub-
+problems, (1) building the geometry at the initial frame, and
+(2) computing the deformation using an embedded defor-
+mation graph, namely ED-graph. This paradigm was also
+followed in OcclusionFusion [7]. Inspired by these previous
+efforts, a solution to the general STAR problem was recently
+proposed in STAR-no-prior [2]. In contrast to SLAM-based
+methods, STAR-no-prior reverses the order of segmentation
+and reconstruction. The entire scene is first reconstructed and
+then segmented into different objects based on topology. By
+doing so, STAR-no-prior outperforms previous state-of-the-
+art methods such as [4] and MidFusion [5].
+However, a major limitation of STAR-no-prior is its re-
+liance on a system of multiple cameras surrounding the
+scene, making it impractical for a mobile robot. To address
+this shortcoming, we propose Mono-STAR, a mono-camera
+STAR solution. Switching from a multi-camera system to
+a mono-camera setting requires solving several non-trivial
+problems. Notably, STAR-no-prior relies on the multi-camera
+system to overcome the plane-based-ICP constraint that it
+inherits from DynamicFusion, which supports tracking of
+only slow motion along the camera view. The use of multiple
+cameras can guarantee that any motion has at least one non-
+zero projection to a camera view. However, the mono-camera
+setting does not have such a guarantee and therefore requires
+a new solution. Occlusion Fusion [7] adds a 2D constraint
+using optical-flow (RAFT [8] or GMA [9]) to deal with fast
+motions. Inspired by this, we propose a new 2D loss to track
+motions that are perpendicular to the camera view, which not
+only stabilizes tracking performance under a single view but
+also improves our system’s ability to handle fast motion.
+Furthermore, STAR-no-prior does not take advantage of
+semantic labels. We, therefore, combine the semantic infor-
+mation with the embedded deformation graph mechanism
+and propose a Semantic-aware Adaptive Deformation graph,
+SAD-graph, which is an extension of ED-graph. With just lit-
+tle extra computation, SAD-graph can easily handle topology
+changes across distinct semantic classes and assign different
+levels of rigidness for each type of object. To the best of our
+knowledge, Mono-STAR is the first single-view real-time 3D
+reconstruction system that can simultaneously handle seman-
+tic fusion, fast motion tracking, non-rigid object deformation,
+and topological change under one unified framework.
+II. RELATED WORKS
+Simultaneous Tracking and Reconstruction. Simulta-
+neous 6D tracking and 3D reconstruction was typically
+arXiv:2301.13244v1  [cs.RO]  30 Jan 2023
+
+Method
+Semantic
+Fast
+Non-rigid
+Topology
+Single
+motion
+objects
+change
+view
+SLAM++
+
+
+
+
+
+DynamicFusion [6]
+
+
+
+
+
+Volume Deform
+
+
+
+
+
+SurfelWarp [10]
+
+
+
+
+
+TCAFusion [11]
+
+
+
+
+
+Co-fusion [3]
+
+
+
+
+
+Fusion4D [12]
+
+
+
+
+
+Motion2Fusion [13]
+
+
+
+
+
+Functon4D [14]
+
+
+
+
+
+MaskFusion [4]
+
+
+
+
+
+RigidFusion [15]
+
+
+
+
+
+MidFusion [5]
+
+
+
+
+
+OcclusionFusion [7]
+
+
+
+
+
+STAR-no-prior [2]
+
+
+
+
+
+Mono-STAR
+
+
+
+
+
+TABLE I: Taxonomy of the-state-of-art scene-level fusion systems.
+regarded in previous works as a multiple-instance dynamic
+SLAM problem. Many works such as Co-fusion [3], Mask-
+Fusion [4], and RigidFusion [15] proposed to divide the
+scene into multiple rigid objects and track each object
+individually. More recently, STAR-no-prior [2] formalized
+the STAR problem as a scene-level non-rigid reconstruction
+problem. Our mono-camera system eliminates the multi-
+camera requirement of STAR-no-prior by adding a new
+optical-flow-based 2D constraint and a novel semantic-aware
+adaptive deformation graph.
+Dynamic Scene Reconstruction. Dynamic scene recon-
+struction [16], [17] is the problem of reconstructing the
+geometry and recording the deformation of a scene with
+moving objects. DynamicFusion [6] was the first real-time
+GPU-based solution for solving this problem. It adopts a
+TSDF-based geometry as the canonical model and an embed-
+deformation graph (ED-graph) to describe the deformation
+of the whole scene. A drawback of this method is that the
+combination of TSDF and ED-graph cannot handle topology
+changes. Many recent techniques such as Fusion4D [12],
+Motion2Fusion [13], [11], [18] have attempted to address this
+problem. However, these methods require significantly more
+computation or rely on expensive sensors. SurfelWarp [10]
+demonstrated that a Surfel-based representation can be used
+to tackle topology changes. Therefore, our proposed system
+also adopts a Surfel-based representation.
+III. PROBLEM FORMULATION AND BACKGROUND
+A. Problem formulation
+Given a sequence of RGB-D images of a given dynamic
+scene taken from a single fixed camera, we consider the
+problem of simultaneous tracking and reconstruction of all
+the objects visible in the scene. The number of objects is
+unknown. The objects can be non-rigid. Measurement, Mt
+can defined as set of measurement surfels mi at time-step
+t, generated from the RGB-D input. mi = (vi,ni,ci), where
+v,n,c are 3D coordinates, normal and color respectively.
+The proposed system returns at each time-step t a Surfel-
+based geometry St (the reconstructed scene) for the entire
+scene and its corresponding deformation graph Gt. Surfel-
+based geometry St is a set of surfels si. si = (vi,ni,ci,ri,lsi),
+where vi,ni,ci,ri,lsi are respectively the 3D coordinates,
+normal, color, radius and semantic label of surfel si ∈ St. We
+assume that there is a maximum of H pre-defined different
+semantic categories {1,2,...,H}. If a surfel does not belong
+to any pre-defined category, it will be labeled as H +1 (i.e.,
+unrecognized). Deformation graph Gt is defined by a set of
+nodes {gi}. Each node gi has a semantic label lgi, and is
+connected to its nearest-neighbor nodes, denoted as NG(gi),
+in the 3D space. Deformation graph Gt is associated with
+a warp field Wt, defined as W = {[pi ∈ R3,δi ∈ R+,Ti ∈
+SE(3)]}, wherein i is the index of a node in Gt, pi is the 3D
+point that corresponds to node gi, δi is the node’s radius of
+influence, and Ti is the 6D transformation defined on node
+gi. Ti is represented by a dual quaternion qi for smooth
+interpolation [19]. Warp field W describes the deformation
+between two consecutive time steps. For each surfel s =
+(v,n,c,r,l) ∈ S, we compute its 6D transformation
+¯W(s)
+based on warp field W,
+¯W(s) = normalize( ∑
+k∈NG(s)
+w(v, pk)qk),
+(1)
+wherein NG(s) denotes the neighbors nodes of surfel s,
+w(s) is an interpolation parameter, defined as w(s) =
+exp
+�
+∥v− pk∥2
+2 /(2δ 2
+k )
+�
+, and v is the 3D position of surfel
+s. The local transformation ¯W(s) is then used to describe the
+deformation of surfel s as follows:
+vwarp = ¯W(s)v
+nwarp = rotation
+� ¯W(s)
+�
+n.
+(2)
+Here, v,n are the vertex and normal of s before warping, and
+vwarp,nwarp are the vertex and normal after the deformation.
+Symbol
+Meaning
+Definition
+Mt
+Measurement at time t.
+III-A, IV-A.1
+St−1
+Surfel geometry from t-1.
+III-A
+Ra
+t−1
+2D maps rendered from St−1.
+IV-B.1
+Swarp
+t−1
+Warped geometry after non-rigid alignment. IV-B.4
+Rg
+t−1
+2D maps rendered from Swarp
+t−1 .
+IV-B.1
+TABLE II: Notation sheet.
+IV. PROPOSED APPROACH
+An overview of the proposed method is shown in Fig. 1.
+Mono-STAR uses two parallel threads, a measurement thread,
+and a geometry thread. The first thread is responsible for
+loading measurements Mt and generating semantic labels
+Lm
+t . The geometry thread uses this measurement Mt and the
+alignment rendering Ra
+t−1 to compute an optical-flow OFt.
+Then, Mt,Ra
+t−1,OFt are given to the optimization module that
+then computes the non-rigid deformation Wt. After the non-
+rigid alignment, previous geometry St−1 is warped to Swarp
+t−1 ,
+and the geometry rendering Rg
+t−1 is generated. Finally, Rg
+t−1,
+deformation Wt, warped geometry Swarp
+t−1 , and semantic labels
+Lm
+t are combined to generate the latest geometry St.
+Noticeably, loading Mt takes less time than updating
+geometry St−1 and rendering Ra
+t−1. Semantic segmentation is
+also faster than the combined process of generating optical-
+flow OFt and the non-rigid alignment. Thus, the geometry
+thread fully hides the latency of the measurement thread.
+
+Fig. 1: Overview of the proposed system. The system runs in two parallel threads, one for measurement and one for geometry. In each
+time-step t, the measurement thread loads a measurement Mt from images or a camera buffer. Then, a segmentation network generates a
+set of semantic labels Lmt . Once the measurement is loaded on the GPU memory, Mt and previous alignment rendering Ra
+t−1 are fed into
+an optical-flow network to generate the optical-flow OFt from previous geometry St−1 to measurement Mt. Optical-flow OFt, geometry
+rendering Rt and measurement Mt are used to compute warp-field Wt with non-rigid alignment. After the alignment, previous geometry
+St−1 will be warped to Swarp
+t−1 . The fusion rendering map Rg
+t−1 is then rendered from Swarp
+t−1 . Rg
+t−1, Swarp
+t−1
+and semantic labels Lmt are used
+to generate the updated geometry St, deformation graph Gt and the surfel semantic label Lst .
+A. Measurement Thread
+1) Measurement: We use one Intel RealSense-415 camera
+to collect RGB-D images. Depth images are denoised with a
+Gaussian filter. The maximum frame rate for this module is
+limited to 20 fps to coordinate with other modules. We use
+a double-buffer strategy to hide latency. Specifically, we use
+two buffers B0 and B1 to store measurements. When B0 is
+used by other threads, B1 can read images simultaneously.
+The filtered images are used to construct three maps, V m
+t ,
+Nm
+t , Cm
+t , storing 3D coordinates vm
+i , normal nm
+i , and color cm
+i ,
+respectively, for surfel mi of each pixel in the measurement.
+2) Segmentation: The segmentation module receives the
+color map Cm
+t
+and returns a semantic label map Lm
+t
+of H
+pre-defined semantic classes. Here, we use two different
+segmentation models, a transformer-based Segmenter Mask
+[20], and a more traditional MaskRCNN [21]. The two
+models are pre-trained on two different datasets, ADE20K
+[22], [23] and COCO-Stuff [24] respectively. We do not
+further train these models on any other dataset. We select
+which one to use based on the types of objects in the scene.
+B. Geometry Thread
+1) Geometry Rendering: The input of the geometry ren-
+dering pipeline is a geometry S, and the output is the
+rasterized rendering R for geometry S from the current
+camera view. The rendering process to generate R follows
+the classical point cloud rasterizing process [25], where every
+surfel is projected to its nearest pixel position on the camera
+plane based on its 3D coordinates. Each rendering map R
+is composed of four 2D maps. Rt = {Ct,Vt,Nt,It}, where
+Vt,Nt,Ct,It are respectively the vertex map, the normal map,
+the color map and the index map. These maps store the 3D
+coordinates vi, the normal ni, the color ci and the surfel index
+i of the projected surfel si at each pixel.
+At each time-step, the geometry rendering pipeline is
+called twice; once to generate Ra
+t−1 from previous geometry
+St−1 for non-rigid alignment, and once to get Rg
+t−1 with
+warped geometry Swarp
+t−1 for updating the geometry. Rendering
+Rg
+t−1 used for updating the geometry operates on surfel-
+level granularity, whereas Ra
+t−1 used for geometry alignment
+rendering operates on deformation node granularity. Another
+difference between Rg
+t−1 and Ra
+t−1 is resolution, Rg
+t−1 is up-
+sampled by 4 × 4 compared to Ra
+t−1 to prevent different
+surfels from being projected onto the same pixel. Rg
+t−1
+requires a higher resolution for accurate geometry update
+and Ra
+t−1 has a lower resolution for faster optimization.
+2) Optical Flow:
+The optical flow module receives
+Ca
+t−1,V a
+t−1 from geometry rendering Ra
+t−1, and Cm
+t−1,V m
+t−1 from
+measurement Mt, and generates an optical-flow map OFt.
+OFt predicts the optical-flow from previous geometry St−1 to
+the latest measurement Mt. This prediction is later used for
+registration through non-rigid alignment. We generate OFt
+using a neural network based on the RAFT architecture [8],
+along with additional global motion features as performed
+in GMA [9]. The global motion features provide stability
+for predicting motion features, even in occluded scenes.
+Both RAFT and GMA models were originally trained using
+only RGB images. The optical flow model used in [7]
+shows that using RGB-D images for training provides a
+far more stable flow, even with motion blurring. Thus, our
+model is also trained on RGB-D images from the datasets
+FlyingThings3D [26], Monkaa [26] and Sintel [27], [28].
+3) Non-rigid alignment:
+Non-rigid alignment is per-
+formed in order to compute non-rigid deformation Wt. This
+step solves a massive optimization problem to warp the
+previous geometry St−1 to a geometry Swarp
+t−1 that fits current
+measurement Mt. We use a Gauss-Seidel solver implemented
+with CUDA to solve this problem, which is summarized as
+minWEtotal(W) = wpicpEpicp(W)+w2DE2D(W)
++waregEareg(W),
+
+Mt
+Mt
+Lm
+Input
+Output
+Sync
+Sync
+Measurement
+St
+(St Gt L)
+-warp
+Lm
+-warp
+R
+Geometry
+St-1
+St-1
+warp
+Ls
+Gt
+Normal
+Vertex
+Vertex
+Normal
+va-1
+Ng
+Sync
+Sync
+t-1
+Color
+Index
+Color
+Index
+Cf-1
+19-1where wpicp,wof ,wareg are the weights of terms Epicp(W),
+E2D(W) and Eareg(W), explained in the following.
+Registration. Let u = (x,y) be a pixel in measurement map
+u, and let mi = M(u) bet its associated surfel. Let (mi,sΠ(i))
+denote a pair of registered measurement and geometry surfel.
+Π(i) is defined as Π(i) := Ia
+t−1(x − of x
+t ,y − of y
+t ), wherein
+(x,y) = u,(o f x
+t ,o f y
+t ) = OFt(u). Ia
+t−1 ∈ Ra
+t−1 is the index map
+of the rendered geometry.
+PICP Loss. Point-to-point ICP loss is sensitive to dis-
+turbance and outliers, which limits its utility in real-world
+applications. Instead, we use a plane-based ICP (PICP) loss
+to align the differences along the depth direction as follows,
+Epicp(W) = ∑
+mi∈M
+nm
+i ·( ¯W(sΠ(i))vs
+Π(i) −vm
+i ),
+(3)
+wherein vs
+Π(i) is the 3D coordinates of surfel sΠ(i), vm
+i ,nm
+i are
+the 3D coordinates and normal of measurement surfel mi. ¯W
+is defined in Eq. 2.
+2D Loss. One limitation of the PICP loss is that it cannot
+correctly capture motions within the same plane, such as
+the moving calendar shown in Fig. 6. We thus add to the
+objective function a 2D loss E2D defined as follows,
+P =
+�
+1
+0
+0
+0
+1
+0
+�
+(4)
+E2D(W) = ∑
+mi∈M
+∥P( ¯W(sΠ(i))vs
+Π(i) −vm
+i )∥2.
+(5)
+Here, P is a projection matrix, projecting the 3D difference
+to the camera X-Y plane. This term constrains mi and sΠ(i)
+to be as close as possible on the camera X-Y plane. It is
+worth noting that our proposed 2D loss is different from the
+one proposed in OcclusionFusion [7], where pixel differences
+are used to calculate the 2D loss. The influence of pixel
+differences scales with the distance to the camera, which
+makes the optimization parameters harder to tune.
+Semantic-aware Adaptive Deformation Graph. The tra-
+ditional Embedded Deformation graph (ED-graph) has been
+widely used in non-rigid tracking and non-rigid reconstruc-
+tion. It can describe complicated warping fields with a simple
+data structure and an interpolation strategy. Moreover, the as-
+rigid-as-possible (ASAP) regulation term defined on defor-
+mation nodes provides a continuity guarantee for neighboring
+nodes. However, the ED-graph cannot handle the topology
+changes of different nodes. For example, if we use an ED
+graph to describe a cup being lifted up from a table, as shown
+in Fig. 2 (a), the motion of the cup’s nodes also influences
+and propagates to the table’s nodes. Many previous works
+have attempted to overcome this limitation of the ED-graph
+by proposing a dual deformation graph [18] or a level-
+set-based TSDF fusion mechanism [11]. However, these
+approaches are too complicated or introduce too much over-
+head computation. We propose the Semantic-aware Adaptive
+Deformation Graph (SAD-graph) to address this issue of
+topological changes among objects with different semantic
+classes. More importantly, the proposed algorithm is intuitive
+and requires little extra computation compared with ED-
+graph. Another advantage is that existing ED-graph based
+approaches can be easily upgraded to support SAD-graph.
+The core idea of SAD-graph is that instead of imposing a
+uniform regulation continuity constraint on all deformation
+nodes, constraints of varying adaptive strengths are imposed
+on different edges. A variable weight ωi, j is associated with
+the constraint (defined in Eq. 7) between neighboring nodes
+gi and gj, and the strength of the constraint is systematically
+adjusted. Constraint weight ωgi,gj is a function of lgi,lg j, the
+semantic labels of nodes gi and gj. It is defined as follows,
+ωgi,g j(lgi,lg j) =
+�
+0.1,
+if lgi ̸= lg j
+δ k,
+if lgi = lg j = k,k ∈ [1,...,H +1]
+(6)
+where δ k is a constant describing the average rigidness of
+objects belonging to semantic category k, e.g., δtable = 1.0,
+δ human = 0.3, etc. For example, in Fig. 2 (b), since the
+internal rigidity constraint within the cup or the table is much
+larger than the constraint between them, their geometries can
+be accurately reconstructed during the topology separation.
+It is important to note that these constraints are not hard
+because the semantic labels obtained from a neural network
+detector are error-prone.
+Fig. 2: Illustration of the proposed Semantic-aware Adaptive De-
+formation Graph (SAD-graph). The scene describes a cup (brown)
+being lifted up from the table (green). Black edges indicate strong
+continuity constraints, while grey edges indicate weak constraints.
+Adaptive Regulation Loss. We introduce a deformation
+graph, SAD-graph, and a new regulation term, adaptive
+regulation Eareg(W). A semantic-related connection weight
+ω is used to adjust the regulation strength among and within
+different semantic classes as follows,
+Eareg(W) = ∑
+g j∈G
+∑
+gi∈NG(g j)
+ωgi,gj
+��Tjpj −Tipi
+��2
+2 ,
+(7)
+wherein G is the deformation graph, NG(gj) refers to the set
+of neighbors of node gj in the deformation graph, Tj and Ti
+are the transformations defined on nodes gi and gj. pi and pj
+are the 3D coordinate of gi and gj, and ωgi,g j is the weight
+of the connection between nodes gi and gj, defined in Eq. 6.
+4) Geometry and Graph Update: Once the non-rigid
+deformation is computed, the geometry update process of
+Mono-STAR is similar to SurfelWarp [10]. Thus, we only
+briefly describe that process and we focus on the semantic
+update. This step returns the updated geometry St (the
+reconstructed scene) and the updated graph Gt, both of which
+are needed for processing the scene in the next time-step.
+Updating the Geometry. The previous geometry St−1 is
+warped to Swarp
+t−1 after the non-rigid alignment step. Although
+Swarp
+t−1
+is already close enough to measurement Mt, there
+
+Before deformation
+After deformation
+Before deformation
+After deformation
+区
+X
+区
+(a) ED-graph
+(b) SAD-graphstill exists a discrepancy between them due to measurement
+noises, emerging surfaces, topology changes, or even track-
+ing failures. The geometry is updated to address this gap
+between the warped geometry Swarp
+t−1
+and measurement Mt.
+There are four steps in total in this process.
+1. Registration: A projective registration is made between
+measurement Mt and warped geometry Swarp
+t−1
+according to
+rendering map Rg
+t−1.
+2. Fusion: If a surfel mi ∈ Mt is mapped to s j ∈ Swarp
+t−1
+in
+the registration, mi is merged into sj to average measurement
+noises. The semantic label ls j of sj is defined as a probability
+distribution psj. When mi is fused into sj, ps j is also updated
+by lmi. The update formula for psj is:
+psj(k) = (psj(k)+δm)/∑
+k′
+psj(k′),if k = lmi
+(8)
+psj(k) = ps j(k)/∑
+k′
+psj(k′),otherwise.
+(9)
+Here, δm is the confidence of the measurement.
+3. Append: If there are no surfels in Swarp
+t−1
+that can be
+registered to mi, mi must belong to a newly observed surface
+or be noise. In the first case, mi will be appended to Swarp
+t−1 .
+The semantic label distribution ps
+i of mi is initialized as:
+ps
+i(k) = δm,if k = lmi; ps
+i(k) = 0,otherwise,
+(10)
+4. Removal: After each mi ∈ Mt is either fused or ap-
+pended, some surfels sj ∈ Swarp
+t−1
+are left with no correspon-
+dence. A geometry violation test is performed on the remain-
+ing surfels, and those that fail the test are removed [10].
+After the four steps given above, we get the updated
+geometry St for time-step t.
+Updating the Graph. The update of the SAD-graph is
+identical to the update of the traditional ED-graph. The
+update appends new nodes but does not remove existing ones.
+Let Sappend be the set of the appended surfels during the
+geometry update. We first compute the distances between
+every surfel s ∈ Sappend and every node g ∈ Gt−1. Let
+D(s,G) = ming∈G distance(s, g). A surfel s is said to be
+unsupported if D(s,G) > σ, for some threshold σ. We per-
+form a spatially uniform sampling from all the unsupported
+surfels. Sampled surfels are appended to graph G as new
+nodes. The semantic label of node gi, lgi is updated according
+to the semantic labels of NS(gi), neighbor surfels of gi.
+lgi = argmaxk{∑sj∈Ns(gi) δ(lsj,k)}. Here δ(ls j,k) = 1,if lsj =
+k;δ(ls
+j,k) = 0,otherwise.
+V. EXPERIMENTS
+We
+test
+our
+technique
+on
+a
+dataset
+we
+collected
+and a public dataset VolumeDeform [29]. An ablation
+study
+and
+comparisons
+with
+SoTA
+methods
+such
+as
+STAR-no-prior [2] and MaskFusion [4] on challenging
+scenes are presented in this section. Since collecting
+ground-truth geometry and deformation for non-rigid ob-
+jects is extremely challenging, experiments and compar-
+isons in this area are limited to qualitative results [29].
+Supplementary results and resources can be found at
+https://github.com/changhaonan/Mono-STAR-demo.
+A. Performance
+We tested our system on a desktop machine with a
+GeForce RTX 3090 and an AMD-Ryzen 9 5900X. On aver-
+age, measurement loading takes 4 ms and segmentation costs
+10 ms (Segmenter Mask [20]). The optimization module uses
+20 ms. The geometry update uses 7 ms. The major bottleneck
+is the optical-flow network, which takes 60 ms. Since the
+latency for the measurement thread is fully hidden by the
+geometry thread, our entire system runs in 11 Hz. If the
+optical-flow runs on a separate graphic card, it would take
+only 26 ms [7], which would double the speed of our system.
+B. Qualitative Results
+1) Soft objects: Fig. 3 illustrates the non-rigid deforma-
+tion ability of MaskFusion and Mono-STAR. We can clearly
+see that MaskFusion fails to track the deformations of the
+pillow and umbrella, while Mono-STAR correctly captures
+both of them in the reconstructed model, which shows the
+advantage of our technique over MaskFusion in handling
+non-rigid deformation.
+Fig. 3: Comparison on deformable objects with MaskFusion on
+our recorded dataset (top) and VolumeDeform dataset (bottom).
+2) Fast Motion: Fig. 4 demonstrates Mono-STAR’s abil-
+ity to handle fast motions. The top scene in Fig. 4 shows
+an accident that was recorded during our data collection.
+While we were pushing a cup on the table, the cup hit a
+bump and fell down. The bottom scene is about passing a
+basketball between two hands. Objects in both scenes moved
+very fast. One is 18 frames, and the other is 30 frames.
+Significant motion blur can be observed in both middle
+images. However, Mono-STAR can still capture these fast
+motions and correctly reconstruct the objects at each frame.
+
+=
+=
+150
+Measurement
+MaskFusion
+OursFig. 4: Experiment on fast motion. Pushing down a coffee cup
+(top). Passing a basketball between two hands (bottom). The second
+and fourth row are our 3D scene reconstruction results.
+3) Resilience to Semantic Segmentation Noises: Fig. 5
+shows how our proposed method can resist noise in semantic
+segmentation. The figures on the left are the RGB measure-
+ment from the beginning and the end frames. The right side
+compares the segmentation from the measurement and the
+segmentation from our reconstruction. Although the ground-
+truth measurement suffers from major segmentation errors,
+where the cup label is completely lost for t > 0, Mono-STAR
+still maintains the correct semantic labels in its reconstruction
+result through semantic fusion.
+Fig. 5: Resilience to semantic label noise. The top sequence is the
+segmentation map Lmt of the measurement. The bottom sequence is
+the segmentation map Lst from our reconstruction technique.
+C. Ablation Study
+1) 2D Loss: We test Mono-STAR with and without the 2D
+loss on the “adventcalender” dataset from VolumeDeform.
+Fig. 6 shows that the proposed 2D loss E2D can efficiently
+track the motions within a plane. In contrast, tracking without
+E2D fails in this type of motion, which clearly shows the
+effectiveness of the proposed 2D loss.
+2) SAD-graph: In Fig. 7, we compare the ED-graph with
+the topology-aware ED-graph (STAR-no-prior) and the SAD-
+graph (Mono-STAR). We can see that the ED-graph fails
+Fig. 6: Ablation study on the 2D loss.
+to support the topology change that results from lifting the
+object from the table. Topology-aware ED-graph can separate
+the topology, but it also generates many outliers on the table.
+With the help of the proposed SAD-graph, Mono-STAR can
+conduct a smoother and cleaner separation.
+Fig. 7: Comparing ED-graph (left), topology-aware ED-graph
+(middle, STAR-no-prior [2]), and SAD-graph (right, ours). The
+scene shows a plushy toy being lifted up from the table.
+D. Discussion of Limitations
+Although Mono-STAR shows great potential in many dif-
+ferent aspects, it still has two limitations. First, it relies on the
+optical flow to track fast motions. However, even the state-of-
+art optical flow detector GMA [9] is not always accurate, es-
+pecially when the motion is too fast and the tracked surfaces
+are heavily occluded. Our system can tolerate some noise
+from the GMA optical-flow module. However, if the optical
+flow provides inaccurate predictions for multiple consecutive
+frames, the tracking of the corresponding object may still
+fail. Another drawback of our system is the incompleteness
+of the reconstructed geometry. Our reconstructed geometries
+usually have holes and are not as smooth as TSDF-based
+geometry. The reason is that Surfel-based geometry, unlike
+TSDF-based geometry, is discrete by default. Therefore,
+it is difficult to maintain the smoothness of Surfel-based
+geometry in highly dynamic scenes. These two challenges
+can be addressed in future works.
+VI. CONCLUSION
+We presented Mono-STAR, a single-view solution for
+the semantic-aware STAR problem. Mono-STAR uses a
+novel semantic-aware and adaptive deformation graph for
+simultaneous tracking and reconstruction, and can handle
+topology changes as well as semantic fusion. Experiments
+show that Mono-STAR achieves promising results in non-
+rigid object reconstruction, while resisting to semantic seg-
+mentation errors, and capturing fast motions on various chal-
+lenging scenes. We believe that this system can inspire and
+boost more future research on imitation learning, dexterous
+manipulation, and many other relevant robotics problems.
+
+t = 47
+t = 58
+t = 65
+t = 15
+t = 26
+t = 45Human
+Cup
+Background
+Measurement
+t=0
+t=0
+t = 102
+t = 162
+Reconstruction
+t = 162
+0=1
+t = 102
+t = 162t= 1
+t = 41
+t = 41
+Initial
+without 2D Loss
+Ourst =31
+t =31
+t = 38
+t = 38
+ED-graph
+STAR-no-prior
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+arXiv:2301.00522v1  [math.RT]  2 Jan 2023
+Irreducible module decompositions of rank 2
+symmetric hyperbolic Kac-Moody Lie algebras by
+sl2 subalgebras which are generalizations of
+principal sl2 subalgebras
+TSURUSAKI Hisanori∗
+Abstract
+There exist principal sl2 subalgebras for hyperbolic Kac-Moody Lie
+algebras.
+In the case of rank 2 symmetric hyperbolic Kac-Moody Lie
+algebras, certain sl2 subalgebras are constructed. These subalgebras are
+generalizations of principal sl2 subalgebras.
+We show that the rank 2
+symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly
+decomposed under the action of this sl2 subalgebras. Furthermore, we
+classify irreducible components of the decomposition. In particular, we
+obtain multiplicities of unitary principal series and complementary series.
+1
+Introduction
+A nilpotent orbit in a finite dimensional simple Lie algebra g0 is an orbit ob-
+tained by acting on the nilpotent element x of g0 by inner automorphisms. In
+[Dyn57], these are classified by weighted Dynkin diagrams. From the Jacobson-
+Morozov theorem, for a nilpotent element x of g0, we can construct a sl2-triple
+with x as a nilpositive element ([CM93, Theorem 3.3.1]). This makes it equiva-
+lent to classify nilpotent orbits of g0 and to classify sl2 triples in g0 up to inner
+automorphisms. Among the nilpotent orbits of a finite dimensional simple Lie
+algebra, the one whose dimension as an algebraic variety is maximal is called the
+principal nilpotent orbit. Correspondingly, we can construct a principal SO(3)
+subalgebra that is compatible with compact involution ([Kos59]).
+Kac-Moody Lie algebras are generalizations of finite-dimensional simple Lie
+algebras.
+They are classified into three types: finite type, affine type, and
+indefinite type. The finite type Kac-Moody Lie algebras are finite dimensional
+simple Lie algebras. Within indefinite Kac-Moody Lie algebra, there is a class
+called hyperbolic Kac-Moody Lie algebra. A hyperbolic Kac-Moody Lie algebra
+∗Graduate
+School
+of
+Mathematical
+Sciences,
+University
+of
+Tokyo
+,
+htsu-
+rusaki1929@gmail.com
+1
+
+is an indefinite type Kac-Moody Lie algebra such that any true subdiagram of
+its Dynkin diagram is of finite or affine type.
+By analogy with the above theory, in [NO01], for a hyperbolic Kac-Moody
+Lie algebra, its principal SO(1, 2) subalgebra was constructed. Note that [GOW02]
+shows that it is possible to construct a principal SO(1, 2) subalgebra for certain
+indefinite Kac-Moody Lie algebra that is not hyperbolic.
+Corresponding to this principal SO(1, 2) subalgebra, we can construct a
+principal sl2-subalgebra in a hyperbolic Kac-Moody Lie algebra. In [Tsu], for
+the rank 2 symmetric hyperbolic Kac-Moody Lie algebras g, the following re-
+sult is obtained. Let the space that the positive real root vectors span be Rg.
+we consider sl2 subalgebras whose nilpositive element exists in Rg. Then we
+can construct certain sl2 subalgebras. These subalgebras are generalizations of
+principal sl2 subalgebra.
+In this paper, for an sl2 subalgebra of rank 2 symmetric hyperbolic Kac-
+Moody Lie algebra g constructed in [Tsu], we show g is decomposed into irre-
+ducible sl2-modules by its action on g.
+We are going to more details. Let s be an sl2 subalgebra constructed in
+[Tsu]. Let H, X, Y be an sl2 triple and assume that s is spanned by H, X, Y .
+Let the Chevalley generators of g be ei, fi, hi, (i = 0, . . . , n − 1). Let hR be the
+R-span of Chevalley generators. From [Kac90, Theorem 2.2], g has a C-valued
+nondegenerate invariant symmetric bilinear form (· | ·) called the standard form.
+An antilinear automorphism ω0 of g, called compact involution, is determined
+by
+ω0(ei) = −fi,
+ω0(fi) = −ei
+(i = 0, . . . , n − 1),
+ω0(h) = −h
+(h ∈ hR).
+From [Kac90, §2.7], we can determine a nondegenerate Hermitian form (· | ·)0
+on g with (x | y)0 = −(ω0(x) | y).
+s-module V is called unitarizable if the following conditions are satisfied.
+(1) (· | ·)0 on V is positive definite.
+(2) for v1, v2 ∈ V , the conditions as follows are satisfied.
+([X, v1], v2)0 = −(v1, [Y, v2])0,
+([H, v1], v2)0 = −(v1, [H, v2])0.
+Theorem 1.1 (Theorem 4.5). g can be decomposed into a direct sum of irre-
+ducible s-modules such that s itself is one of the direct summand. All of these
+modules except for s are unitarizable.
+Also, we classify how many highest weight modules, lowest weight modules,
+and modules that are neither highest weight module nor lowest weight module
+appear in this decomposition. We regard a root sα1 + tα2 as a point (s, t) in
+xy-plane, and We define a region L, −L in xy-plane in §5. If a root α satisfies
+α(H) ∈ (0, 2), α ∈ L. If a root α satisfies α(H) ∈ (−2, 0), α ∈ −L.
+2
+
+Theorem 1.2 (Theorem 7.1). We consider an irreducible decomposition of g
+by the action of s.
+(1) Let M is an irreducible component of decomposition of g, which contain
+a root space for a real root in L. Then, M is an unitary principal or
+complementary series representation.
+(2) (cf. [Tsu, Proposition 7.3]) There is an unitary principal series represen-
+tation containing an 1-dimensional space in h.
+(3) g is decomposed into a direct sum of s-submodules described in (1) and (2)
+above, s itself, irreducible lowest weight modules, and irreducible highest
+weight modules.
+We also discuss how to calculate multiplicities of irreducible highest or low-
+est modules (§7). Furthermore, we classified irreducible components which are
+neither highest weight modules nor lowest weight modules, as either unitary
+principal or complementary series representations.
+Theorem 1.3 (Theorem 8.11). We consider irreducible components which are
+neither highest weight modules nor lowest weight modules and contain root
+vectors about real roots in L, obtained by Theorem 7.1. The irreducible com-
+ponents are complementary series representations, except those described in
+Lemma 8.5 and Lemma 8.10. For the exceptions, the irreducible components
+are unitary principal series representations.
+2
+General theory of Kac-Moody Lie algebras
+Let g be a symmetrizable Kac-Moody Lie algebra on C. Let A be the Cartan
+matrix of g and let A be an n × n matrix. Let h be a Cartan subalgebra of g.
+Let the Chevalley generators of g be ei, fi, hi, (i = 0, . . . , n − 1). Let hR be the
+R-span of Chevalley generators.
+From [Kac90, Theorem 2.2], g has a C-valued nondegenerate invariant sym-
+metric bilinear form (· | ·) called the standard form.
+An antilinear automorphism ω0 of g, called compact involution, is deter-
+mined by
+ω0(ei) = −fi,
+ω0(fi) = −ei
+(i = 0, . . . , n − 1),
+ω0(h) = −h
+(h ∈ hR).
+From [Kac90, §2.7], we can determine a nondegenerate Hermitian form (· | ·)0
+on g with (x | y)0 = −(ω0(x) | y).
+Write n+ for a subalgebra of g generated by ei’s and n− for a subalgebra of
+g generated by fi’s.
+We can construct a 3-dimensional subalgebra of g which is spanned by three
+non-zero elements J+ ∈ n+, J− ∈ n−, J3 ∈ h. J+, J− and J3 satisfy
+[J3, J±] = ±J±,
+3
+
+[J+, J−] = −J3.
+This subalgebra is called SO(1, 2) subalgebra of g.
+A representation of SO(1, 2) subalgebra is called unitary if the representation
+space V has a Hermitian scalar product (·, ·) and the following two conditions
+are satisfied.
+(1) The actions of J+ and J− are adjoint each other, and the action of J3 is
+self-adjoint. That is, for x, y ∈ V , we have
+([J+, x], y) = (x, [J−, y]),
+([J3, x], y) = (x, [J3, y]).
+(2) Hermitian scalar product (·, ·) is positive definite.
+When considering the adjoint action of an SO(1, 2) subalgebra of g to g, from
+[Tsu, Lemma 3.1, Lemma 3.2], we can see that the adjoint action satisfying the
+condition (1) to be unitary and J− = −ω0(J+) are equivalent. In [NO01], prin-
+cipal SO(1, 2) subalgebras for hyperbolic Kac-Moody Lie algebras are studied.
+Principal SO(1, 2) subalgebra satisfies that J− = −ω0(J+).
+When three non-zero elements X ∈ n+, Y ∈ n−, H ∈ h of g satisfy
+[H, X] = 2X,
+[H, Y ] = −2Y,
+[X, Y ] = H,
+these three elements are called sl2-triple of g. A g-subalgebra that these elements
+span is called sl2 subalgebra. The SO(1, 2) subalgebras and the sl2 subalgebras
+can be converted by
+J+ =
+1
+√
+2X,
+J− = − 1
+√
+2Y,
+J3 = 1
+2H.
+The condition J− = −ω0(J+) in SO(1, 2) subalgebra is converted to Y =
+ω0(X) in sl2 subalgebra. In the following paper, we consider sl2 subalgebra
+that satisfies Y = ω0(X).
+4
+
+3
+sl2-triples of rank 2 hyperbolic symmetric Lie
+algebra that is compatible to compact involu-
+tion
+Let a be an integer that satisfies a ≥ 3, and let g be a hyperbolic Kac-Moody
+Lie algebra on C such that the Cartan matrix of g is
+�
+2
+−a
+−a
+2
+�
+.
+Let α0, α1 be the simple roots of g.
+Let {Fn} be the sequence of numbers
+determined by F0 = 0, F1 = 1, Fk+2 = aFk+1 − Fk.
+Lemma 3.1 ([KM95, Proposition 4.4]). The real positive roots of g are of the
+form
+α = Fk+1α0 + Fkα1
+or
+β = Fkα0 + Fk+1α1.
+We distinguish these roots as type α and type β, and we also distinguish
+root vectors belonging to each root as type α and type β (cf. [Tsu, §4]).
+Let X be an element of the space which real positive root vectors span. Then
+X can be written as
+X =
+�
+k
+ckEk,
+(k ∈ {0, . . ., nX − 1}, ck ∈ C, ck ̸= 0, Ek ∈ gβk, Ek ̸= 0)
+where βk (k ∈ {0, . . . , nX − 1}) are distinct real roots and nX is a positive
+integer.
+We call this nX the length of X. Then the following holds.
+Lemma 3.2 ([Tsu, Theorem 5.8]). Let X be an element in the space which
+real positive root vectors span.
+(1) When the length of X is 1 or more than 3, X, Y = ω0(X), H = [X, Y ]
+do not form sl2-triple.
+(2) Suppose the length of X is 2 and E0, E1 are real positive root vectors
+of different types (in the sense of α-type and β-type). Then, taking the
+appropriate c0, c1 ∈ C, X = c0E0 + c1E1, Y = ω0(X), and H = [X, Y ]
+form sl2-triple. In particular, c0, c1 can be chosen so that c0, c1 ∈ R.
+Lemma 3.3 ([Tsu, Theorem 6.4]). Take ⟨H, X, Y ⟩ in Lemma 3.2, (2).
+Let
+X = c0E0 +c1E1, where E0 is type α and E1 is type β. From Lemma 3.1, using
+integers i, j ≥ 0, we can write E0 ∈ gFi+1α0+Fiα1, E1 ∈ gFjα0+Fj+1α1. If and
+only if i = j − 1, j, j + 1, H is dominant.
+5
+
+4
+Irreducible decomposition of g as an sl2 mod-
+ule
+In this section, we consider an sl2-subalgebra s = ⟨H, X, Y ⟩ of g, which satisfies
+the following conditions.
+(1) H ∈ h and H is dominant.
+(2) X is in the space which is spanned by positive root vectors.
+(3) Y = ω0(X).
+We show that g is decomposed to irreducible modules by the action of s.
+s-module V is called unitarizable if the following conditions are satisfied.
+(1) (· | ·)0 on V is positive definite.
+(2) for v1, v2 ∈ V , the conditions as follows are satisfied.
+([X, v1], v2)0 = −(v1, [Y, v2])0,
+([H, v1], v2)0 = −(v1, [H, v2])0.
+When Y = ω0(X), the condition (2) are automatically satisfied.
+Therefore,
+(· | ·)0 is positive definite on V if and only if V is unitarizable.
+First, we put
+U = {x ∈ g | ∀y ∈ s (x | y)0 = 0}.
+U is closed under the action of s, and g = s ⊕ U.
+Lemma 4.1. (· | ·)0 is positive definite on U.
+Proof. From [Kac90, Theorem 11.7], (· | ·)0 is positive definite on n+ ⊕ n−. The
+sign of (· | ·)0 on h is (n − 1, 1). Since s itself is not unitarizable, when we write
+h = s ⊕ h′, (· | ·)0 is not positive definite on s. Therefore, (· | ·)0 is positive
+definite on h′. Since U = h′ ⊕ n+ ⊕ n−, (· | ·)0 is positive definite on U.
+Lemma 4.2. Consider a subspace V of U that is closed under the action of H.
+Let V ⊥ be the subspace of U orthogonal to V with respect to the Hermitian
+form (· | ·)0. Then U = V ⊕ V ⊥.
+Proof. We consider the eigenspace decomposition of U by H. Let Uλ be the
+eigenspace for λ and write
+U =
+�
+λ∈C
+Uλ.
+Since H is a Hermitian operator on (· | ·)0, Uλ and Uµ are orthogonal with
+respect to this inner product if λ ̸= µ.
+Since H is dominant, Uλ is finite-
+dimensional. For each λ, V also inherits the eigenspace decomposition of U.
+6
+
+Let Vλ be an eigenspace of V for λ, and V can be written as a direct sum of
+Vλ’s. Let
+V ′
+λ = {v ∈ Vλ | ∀x ∈ Vλ (v | x)0 = 0},
+and
+V ′ =
+�
+λ∈C
+V ′
+λ.
+Vλ is finite dimensional. From Lemma 4.1, (· | ·)0 is positive definite on U. Thus
+we have Uλ = Vλ ⊕ V ′
+λ. Therefore, we have U = V ⊕ V ′ and V ′ = V ⊥.
+In the following, we show that U can be decomposed into irreducible modules
+by the action of s.
+Lemma 4.3. Any non-zero sl2-submodule V of U includes an irreducible sub-
+module.
+Proof. Take the eigenspace decomposition of U by the action of H. V is also
+decomposed into eigenspaces with this decomposition, and each eigenspace of
+V is finite-dimensional. We regard H as a linear transform on V and take some
+eigenvalue λ of H on V . Let U(sl2) be an universal enveloping algebra of sl2.
+Considering the Casimir element C of U(sl2), it preserves Vλ. Since Vλ is finite-
+dimensional, there exists an eigenvector of C. Let v denote this. Consider the
+sl2-submodule generated by v, which includes an irreducible submodule.
+Theorem 4.4. U can be decomposed into direct sum of irreducible s-modules,
+and all of these modules are unitarizable.
+Proof. We consider a set of irreducible submodules of U such that these sub-
+modules are orthogonal to each other with respect to (· | ·)0. Let T be the
+set. We order the elements of T by inclusion. Then T is non-empty and in-
+ductively ordered. Therefore, from Zorn’s lemma, T has a maximal element.
+Take a maximal element of T and denote it by M. Consider the direct sum
+of all submodules belonging to M. Let M denote this sum. Suppose U ̸= M,
+we derive the contradiction. Since M is a subspace of U which is closed by the
+action of H, from Lemma 4.2, we have U = M ⊕ M ⊥. Since M ⊥ is non-zero
+sl2 submodule of U, from Lemma 4.3, M ⊥ includes an irreducible submodule.
+Let W denote this. we have M ∪ {W} ∈ T , that is contradict the maximality
+of M. Therefore, we have U = M, and U can be decomposed into direct sum
+of irreducible submodules. Combining this with Lemma 4.1, we can also see the
+unitarizability of the modules.
+Theorem 4.5. g can be decomposed into direct sum of irreducible s-modules,
+which consists s itself. All of these modules except for s are unitarizable.
+7
+
+5
+sl2 modules in g
+In the following, we consider what kind of modules appear when g is decom-
+posed into irreducible s-modules. In particular, we consider how many unitary
+principal or complementary series representations.
+For a lie algebra a, Let U(a) be the universal enveloping algebra of a. Let
+V be an irreducible s-module which is an irreducible component of g.
+The
+Casimir element C of U(s) acts on V by constant multiplication. Let µ be this
+constant. From [HT92, Chapter II, Corollary 1.1.11], for an eigenvalue λ0 ∈ C
+of H on V , some interval I ⊂ Z exists, and V can be expressed as a direct sum
+of 1-dimensional eigenspaces such that the eigenvalues of H are λk = λ0 + 2k
+(k ∈ I). From [HT92, Chapter II, Theorem 1.1.13], for an eigenvalue λ of H on
+V , we define s1(k) for an integer k as
+s1(k) = 8µ − (λ + 2k − 1)2 + 1
+4
+.
+(A)
+We take an element vk of the eigenspace of V with respect to an eigenvalue
+λ + 2k. Then we have X(Y vk) = s1(k)vk. If k ∈ Z such that s1(k) = 0 does
+not exist, then V is an irreducible module that is neither highest weight module
+nor lowest weight module. If there exists a k ∈ Z such that s1(k) = 0, V is a
+highest weight module or a lowest weight module.
+Let W be the Weyl group of g. Using Lemma 3.2, we may write H, X, Y in
+s as follows.
+X = c0w0(ep) + c1w1(eq)
+(c0, c1 ∈ R, w0, w1 ∈ W, (p, q) ∈ {(0, 1), (0, 0), (1, 1)}),
+Y = −c0w0(fp) − c1w1(fq),
+H = −c0w0(hp) − c1w1(hq).
+Let ks, ls, ms, ns be real numbers such that c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈
+gmsα0+nsα1. From Lemma 3.3, we can write ks = Fi+1, ls = Fi, ms = Fj, ns =
+Fj+1 with integers i, j ≥ 0, and furthermore, i ∈ {j − 1, j, j + 1}.
+When we take the root vector E ∈ gsα0+tα1 with s, t ∈ Z, we want to find
+out which of the three types of modules E generates under the action of s.
+We define L in the xy-plane as follows. L is a region satisfying x ≥ 0, y ≥
+0, (x, y) ̸= (0, 0), x2 − axy + y2 ≤ 1 and the following conditions.
+x < ks = Fi+1,
+(when i = j − 1)
+x + y < ks + ls = Fi + Fi+1,
+(when i = j)
+y < ls = Fi.
+(when i = j + 1)
+If we take the root sα0 + tα1 with s, t ∈ Z, then from [KM95, Cor 4.3], the
+point in the xy-plane given by (s, t) is in the interior or on the boundary of the
+hyperbola x2 − axy + y2 = 1. Let hC be this hyperbola. Let λ ∈ R as the value
+for which HE = λE. We have λ = (sα0 + tα1)(H). λ ∈ (0, 2) if and only if
+(s, t) ∈ L. In the following, we regard a root sα0 + tα1 as a point (s, t) in the
+xy-plane.
+8
+
+Figure
+1: Imaginary roots and real roots in L, a = 3, X = c0r0r1(e0) +
+c1r1r0(e1)
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+x
+y
+imaginary roots
+real roots
+Lemma 5.1. We consider the hyperbola hC on the xy-plane.
+The hC was
+represented by x2 − axy + y2 = 1. Let lb be a line represented by the function
+y = −x+ b with some real number b ≥ 0. There are two intersections of hC and
+lb. Let p1 and p2 be these points. Let db be a distance between p1 and p2. db is
+strictly monotonically increasing with respect to b ≥ 0. The same result holds
+when lb is a line represented by y = b or x = b.
+Proof. First, we consider the case where lb is represented by y = −x + b. Cal-
+culating the y-coordinates of p1, p2 gives
+y = (a + 2)b ±
+�
+(a + 2)(a − 2)b2 + 4(a + 2)
+2(a + 2)
+.
+Therefore, we have
+db =
+√
+2 ·
+�
+(a + 2)(a − 2)b2 + 4(a + 2)
+a + 2
+.
+This db is strictly monotonically increasing with respect to b ≥ 0.
+Next, we consider the case where lb is represented by y = b. Calculating the
+x-coordinates of p1, p2 gives
+x = ab ±
+�
+(a2 − 4)b2 + 4
+2
+.
+Therefore, we have
+db =
+�
+(a2 − 4)b2 + 4.
+This db is strictly monotonically increasing with respect to b ≥ 0. The same
+argument is presented when lb is a line represented by x = b.
+9
+
+Let R be the interior of hC and hC itself. For s, t ∈ Z, (s, t) is a root if and
+only if (s, t) ∈ R.
+Lemma 5.2. If (x, y) ∈ L ∪ −L, then neither (x + ks − ms, y + ls − ns) nor
+(x − ks + ms, y − ls + ns) are roots.
+Proof. First we assume (x, y) ∈ L. The points (ks, ls) and (ms, ns) are on the
+hyperbola hC. Let l1 be the line connecting these two points. Using some real
+number b > 0, l1 is represented by y = −x+ b when i = j, y = b when i = j − 1,
+and x = b when i = j + 1. Let l2 be a line parallel to l1 and passing through
+(x, y). Using some real number 0 < b′ < b, l2 is represented by y = −x + b′
+when i = j, y = b′ when i = j − 1, and x = b′ when i = j + 1. Let p11, p12
+be intersections of hC and l1. Let d1 be the distance between p11 and p12. Let
+p21, p22 be intersections of hC and l2. Let d2 be the distance between p21 and p22.
+From Lemma 5.1, we have d1 > d2. The distance between (ks, ls) and (ms, ns)
+is d1. The distance between (x, y) and (x+ks −ms, y+ls −ns) is also d1. These
+two points are on l2. The length of the part of l2 that is inside the hyperbola
+is d2 < d1. From the fact that (x, y) is inside hC, (x + ks − ms, y + ls − ns) is
+outside the hyperbola. Therefore, (x + ks − ms, y + ls − ns) is not in R. The
+same argument for (x − ks + ms, y − ls + ns) shows that it is not in R. From
+symmetry, the case when (x, y) ∈ −L is also shown.
+Lemma 5.3. For a point (s, t) ∈ L corresponding to the root, we consider the
+root vector E ∈ gsα0+tα1. Then [X, [Y, E]] ∈ gsα0+tα1.
+Proof. We have X = c0w0(ep) + c1w1(eq), Y = −c0w0(fp) − c1w1(fq). Also we
+have c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈ gmsα0+nsα1. Then we have
+[X, [Y, E]] ∈ gsα0+tα1 + g(s−ks+ms)α0+(t−ls+ns)α1 + g(s−ms+ks)α0+(t−ns+ls)α1.
+Since (s, t) is a root, from Lemma 5.2, (s − ks + ms, t − ls + ns) and (s − ms +
+ks, t − ns + ls) are not roots. Therefore, we have g(s−ks+ms)α0+(t−ls+ns)α1 +
+g(s−ms+ks)α0+(t−ns+ls)α1 = 0, and [X, [Y, E]] ∈ gsα0+tα1.
+We consider the Casimir element C of U(s). We can write C = 1
+8H2 − 1
+4H +
+1
+2XY .
+Lemma 5.4. C acts on a root space as endomorphism. The action is diago-
+nalizable.
+Proof. From Lemma 5.3, C acts on the root spaces as endomorphism. Since g
+is completely reducible as an s-modules, the action on the root space is diago-
+nalizable.
+Lemma 5.5. For a point (s, t) ∈ L corresponding to the root, we can take the
+root vector E ∈ gsα0+tα1 such that E is an eigenvector of the Casimir element
+C, and E generates an irreducible s-module.
+Proof. From Lemma 5.4, we have the lemma.
+10
+
+From Lemma 5.5, if we decompose g by the action of s, the decomposition
+is compatible with the root space decomposition in the root in L.
+We consider how many unitary principal or complementary series represen-
+tations appear in the decomposition of g. Since the set of eigenvalues of unitary
+principal or complementary series representations is {λ + 2k | k ∈ Z} for some
+λ, such a module must contain an eigenspace such that its eigenvalue lie on
+[0, 2). Therefore, we consider the root vector of H such that the eigenvalue λ of
+H satisfies λ ∈ [0, 2).
+If λ = 0, i.e., s = t = 0, Since the dimension of h is 2, there are two
+irreducible components of V which have 0-eigenspace (cf. [Tsu, §7]). Since one
+is sl2 itself, we consider the other module. The casimir element C acts on this
+module by a constant multiple (let µ times). If k satisfies s1(k) = 0, we get
+8µ+ 1 = (2k − 1)2. Since µ < −1 from [Tsu, Proposition 7.3], the left hand side
+is less than 0. Therefore, there is no integral solution to s1(k) = 0, and this is
+an irreducible module that is neither highest weight module nor lowest weight
+module. In particular, this module is an unitary principal series representation.
+In the following, we consider the case of λ ∈ (0, 2). In this case, (s, t) is a
+root in L. We compute [X, [Y, E]]. Since Y = −c0w0(fp) − c1w1(fq), we have
+[Y, E] = [−c0w0(fp), E] + [−c1w1(fq), E].
+We have also
+[−c0w0(fp), E] ∈ g(s−ks)α0+(t−ls)α1,
+[−c1w1(fq), E] ∈ g(s−ms)α0+(t−ns)α1.
+If [−c0w0(fp), E] and [−c1w1(fq), E] are not 0, then the eigenvalue of H for them
+must be in the (−2, 0) interval. we consider root vectors which the eigenvalue
+of H are in the (−2, 0). Since R = −R, the roots with respect to these root
+vectors are −L. From Lemma 5.2, if we take two points such that the difference
+is (ks − ms, ls − ns) and one of which is a root in −L, then the other is not a
+root. Now we have ((s − ms) − (s − ks), (t − ns) − (t − ls)) = (ks − ms, ls − ns).
+Therefore, we know that at least one of [−c0w0(fp), E], [−c1w1(fq), E] is zero.
+When both of these are 0, we have [Y, E] = 0 and from the fact that C =
+1
+8H2 − 1
+4H + 1
+2XY , we can write 8µ = λ2 − 2λ.
+When [−c0w0(fp), E] ̸= 0, i.e., (s − ks, t − ls) ∈ R, we have
+[X, [Y, E]] = [c0w0(ep), [−c0w0(fp), E]]
+= [E, [−c0w0(fp), c0w0(ep)]] + [−c0w0(fp), [c0w0(ep), E]].
+We define ps ∈ C by [−c0w0(fp), [c0w0(ep), E]] = psE, then we have
+[X, [Y, E]] = [E, c2
+0w0(hp)] + psE.
+When ps = 0, we have
+[X, [Y, E]] = −[c2
+0w0(hp), E].
+11
+
+Therefore in this case, if we let −[c2
+0w0(hp), E] = k0E, then we have 8µ =
+λ2 − 2λ + 4k0.
+When [c0w0(ep), E] = 0, i.e., (s + ks, t + ls) ̸∈ R, we have ps = 0.
+To summarize the above, we take an irreducible decomposition of g by s. let
+sα0 +tα1 be a root in L. Let E ∈ gsα0+tα1 such that E generates an irreducible
+component of g. Let C be the Casimir element of U(s), and Let µ be a complex
+number such that CE = µE. Let k0 and ps be complex numbers satisfying
+[−c2
+0w0(hp), E] = k0E,
+[−c0w0(fp), [c0w0(ep), E]] = psE.
+If (s − ms, t − ns) ̸∈ R, we have
+8µ =
+
+
+
+
+
+λ2 − 2λ
+((s − ks, t − ls) ̸∈ R) ,
+λ2 − 2λ + 4k0
+((s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R) ,
+λ2 − 2λ + 4k0 + ps
+((s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R) .
+If (s − ks, t − ls) ̸∈ R and not necessarily (s − ms, t − ns) ̸∈ R, we have
+8µ =
+
+
+
+
+
+λ2 − 2λ
+((s − ms, t − ns) ̸∈ R) ,
+λ2 − 2λ + 4k0
+((s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R) ,
+λ2 − 2λ + 4k0 + ps
+((s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R) .
+Solving
+s1(k) = 8µ − (λ + 2k − 1)2 + 1
+4
+= 0
+for k on R, we obtain that
+k =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0, 1 − λ
+((s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R) ,
+1 − λ ±
+�
+(λ − 1)2 + 4k0
+2
+
+
+
+(s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R
+or
+(s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R
+
+
+ ,
+1 − λ ±
+�
+(λ − 1)2 + 4k0 + ps
+2
+
+
+
+(s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R
+or
+(s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R
+
+
+ .
+When (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R, since (s, t) ∈ L, we have
+1 − λ ∈ (−1, 1). Therefore, we know that the only integral solution of s1(k) = 0
+is 0. In this case E belongs to an irreducible lowest weight module.
+6
+Classification by roots
+Based on the previous section, we classify the root (s, t) in L. We define the
+types of roots as follows.
+12
+
+(1) We say that (s, t) is of type A when (s−ks, t−ls) ̸∈ R and (s−ms, t−ns) ̸∈
+R.
+(2) We say that (s, t) is of type B when
+
+
+
+
+
+(s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R
+or
+(s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R
+
+
+
+
+
+.
+(3) We say that (s, t) is of type C when
+
+
+
+
+
+(s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R
+or
+(s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R
+
+
+
+
+
+.
+All roots belong to one of the above types. We put f(x, y) = x2 − axy + y2 for
+x, y ∈ R. From [KM95, Cor 4.3], for s, t ∈ Z, (s, t) ̸= (0, 0), (s, t) is a real root if
+and only if f(s, t) = 1, and (s, t) is an imaginary root if and only if f(s, t) < 1.
+Lemma 6.1. For x, y, x′, y′ ∈ R, if there exists w ∈ W such that (x′, y′) =
+w(x, y), then f(x′, y′) = f(x, y).
+Proof. It is sufficient to check the case w = r0 and the case w = r1. From the
+symmetry, it is sufficient to check the case w = r0. In this case, from the fact
+that x′ = ay − x and y′ = y, we have
+f(x′, y′) = f(ay − x, y)
+= (ay − x)2 − ay(ay − x) + y2
+= x2 − axy + y2
+= f(x, y).
+First, we know the following results on real roots.
+Lemma 6.2. If (s, t) is a real root in L and s > t, then f(s − ks, t − ls) ≤ 0.
+Also, If (s, t) is a real root in L and s < t, then f(s − ms, t − ns) ≤ 0.
+Proof. From symmetry, it is sufficient to show f(s − ks, t − ls) ≤ 0 when s > t.
+We can write s = Fc+1, t = Fc with c ≥ 0 being an integer. Since ks = Fi+1
+and ls = Fi, we have c < i. Let dic = i − c. From Lemma 6.1, by acting r0 and
+r1 on (s − ks, t − ls), we know that
+f(s − ks, t − ls) = f(Fc+1 − Fi+1, Fc − Fi)
+= f(r0(Fc+1 − Fi+1, Fc − Fi))
+= f(Fc−1 − Fi−1, Fc − Fi)
+= f(r1(Fc−1 − Fi−1, Fc − Fi))
+= f(Fc−1 − Fi−1, Fc−2 − Fi−2)
+= · · ·
+13
+
+=
+�
+f(F1 − Fdic+1, F0 − Fdic)
+(when c is even)
+f(F0 − Fdic, F1 − Fdic+1)
+(when c is odd)
+= f(F1 − Fdic+1, F0 − Fdic).
+Since F1 = 1, F0 = 0, we have
+f(s − ks, t − ls) = f(1 − Fdic+1, −Fdic)
+= 2 − aFdic + 2Fdic−1
+< 2 − 2(Fdic − Fdic−1)
+≤ 0.
+Lemma 6.3. If (s, t) is a real root in L, then (s, t) is of type B.
+Proof. First we show that (s, t) is not of type A. From the fact that (s, t) is
+a real root and from symmetry, we can write s = Fc+1, t = Fc with c ≥ 0
+being an integer. From ks = Fi+1, ls = Fi, we have c < i. From Lemma 6.2,
+f(s − ks, t − ls) ≤ 0. Therefore, (s − ks, t − ls) ∈ R and so we know that (s, t)
+is not of type A.
+Next, we show that (s, t) is of type B. To show this, we need to show that
+(s + ks, t + ls) ̸∈ R. We show f(s + ks, t + ls) > 1. Let dic = i − c. From
+Lemma 6.1, by acting r0 and r1 on (s + ks, t + ls), we have
+f(s + ks, t + ls) = f(Fc+1 + Fi+1, Fc + Fi)
+= f(r0(Fc+1 + Fi+1, Fc + Fi))
+= f(Fc−1 + Fi−1, Fc + Fi)
+= f(r1(Fc−1 + Fi−1, Fc + Fi))
+= f(Fc−1 + Fi−1, Fc−2 + Fi−2)
+= · · ·
+=
+�
+f(F1 + Fdic+1, F0 + Fdic)
+(when c is even)
+f(F0 + Fdic, F1 + Fdic+1)
+(when c is odd)
+= f(F1 + Fdic+1, F0 + Fdic)
+= f(1 + Fdic+1, Fdic)
+= 2 + aFdic − 2Fdic−1
+> 2 + 2(Fdic − Fdic−1)
+> 4.
+This shows that (s, t) is of type B.
+We classify also for imaginary roots in L.
+Lemma 6.4. If (s, t), (s′, t′) are imaginary roots, then (s + s′, t + t′) is also
+imaginary root.
+14
+
+Proof. Since f(s, t) ≤ 0, for any r ∈ R, we have f(rs, rt) = r2f(s, t) ≤ 0. It
+shows that the line connecting the origin and (s, t) is inside the asymptotes of
+the hyperbola x2 − axy + y2 = 1. Similarly, the line connecting the origin and
+(s′, t′) is also inside the asymptotes.
+Since (s+ s′, t+ t′) is the midpoint of (2s, 2t) and (2s′, 2t′), this point is also
+inside the asymptotes. Therefore, (s + s′, t + t′) is an imaginary root.
+Lemma 6.5. Let (u, v) ∈ L (u > v) be a real root such that (uα0+vα1)(H) ̸= 0.
+Put (s, t) = (ks−u, ls−v). Then (s, t) is a type C imaginary root in L. Similarly,
+let (u′, v′) ∈ L (u′ < v′) be a real root such that (u′α0 + v′α1)(H) ̸= 0. Put
+(s′, t′) = (ms − u′, ns − v′). Then (s′, t′) ∈ L and (s′, t′) is the imaginary root
+of type C. The other imaginary roots in L are of type A.
+Proof. From Lemma 6.2, f(−s, −t) = f(s, t) ≤ 0.
+It shows that (s, t) is a
+imaginary root. We also see that the eigenvalue of H for (s, t) is in the range
+(0, 2). Therefore, (s, t) ∈ L is shown.
+We show that (s, t) is of type C. To show this, we show that f(s + ks, t +
+ls) ≤ 1. Using c ∈ Z, we can write (u, v) = (Fc+1, Fc). Together this with
+s + ks = 2ks − u, t + ls = 2ls − v, we have
+f(s + ks, t + ls) = f(2Fi+1 − Fc+1, 2Fi − Fc).
+Let dic = i−c > 0. From Lemma 6.1, acting r0, r1 on (s+ks, t+ls), i−c = λ ≥ 1,
+we have
+f(2Fi+1 − Fc+1, 2Fi − Fc) = f(r0(2Fi+1 − Fc+1, 2Fi − Fc))
+= f(2Fi−1 − Fc−1, 2Fi − Fc)
+= f(r1(2Fi−1 − Fc−1, 2Fi − Fc))
+= f(2Fi−1 − Fc−1, 2Fi−2 − Fc−2)
+= · · ·
+=
+�
+f(2Fdic+1 − F1, 2Fdic − F0)
+(when c is even)
+f(2Fdic − F0, 2Fdic+1 − F1)
+(when c is odd)
+= f(2Fdic+1 − F1, 2Fdic − F0)
+= f(2Fdic+1 − 1, 2Fdic)
+= −2aFdic + 4Fdic−1 + 5
+< −6Fdic + 4Fdic−1 + 5
+= (−4Fdic + 4Fdic−1) + (−2Fdic + 5)
+< −4 − 2Fdic + 5
+≤ −1.
+This shows that f(s+ ks, t+ ls) ≤ −1 and that (s, t) is type C. From symmetry,
+we also know that (s′, t′) is in L and is the imaginary root of type C.
+Finally, we show the other imaginary roots in L are of type A. Let (s′′, t′′) ∈ L
+be such an imaginary root. We show (s′′−ms, t′′−ns) ̸∈ R and (s′′−ks, t′′−ls) ̸∈
+15
+
+R. If (s′′ − ms, t′′ − ns) ∈ R or (s′′ − ks, t′′ − ls) ∈ R, (s′′ − ms, t′′ − ns) ∈ −L or
+(s′′−ks, t′′−ls) ∈ −L. Since (s′′−ms, t′′−ns)−(s′′−ks, t′′−ls) = (ks −ms, ls−
+ns), from Lemma 5.2, we know (s′′ − ms, t′′ − ns) ̸∈ R or (s′′ − ks, t′′ − ls) ̸∈ R.
+From symmetry, it is sufficient to consider when (s′′ − ms, t′′ − ns) ̸∈ R.
+Under this assumption, (s′′ − ks, t′′ − ls) is an imaginary root or not a root. If
+(s′′ − ks, t′′ − ls) is imaginary root, then (ks − s′′, ls − t′′) is also imaginary root
+from the symmetry of R. We consider that (ks, ls) = (s′′, t′′) + (ks − s′′, ls − t′′).
+The left hand side is real root and the right hand side is the sum of imaginary
+roots, which contradicts Lemma 6.4. Therefore, (s′′ − ks, t′′ − ls) is not a root
+and (s′′, t′′) is of type A.
+The contents of this section can be summarized as follows.
+Theorem 6.6.
+(1) A real roots in L is of type B.
+(2) We consider an imaginary root that can be written as (ks − s, ls − t) or
+(ms − s, ns − t) where (s, t) is a real root. Such an imaginary root is of
+type C.
+(3) The other imaginary roots are of type A.
+We now summarize the irreducible s-modules through type A and type C.
+For s-modules through type A, we have the following.
+Lemma 6.7. An irreducible s-module containing a root vector about a root of
+type A in L is a lowest weight module which the root vector is the lowest weight
+element.
+Proof. Since (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R for the root (s, t) of
+type A, we know that acting Y on the type A root vector will result in 0. This
+shows the lemma.
+Lemma 6.8. Let M be an irreducible s-module containing a root vector (say
+v) with respect to type C root in L. Then one of the following conditions (1),
+(2), or (3) is hold.
+(1) M is a lowest weight module such that v is a lowest element.
+(2) M is a highest weight module such that v is a highest element.
+(3) M contains a real root vector with respect to a real root in −L.
+Proof. The type C root (s, t) can be written with some real root (sr, tr) that
+(ks − sr, ls − tr) or (ms − sr, ns − tr). Therefore, the root vector E of type C
+becomes either zero or a real root vector when Y act on it. If E becomes 0
+under the action of Y , then E generates an irreducible lowest weight module.
+If E becomes a real root vector, then the real root for this vector is in −L, and
+this lemma is shown.
+16
+
+We also give the type A, B, C distinction to the root of −L by defining
+Theorem 6.6.
+Then, if there is a unitary principal or complementary series
+representation that passes through a root vector of type C in L, −L, it will also
+pass through the root vector of type B in −L, L. Therefore, We have only to
+classify the modules that contains a type B root space.
+Figure 2: a = 3, X = c0r0(e1) + c1r1(e0)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+0
+1
+2
+3
+x
+y
+type A
+type B
+type C
+roots of X
+Figure 3: a = 3, X = c0r0(e1) + c1r1r0(e1)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+0
+2
+4
+6
+8
+x
+y
+type A
+type B
+type C
+roots of X
+17
+
+7
+Irreducible modules which contains a root space
+with respect to a type B root
+We consider an irreducible decomposition of g by s, and we consider an irre-
+ducible component M containing a type B root space. The multiplicity of a real
+root space is 1. We can take 0 < λ < 2 such that {λ + 2k′ | k′ ∈ Z} is the set
+of the eigenvalues of H in M. We consider the H eigenspace of M such that
+the eigenvalue is λ. We assume this eigenspace is gsα0+tα1 such that (s, t) ∈ L,
+and (s, t) is real root. We consider k such that s1(k) = 0 in (A) in §5. We show
+that it is not an integer.
+Let e0, e1, f0, f1, h0 and h1 be Chevalley generators. Using some c0, c1 ∈
+R, w0, w1 ∈ W, and (p, q) ∈ {(0, 1), (0, 0), (1, 1)}, let X = c0w0(ep) + c1w1(eq).
+Suppose s > t. We take the root vector E with respect to the root sα0+tα1. We
+define λ by HE = λE, and define k0 by [−c2
+0w0(hp), E] = k0E. Thus s1(k) = 0
+implies
+k = 1 − λ ±
+�
+(λ − 1)2 + 4k0
+2
+.
+We put
+k+ = 1 − λ +
+�
+(λ − 1)2 + 4k0
+2
+,
+k− = 1 − λ −
+�
+(λ − 1)2 + 4k0
+2
+and we show that k± ̸∈ R or 0 < k± < 1.
+When (λ − 1)2 + 4k0 < 0 or (λ − 1)2 + 4k0 ̸∈ R, k± are imaginary numbers.
+Therefore we can assume (λ − 1)2 + 4k0 ≥ 0. From 0 < λ < 1, it is clear that
+k+ > 0 and k− < 1. To show k+ < 1, we need to show
+1 − λ +
+�
+(λ − 1)2 + 4k0 < 2.
+we can easily show that it is reduced to k0 < λ. Also, to show that k− > 0, we
+need to show
+1 − λ −
+�
+(λ − 1)2 + 4k0 > 0.
+we can easily show that it is reduced to k0 < 0. In summary, we have only to
+show that k0 < 0.
+First, consider the case (s, t) = (1, 0), i.e., E ∈ gα0.
+In this case, from
+Lemma 3.3, we have c0w0(ep) ∈ gFi+1α0+Fiα1 and i ≥ 1. Since
+k0E = [−c2
+0r0r1r0 . . . r1−p(hp), E]
+= [−c2
+0(Fi+1h0 + Fih1), E]
+= −c2
+0(2Fi+1 − aFi)E
+= −c2
+0(Fi+1 + Fi−1)E,
+18
+
+we have k0 < 0. When (s, t) = (0, 1), we can show that k0 < 0 by replacing i
+with j, p with q and making the same argument.
+If (s, t) is general and s > t, we can write (s, t) = (Fi′+1, Fi′). Let p′ be 0 or
+1, we can write E = r0r1r0 . . . r1−p′(ep′). From this, we have
+[−c2
+0w0(hp), E] = −c2
+0[r0r1r0 . . . r1−p(hp), r0r1r0 . . . r1−p′(ep′)]
+= −c2
+0r0r1r0 . . . r1−p′[rp′r1−p′rp′ . . . r1−p(hp), ep′].
+We consider k0 and c0 when i is replaced by i−i′, and rewrite them as k′
+0 and c′
+0.
+Considering (s, t) = (1, 0) or (0, 1) cases, we have [rp′r1−p′rp′ . . . r1−p(hp), ep′] =
+− k′
+0
+c′
+0 ep′. That is, k0 = c2
+0
+c′2
+0 k′
+0. Since k′
+0 < 0, we have k0 < 0. When s < t, we can
+show that k0 < 0 as well.
+From the above, it can be shown that k0 < 0 in any case, i.e., k is not an
+integer. From this, we can see the following.
+Theorem 7.1. We consider an irreducible decomposition of g by the action of
+s.
+(1) Let M is an irreducible component of decomposition of g, which contain
+a root space for a type B root sα0 + tα1. Then, M is an unitary principal
+or complementary series representation.
+(2) (cf. [Tsu, Proposition 7.3]) There is an unitary principal series represen-
+tation containing an 1-dimensional space in h.
+(3) g is decomposed into a direct sum of s-submodules described in (1) and (2)
+above, s itself, irreducible lowest weight modules, and irreducible highest
+weight modules.
+From [KM95, §3], the multiplicity of each root of g is calculated. Using
+this, we can find how many modules appear such that the following condition is
+satisfied: the modules are highest or lowest modules, and eigenvalues of H for
+root vectors with the highest or the lowest roots are certain value.
+First, the modules which contain root spaces in L and −L can be seen from
+previous contents. Among the positive root spaces not in L, those with the
+smallest eigenvalue in H are considered together. Let λH be their eigenvalue
+and dH be their dimensions. Suppose pH modules which contain space with
+eigenvalue λH that also contain the root spaces already obtained. Then there
+are dH−pH lowest weight modules with the root with eigenvalue λH as the lowest
+root. The multiplicities of modules can be obtained inductively by replacing λH
+with the next smallest eigenvalue of H and performing the same calculation.
+Negative root spaces can be classified by the same calculation.
+8
+Unitary principal series representation and com-
+plementary series representation
+In this section, we consider a module (say M) that is neither highest weight
+module nor lowest weight module containing a root vector about the root of type
+19
+
+B. We compute whether the module is a unitary principal series representation
+or a complementary series representation. First, we state the following lemma.
+Lemma 8.1. If 8µ ≤ −1, then M is a unitary principal series representation.
+If 8µ > −1, then M is a complementary series representation.
+Proof. From [HT92, §II 1.2], M is isomorphic to U(ν+, ν−). U(ν+, ν−) is a
+sl2-module with H eigenvectors {vn | n ∈ Z} as a basis of linear space, such
+that
+Hvn = (ν+ − ν− + 2j)vn
+(n ∈ Z),
+e+vn = (ν+ + n)vn+1,
+e−vn = (ν− − n)vn−1,
+8µ = (ν+ + ν− − 1)2 − 1.
+From [HT92, §III Theorem 1.1.3], if ν+ + ν− = 1, U(ν+, ν−) is a unitary
+principal series representation. When 8µ ≤ −1, from
+λ = ν+ − ν− ∈ R,
+8µ = (ν+ + ν− − 1)2 − 1 < −1,
+using b ∈ R we can write
+ν+ − ν− = λ,
+ν+ + ν− = 1 + bi.
+(i =
+√
+−1)
+In this case, we have
+ν+ + ν− = λ + 1
+2
++ b
+2i + −λ + 1
+2
+− b
+2i
+= 1.
+Therefore, M is a unitary principal series representation.
+Consider the case when 8µ > −1.
+From [HT92, §III Theorem 1.1.3], if
+ν± ∈ R and if ν− − 1 and −ν+ are both contained in the interval (l − 1, l) with
+some l ∈ Z, then U(ν+, ν−) is a complementary series representation. From
+8µ > −1, we have
+ν+ + ν− = 1 ±
+�
+8µ + 1,
+ν+ − ν− = λ.
+Therefore, we have
+−ν+, ν− − 1 = −λ − 1 ± √8µ + 1
+2
+.
+We show that they are in (−1, 0).
+20
+
+We show first that 0 < λ < 1. Let n, m be integers such that n > m ≥ 0.
+We can write
+λ = 2(Fm+1 + Fm)
+Fn+1 + Fn
+.
+It is clear that λ > 0. From a ≥ 3, for integer z ≥ 0, we have
+Fz+2 = aFz+1 − Fz
+> (a − 1)Fz+1
+≥ 2Fz+1.
+Hence we have
+Fm+1 + Fm
+Fn+1 + Fn
+< 1
+2.
+Therefore, we have λ < 1. We show that
+−1 < −λ − 1 + √8µ + 1
+2
+.
+From λ < 1, we have −1 < −λ−1
+2
+. Therefore, this inequality is shown. Next we
+show
+−λ − 1 + √8µ + 1
+2
+< 0.
+We have 8µ = λ(λ − 2) + 4k0. From 0 < λ < 1, we have λ(λ − 2) < 0. Also,
+since k0 < 0, we have 8µ < 0. Therefore, we have √8µ + 1 < 1. Using 0 < λ
+again, we know that
+−λ − 1 + √8µ + 1
+2
+< 0.
+For
+−λ − 1 − √8µ + 1
+2
+< 0,
+this is clear from λ > 0. Finally, we show
+−1 < −λ − 1 − √8µ + 1
+2
+.
+From k0 < 0 and 8µ = λ2 − 2λ + 4k0, we have λ2 − 2λ > 8µ. From this and
+λ < 1 we get 1 − λ > √8µ + 1, which can be transformed to
+−1 < −λ − 1 − √8µ + 1
+2
+.
+From the above, −λ−1±√8µ+1
+2
+are both in (−1, 0). Therefore, M is a comple-
+mentary series representation.
+21
+
+Hereafter, we want to determine when M is complementary series. First, we
+consider the case where i = j. we have
+8µ = λ2 − 2λ + 4k0,
+λ = 2(Fn+1 + Fn)
+Fi+1 + Fi
+,
+k0 =
+−2(2Fi+1 − aFi)
+a(F 2
+i + F 2
+i+1) − 4FiFi+1 − 2,
+(*)
+where n is an integer such that i > n ≥ 0. That is, 8µ is determined by i, n,
+and a. We show that 8µ is greater than −1 with finite exceptions.
+Lemma 8.2. We assume i = j. If we consider 8µ to be a function of n by (*),
+8µ is monotonically decreasing with respect to n.
+Proof. k0 is independent on n. λ is monotonically increasing with respect to n.
+Since 8µ = λ(λ − 2) + 4k0 and 0 < λ < 1, we know that 8µ is monotonically
+decreasing with respect to n.
+To show that 8µ is greater than −1 with finite exceptions, we need to examine
+when n is large.
+Lemma 8.3. We assume i = j, n = i − 1. If we consider 8µ to be a function
+of i by (*), 8µ is monotonically increasing with respect to i.
+Proof. First we write {Fi} explicitly as follows. The real solutions of x2 − ax +
+1 = 0 are x = a±
+√
+a2−4
+2
+. As α = a−
+√
+a2−4
+2
+, β = a+
+√
+a2−4
+2
+, we can write
+Fi = βi − αi
+β − α .
+From n = i − 1, we have
+λ = 2(Fi + Fi−1)
+Fi+1 + Fi
+,
+k0 =
+−2(2Fi+1 − aFi)
+a(F 2
+i + F 2
+i+1) − 4FiFi+1 − 2.
+Let t be a real variable. We define the functions Λ and K0 as follows.
+Λ(t) = 2(βt − αt + βt−1 − αt−1)
+βt+1 − αt+1 + βt − αt
+,
+K0(t) =
+−2(β − α)(2(βt+1 − αt+1) − a(βt − αt))
+a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2,
+We have λ = Λ(i) and k0 = K0(i). Using these function, we can calculate as
+follows.
+d
+dtΛ =
+4 log β(a + 2)(β − α)
+(βt+1 − αt+1 + βt − αt)2
+22
+
+d
+dt(Λ2 − 2Λ) = 8 log β(a + 2)(β − α)
+�
+(1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1�
+(βt+1 − αt+1 + βt − αt)3
+d
+dtK0 =
+2(β − α)(a2 − 4) log β
+(a2 − 4)2(β2t+1 + α2t+1 − 2)2
+·
+�
+2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1
++(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt�
+d
+dt(Λ2 − 2Λ + 4K0) = 8 log β(β − α) ·
+�
+(a + 2)
+�
+(1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1�
+(βt+1 − αt+1 + βt − αt)3
++
+1
+(a2 − 4)(β2t+1 + α2t+1 − 2)2
+·
+�
+2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1
++(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt�
+�
+Clearing the denominator, we can calculate as follows.
+(βt+1 − αt+1 + βt − αt)3(a − 2)(β2t+1 + α2t+1 − 2)2
+8(a + 2) log β(β − α)
+· d
+dt(Λ2 − 2Λ + 4K0)
+= (β6t+3 − α6t+3) − (β6t+2 − α6t+2)
++ (a − 2)(1 − a)(β5t+2 − α5t+2) + 3(a − 2)(β5t+1 − α5t+1)
++ 2(β4t+3 − α4t+3) + (a − 4)(β4t+2 − α4t+2)
++ (11 − 2a)(a − 2)(β3t+2 − α3t+2) − (8a + 1)(a − 2)(β3t+1 − α3t+1)
++ (−11a + 5)(β2t+1 − α2t+1) + 17(β2t − α2t)
++ (8a − 14)(a − 2)(βt+1 − αt+1) + (12a − 6)(a − 2)(βt − αt)
++ 14(β − α)
+The coefficient on the left hand side is positive. Using the fact that βt − αt
+is monotonically increasing, we can calculate that the right hand side is also
+positive. This shows that 8µ = (Λ2 − 2Λ + 4K0)(i) is monotonically increasing
+with respect to i.
+From Lemma 8.3, we consider the case when i = 1, n = 0.
+Lemma 8.4. We assume i = j = 1 and n = 0. If we consider 8µ to be a
+function of a by (*), 8µ is monotonically increasing with respect to a.
+Proof. Under this assumption, we have
+8µ =
+−4a2
+a3 − 3a − 2.
+Differentiating this as a function of the real variable a, from a ≥ 3, we know
+that 8µ is monotonically increasing with respect to a.
+23
+
+Lemma 8.5. When i = j, we consider s-modules of g that are neither a highest
+weight module nor a lowest weight module containing a root vector about the
+root of type B obtained by Theorem 7.1. The modules are complementary series
+representations, except for the following five types. For these exceptions, the
+modules are unitary principal series representations.
+(a, i, n) = (4, 1, 0), (3, 1, 0), (3, 2, 1), (3, 3, 2), (3, 4, 3)
+Proof. We use Lemma 8.6, Lemma 8.3, and Lemma 8.4.
+First, when a = 5, i = 1, n = 0, we have 8µ = − 25
+27 > −1. Therefore, when
+a ≥ 5, for any i, n, the module for a, i, n is a complementary series representa-
+tion.
+Next, when a = 4, i = 1, n = 0, we have 8µ = − 32
+25 < −1. Hence the module
+for this is a unitary principal series representation. On the other hand, when
+a = 4, i = 2, n = 1, we have 8µ > −1. Therefore, when a = 4, the module for
+a, i, n is a complementary series representation except when i = 1, n = 0.
+Finally, when a = 3, 8µ < −1 when i = 1, 2, 3, 4 and n = i − 1, and in
+these four cases the module is a unitary principal series representation. When
+n = i − 2 or i = 5, we have 8µ > −1. Therefore, we know that the module is a
+complementary series representation in other cases.
+From the above, with five exceptions, neither a highest weight module nor
+a lowest weight module containing a root vector about the root of type B is a
+complementary series representation.
+Next, we consider the case i = j − 1 or i = j + 1. From symmetry, it is
+sufficient to consider the case i = j − 1. In this case, λ can be written
+λ = 2Fn
+Fi+1
+with n as an integer such that i ≥ n ≥ 0. On the other hand, for k0, we have
+k0 =
+−2(2Fi+1 − aFi)
+a(F 2
+i + F 2
+i+1) − 4FiFi+1 − 2
+as for i = j. As with i = j, 8µ is determined by i, n and a. The next lemma is
+obvious.
+Lemma 8.6. We assume i = j − 1. If we consider 8µ to be a function of n by
+(*), 8µ is monotonically decreasing with respect to n.
+In the following, we consider whether 8µ is monotonically increasing with
+respect to i when n = i. In this case, we have
+λ = 2Fi
+Fi+1
+.
+When i = j, n = i − 1, we already know
+λ = 2(Fi + Fi−1)
+Fi+1 + Fi
+.
+24
+
+We rewrite as
+λ1 = 2Fi
+Fi+1
+,
+λ2 = 2(Fi + Fi−1)
+Fi+1 + Fi
+.
+Let t be a real variable. We define Λ1, Λ2 and K0 as follows.
+Λ1(t) = 2(βt − αt + βt−1 − αt−1)
+βt+1 − αt+1 + βt − αt
+,
+Λ2(t) = 2(βt − αt)
+βt+1 − αt+1 ,
+K0(t) =
+−2(β − α)(2(βt+1 − αt+1) − a(βt − αt))
+a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2.
+We have λ1 = Λ1(i), λ2 = Λ2(i), k0 = K0(i), and 8µ = (Λ2
+1 − 2Λ1 + 4K0)(i).
+When i = j, 8µ = (Λ2
+2 − 2Λ2 + 4K0)(i). We compare
+d
+dtΛ1 and
+d
+dtΛ2. Since
+d
+dt(Λ2
+1 − 2Λ1 + 4K0) = 2(Λ1 − 1) d
+dtΛ1 + 4 d
+dtk0 and 0 < Λ1 − 1 < 1, the smaller
+the value of
+d
+dtΛ1, the larger the value of
+d
+dt(Λ2
+1 − 2Λ1 + 4K0). We know that
+d
+dt(Λ2
+2−2Λ2+4K0) > 0. If we show d
+dtΛ1 > d
+dtΛ2, we also know (Λ2
+2−2Λ2+4K0)
+is monotonically increasing with respect to t. Therefore, we know 8µ is also
+monotonically increasing with respect to i when i = j − 1, n = i.
+Lemma 8.7.
+d
+dtΛ1 > d
+dtΛ2.
+Proof. we show that
+d
+dtΛ1 − d
+dtΛ2 > 0. we have
+d
+dtΛ1 − d
+dtΛ2 =
+4 log β(a + 2)(β − α)
+(βt+1 − αt+1 + βt − αt)2 − 4 log β(β − α)
+(βt+1 − αt+1)2 .
+Calculating this, we have
+(βt+1 − αt+1 + βt − αt)2(βt+1 − αt+1)2
+4 log β(β − α)
+� d
+dtΛ1 − d
+dtΛ2
+�
+=(β2t+3 + β2t+3) + (β2t+2 + α2t+2) − (β2t+1 + α2t+1) − (β2t + α2t).
+The coefficient on the left hand is positive. We can easily calculate to know
+that βt + αt is monotonically increasing with respect to t. From this, we know
+the right hand side is also positive. Therefore,
+d
+dtΛ1 > d
+dtΛ2.
+Lemma 8.8. We assume i = j − 1, n = i. If we consider 8µ to be a function
+of i by (*), 8µ is monotonically increasing with respect to i.
+Lemma 8.9. We assume i = 0, j = 1, and n = 0. If we consider 8µ to be a
+function of a by (*), 8µ is monotonically increasing with respect to a.
+25
+
+Proof. Under this assumption,
+8µ =
+−4
+a − 2.
+This is monotonically increasing with respect to a ≥ 3.
+Lemma 8.10. When i = j −1, We consider s-modules containing a root vector
+about the root of type B that are neither highest weight modules nor lowest
+weight modules obtained by Theorem 7.1.
+The modules are complementary
+series representations, except for the following 23 types. For these exceptions,
+the modules are unitary principal series representations.
+(a, i, n) =(a′, 0, 0) (6 ≤ a′ ≤ 18),
+(5, 0, 0), (5, 1, 1),
+(4, 0, 0), (4, 1, 1),
+(3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 2, 2), (3, 3, 3), (3, 4, 4)
+Proof. We use Lemma 8.6, Lemma 8.8, and Lemma 8.9.
+First, when a =
+18, i = 0, n = 0, 8µ = −1. Therefore, when a ≥ 18, the modules for a, i, n
+are complementary series representations except when (a, i, n) = (18, 0, 0).
+Then, when 6 ≤ a ≤ 17, i = 0, n = 0, from 8µ = − 4
+3 > −1, the module for
+this pair is a unitary principal series representation. On the other hand, when
+6 ≤ a ≤ 17, 8µ < −1 except when (a, i, n) = (a, 0, 0), i.e., the module about
+a, i, n is a complementary series representation.
+When a = 5, if (a, i, n) = (5, 0, 0), (5, 1, 1), then the modules are unitary
+principal series representations, and the others are complementary series repre-
+sentations.
+When a = 4, if (a, i, n) = (4, 0, 0), (4, 1, 1), then the modules are unitary
+principal series representations, and the others are complementary series repre-
+sentations.
+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),
+then the modules are unitary principal series representations, and the others
+are complementary series representations.
+From the above, 23 unitary principal series representations are obtained, and
+the rest are all complementary series representations.
+Theorem 8.11. We consider modules obtained by (1) of Theorem 7.1. The
+modules are neither highest weight modules nor lowest weight modules and
+contain root vectors about roots of type B. The modules are complementary
+series representations, except those enumerated by Lemma 8.5 and Lemma 8.10.
+For the exceptions, the modules are unitary principal series representations.
+Proof. It can be shown from Lemma 8.5 and Lemma 8.10.
+26
+
+Acknowledgements
+I would like to express my appreciation to my supervisor, Prof. Hisayosi Matu-
+moto for his thoughtful guidance.
+References
+[CM93] D. H. Collingwood, W. M. McGovern, Nilpotent Orbits in Semisimple
+Lie Algebras, Van Nostrand Reinhold, 1993
+[Dyn57] E. Dynkin, Semisimple subalgebras of simple Lie algebras, American
+Mathematical Society Translations: Series 2, 6, 1957, pp. 111–245
+[GOW02] M. R. Gaberdiel, et al., A class of Lorentzian Kac-Moody algebras,
+Nuclear Physics B, 645, 2002, pp. 403–437
+[Kac90] V. G. Kac, Infinite dimensional Lie algebras 3rd edition, Cambridge
+university press, 1990
+[KM95] S-J. Kang, D. J. Melville, Rank 2 symmetric hyperbolic Kac-Moody
+algebras, Nagoya Mathematical Journal, 140, 1995, pp. 41–75
+[Kos59] B. Kostant, The principal three-dimensional subgroup and the Betti
+numbers of a complex simple Lie group, American Journal of Mathemat-
+ics, 81, 1959, pp. 973–1032
+[HT92] R. Howe, E. C. Tan, Non-Abelian Harmonic Analysis, Springer-Verlag,
+1992
+[NO01] H. Nicolai, D. I. Olive, The Principal SO(1, 2) Subalgebra of a Hy-
+perbolic Kac Moody Algebra, Letters in Mathematical Physics, 2001, pp.
+141–152
+[Tsu] H. Tsurusaki, sl2 triples whose nilpositive elements are in a space which is
+spanned by the real root vectors in rank 2 symmetric hyperbolic Kac-Moody
+Lie algebras, Publications of the Research Institute for Mathematical Sci-
+ences, to appear
+27
+

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+Application of Causal Inference Techniques to the
+Maximum Weight Independent Set Problem
+Jianfeng Liu †§
+Sihong Shao‡∗
+Chaorui Zhang§
+January 16, 2023
+Abstract
+A powerful technique for solving combinatorial optimization problems is
+to reduce the search space without compromising the solution quality by ex-
+ploring intrinsic mathematical properties of the problems. For the maximum
+weight independent set (MWIS) problem, using an upper bound lemma which
+says the weight of any independent set not contained in the MWIS is bounded
+from above by the weight of the intersection of its closed neighbor set and the
+MWIS, we give two extension theorems — independent set extension theorem
+and vertex cover extension theorem. With them at our disposal, two types
+of causal inference techniques (CITs) are proposed on the assumption that a
+vertex is strongly reducible (included or not included in all MWISs) or re-
+ducible (contained or not contained in a MWIS). One is a strongly reducible
+state-preserving technique, which extends a strongly reducible vertex into a
+vertex set where all vertices have the same strong reducibility. The other, as a
+reducible state-preserving technique, extends a reducible vertex into a vertex
+set with the same reducibility as that vertex and creates some weighted pack-
+ing constraints to narrow the search space. Numerical experiments show that
+our CITs can help reduction algorithms find much smaller remaining graphs,
+improve the ability of exact algorithms to find the optimal solutions and help
+heuristic algorithms produce approximate solutions of better quality. In par-
+ticular, detailed tests on 12 representative graphs generated from datasets in
+†Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
+‡CAPT, LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, P.R.
+China
+§Theory Lab, Central Research Institute, 2012 Labs, Huawei Technologies Co., Ltd.
+∗Author to whom correspondence should be addressed: sihong@math.pku.edu.cn
+1
+arXiv:2301.05510v1  [math.OC]  13 Jan 2023
+
+Network Data Repository demonstrate that, compared to the state-of-the-art
+algorithms, the size of remaining graphs is further reduced by more than 32.6%,
+and the number of solvable instances is increased from 1 to 5.
+AMS subject classifications: 05C69; 68W40; 90C06; 90C27; 90C57
+Keywords: maximum weight independent set; independent set extension;
+vertex cover extension; causal inference techniques; reduction algorithm; exact
+algorithm; heuristic algorithm; Network Data Repository.
+1
+Introduction
+Let G = (V, E, w) be an undirected vertex-weighted graph, where each vertex v ∈ V
+is associated with a weight w(v) ∈ R+. A subset I ⊆ V is called an independent set
+if its vertices are pairwise non-adjacent, and the vertex cover of graph G is a subset
+of vertices V C ⊆ V such that every edge e ∈ E is incident to at least one vertex
+in subset V C. Independent set and vertex cover are two complementary concepts
+in graph and can be transformed into each other on demand [29]. The maximum
+weight independent set (MWIS) problem is to find the independent set of largest
+weight among all possible independent sets and the weight of a MWIS of graph G is
+denoted by αw(G), while the minimum weight vertex cover (MWVC) problem asks for
+the vertex cover with the minimum weight. Furthermore, if subset I ⊆ V is a MWIS,
+then subset V C = V \I is a MWVC, and vice versa [6, 29]. The MWIS problem is an
+extension of the maximum independent set (MIS) problem, which is a classic NP-hard
+problem [13, 9]. It can be applied to various real-world problems, such as information
+retrieval [4], computer vision [12], combinatorial auction problem [29] and dynamic
+map labeling problem [17].
+Due to its wide range of practical applications, the
+research on efficient algorithms for computing the MWIS is of great significance.
+Most previous work are focused on heuristic algorithms to find near-optimal solutions
+in reasonable time [24, 20, 6, 18], while exact algorithms, usually referring to Branch-
+and-Bound (B&B) methods [3, 26, 2, 22], become infeasible when the size of problem
+increases.
+Recently, it has been well demonstrated that reduction rules (a.k.a. kerneliza-
+tion) are very effective in practice for solving the MIS problem [25]. These rules mine
+the structural properties of underlying graph and reduce the search space by such
+as removing vertices, contracting subgraphs, restricting the set of independent sets,
+etc., to produce a smaller kernel graph such that the MIS of the original graph can be
+recovered from the MIS of the kernel. After integrating them, some state-of-the-art
+exact solvers are able to solve the MIS problem on many large real networks [11].
+2
+
+These solvers can be usually divided into two types: One performs the kernelization
+only once and runs the B&B algorithm [23, 16] on the kernelized instance, while
+the other joins hands with the Branch-and-Reduce (B&R) algorithm [19] and per-
+forms reduction in every branch of the search tree. As for those instances that can’t
+be solved exactly, high-quality solutions can be found by combining kernelization
+with local search [8, 10]. Moreover, when a vertex is selected for branching in the
+branching process of the B&R algorithm, if it is assumed to be in all MISs, then
+its satellite set will also be in all MISs [14], while its mirror set will be removed
+directly from the graph, if it is assumed not to be in all MISs [13]. Further, a conflict
+analysis on the assumption that a vertex is in all MISs can be also plugged in to
+find some contradictions and the concept of “unconfined/confined vertices” was in-
+troduced [28]. Later, an auxiliary constraint called packing constraint was proposed
+to accelerate the B&R algorithm by simply exploring branches that satisfy all pack-
+ing constraints [1]. The central idea behind all these attempts for the MIS problem
+involves a state-preserving technique which starts from a vertex, named the starting
+vertex for convenience, and then finds a vertex set with the same state as the starting
+vertex to reduce the search space, thereby implying that some subsequent operations
+can be implemented on the resulting vertex set instead of only on the starting vertex.
+For the MWIS problem, similar state-preserving techniques are rarely used except
+for a recent work using unconfined/confined vertices [27], though some simple and
+fast reduction rules have been used in B&R algorithms [15, 27]. To this end, we
+devote ourselves into developing state-preserving techniques for the MWIS problem
+in this work. The state of the starting vertex we consider can be
+• strongly reducible, meaning that the vertex is included in all MWISs/MWVCs;
+or
+• reducible, meaning that the vertex is contained in a MWIS/MWVC.
+Considering that the assumed state of the starting vertex must be used to analyze its
+local structure to obtain inference results, these targeted state-preserving techniques
+are called causal inference techniques (CITs). Inspired by their success in solving the
+MIS problem, we will systematically develop CITs to solve the MWIS problem by
+analyzing intrinsic mathematical properties of underlying graph. More specifically,
+our main contributions are in three aspects as follows.
+First, by virtue of the upper bound lemma, i.e., the weight of any independent set
+not contained in the MWIS is bounded from above by the weight of the intersection
+of its closed neighbor set with the MWIS, two extension theorems are developed.
+With them, we propose a series of CITs which have been rarely used previously in
+3
+
+the MWIS problem. According to the state of the starting vertex, our CITs can be
+divided into two categories. The first type is a strongly reducible state-preserving
+technique. We first assume that the starting vertex is strongly reducible, and then
+try to extend this vertex to obtain a vertex set with the same strong reducibility.
+If the upper bound lemma is not satisfied in this process, then this contradicts
+the assumption, and the starting vertex can be removed from the graph directly.
+Otherwise, combined with the state-preserving result obtained from the previous
+process, we continue to search for a set called the simultaneous set, which is either
+included in a MWIS or contained in a MWVC. The second type is a reducible state-
+preserving technique. Under the assumption that the starting vertex is reducible, a
+vertex set with the same reducibility can be obtained by extending from this vertex.
+Moreover, if this vertex is selected for branching in the B&R algorithm, with the
+upper bound lemma, an inequality constraint called weight packing constraint will
+be created to restrict subsequent searches.
+Next, according to the characteristics of the proposed CITs, we integrate them
+into the existing algorithmic framework. The first type of CIT can be used to design
+reduction rules to simplify graph.
+These reduction rules are integrated into the
+existing reduction algorithm. In the B&R algorithm, when a vertex is selected to
+branch, a vertex set and a weight packing constraint depending on the assumed
+state of the vertex can be obtained from state-preserving results of two types of
+CITs. The vertex set is used to further simplify the corresponding branch, while we
+can prune branches that violate constraints and simplify the graph by maintaining all
+created weight packing constraints. During the local search process of the heuristic
+algorithm, when the state of a vertex needs to be changed, all vertex states in the
+vertex set obtained by the second type of CIT will also be modified to be the same
+as that vertex, which expands the area of local search and improves the ability of
+local search to find better local optima.
+Numerical experiments on 12 representative graphs generated from datasets in
+Network Data Repository show that the performance of various algorithms is greatly
+improved after integrating our CITs. The size of the kernel obtained by the resulting
+reduction algorithm is greatly reduced. In addition, compared to the state-of-the-art
+exact algorithm, the number of solvable instances have been increased from 1 to 5.
+And the ability of the heuristic algorithm to find better local optimal solutions is
+significantly improved. These experimental results form the third major contribution
+of this paper.
+Relevant notations used in this work are given in Table 1 and the rest of the paper
+is organized as follows. We present two extension theorems in Section 2 and detail
+CITs in Section 3. How the CITs are combined with existing algorithmic frameworks
+4
+
+G = (V,E,w)
+an undirected vertex-weight graph G with vertex set V , edge set E and vertex weight function w : V → R+
+N(v) = {u ∈ V |{u,v} ∈ E}
+the neighbor set of vertex v
+N[v] = N(v) ∪ {v}
+the closed neighbor set of vertex v
+N(S) = (
+�
+v∈S
+N(v))\S
+the open neighbor set of set S
+N[S] = N(S) ∪ {S}
+the closed neighbor set of set S
+|S|
+the size of set S
+w(S) =
+�
+v∈S
+w(v)
+the weight of all vertices in set S
+d(v)
+the degree of a vertex v
+dist(u,v)
+the minimum number of edges in the path from vertex u to vertex v
+Nl(v) = {u| dist(u,v) = l}
+the set of vertices at distance l from vertex v,
+in particular, N1(v) = N(v)
+G[S] = (S,ES,w),
+∀e = {u,v} ∈ ES, u,v ∈ S
+the subgraph induced by a non-empty vertex subset S of V
+α(G)
+the size of a MIS of unweight graph G
+αw(G)
+the weight of a MWIS of graph G
+AI
+the set of all MWISs in graph G
+AC
+the set of all MWVCs in graph G
+S ◀ AI
+set S is an independent set and is included in all MWISs
+C ◁ AC
+set C is contained in all MWVCs
+vertex v is strongly reducible
+vertex v is included in all MWISs/MWVCs
+vertex v is reducible
+vertex v is contained in a MWIS/MWVC
+vertex v is strongly inclusive
+vertex v is included in all MWISs
+vertex v is strongly sheathed
+vertex v is contained in all MWVCs
+vertex v is inclusive
+vertex v is included in a MWIS
+vertex v is sheathed
+vertex v is contained in a MWVC
+set S is strongly inclusive
+set S is an independent set and is included in all MWISs
+set S is strongly sheathed
+set S is contained in all MWVCs
+set S is inclusive
+set S is an independent set and is included in a MWIS
+set S is sheathed
+set S is contained in a MWVC
+independent set S is strongly exclusive
+independent set S is not contained in all MWIS
+independent set S is exclusive
+independent set S is not contained in a MWIS
+a set S called a simultaneous set
+set S is either included in a MWIS or contained in a MWVC
+Table 1: Notations used throughout the paper.
+is described in Section 4. Extensive numerical tests are carried out in Section 5 to
+verify the performance improvement of integrating our CITs into existing algorithmic
+frameworks in terms of efficiency and accuracy. The paper is concluded in Section 6
+with a few remarks.
+2
+Two Extension Theorems
+The theoretical cornerstones of CITs in this paper are two extension theorems: inde-
+pendent set extension theorem and vertex cover extension theorem. Before delineat-
+ing them, we need to have a deep understanding of the local structure of the MWIS
+and first give the upper bound lemma.
+Lemma 2.1 (upper bound lemma). Let set IC be an independent set in the graph.
+(a) Suppose there is an Iw ∈ AI such that IC ̸⊆ Iw, then w(Iw ∩ N[IC]) ⩾ w(IC)
+holds.
+(b) Assume that IC ̸⊆ I, ∀I ∈ AI holds, then it satisfies: w(IC) < w(I∩N[IC]), ∀I ∈
+AI.
+Proof. Proof We first prove (a) by contradiction. If not, we can obtain an indepen-
+dent set I′
+w = (Iw\N[IC])∪(IC) such that w(I′
+w) = w(Iw)+w(IC)−w(Iw ∩N[IC]) >
+w(Iw), a contradiction.
+Next, we consider (b). If there is an I1 ∈ AI such that w(IC) ⩾ w(I1 ∩ N[IC])),
+we can construct an independent set I′
+1 = (I1\N[IC])∪IC satisfying w(I′
+1) = w(I1)+
+w(IC) − w(I1 ∩ N[IC]) ⩾ w(I1).
+Then I′
+1 ∈ AI and IC ⊆ I′
+1, which leads to a
+contradiction.
+5
+
+The upper bound lemma describes such a property: For any independent set that
+is (strongly) exclusive, the weight of the intersection of its closed neighbor set with
+the MWIS is the upper bound on its weight. With it, the independent set extension
+theorem can be introduced as follows.
+Theorem 2.2 (Independent Set Extension Theorem). Let sets IS and S be two
+independent sets in the graph.
+(a) Assume that there exists an Iw ∈ AI such that IS ⊆ Iw.
+If there is an
+independent set IS′ ⊆ N(IS) such that w(IS′) > w(IS ∩ N(IS′)), then
+there exists an independent set IS′′ ⊆ N(IS′)\N[IS] satisfying the inequal-
+ity: w(IS′) ⩽ w(IS ∩ N(IS′)) + w(IS′′). In addition, IS ∪ IS′′ ⊆ Iw if such
+IS′′ is unique.
+(b) Suppose S ◀ AI, then for any independent set S′ ⊆ N(S), there is an indepen-
+dent set S′′ ⊆ N(S′)\N[S] such that w(S′) < w(S ∩ N(S′)) + w(S′′). Besides,
+if such S′′ is unique, then S ∪ S′′ ◀ AI.
+Proof. Proof We first consider the proof of (a), and it is obvious that IS′ ̸⊆ Iw. In
+view of the fact that the relationship between Iw and N[IS′] satisfies: Iw ∩ N[IS′] =
+Iw∩N(IS′) = (IS∩N(IS′))∪(Iw∩(N(IS′)\N[IS])) and by the upper bound lemma,
+we can get: w(IS ∩ N(IS′)) + w(Iw ∩ (N(IS′)\N[IS])) = w(Iw ∩ N(IS′)) = w(Iw ∩
+N[IS′]) ⩾ w(IS′). Thus, the existence of such IS′′ is proved. Furthermore, assuming
+that such IS′′ is unique, then IS′′ = Iw ∩ (N(IS′)\N[IS]) and IS ∪ IS′′ ⊆ Iw.
+Similar ideas can be used to prove (b). Obviously S′ ̸⊆ I, ∀I ∈ AI holds, so from
+the upper bound lemma, it can be directly obtained: ∀I ∈ AI, w(I ∩N[S′]) > w(S′).
+Further, by considering that the relationship between I and N[S′] satisfies: I ∩
+N[S′] = I ∩N(S′) = (S∩ N(S′))∪(I ∩(N(S′)\N[S])), we prove the existence of such
+S′′. Also, if such S′′ is unique, the following result holds: S′′ = I∩(N(S′)\N[S]), ∀I ∈
+AI, and then S ∪ S′′ ◀ AI.
+The independent set extension theorem gives a method for extending independent
+set that is (strongly) inclusive: Try to find an independent set to add to the extended
+independent set, and that independent set is the only one that guarantees that the
+upper bound lemma is satisfied in the local structure of the extended independent
+set.
+Next, with the help of the upper bound lemma, the vertex cover extension
+theorem is given below.
+Theorem 2.3 (Vertex Cover Extension Theorem). Let sets IC and C be two vertex
+subsets in the graph.
+6
+
+(a) Suppose set IC ⊆ V Cw, then the vertices in IC have the property: ∀p ∈ IC,
+w(p) ⩽ αw(G[N(p)\IC]). Also, for a vertex v ∈ IC and a vertex u ∈ N 2(v),
+IC ∪ {u} ⊆ V Cw holds if the inequality w(v) > αw(G[N(v)\(IC ∪ N(u))]) is
+satisfied.
+(b) Assume that set C ◁ AC, then ∀p ∈ C, w(p) < αw(G[N(p)\C]) is always
+satisfied. In addition, if there exists a vertex v ∈ C and a vertex u ∈ N 2(v)
+such that w(v) ⩾ αw(G[N(v)\(C ∪ N(u))]), then C ∪ {u} ◁ AC.
+Proof. Proof We first consider (a) and let set Iw = V \V Cw. From the upper bound
+lemma, these results can be directly obtained: ∀p ∈ IC, w(p) ⩽ w(Iw ∩ N[p]) =
+w(Iw ∩ N(p)) ⩽ αw(G[N(p)\IC]). Also, based on the assumption about u in (a),
+if u ∈ Iw, then w(v) ⩽ w(Iw ∩ N[v]) = w(Iw ∩ N(v)) ⩽ αw(G[N(v)\(IC ∪ N(u))]),
+which leads to a contradiction.
+Similar methods can be used to prove (b).
+First, ∀p ∈ C, ∀I ∈ AI, w(p) <
+w(I ∩N[p]) = w(I ∩N(p)) ⩽ αw(G[N(p)\C]) can be obtained from the upper bound
+lemma. Besides, under given conditions about u in (b), if there is an I∗ ∈ AI such
+that u ∈ I∗, a contradiction is deduced from w(p) < w(I∗ ∩ N[p]) = w(I∗ ∩ N(p)) ⩽
+αw(G[N(p)\(C ∪ N(u))]).
+The vertex cover extension theorem describes how to expand a set that is (strongly)
+sheathed: Attempt to find a vertex that satisfies the condition that after removing
+its neighbor set, the upper bound lemma is not satisfied in the local structure of the
+expanded set. If such a vertex is found, it is directly added to the expanded set.
+3
+Causal Inference Techniques
+In this section, with the help of the upper bound lemma and two extension theorems,
+we give the CITs used in this paper. Our CITs can be divided into two types: The
+first type is a strongly reducible state-preserving technique introduced in Section 3.1,
+while the second type is a reducible state-preserving technique shown in Section 3.2.
+3.1
+Strongly reducible state-preserving technique
+The strongly reducible state-preserving technique exploits the assumption that a
+vertex is strongly reducible, and the assumed state of the vertex can be divided into
+two cases: The vertex is assumed to be strongly inclusive or is assumed to be strongly
+sheathed. We first consider the assumption that a vertex is strongly inclusive and
+give the following definition.
+7
+
+Definition 3.1. Let set S be an independent set in the graph. If a vertex u ∈ N(S)
+such that w(u) ⩾ w(S ∩ N(u)), we call it a child of set S. A child u is called an
+extending child if and only if there exists a unique independent set S∗ ⊆ N(u)\N[S]
+such that w(u) < w(S ∩ N(u)) + w(S∗) and vertex set S∗ is called a satellite set of
+set S.
+On the basis of Definition 3.1, with the assumption that a vertex is strongly inclu-
+sive, the concept of ‘confined/unconfined vertices’ is given by the following conflict
+analysis process:
+Definition 3.2. Let v be a vertex in the graph. Suppose set S := {v} ◀ AI, repeating
+(i) until (ii) or (iii) holds:
+(i) As long as set S has an extending child in N(S), set S is extended by including
+the corresponding satellite set into set S.
+(ii) If a child u such that w(u) ⩾ w(S ∩N(u))+αw(G[N(u)\N[S]]) could be found,
+that is, the upper bound lemma is not satisfied in the local structure of set S,
+then halt and vertex v is called an unconfined vertex.
+(iii) If any child is not an extending child, then halt and return set Sv = S. In this
+case, vertex v is called a confined vertex and the set Sv is called the confining
+set of vertex v.
+Some examples of unconfined vertex are given in Figure 1. By means of the con-
+flict analysis process in Definition 3.2, vertices a and h can be found to be unconfined
+vertices. It is also worth noting that, by the definition of unconfined vertex given
+in [27], in Figure 1, only vertex a can be found to be an unconfined vertex. The
+reason for this is that we further generalize the concept of confined/unconfined ver-
+tices in this work. Compared with the definition of extending child u in [27], which
+requires |N(u)\N[S]| = 1 and w(u) < w(N(u)\N(S)), we can consider the more
+general case where N(u)\N[S] is an independent set rather than a single vertex,
+helping us find more unconfined vertices.
+Next, we will explore the properties of confined/unconfined vertices.
+By the
+conflict analysis process in Definition 3.2 and the independent set extension theorem,
+set S can be extended under the assumption: set S := {v} ◀ AI, and set S ◀ AI is
+always satisfied. If vertex v is a unconfined vertex, then the upper bound lemma is not
+satisfied in the local structure of set S, which contradicts set S ◀ AI. Thus, vertex v
+is sheathed. Otherwise, then there is a state-preserving result, i.e., the corresponding
+confining set Sv ◀ AI holds. Furthermore, suppose two confined vertices u, v and the
+corresponding confining sets Su, Sv such that u ∈ Sv and v ∈ Su. If {v} ◀ AI, then
+8
+
+Figure 1:
+Some examples of unconfined vertices, and a MWIS in this graph
+is {b, d, g, i}.
+Let set S := {a}, from Definition 3.1, vertex b is an extend-
+ing child of set S and set {c} is a satellite set of set S.
+Thus, set S can be
+extended as:
+{a, c}.
+At this time, it can be found that a child d such that
+w(d) ⩾ w(S ∩ N(d)) + αw(G[N(d)\N[S]]), then halt and conclude that vertex a
+is an unconfined vertex.
+Similarly, let set S := {h}, then it can be found that
+vertex g is an extending child of set S and set {f, l} is a satellite set of set S.
+So set S can be further expanded as: {h, l, f}.
+After that, the child i satisfied:
+w(i) ⩾ w(S ∩ N(i)) + αw(G[N(i)\N[S]]), hence, vertex h is an unconfined vertex.
+obviously {u} ◀ AI holds. If not, vertex v is sheathed in graph G. Since v ∈ Su,
+then vertex v is included in the satellite set of an intermediate state set S′ of Su,
+which means that in graph G[V \{v}], the upper bound lemma is not satisfied in the
+local structure of set S′. Thus, by Definition 3.2, vertex u is an unconfined vertex
+of graph G[V \{v}] and is sheathed in this graph. From these analysis results and
+the symmetry of the relationship between vertex v and vertex u, we can know that
+vertex set {u, v} is a simultaneous set. Therefore, the following properties can be
+obtained:
+Corollary 3.3. Let v is a vertex in the graph.
+(a) If vertex v is an unconfined vertex, then it is sheathed and after deleting it from
+the graph, the weight of the MWIS in the remaining graph remains unchanged.
+(b) Suppose vertex v is a confined vertex, then either it is sheathed or the corre-
+sponding confining set Sv ◀ AI. Moreover, if a vertex u ∈ Sv is also a confined
+vertex with the corresponding confining set Su and v ∈ Su, then vertex set {u, v}
+is a simultaneous set.
+From Corollary 3.3, it can be known that the conflict analysis process in Defini-
+tion 3.2 can be used to find the vertex that is sheathed or a simultaneous set. These
+9
+
+h
+a
+:
+4
+3
+b
+6
+11
+2
+g
+d
+9
+10
+m
+c
+5
+5
+k
+5
+f
+3
+8
+4
+eCITs will be used to design reduction rules in Section 4.1. In addition, by the prop-
+erty of confined vertex, a fact is obvious: If confined vertex v such that {v} ◀ AI,
+then the corresponding confining set Sv ◀ AI. We will exploit this state-preserving
+result in the B&R algorithm to design a branching rule to search for a solution in
+Section 4.2.
+Next, we proceed to consider the assumption that a vertex is strongly sheathed.
+In the MIS problem, the notion of mirror is given by means of such an assumption
+and is very useful in practice [1]. We will generalize the notion of mirror to the
+MWIS problem: For a vertex v ∈ V , a mirror of vertex v is a vertex u ∈ N 2(v) such
+that w(v) ⩾ αw(G[N(v)\N(u)]).
+Remark 3.4. When the weight of all vertices in the graph is 1, then α(G[N(v)\N(u)]) =
+αw(G[N(v)\N(u)]) ⩽ w(v) = 1. This means that N(v)\N(u) induces a clique or is
+an empty set, and this is exactly the definition that vertex u is the mirror of vertex
+v in the MIS problem.
+To make the concept of mirror more practical, we further generalize it to the case
+of set, which leads to the following definitions:
+Definition 3.5. Let set C be a vertex subset in the graph. If a vertex v ∈ C satisfies
+the inequality: w(v) < αw(G[N(v)\C]), we call it a father of set C. Furthermore,
+if there exists a vertex u ∈ N 2(v) such that w(v) ⩾ αw(G[N(v)\(C ∪ N(u))]), then
+the father v is called an extending father of set C and vertex u is called a mirror of
+vertex v. We use M(v) to denote the set of mirrors of vertex v.
+By means of Definition 3.5, and under the assumption that a vertex is strongly
+sheathed, the concept of ‘covered/uncovered vertices’ is given by the following conflict
+analysis process:
+Definition 3.6. Let v be a vertex in the graph. At the beginning, suppose set C :=
+{v} ◁ AC and repeating (i) until (ii) or (iii) are met:
+(i) When set C has an extending father, extend set C by including the correspond-
+ing set of mirrors to set C.
+(ii) If there is a vertex u ∈ C such that w(u) ⩾ αw(G[N(u)\C]), in this case, the
+upper bound lemma is not satisfied, then halt and vertex v is called an uncovered
+vertex.
+(iii) If set C has no extending father, then halt and return set Cv = C. In this case,
+vertex v is called a covered vertex and vertex set Cv is called the covering set
+of vertex v.
+10
+
+Figure 2: An example of uncovered vertex and a MWIS of this graph is {a, e, g, h, j, l}.
+Starting with set C := {a}, from Definition 3.5, it can be seen that vertex a is an
+extending father of set C and set {e, g, h} is the mirrors set of vertex a. Thus, set C
+can be extended to: {a, e, g, h}. Then, vertex h is also an extending father of set C
+and set {j, k, l} is the mirrors set of vertex h. So set C can be further expanded as:
+{a, e, g, h, j, k, l}. At this time, we find that w(l) ⩾ αw(G[N(l)\C]), then halt and
+conclude that vertex a is uncovered.
+An example of uncovered vertex is given in Figure 2 and we find that vertex a
+is an uncovered vertex. In addition, the properties of uncovered/covered vertices
+are worth further study. From the vertex cover extension theorem, in the conflict
+analysis process of Definition 3.6, for any extending father f of set C, ∀u ∈ M(f),
+if set C ◁ AC, set C ∪ {u} ◁ AC always holds. Thus, under the assumption set
+C := {v} ◁ AC, if vertex v is not an uncovered vertex, then a state-preserving
+result can be obtained: The corresponding covering set Cv ◁ AC. Otherwise, the
+upper bound lemma is not satisfied in the local structure of set C, which contradicts
+hypothesis set C ◁ AC. So vertex v is inclusive. Also, assume that the two covered
+vertices u, v and the corresponding covering set Cu, Cv satisfy: v ̸∈ N(u), u ∈ Cv and
+v ∈ Cu. If vertex v is inclusive, we first remove N[v] from graph G. Since v ∈ Cu,
+then vertex v is a mirror of an extending father of an intermediate state set C′ of
+set Cu and the upper bound lemma cannot be satisfied in graph G[V \N[v]] at this
+time. Thus, vertex u is an uncovered vertex of graph G[V \N[v]] and is inclusive in
+this graph. So there exists a MWIS in graph G containing both vertex v and vertex
+u. Moreover, if {v} ◁ AC, {u} ◁ AC is clearly satisfied. Thus, from the symmetry
+of the relationship between vertex u and vertex v, it can be known that vertex set
+{u, v} is a simultaneous set. These properties are summarized as follows.
+Corollary 3.7. Let v be a vertex in the graph G.
+(a) If vertex v is an uncovered vertex, then it is inclusive. After deleting N[v] from
+the graph, the weight of the MWIS in the remaining graph satisfies: αw(G) =
+11
+
+h
+d
+b
+a
+J
+6
+3
+5
+3
+4
+3
+g
+c
+k
+e
+2
+8
+6
+9
+4
+fαw(G[V \N[v]]) + w(v).
+(b) If vertex v is a covered vertex. Then, either vertex v is inclusive or the cor-
+responding covering set Cv ◁ AC. Also, if another covered vertex u with the
+corresponding covering set Cu satisfies: v ̸∈ N(u), u ∈ Cv and v ∈ Cu, then
+vertex set {u, v} is a simultaneous set.
+Corollary 3.7 gives the following results: The conflict analysis process in Defi-
+nition 3.6 can be applied to find the vertex that is inclusive or a simultaneous set.
+In Section 4.1, we will use these CITs to design reduction rules. Besides, by the
+property of covered vertex in (b) of Corollary 3.7, we can know a state-preserving
+result: if the covered vertex v such that {v} ◁ AC, then the corresponding covering
+set Cv ◁ AC.
+3.2
+Reducible state-preserving technique
+Similar to the first type of CIT, the reducible state-preserving technique utilizes the
+assumption that a vertex is reducible, that is, assumes that a vertex is inclusive or
+sheathed. With these assumptions, we can give state-preserving results similar to
+the first type of CIT. Before that, we give the following definition.
+Definition 3.8. Let sets IS and IC be two vertex subsets in the graph and set IS
+is an independent set.
+(a) A vertex u ∈ N(IS) is called an inferred child of set IS if it holds that
+w(u) > w(IS ∩ N(u)).
+Further, if there is only a unique independent set
+IS∗ ⊆ N(u)\N[IS] that satisfies the inequality: w(u) ⩽ w(IS∩N(u))+w(IS∗),
+we call the inferred child u an inferred extending child of set IS and vertex set
+IS∗ is called an inferred satellite set of set IS.
+(b) A vertex v ∈ IC is called an inferred father of set IC if it holds that w(v) ⩽
+αw(G[N(v)\IC]). An inferred father v is called an inferred extending father of
+set IC if there exists a vertex u ∈ N 2(v) such that w(v) > αw(G[N(v)\(IC ∪
+N(u))]) and vertex u is called an inferred mirror of vertex v. Also, IM(v) is
+used to denote its set of inferred mirrors.
+By virtue of Definition 3.8 and the assumption that a vertex is inclusive or
+sheathed, we can directly give the definitions of inferred confining set and inferred
+covering set accordingly.
+Definition 3.9. Suppose there are no unconfined vertex in the graph. Let v be a
+vertex in the graph. Beginning with the assumption set IS := {v} ⊆ Iw.
+12
+
+(i) Only if set IS has an inferred extending child in N(IS), set IS can be extended
+by including the corresponding inferred satellite set to set IS.
+(ii) The above process halts if set IS has no inferred extending child in N(IS) and
+return set ISv = IS. We call vertex set ISv is the inferred confining set of
+vertex v.
+Definition 3.10. We assume that there are no uncovered vertex in graph. Let v be
+a vertex in the graph. Starting with the assumption set IC := {v} ⊆ V Cw.
+(i) While set IC has an inferred extending father, extend set IC by including the
+corresponding set of inferred mirrors to set IC.
+(ii) The above process halts if set IC has no inferred extending father and return
+set ICv = IC. We call vertex set ICv is the inferred covering set of vertex v.
+Examples of inferred confining set and inferred covering set are given in Fig-
+ure 3.
+By the process in Definition 3.9, we can find the inferred confining set
+ISa = {a, c, e, j, g, h, k} of vertex a.
+Similarly, according to the process in Defi-
+nition 3.10, we can find the inferred covering set ICd = {b, d, f, i, l} of vertex d.
+Moreover, from the independent set extension theorem and the vertex cover exten-
+sion theorem, we can directly obtain the following Corollary:
+Corollary 3.11. Let v be a vertex in the graph.
+(a) If {v} ⊆ Iw, then the corresponding inferred confining set ISv ⊆ Iw.
+(b) Suppose {v} ⊆ V Cw, then the corresponding inferred covering set ICv ⊆ V Cw.
+From (a) of Corollary 3.11, under the premise {v} ⊆ Iw, the state-preserving
+result can be obtained: ISv ⊆ Iw. We will integrate this result into the local search
+process of heuristic algorithm in Section 4.3. In addition, (b) of Corollary 3.11 also
+gives a similar state-preserving result result: If {v} ⊆ V Cw, then the corresponding
+inferred covering set ICv ⊆ V Cw. This result can be used to design a branching rule
+to search for a solution in Section 4.2.
+Furthermore, during the branching process of the B&R algorithm, it is assumed
+that a vertex v is selected for branching. Inspired by the successful application of
+packing constraints in the MIS problem, we extend them to the MWIS problem and
+propose the concept of “weight packing constraint”.
+When assuming that vertex v is inclusive, if ∃u ∈ N(v) such that w(u) ⩾ w(v),
+let N +(u) = N(u)\N[v]. To avoid obtaining another MWIS by adding vertex u to
+the independent set and removing vertices in N(u) from the independent set, by the
+13
+
+Figure 3: Examples of inferred confining set and inferred covering set. A MWIS for
+this graph is {a, c, e, g, h, j, k}. We first search for the inferred confining set ISa of
+vertex a. Let set IS := {a}, it can be seen from (a) of Definition 3.8 that vertex b
+is an inferred extending child of set IS and set {c} is an inferred satellite set of set
+IS. Thus, set IS can be extended to: {a, c}. Further, vertex d is also an inferred
+extending child of set IS and set {e, j} is the corresponding inferred satellite set. So
+set IS can be further extended to: {a, c, e, j}. At this time, it can be found that both
+vertex f and vertex i are inferred extending children of set IS. Then, both vertex
+set {g, h} and vertex set {k} are the corresponding inferred satellite sets. Finally,
+the inferred confining set of vertex a can be found as: ISa = {a, c, e, j, g, h, k}.
+Furthermore, we continue to search the inferred covering set ICd of vertex d. Let
+set IC := {d}, according to (b) of Definition 3.8, vertex d is an extending father of
+set IC and set {b, f, i} is its inferred mirrors set. Then, set IC can be extended as:
+{b, d, f, i}. Next, it can be found that vertex b is an extending father of set IC and set
+{l} is its inferred mirrors set. Thus, set IC can be further extended as: {b, d, f, i, l}.
+Finally, the inferred covering set of vertex d can be found as: ICd = {b, d, f, i, l}.
+.
+upper bound lemma, the following state-preserving result needs to be guaranteed to
+hold:
+w(v) +
+�
+z∈N+(u)
+w(z)(1 − xz) > w(u).
+The 0-1 integer variable xz is used to indicate whether vertex z ∈ N +(u) is in the
+independent set, and xz = 0 means it is in the independent set, otherwise it is not.
+Thus, a weight packing constraint can be created as shown below:
+�
+z∈N+(u)
+w(z)xz <
+�
+z∈N+(u)
+w(z) − (w(u) − w(v)).
+(3.1)
+When assuming that vertex v is sheathed, to avoid that a MWIS containing it can
+be found by modifying its state, by means of the upper bound lemma, the following
+14
+
+d
+C
+e
+9
+b
+9
+3
+3
+11
+4
+a
+8
+10
+5
+3
+6
+3
+10
+h
+！
+kstate-preserving result needs to be satisfied:
+�
+z∈N(v)
+w(z)(1 − xz) > w(v).
+So a weight packing constraint can also be created as follows:
+�
+z∈N(v)
+w(z)xz <
+�
+z∈N(v)
+w(z) − w(v).
+(3.2)
+These constraints will be kept and managed while the algorithm is searching for
+a solution, and we only need to search all branches satisfying these constraints, since
+no better solution exists in the remaining branches, thus narrowing the search space.
+Let �
+z∈S
+w(z)xz < k be a weight packing constraint such that set S is non-empty.
+When a vertex z is found to be inclusive, for each constraint that includes variable
+xz, we delete the variable on the left side of the constraint and keep the right side
+of the constraint unchanged. When a vertex z is inferred to be sheathed, for each
+constraint that contains variable xz, we delete the variable on the left side of the
+constraint and decrease the weight of vertex z on the right side of the constraint.
+In the process of keeping and managing these constraints, some properties of causal
+inference are mined, which can be divided into the following three cases.
+(a) When there is a constraint whose right-hand term k is less than or equal to 0,
+then we can directly prune subsequent searches from the current branch vertex.
+(b) When there is a constraint whose right-hand term k is less than or equal to
+the weight of any vertex in set S, if this set is not an independent set, we can
+prune subsequent searches from the current branch vertex. If not, the vertices
+in set S will be included in the independent set.
+In addition, some new weight packing constraints can also be introduced. Sup-
+pose there is a vertex p ∈ N(S) such that w(p) ⩾ w(N(p) ∩ S), let N +(p) =
+N(p)\N[S], by the upper bound lemma, the following state-preserving result
+needs to be guaranteed:
+w(N(u) ∩ S) +
+�
+z∈N+(u)
+w(z)(1 − xz) > w(u).
+Therefore, we can introduce the following weight packing constraint:
+�
+z∈N+(p)
+w(z)xz <
+�
+z∈N+(p)
+w(z) − (w(p) − w(N(p) ∩ S)).
+(3.3)
+15
+
+(c) When there is a constraint whose right-hand term k > 0 and there is vertex
+u ∈ N(S) such that
+�
+z∈N(u)∩S
+w(z) ⩾ k, it can be inferred that vertex u is
+sheathed to ensure that this constraint holds. In addition, in order to ensure
+that the current state-preserving result is valid, similar to constraint (3.2), the
+following constraint needs to be introduced:
+�
+z∈N(u)
+w(z)xz <
+�
+z∈N(u)
+w(z) − w(u).
+(3.4)
+The above properties of causal inference provide new pruning search techniques for
+the B&R algorithm and can simplify the graph. We will integrate these techniques
+into B&R algorithm in Section 4.2.
+4
+Integrate CITs into Existing Algorithmic Frame-
+works
+We next describe in detail how CITs in Section 3 are integrated into the existing
+algorithmic frameworks. Section 4.1 introduces how to apply the first type of CIT
+to the reduction algorithm. Further, integrating the resulting reduction algorithm
+and the state-preserving results of two types of CITs into B&R algorithm will be
+presented in Section 4.2, and Section 4.3 will introduce the application of the state-
+preserving results of the second type of CIT to the local search process of heuristic
+algorithm.
+4.1
+The Causal Reduce
+We first introduce how to design reduction rules with the first type of CIT and
+how to integrate them into the existing reduction algorithm. From the property of
+unconfined vertex in Corollary 3.3 and the property of uncovered vertex in Corol-
+lary 3.7, the following reduction rules that can directly determine whether a vertex
+is reducible are given first:
+• Rule I: Check whether a vertex v is unconfined or confined by the procedure
+in Definition 3.2, and if it is unconfined, remove vertex v directly from the
+graph.
+• Rule II: Use the procedure in Definition 3.6 to check whether a vertex v is cov-
+ered or uncovered, and if it is uncovered, include vertex v into the independent
+set and remove N[v] from the graph.
+16
+
+Before further introducing how to utilize the first type of CIT to design reduction
+rules, we first give an important property about simultaneous set mentioned in [27]:
+A simultaneous set S can be contracted by removing all vertices in set S from the
+graph and introducing a vertex v∗ such that it is adjacent to all vertices in N(S)
+with weight w(v∗) = w(S), while the weight of the MWIS in the remaining graph
+remain unchanged.
+Next, we will design reduction rules on simultaneous set through the first type of
+CIT, and give the following definitions by the results of the simultaneous set given
+in (b) of Corollary 3.3 and (b) of Corollary 3.7.
+Definition 4.1. Let u, v be two vertices in the graph.
+(a) Suppose vertices u and v be two confined vertices with confining set Su and Sv.
+If u ∈ Sv and v ∈ Su, then set {u, v} is called a confining simultaneous set.
+(b) Assume that vertices u and v be two covered vertices with covering set Cu and
+Cv. Set {u, v} is called a covering simultaneous set if u ∈ Cv and v ∈ Cu.
+From Definition 4.1, we have the following rules:
+• Rule III: If there are two confined vertices that constitute a confining simul-
+taneous set, then merge them.
+• Rule IV: Merge two covered vertices u and v if they form a covering simulta-
+neous set.
+Next, we will describe how to integrate our reduction rules into an existing re-
+duction algorithm—Reduce proposed by [27]. Reduce consists of seven steps. The
+reduction rules used in these steps exploit the sufficient conditions that a vertex is
+reducible. It executes these steps incrementally, which means that the next step is
+only executed when all previous steps are no longer applicable. Thus, if the graph
+is changed, it will go back to the first step. Notably, our reduction rules I and III
+are further generalization of the reduction rules used in step 5 of Reduce. So, we
+can combine our reduction rules I and III into one step to replace step 5 in Reduce
+and label this step as Remove Unconfined & Contract Confining. Similarly,
+we can also integrate our reduction rules II and IV into another new step in the
+reduction algorithm, called Remove Uncovered & Contract Covering.
+• Remove Unconfined & Contract Confining: Check whether a vertex is
+unconfined or confined. If it is confined, apply Rule I to remove it; If not, use
+Rule III to contract the corresponding confining simultaneous set when it can
+be found.
+17
+
+• Remove Uncovered & Contract Covering: If a vertex is checked to be
+uncovered, use Rule II to reduce it. Otherwise, if the corresponding covering
+simultaneous set can be found, use Rule IV to merge it.
+Figure 4: Casual Reduce: Given an input graph G, each step of the algorithm
+is executed sequentially and the graph changes, immediately go back to the first
+step. When all steps are completed and the graph no longer changes, return to the
+remaining graph kernel.
+Thus, a new reduction algorithm called Causal Reduce can be obtained by using
+Remove Unconfined & Contract Confining to replace step 5 of Reduce and
+adding Remove Uncovered & Contract Covering between Remove Uncon-
+fined & Contract Confining and step 6 of Reduce, which is shown in Figure 4.
+We will use Causal Reduce(G) = (K, c) to represent the processing of this algo-
+rithm on a given input graph G. The processing result of this algorithm consists of
+two parts: One is the remaining graph called kernel K and the other is the weight of
+the vertex set contained in the MWIS obtained by inference. It’s worth noting that
+the reduction algorithm Causal Reduce may not resolve all instances directly, but
+it can be used as a preprocessing for heuristic and exact algorithm.
+4.2
+The Causal B&R Solver
+Before introducing how to integrate our CITs into B&R algorithm, we briefly intro-
+duce the state-of-the-art exact algorithm Solve proposed by [27]. Solve is based on
+the idea of B&R algorithm, which first apply reduction algorithm Reduce to reduce
+the instance. Then, apply branching rule by virtue of the property of the confining
+set and perform reduction algorithm Reduce in every branch of the search tree to
+18
+
+Step 4 of Reduce
+Remove Unconfined & Contract Confining
+Step 3 of Reduce
+Remove Uncovered& Contract Covering
+Graph Change
+Step 2 of Reduce
+Step 6 of Reduce
+Step 1 of Reduce
+Step 7 of Reduce
+G = (V, E, w)
+Kernel Kfind a solution. During the searching, it uses a standard technique based on finding
+upper and lower bounds to prune the search tree and take the best solution weight
+Wb currently found in the algorithm as the lower bound. Initially, let Wb be the
+weight of the solution obtained by heuristic algorithm on the kernel K, and update
+Wb once a better solution is obtained in the algorithm. The heuristic algorithm,
+denoted by Greedy(G), is a greedy algorithm that iteratively selects a vertex in
+order of some measure and removes its closed neighbor set from the graph. In each
+searching branch, it uses a heuristic method to find an upper bound Wub of the op-
+timal solution weight of the current graph, which is based on weight clique covers
+and is denoted by UpperBound(G). If the current best solution weight Wb is not
+smaller than Wub, then there is no better solution in this searching branch and it can
+be discarded directly.
+Algorithm 1 The Causal B&R Solver(G)
+Require: A vertex weight graph G = (V, E, w);
+Ensure: The weight of a MWIS of G.
+1: Initialization of global variable Wb: Wb ← 0;
+2: if weight packing constraints have been created then
+3:
+while True do
+4:
+(K, c) ← Causal Reduce(G);
+5:
+check constraints();
+6:
+if existence constraints are not satisfied then
+7:
+return Wb;
+8:
+else if graph is simplified then
+9:
+continue;
+10:
+else
+11:
+break;
+12: else
+13:
+(K, c) ← Causal Reduce(G);
+14: Wb ← max{Wb, c + Greedy(K)};
+15: if c + UpperBound(G) ⩽ Wb then
+16:
+return Wb;
+17: Pick up a vertex v of maximum degree and compute the confining set Sv and the inferred covering set ICv;
+18: create weight packing constraint (3.1) and Wb ← max{Wb, c + w(Sv) + Causal B&R Solver(K − N[Sv])};
+19: create weight packing constraint (3.2) and Wb ← max{Wb, c + Causal B&R Solver(K − ICv)};
+20: return Wb;
+Our CITs will be integrated into two parts of Solve, resulting in a new exact al-
+gorithm called Causal B&R Solver. The first part is that we will use our reduction
+algorithm Causal Reduce to reduce the instance to get the kernel K, and perform
+the reduction algorithm on each branch of the search tree. The second part is that
+we will make use of the state-preserving results of two types of CITs during the
+branching process. Similar to the idea of Solve in [27], using property of confining
+set to the branching process, when choosing a vertex with the maximum degree to
+branch, the state-preserving results of (b) of Corollary 3.11 and (b) of Corollary 3.3
+19
+
+will be used in this part. This means that during branching, we either remove the
+inferred covering set of the branching vertex from the graph or include the confining
+set of the branching vertex into the independent set. Furthermore, we will create
+weight packing constraint (3.2) while removing the inferred covering set of branching
+vertex. Similarly, we will also create weight packing constraint (3.1) when includ-
+ing the confining set of branching vertex into the independent set. We will keep and
+manage these weight packing constraints when searching for solutions in each branch
+of the search tree. Specifically, another step called check constraints is added after
+the last step of Casual Reduce. In this step, for each weight packing constraint, we
+will check whether the constraint holds and whether the graph can be simplified by
+the causal inference properties of that constraint. If any constraint is violated, the
+searching branch will be skipped. If the graph can be simplified, Causal Reduce
+will continue to execute after reducing the graph. If none of the above conditions are
+met, the subsequent process will be performed. The main steps of Causal B&R
+Solver are listed in Algorithm 1.
+4.3
+The Causal Search
+After taking our reduction algorithm Causal Reduce as a preprocessing, we ap-
+ply the state-preserving result of second type of CIT to the local search process
+of heuristic algorithm DynWVC2 [6] to solve the complementary problem of the
+MWIS problem—the MWVC problem, which leads to a new algorithm called Causal
+Search.
+Algorithm 2 The basic framework of DynWVC2 algorithm.
+Require: A vertex weight graph G = (V, E, w), the cutoff time of the running T;
+Ensure: A vertex cover of G.
+1: V C ← Construct();
+2: V C∗ ← V C;
+3: while elapsed time<T do
+4:
+R ← RemoveVertices(VC)
+5:
+while some edge is uncovered by V C do
+6:
+choose a vertex v from N(R);
+7:
+V C ← V C ∪ {v};
+8:
+remove redundant vertices from V C;
+9:
+if w(V C)<w(V C∗) then
+10:
+V C∗ ← V C
+The DynWVC2 algorithm proposed by [6], is the state-of-the-art heuristic algo-
+rithm for solving MWVC problem. The basic framework of this algorithm is shown in
+Algorithm 2. The local search process of this algorithm mainly consists of a removing
+phase and an adding phase, and the specific process can be found in [6].
+20
+
+Our CITs will be considered in the removing phase of the algorithm — Re-
+moveVertices function. In this function, there are two scoring functions loss and
+valid score used to select the vertices to remove from the vertex cover V C. The
+specific definition of these two scoring functions can be seen in [6]. The loss and
+valid score functions have fundamentally different effects on the behavior of the al-
+gorithm. Vertex selection using loss function is an “exploratory” selection; in other
+words, it is quite possible that such a chosen vertex is good for the quality of the
+solution, but this cannot be determined. Different from “exploratory” vertex selec-
+tion, valid score is a “deterministic” selection, that is, we can determine whether
+removing a vertex will have a positive impact on the quality of the solution. For
+example, if a vertex has a negative valid score value, this means that after removing
+this vertex and adding its adjacent uncovering vertices, a vertex cover with lower
+weight than the current vertex cover can be obtained [6].
+In removing phase, the vertex with the minimum loss is removed from vertex
+cover V C first, and then the second removed vertex is selected by a dynamic vertex
+selection strategy. The details of dynamic vertex selection strategy can be learned
+in [6]. After removing the two vertices, if the total degree of the removed vertices
+does not reach a predetermined value (which is set to 2 times average degree of
+the graph), another vertex to be selected with the BMS strategy [5], which samples
+t (t = 50) vertices from vertex cover V C and chooses the one with the minimum
+loss, will be removed to expand the search region. In this way, it solves the problem
+that when removing two vertices the resulting search area is too small and limits the
+ability of the adding phase to find better local optima. If the search area obtained
+by removing two vertices is large enough, in order to balance the search time and
+search quality, the third vertex will not be selected for removing.
+The state-preserving result of second type of CIT will be applied to the dynamic
+vertex selection strategy for selecting the second vertex to be removed. The dynamic
+vertex selection strategy consists of a primary vertex scoring function valid score
+and a secondary scoring function loss. When the removed vertex v is selected by
+valid score function, it can be seen from the nature of the valid score function:
+There is a high probability that there exists a MWIS I containing it. If the vertex v
+is indeed included in I, by (a) of Corollary 3.11, the corresponding inferred confining
+set ISv also contained in I. Inspired by this result, when selecting the second removed
+vertex v by scoring function valid score, we will remove the vertices in the inferred
+confining set ISv from the vertex cover V C. In this way, the search region can be
+expanded and the number of times to continue to use the third removed vertex to
+expand the search area is reduced, which means that the ability of local search to
+find better local optima is improved. An example of our CITs applied to the vertex
+21
+
+removing process is presented in Figure 5.
+Figure 5: Example of our CITs applied to the vertex removing process: When we uti-
+lize valid score to select the removed vertex c from the vertex cover V C = {a, c, d},
+we can compute the corresponding inferred confining set ISc = {a, c, d} of vertex c
+and remove set ISc from vertex cover V C.
+In addition, it can be seen from the calculation process of Definition 3.9 about
+the inferred confining set: the computational complexity of ISv for each vertex v is
+O(|N(ISv)||ISv|). This means that in the actual application process, since the size
+of the generally obtained inferred confining set is relatively small, its computational
+cost is very small. Thus, our CITs is helpful for improving the performance of local
+search process.
+5
+Experiments
+We will conduct four experiments to verify the effect of integrating our CITs into
+current algorithmic frameworks. The first experiment is used to analyze the impact
+of our CITs for the reduction algorithm. The examination of the performance gain
+of our CITs in the B&R algorithm is shown in the second experiment. The third
+experiment is used to test the ability of our Causal Reduce as a preprocessing
+to improve the performance of the heuristic algorithm. The last experiment is con-
+ducted to verify the effect of adding our CITs to the local search process of heuristic
+algorithm.
+Experiment environment Setup. All of our algorithms are implemented in C++,
+and compiled by g++ with ‘-O3’ option. All experiments are run on a platform with
+128G RAM and one Intel(R) Xeon(R) Gold 5117 CPU @ 2.00GHz.
+Compared Algorithms. In previous studies, most of them only use some sim-
+ple rules as preprocessing to reduce problem instances, and do not pay attention to
+the performance of preprocessing. Two recent papers [15, 27] have studied in depth
+the reduction rules for the MWIS and analyzed their performance. Since the algo-
+rithm Reduce in [27] outperforms the algorithm in [15] and our Causal Reduce
+22
+
+a
+a
+6
+Compute the corresponding
+5
+inferred confining set
+12
+12
+5
+b
+C
+b
+4
+e
+eis obtained by integrating our CITs into Reduce, in this paper, we only use it as a
+baseline to analyze the impact of our CITs for the reduction algorithm. Additionally,
+in order to fully understand the role of different CITs on the reduction algorithm, we
+control the application of CITs in Reduce and conduct comparative experiments.
+Similar to the Causal Reduce shown in Figure 4, we use Re-Confin to represent
+the algorithm obtained after replacing the step 5 of Reduce with Remove Uncon-
+fined & Contract Confining and Re-Cover to denote the algorithm obtained
+by adding Remove Uncovered & Contract Covering between step 5 and 6 of
+Reduce.
+On the basis of the reduction algorithm Reduce, the authors of [27] also de-
+veloped a fast exact algorithm Solve, which is the state-of-art exact algorithm in
+previous work, and our Causal B&R Solver is obtained by applying our CITs
+into it, so it will be used as a baseline to verify the performance improvement of
+our CITs for the B&R algorithm. Furthermore, we use Solve-CR to identify the
+algorithm obtained by replacing Reduce with Causal Reduce in Solve, Solve-
+CR-IC refers to the algorithm obtained by further simplifying the branch by using
+the inferred covering set of the branching vertex in the branching process on the basis
+of Solve-CR, and Solve-Packing to represent the algorithm obtained by applying
+our weight packing constraints to the branching process of Solve. We will conduct
+comparative experiments on these algorithms to clarify the impact of different CITs
+on the B&R algorithm.
+Two state-of-the-art heuristic algorithms FastWVC (Fast) [7] and DynWVC2 (Dyn) [6]
+will be used to verify that our Causal Reduce as preprocessing improves the per-
+formance of the heuristic algorithm. We will use Causal Re + Fast and Causal
+Re + Dyn to denote applying our Causal Reduce as preprocessing before execut-
+ing FastWVC and DynWVC2. In addition, to further verify the superiority of our
+Causal Reduce as preprocessing for improving the performance of the heuristic al-
+gorithm, we also conduct comparative experiments using Reduce as a preprocessing
+of the heuristic algorithm. Likewise, we use Re + Fast and Re + Dyn to indicate
+the application of the previous reduction algorithm Reduce before FastWVC and
+DynWVC2 are executed. Moreover, our Causal Search is obtained by integrating
+CITs into the local search process of DynWVC2. Therefore, we can verify the effect
+of this operation by comparing DynWVC2 with Causal Search.
+Instances. We evaluate all algorithms on six real graphs which are most representa-
+tive and most difficult graphs from different domains. These graphs are downloaded
+from Network Data Repository [21]. All of them have 100 thousands to millions of
+vertices, and dozens of millions of edges. These instances become popular in recent
+works for the MWIS problem. Statistics of these graphs are shown in Table 2. In our
+23
+
+experiment, the weight of each vertex in the graph will have two random allocation
+mechanisms ∗, which are commonly used in previous work [15, 27, 6, 7]. The first
+allocation mechanism is that the weight of each vertex in the graph is obtained from
+[1, 200] uniformly at random, we will number the six datasets with 1−6. The second
+allocation mechanism is that the weight of each vertex in the graph follows a random
+uniform distribution of [20, 100], and 7 − 12 will be used to number the six datasets.
+inf-road-usa
+soc-livejournal
+sc-ldoor
+tech-as-skitter
+sc-msdoor
+inf-roadNet-CA
+Vertices
+23947347
+4033137
+952203
+1694616
+415863
+1957027
+Edges
+28854312
+27933062
+20770807
+11094209
+9378650
+2760388
+NO.
+1, 7
+2, 8
+3, 9
+4, 10
+5, 11
+6, 12
+Table 2: All graphs are sorted in descending order regarding the number of edges.
+In the row headed by “NO.”, each number is used to represent the corresponding
+number of the dataset generated by the graph according to the corresponding vertex
+weight allocation mechanism.
+5.1
+Impact of CITs on Reduction Algorithm
+We first analyze the impact of our CITs for the reduction algorithm and evaluate the
+performance of all reduction algorithms by measuring the running time, the size of
+the remaining graphs (kernel size), and the ratio of the kernel size to the number of
+vertices in the original graph (We simply refer to it here as the ratio for convenience.).
+The experimental results of all algorithms are output in Table 3. We can know
+that all reduction algorithms can significantly simplify the graph, and even reduce
+the graph to less than 0.1% of the original size. Besides, we can see that our Causal
+Reduce achieves best reduction effect in all datasets, that is, our Causal Reduce
+results in a much smaller kernel size than other algorithms. Moreover, compared
+with Reduce, Re-Confin can achieve better reduction effect in all datasets, while
+Re-Cover has basically no performance improvement. This shows that replacing the
+step 5 of Reduce with Remove Unconfined & Contract Confining plays a key
+role in improving the performance of the reduction algorithm, and combined with
+Remove Uncovered & Contract Covering, the performance of the reduction
+∗All datasets obtained through these two random assignment mechanisms can be found at http:
+//lcs.ios.ac.cn/~caisw/graphs.html.
+24
+
+Reduce
+Re-Confin
+Re-Cover
+Causal Reduce
+NO.
+|V |
+Time(S)
+Kernel Size
+Ratio(%)
+Time(S)
+Kernel Size
+Ratio(%)
+Time(S)
+Kernel Size
+Ratio(%)
+Time(S)
+Kernel Size
+Ratio(%)
+1
+23947347
+72.00
+431891
+1.80
+71.66
+428137
+1.79
+74.03
+431888
+1.80
+55.88
+275082
+1.15
+2
+4033137
+12.93
+7273
+0.18
+16.57
+5960
+0.15
+16.20
+7261
+0.18
+17.92
+3281
+0.08
+3
+952203
+3.08
+6492
+0.68
+3.89
+2780
+0.29
+3.89
+6447
+0.68
+3.84
+1682
+0.18
+4
+1694616
+1.78
+5904
+0.35
+2.32
+5613
+0.33
+2.26
+5909
+0.35
+5.03
+3974
+0.23
+5
+415863
+1.66
+6570
+1.58
+1.84
+3166
+0.76
+2.04
+6570
+1.58
+1.86
+2162
+0.52
+6
+1957027
+76.14
+305470
+15.61
+63.65
+300135
+15.34
+76.57
+305470
+15.61
+41.12
+202885
+10.37
+7
+23947347
+117.54
+437993
+1.83
+94.52
+434606
+1.81
+113.07
+438004
+1.83
+58.02
+235243
+0.98
+8
+4033137
+11.68
+7620
+0.19
+15.90
+6371
+0.16
+15.65
+7576
+0.19
+23.38
+3806
+0.09
+9
+952203
+8.05
+29116
+3.06
+5.08
+7906
+0.83
+8.88
+29116
+3.06
+5.06
+4628
+0.49
+10
+1694616
+1.94
+6999
+0.41
+2.55
+6623
+0.39
+2.53
+6991
+0.41
+5.19
+4429
+0.26
+11
+415863
+4.09
+24736
+5.95
+2.75
+10098
+2.43
+4.24
+24736
+5.95
+2.66
+7081
+1.70
+12
+1957027
+19.01
+131498
+6.72
+17.00
+128637
+6.57
+19.54
+131498
+6.72
+8.64
+65124
+3.33
+Table 3: Impact of CITs for the the reduction algorithm. The bold and underlined
+numbers are the minimum kernel size and shortest running time, respectively.
+algorithm will be greatly improved, but only adding Remove Uncovered & Con-
+tract Covering can hardly improve the performance of the reduction algorithm.
+More notably, our Causal Reduce take less time than other algorithms on half
+of the datasets. On the rest of the datasets, our Causal Reduce only takes a few
+seconds longer than other algorithms. These phenomena show that integrating our
+CITs into the reduction algorithm can significantly improve the performance of the
+algorithm, but the increase in time cost is very small, and they can even reduce the
+time cost.
+5.2
+Performance Gain of CITs on the B&R Algorithm
+We will examine the performance gain of our CITs for B&R algorithm. The running
+time bound is set as 1, 000 seconds for all algorithms, and if the algorithm cannot
+find the optimal solution within the time bound, the best solution found in all search
+branches is output.
+We output the numerical results and running times of all algorithms in Table 4. It
+can be seen from Table 4 that Solve-CR and Solve-CR-IC, like our Causal B&R
+Solver, can obtain the optimal solution in five data sets, while Solve-Packing,
+like Solve, can only obtain the optimal solution in one data set. In addition, on
+those datasets where the optimal solution cannot be solved within 1000 seconds, our
+Causal B&R Solver can basically obtain better numerical solutions than Solve-
+CR-IC, and Solve-CR-IC can obtain numerical results that are slightly better than
+Solve-CR, while Solve-Packing can generally get better numerical solutions than
+Solve. These results demonstrate that our reduction algorithm, Causal Reduce,
+is critical for the B&R algorithm to obtain optimal solutions on more datasets.
+25
+
+Solve
+Solve-CR
+Solve-Packing
+Solve-CR-IC
+Causal B&R Solver
+NO.
+Time(S)
+Result
+Time(S)
+Result
+Time(S)
+Result
+Time(S)
+Result
+Time(S)
+Result
+1
+1000
+1380579565
+1000
+1380810330
+1000
+1380579506
+1000
+1380980673
+1000
+1380980749
+2
+1000
+232813323
+28.5058
+232828253
+1000
+232814520
+28.8413
+232828253
+25.4889
+232828253
+3
+3.8108
+10303506
+5.3508
+10303506
+4.7170
+10303506
+4.1507
+10303506
+3.9148
+10303506
+4
+1000
+124020452
+1000
+124020466
+1000
+124020706
+1000
+124021474
+1000
+124022398
+5
+1000
+3904552
+4.2820
+3916599
+1000
+3904544
+4.3919
+3916599
+4.33729
+3916599
+6
+1000
+100956490
+1000
+101259145
+1000
+100957090
+1000
+101259161
+1000
+101288073
+7
+1000
+798872105
+1000
+799021102
+1000
+798911163
+1000
+799021209
+1000
+799021209
+8
+1000
+134613130
+34.0139
+134621271
+1000
+134613130
+34.4151
+134621271
+32.1868
+134621271
+9
+1000
+7237240
+6.8162
+7273973
+1000
+7237411
+6.8372
+7273973
+6.9285
+7273973
+10
+1000
+71945454
+1000
+71944343
+1000
+71946049
+1000
+71944343
+1000
+71945241
+11
+1000
+2707746
+1000
+2743962
+1000
+2707846
+1000
+2743962
+1000
+2743962
+12
+1000
+61702804
+1000
+61818326
+1000
+61702794
+1000
+61819495
+1000
+61818234
+Table 4: Performance gain of CITs on B&R algorithm. The bold and underlined
+numbers are the best numerical results of all algorithms and the shortest running
+time of all algorithms to find the optimal solution, respectively.
+Moreover, both the inferred covering set of the branching vertex and the weighted
+packing constraints can help B&R algorithm find more promising branches and find
+better solutions.
+5.3
+Causal Reduce’s Improvement on Heuristic Algorithm
+Next, we will verify the superiority of our Causal Reduce as a preprocessing for
+improving the heuristic algorithm.
+Table 5 presents the running time (including
+preprocessing time) and numerical results. We find that the preprocessed heuristic
+algorithm with Causal Reduce usually stop execution after running for a short
+time, while the rest of the heuristic algorithms are allowed to run for 1000 seconds.
+Meanwhile, it can be observed from Table 5 that adding the reduction algorithm as
+preprocessing is obvious for improving the performance of the heuristic algorithm,
+and our Causal Reduce helps heuristics find better solutions on all instances in less
+time (essentially within 100 seconds) than Reduce. Thus, although our Causal Re-
+duce takes no more than 12 seconds longer than Reduce on half of the datasets (as
+can be known from the numerical results in Section 5.1), it can further reduce the size
+of remaining graph by more than 32.6%, which is critical for subsequent processing
+of the problem (also be mentioned in Section 5.2), so such processing time cost is
+worth it!
+26
+
+Fast
+Re + Fast
+Causal Re + Fast
+Dyn
+Re + Dyn
+Causal Re + Dyn
+NO.
+Time(S)
+Result
+Time(S)
+Result
+Time(S)
+Result
+Time(S)
+Result
+Time(S)
+Result
+Time(S)
+Result
+1
+1000
+1308864893
+1000
+1381212394
+250
+1381215439
+1000
+1310465732
+1000
+1381174854
+100
+1381178355
+2
+1000
+227121250
+1000
+232826881
+20
+232827816
+1000
+229769205
+1000
+232827891
+20
+232828157
+3
+1000
+10044429
+1000
+10302725
+5
+10303465
+1000
+10224463
+1000
+10303168
+5
+10303476
+4
+1000
+122468973
+1000
+124025600
+6
+124026219
+1000
+123179849
+1000
+124026286
+6
+124026433
+5
+1000
+3823908
+1000
+3916222
+3
+3916534
+1000
+3894401
+1000
+3916381
+3
+3916568
+6
+1000
+97155884
+1000
+101739247
+275
+101745579
+1000
+99122831
+1000
+101740261
+250
+101744463
+7
+1000
+756581612
+1000
+799087671
+150
+799111311
+1000
+757212666
+1000
+799058579
+70
+799083159
+8
+1000
+131848587
+1000
+134620719
+25
+134621064
+1000
+132970128
+1000
+134621142
+25
+134621249
+9
+1000
+7131770
+1000
+7273626
+6
+7273655
+1000
+7252153
+1000
+7273750
+6
+7273931
+10
+1000
+70966490
+1000
+71946839
+6
+71947488
+1000
+71459210
+1000
+71947516
+6
+71947590
+11
+1000
+2682982
+1000
+2748925
+25
+2748945
+1000
+2743648
+1000
+2748982
+10
+2749005
+12
+1000
+59619787
+1000
+61850413
+50
+61852628
+1000
+60802433
+1000
+61855209
+75
+61857313
+Table 5: A comparative experiment of the effect of Causal Reduce on improving
+the heuristic algorithm. The numbers in bold and underlined are the correspond-
+ing experimental results and running time (including preprocessing time) of each
+heuristic algorithm using our Causal Reduce as preprocessing, respectively.
+5.4
+Comparative Experiment on Causal Search
+On the basis of preprocessing the input graph with Causal Reduce, we will compare
+our Causal Search with DynWVC2 to verify the effect of adding CITs to the local
+search process of DynWVC2 algorithm. The running time for both algorithms (in-
+cluding pre-processing time) is set to 1000 seconds. To avoid randomness, we run
+each instance 5 times and record the mean and maximum values. Furthermore, in
+order to estimate the gap between the results obtained by these two algorithms and
+the MWIS, we need to calculate the upper bound of each instance. The upper bound
+for the 2nd, 3rd, 5th, 8th, 9th instance is nothing but the weight of the optimal so-
+lution obtained by Causal B&R Solver, and for the rest of the instances, it is
+obtained by applying the weighted clique cover method mentioned in Section 4.2 to
+the remaining graph obtained by Causal Reduce. Table 6 outputs the numerical
+results and the estimated gap. The small gaps there demonstrate that after prepro-
+cessing with our Causal Reduce, both algorithms can obtain numerical results very
+close to the optimal solution. In particular, for those instances where the optimal
+solution is obtained, their gap can basically reach 10−6 ∼ 10−7, and in the remaining
+instances, the estimated gap can basically reach 10−4 ∼ 10−2. Besides, from the
+mean and maximum values, our Causal Search can basically achieve better perfor-
+mance than DynWVC2, thereby implying that our CITs can help local search find
+better local optima.
+27
+
+Dyn
+Causal Search
+NO.
+Upper Bound
+Mean
+Gap
+Max
+Gap
+Mean
+Gap
+Max
+Gap
+1
+1384376268
+1381464698.6
+2.103 × 10−3
+1381467306
+2.101 × 10−3
+1381466033.6
+2.102 × 10−3
+1381470750
+2.099 × 10−3
+2
+232828253
+232828153.4
+4.278 × 10−7
+232828171
+3.522 × 10−7
+232828159.2
+4.029 × 10−7
+232828188
+2.792 × 10−7
+3
+10303506
+10303485.2
+2.019 × 10−6
+10303491
+1.456 × 10−6
+10303488.6
+1.689 × 10−6
+10303494
+1.165 × 10−6
+4
+124076790
+124026438.6
+4.058 × 10−4
+124026451
+4.057 × 10−4
+124026444.8
+4.058 × 10−4
+124026449
+4.057 × 10−4
+5
+3916599
+3916582.2
+4.289 × 10−6
+3916583
+4.085 × 10−6
+3916582.6
+4.187 × 10−6
+3916584
+3.830 × 10−6
+6
+103562461
+101846521.0
+1.657 × 10−2
+101848242
+1.655 × 10−2
+101847524.8
+1.656 × 10−2
+101849650
+1.654 × 10−2
+7
+800748442
+799264479.2
+1.853 × 10−3
+799265827
+1.852 × 10−3
+799266278.8
+1.851 × 10−3
+799267573
+1.849 × 10−3
+8
+134621271
+134621255.0
+1.189 × 10−7
+134621257
+1.040 × 10−7
+134621256.8
+1.055 × 10−7
+134621265
+4.457 × 10−6
+9
+7273973
+7273939.8
+4.564 × 10−6
+7273945
+3.849 × 10−6
+7273936.8
+4.977 × 10−6
+7273947
+3.574 × 10−6
+10
+71978922
+71947636.0
+4.347 × 10−4
+71947639
+4.346 × 10−4
+71947636.8
+4.346 × 10−4
+71947642
+4.346 × 10−4
+11
+2819343
+2749008.4
+2.495 × 10−2
+2749009
+2.495 × 10−2
+2749010.8
+2.495 × 10−2
+2749019
+2.494 × 10−2
+12
+62276875
+61860510.4
+6.686 × 10−3
+61860616
+6.684 × 10−3
+61860549.0
+6.685 × 10−3
+61860717
+6.682 × 10−3
+Table 6: Compare our Causal Search with the DynWVC2 algorithm. The bold
+and underlined numbers are better maximum and average values, respectively. In
+the column headed by “Upper Bound”, each number is the upper bound of the
+MWIS of the corresponding instance.
+6
+Conclusion and Outlook
+In this paper, we propose a series of causal inference techniques (CITs) for the max-
+imum weight independent set (MWIS) problem by fully exploiting the upper bound
+property of MWIS. After integrating our CITs, the performance of various existing
+algorithms, including the Branch-and-Reduce (B&R) algorithm and some heuristic
+algorithms, is significantly improved. We are now conducting theoretical analysis
+to find some guarantees on solution quality, developing strategies to help the B&R
+algorithm analyze the causes of conflicts and perform more efficient backtracking
+searches, and generalizing the proposed CITs to other combinatorial optimization
+problems.
+Acknowledgements
+This research was supported by the National Key R&D Program of China (Nos.
+2020AAA0105200, 2022YFA1005102) and the National Natural Science Foundation
+of China (Nos. 12288101, 11822102). SS is partially supported by Beijing Academy
+of Artificial Intelligence (BAAI). The authors would like to thank Professor Hao Wu
+for his useful discussions and valuable suggestions.
+28
+
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+31
+

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+⋆ manuscript No.
+(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.e. Morse-Smale diffeo-
+morphisms in dimension 3 whose chain recurrent set consists of four points: one
+source, one saddle and two sinks). Ch. Bonatti and V. 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. They also proved that there are
+infinitely many such classes and that any Hopf knot can be realized by a Pixton
+diffeomorphism. D. 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.e.
+non equivalent to the standard) Hopf knot is strictly larger than the chain recur-
+rent set of the diffeomorphism. The Lyapunov function for Pixton diffeomorphism
+with minimal number of critical points is called the quasi-energy function. In this
+paper we construct a quasi-energy function for Pixton diffeomorphisms defined by
+a generalized Mazur knot.
+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.
+T. Medvedev
+Laboratory of Algorithms and Technologies for Network Analysis; HSE University
+136 Rodionova Street, Niznhy Novgorod, Russia
+E-mail: mtv2001@mail
+O. Pochinka
+International Laboratory of Dynamical Systems and Applications; HSE University, 25/12 Bol-
+shaya Pecherckaya Street, Niznhy Novgorod, Russia
+arXiv:2301.02405v1  [math.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. For two given points x, y ∈ M n a sequence of points
+x = x0, . . . , 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. 1).
+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. 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). The chain recurrent set, denoted by Rf, is the set of all chain recurrent
+points of f. 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. This equivalence relation defines
+equivalence classes called chain components.
+If the chain recurrent set of a diffeomorphism f is finite then it consists of
+periodic points. 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. If absolute values of all these eigenvalues are greater (less) than 1 then
+p is called a sink (a source). Sinks and sources are called knots. If a periodic point
+is not a knot then it is called a saddle.
+Let p be a hyperbolic periodic point of a diffeomorphism f whose chain recur-
+rent set is finite. The Morse index of p, denoted by λp, is the number of eigen-
+values of Jacobian matrix whose absolute values are greater than 1. 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. Stable and unstable manifolds are called
+invariant manifolds. A connected component of the set W u
+p \ p (W s
+p \ p) is called
+a unstable (stable) separatrice of p.
+A diffeomorphism f : M n → M n is called a Morse-Smale diffeomorphism if
+1. its chain recurrent set Rf consists of finite number of hyperbolic points;
+2. for any two points p, q ∈ Rf the manifolds W s
+p , W u
+q intersect transversally.
+
+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;
+– ϕ(f(x)) = ϕ(x) if x ∈ Rf.
+Notice that every Morse-Smale diffeomorphism f has a Morse-Lyapunov func-
+tion 1, i.e. 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), . . . , xn(y)) ∈ Rn and
+φ−1
+p (Ox1 . . . xλp) ⊂ W u
+p , φ−1
+p (Oxλp+1 . . . xn) ⊂ W s
+p .
+(∗)
+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.
+The proof of existence of an energy Morse function for a Morse-Smale diffeo-
+morphism of the circle is an easy exercise. D. Pixton [15] in 1977 proved that
+every Morse-Smale diffeomorphism of a surface has an energy function. There he
+also constructed an example of a Morse-Smale diffeomorphism on the 3-sphere
+which admits no energy function. The obstacle to existence of an energy function
+in his example was the wild embedding of the saddle separatrices in the ambient
+manifold (i.e. the closure of the separatrice is not a submanifold of the ambient
+space). From [11] it follows that there are Morse-Smale diffeomorphisms with no
+energy function on manifolds of any dimension n > 2. 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. Notice
+that ρf is a topological invariant, i.e. 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′ .
+In this paper we give a lower estimate of ρf for Pixton diffeomorphisms. The
+class of Pixton diffeomorphisms P is defined in the following way. 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). Notice that Pixton’s example is a diffeomorphism of this class. 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.e. the knot in the generating class
+of the fundamental group of the manifold S2 × S1 (see Proposition 1). Moreover,
+any Hopf knot can be realized as a Pixton diffeomorphism.
+Recall that a knot in S2 × S1 is a smooth embedding γ : S1 → S2 × S1 or the
+image of this embedding L = γ(S1). Two knots L, L′ are said to be equivalent if
+there is a homeomorphism h : S2 × S1 → S2 × S1 such that h(L) = L′. Two knots
+1 This function can be constructed, for example, by suspension. Consider the topological
+flow ˆft on the manifold Mn × R defined by ˆft(x) = x + t. 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.
+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))). The flow ft is called the suspension over f. By construction the
+chain recurrent set of ft consists of the finite number of periodic orbits δi = pW (Oi × R), i ∈
+{1, . . . , kf} and this means that the suspension ft is a Morse-Smale flow. A Lyapunov function
+for these flows is constructed in [12]. Then the restriction of this function on M is the desired
+Lyapunov function for f.
+
+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. If Γ|S1×{t} is an
+embedding for every t ∈ [0, 1] then the knots are said to be isotopic.
+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. B. 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. 2).
+It follows from the results of [1] that there exists a countable family of pairwise
+Fig. 2 Two non-isotopic and non equivalent Hopf knots L0 and LM: a) the standard Hopf
+knot L0; b) the Mazur knot LM
+non-equivalent Hopf knots Ln, n ∈ N which are generalized Mazur knots (Fig. 3).
+Fig. 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.e. equivalent to the standard one). 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.
+The main result of this paper is the proof of Theorem 1.
+Theorem 1 Let f be a Pixton diffeomorphism (f ∈ P) and let Ln, n ∈ N be
+its knot. Then the number ρf of critical points of a quasi-energy function of f is
+calculated by2
+ρf = 4 + 2n.
+2 Construction of Pixton diffeomorphisms
+In dynamics a wild Artin-Fox arc was for the first time introduced by D. 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. We give the
+modern construction of these diffeomorphisms following Ch. Bonatti and V. Grines
+[2] where Pixton diffeomorphisms were also classified (see also [8], [11]).
+For x = (x1, . . . , xn) ∈ Rn denote ||x|| =
+�
+x2
+1 + · · · + x2n. Let h : R3 → R3
+be the diffeomorphism defined by h(x1, x2, x3) =
+� x1
+2 , x2
+2 , x3
+2
+�
+. Define the map
+p : R3 \ O → S2 × S1 by
+p(x1, x2, x3) =
+� x1
+||x||, x2
+||x||, log2(||x||)
+(mod 1)
+�
+.
+Let L ⊂ (S2 ×S1) be a Hopf knot and let U(L) be its tubular neighborhood. 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. 4).
+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).
+Then there is a diffeomorphism ζ : U(L) → C that conjugates h|U(L) and g = g1|C.
+Define the flow φt on C by:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+˙x1 =
+�
+1 − 1
+9(x2
+1 + x2
+2 + x2
+3 − 4)2,
+x2
+1 + x2
+2 + x2
+3 ⩽ 4
+1,
+x2
+1 + x2
+2 + x2
+3 > 4
+˙x2 =
+�
+�
+�
+�
+�
+x2
+2
+�
+sin
+� π
+2
+�
+x2
+1 + x2
+2 + x2
+3 − 3
+��
+− 1
+�
+,
+2 < x2
+1 + x2
+2 + x2
+3 ⩽ 4
+−x2,
+x2
+1 + x2
+2 + x2
+3 ⩽ 2
+0,
+x2
+1 + x2
+2 + x2
+3 > 4
+˙x3 =
+�
+�
+�
+�
+�
+x3
+2
+�
+sin
+� π
+2
+�
+x2
+1 + x2
+2 + x2
+3 − 3
+��
+− 1
+�
+,
+2 < x2
+1 + x2
+2 + x2
+3 ⩽ 4
+−x3,
+x2
+1 + x2
+2 + x2
+3 ⩽ 2
+0,
+x2
+1 + x2
+2 + x2
+3 > 4.
+By construction the diffeomorphism φ = φ1 has two fixed points: the saddle
+P(1, 0, 0) and the sink Q(−1, 0, 0) (Fig. 5), both being hyperbolic. One unstable
+2 For n = 1 Theorem 1 is proved in [7].
+
+6
+Timur Medvedev, Olga Pochinka
+L
+L
+_
+U(L)
+_
+Fig. 4 Suspension of a Hopf knot
+1
+-1
+2
+O
+Fig. 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
+�
+.
+Notice that φ coincides with the diffeomorphism g = g1 outside the ball {(x1, x2, x3) ∈
+C : x2
+1 + x2
+2 + x2
+3 ⩽ 4}.
+Define the diffeomorphism ¯fL : R3 → R3 so that ¯fL coincides with h outside
+U(L) and it coincides with ζ−1φζ on U(L). Then ¯fL has in U(L) two fixed points:
+the sink ζ−1(Q) and the saddle ζ−1(P), both being hyperbolic. The unstable
+separatrice of the saddle ζ−1(P) lies in L (Fig. 6).
+Now project the dynamics onto the 3-sphere. Denote by N(0, 0, 0, 1) the North
+Pole of the sphere S3 = {x = (x1, x2, x3, x4) : ||x|| = 1}. For every point x ∈ (S3 \
+{N}) there is the unique line passing through N and x in R4. This line intersects
+
+Quasi-energy function for Pixton diffeomorphisms
+7
+R
+Fig. 6 The phase portrait of the diffeomorphism ¯fL
+R3 in exactly one point ϑ+(x) (Fig. 7). The point ϑ+(x) is the stereographic
+projection of the point x. One can easily check that
+ϑ+(x1, x2, x3, x4) =
+�
+x1
+1 − x4 ,
+x2
+1 − x4 ,
+x3
+1 − x4
+�
+.
+Thus, the stereographic projection ϑ+ : S3 \ {N} → R3 is a diffeomorphism.
+N
+x
+��(x)
+Fig. 7 The stereographic projection.
+By construction ¯fL coincides with h in some neighborhood of the point O and
+in some neighborhood of the infinity. Therefore, it induces on S3 the Morse-Smale
+fL(x) =
+�
+ϑ−1
++ ( ¯fL(ϑ+(x))), x ̸= N;
+N, x = N
+.
+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. We say the constructed diffeomorphism to
+be model and it is of Pixton class.
+
+8
+Timur Medvedev, Olga Pochinka
+Proposition 1 ([2])
+– Any diffeomorphism f ∈ P is topologically conjugate to some model diffeomor-
+phism fL.
+– Two model diffeomorphisms fL, fL′ are topologically conjugate if and only if
+their knots L, L′ are equivalent.
+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.
+Let L be a Hopf knot and let ¯L = p−1(L) be its cover in R3\O. 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. The minimally possible genus gL of the secant surface
+is called the genus of the knot L. The secant surface of L of genus gL is said to be
+minimal.
+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.e. 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.
+Proof Let Σ be a minimal secant surface of L and let ¯y be the unique point of
+the intersection ¯L ∩ ¯Σ. Assume the opposite: the surface ¯Σ \ ¯y is compressible in
+R3 \ (O ∪ ¯L). 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]). Then we have two possibilities:
+(int D) ∩
+� �
+k∈Z
+hk( ¯Σ)
+�
+= ∅,
+(1)
+(int D) ∩
+� �
+k∈Z
+hk( ¯Σ)
+�
+̸= ∅.
+(2)
+In case (1) two subcases are possible: (1a) D ⊂ QΣ, (1b) D ⊂ (R3 \ int QΣ). For
+case 1a) let N(D) ⊂ QΣ be a tubular neighborhood of the disk D. Then exactly
+one connected component of the set QΣ \ int N(D) intersects ¯L. According to (1)
+this component is h-compressible and its boundary intersects ¯L at a unique point.
+The projection of this boundary into S2 × S1 is, therefore, the secant surface of
+L of genus less than gL. This contradicts the fact that the surface Σ is minimal.
+In case 1b) let N(D) ⊂ (R3 \ int QΣ) be a tubular neighborhood of D. Then due
+to (1) the set QΣ ∪ N(D) is h-compressible and its boundary intersects ¯L at a
+unique point. 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.
+
+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. Let γ be a curve from Γ. 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. Consider this innermost curve γ ⊂ f k(Σ). There are
+two subcases: a) γ is essential on f k(Σ) and b) γ is contractible on f k(Σ). 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. 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. Consider: b1) Bγ ⊂ f k(QΣ) and b2)
+Bγ ⊂ (R3 \ int f k(QΣ)). For b1) let N(Bγ) ⊂ f k(QΣ) be a tubular neighborhood
+of Bγ. 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.
+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. We get the same result for b2) for the set QΣ ∪ N(Bγ). Thus, iterating the
+process we come either to the case a) or to the case (1) and get a contradiction.
+Lemma 2 For any diffeomorphism f ∈ P the following estimation holds
+ρf ⩾ 4 + 2gLf .
+(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. Denote by ℓ the non-stable separatrice
+of the saddle σ lying in the basin of the sink S. 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. 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.
+Consider an arbitrary Morse-Lyapunov function ϕ : S3 → R of the diffeomor-
+phism fL. To be definite let ϕ(S) = 0, ϕ(σ) = 1 and ϕ(N) = 3. From the definition
+of the Morse-Lyapunov function it follows that ϕ|ℓ monotonically decreases in some
+neighborhood of the saddle σ. 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. Denote this
+point by w1.
+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. Since ϕ decreases along
+the trajectories of f, the values of ϕ on W s
+σ are greater than 1. Therefore, the
+manifold ¯Q1 lies in the manifold W s
+S diffeomorphic to R3. Let the function ϕ| ¯
+Q1
+have kq, q ∈ {0, . . . , 3} critical points of index q. Due to [5, Theorem 6.1] 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. 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. Then the genus
+of any surface of the set ∂ ¯Q1 cannot be greater than g1.
+On the other hand, the number of critical points of ϕ| ¯
+Q1 is not less than k0+k1.
+If k0 ⩾ 1 and g1 = 1 + k1 − k0 then one gets k0 + k1 = g1 + 2k0 − 1 ⩾ g1 + 1. Thus,
+ϕ| ¯
+Q1 has at least g1 + 1 critical points.
+Denote by ¯Σ1 the connected component of ∂ ¯Q1 which intersects the separatrice
+ℓ. Then the surface ¯Σ1 divides the manifold W s
+S ∼= R3 into two parts, one of which
+Q1 being an h-compressible body. This means that Σ1 = pS( ¯Σ1) is the secant
+surface of L and, therefore,
+g1 ⩾ gL.
+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. 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. Denote by ¯Σ2
+the connected component of ∂ ¯Q2 which intersects W s
+σ and denote by g2 its genus.
+The surface ¯Σ2 divides the manifold W u
+N ∼= R3 into two parts, one of which Q2
+being an h−1-compressible body. Arguing as above one comes to conclusion that
+the number of critical points of ϕ|Q2 is at least g2 +1. Therefore, the total number
+of critical points of ϕ is greater or equal to
+g1 + 3 + g2 + 1 ⩾ 4 + 2g1 ⩾ 4 + 2gLf .
+4 The generalized Mazur knot Ln
+In this section we show that the genus gLn of a generalized Mazur knot equals n.
+At first we give a detailed description of construction of Ln.
+4.1 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)
+�
+.
+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).
+
+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, . . . , α2n+1 numbered in counter-clockwise order (Fig. 8).
+Let ai, i ∈ {1, . . . , 2n} be the arc of the circle S1 bounded by αi, αi+1 whose inte-
+rior contains no points of {α1, . . . , α2n+1}. Let B, Ai ⊂ int K, i ∈ {1, . . . , 2n} be
+pairwise disjoint smooth arcs such that:
+1. the boundary points of B are α2n+1, h(α1); 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, . . . , n};
+2. 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;
+3. the arc cl(h(A1) ∪ A2 ∪ · · · ∪ h(A2n−1) ∪ A2n ∪ B) is smooth.
+Fig. 8 Construction of the knot Ln
+Let
+¯Ln =
+�
+k∈Z
+hk(B ∪ A1 ∪ · · · ∪ A2n), Ln = p(¯Ln).
+4.2 The genus of the knot Ln
+Lemma 3 The genus gLn of the knot Ln equals n.
+
+12
+Timur Medvedev, Olga Pochinka
+Fig. 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. 9).
+Now we show that gLn ⩾ n. To that end we prove that for Ln there exists a
+minimal secant surface Σ such that ¯Σ ⊂ K and ¯Ln ∩ ¯Σ = h(α1).
+Indeed, let Σ0 be some minimal secant surface of Ln. 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. 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).
+Denote by k+, k− ⩾ 0 the maximal integers for whichf k( ¯Σ0)∩ ¯Σ0 ̸= ∅, f −k( ¯Σ0)∩
+¯Σ0 ̸= ∅, k ⩾ 0, respectively. If k+ = k− = 0 then ¯Σ0 is the desired surface. 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.
+Notice that ¯Σ0 ∩ f k+(c2j−1) = ∅, j ∈ {1, . . . , n}. Without loss of generality let
+the intersection Γ =
+n�
+j=1
+f k+(d2j−1) ∩ ¯Σ0 be transversal. Let γ be a curve from
+Γ. Then γ bounds the unique disk Dγ ⊂ f k+(d2j−1). There are two possibilities:
+1) b2j−1 /∈ Dγ, 2) b2j−1 ∈ Dγ. 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. Consider this innermost curve γ. 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. Denote by Bγ ⊂ (R3 \(O∪ ¯Ln)) the 3-ball bounded by the 2-sphere
+Dγ ∪ dγ. Consider two subcases: 1a) Bγ ⊂ QΣ0 and 1b) Bγ ⊂ (R3 \ int QΣ0).
+In case 1a) let N(Bγ) ⊂ QΣ0 be a tubular neighborhood of the ball Bγ. 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. 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. For it the
+number of the connected components of the set Γ is less. One gets the same result
+in case 1b) for the set QΣ0 ∪ N(Bγ).
+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). Denote the
+resulting surface again by Σ0. Now the set Γ consists only of the curves γ bounding
+the disk Dγ ⊂ b2j−1 which contains the point b2j−1. Since (b2j−1 ⊔ c2j−1) ⊂
+(R3 \QΣ0), the number of these curves on the disk d2j−1 is even. 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. Denote by ˜κ ⊂ d2j−1 the annulus bounded by the same
+curves on the disk d2j−1. Let ˜Σ0 = ¯Σ0 \ κ ∪ ˜κ. Due to construction the surface
+˜Σ0 is of the same genus as the surface ¯Σ0 and it bounds an h-compressible body.
+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.
+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. Denote this surface again by
+Σ0. Without loss of generality let the intersections of the surface ¯Σ0 with the
+spheres f k(S2) be transversal. Denote by F the set of the connected components
+of the intersection f k+(K) ∩ ¯Σ0. Now we show the way to reduce by 1 the number
+of the components in F using isotopy of the secant surface.
+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, . . . , n}. Then Q is homeomorphic to the direct product
+of the 2-sphere with 2n + 1 deleted points and the segment. 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. Due to [16, Corollary 3.2] 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].
+Then we replace the part F of ¯Σ0 with ˜F. If we continue the process we get the
+desired secant surface Σ ⊂ K.
+Notice (see, for instance, [4, Exercise 2.8.1]) that the fundamental group π1(K\
+¯Ln) has 2n generators γ1, . . . , γ2n, each of which γi, i ∈ {1, . . . , 2n} being the
+generator of the punctured disk di \bi (Fig. 10). 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. 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. In the same way one finds the curves ˜γ2j ⊂
+( ¯Σ \ h(α1)) homotopic to the curves γ2j on the punctured disk d2j \ b2j (Fig. 10).
+Due to Lemma 1 the surface ¯Σ \ h(α1) is non-compressible in K \ ¯Ln. Then the
+curves ˜γ1, . . . , ˜γ2n are pairwise non-homotopic to the generators on the surface
+¯Σ \ h(α1). Therefore, the genus of the surface ¯Σ cannot be less than n.
+
+14
+Timur Medvedev, Olga Pochinka
+Fig. 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.
+Then its non-wandering set Ωf consists of four points: two sinks ω, S, a source N
+and a saddle σ. 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.
+Let ¯Σ be the surface of genus n bounding the handle-body QΣ of the same genus.
+Now we construct for f a Morse-Lyapunov function with 6 + 2n critical points.
+Our construction of a quasi-energy function is analogous to the construction
+of an energy function in [7].
+1. 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 . Let Bω, BS be the 3-balls which are the
+level sets of respective functions ϕω, ϕS and such that BS ⊂ int QΣ. 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. Denote by P + the smoothing of this body by addition of a small
+exterior collar.
+2. Due to [7, Section 4.3] 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. The disks d1, . . . , d2n−1 cut the
+handle-body QΣ making the 3-ball. Denote by BΣ the smoothing of this ball
+by removal of the interior collar. 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, . . . , d2n−1, while the value of ϕ on ∂BΣ is 1/3. Due to [7, Lemma
+
+Quasi-energy function for Pixton diffeomorphisms
+15
+4.2] the function ϕ can be extended to the ball BΣ by an energy function
+with the unique critical point S of Morse index 0. Since f(QΣ) ⊂ int BΣ, the
+constructed function decreases along the trajectories of the diffeomorphism f.
+3. It follows from the definition of the knot Ln that P − = S3 \ int P + is the
+handle-body of genus n. Moreover, the disks d2, . . . , d2n cut P − making the
+3-ball. Denote by B− smoothing of this ball by removal of the interior collar.
+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, . . . , d2n, while the
+value of ϕ on ∂B− is 5/3. According to [7, Lemma 4.2] the function ϕ can be
+extended to the ball B− by an energy function with unique critical point N
+of Morse index 3. 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.
+Conflict of interest
+The authors declare that they have no conflict of interest.
+References
+1. Akhmetiev, P., Medvedev, T., Pochinka, O.: On the number of the classes of topological
+conjugacy of Pixton diffeomorphisms. Qualitative Theory of Dynamical Systems 20(3),
+1–15 (2021)
+2. Bonatti, C., Grines, V.: Knots as topological invariants for gradient-like diffeomorphisms
+of the sphere S3. Journal of Dynamical and Control Systems 6(4), 579–602 (2000)
+3. Conley, C.: Isolated invariant sets and the morse index. American Mathematical Society,
+CBMS, Providence, RI 38 (1978)
+4. Daverman, R.J., Venema, G.: Embeddings in manifolds, vol. 106. American Mathematical
+Soc. (2009)
+5. Fomenko, A.: Differential Geometry and Topology: Additional Chapters. Moscow Univer-
+sity Press (1983)
+6. Grines, V., Laudenbach, F., Pochinka, O.: The energy function for gradient-like diffeomor-
+phisms on 3-manifolds. Doklady Mathematics 78(2), 702–704 (2008)
+7. Grines, V.Z., Laudenbach, F., Pochinka, O.V.: Quasi-energy function for diffeomorphisms
+with wild separatrices. Mathematical Notes 86(1), 163–170 (2009)
+8. Grines, V.Z., Medvedev, T.V., Pochinka, O.V.: Dynamical Systems on 2- and 3-Manifolds,
+Developments in Mathematics, vol. 46. Springer International Publishing (2016). DOI
+10.1007/978-3-319-44847-3
+9. Kirk, P., Livingston, C.: Knot invariants in 3-manifolds and essential tori. Pacific Journal
+of Mathematics 197(1), 73–96 (2001)
+10. Mazur, B.: A note on some contractible 4-manifolds. Annals of Mathematics 79(1), 221–
+228 (1961)
+11. Medvedev, T.V., Pochinka, O.V.: The wild Fox-Artin arc in invariant sets of dynamical
+systems. Dynamical Systems 33(4), 660–666 (2018). DOI 10.1080/14689367.2017.1421903.
+URL https://doi.org/10.1080/14689367.2017.1421903
+12. Meyer, K.R.: Energy functions for morse smale systems. American Journal of Mathematics
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+13. Milnor, J.: Morse theory.(am-51), volume 51.
+In: Morse Theory.(AM-51), Volume 51.
+Princeton university press (2016)
+14. Neumann, W.D.: Notes on geometry and 3-manifolds. Citeseer (1996)
+15. Pixton, D.: Wild unstable manifolds.
+Topology 16, 167–172 (1977).
+DOI 10.1016/
+0040-9383(77)90014-3
+16. Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Annals of Math-
+ematics pp. 56–88 (1968)
+

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+page_content='⋆ manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' Bonatti and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content=' Pochinka  International Laboratory of Dynamical Systems and Applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='02405v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Sinks and sources are called knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Stable and unstable manifolds are called  invariant manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content='  – ϕ(f(x)) = ϕ(x) if x ∈ Rf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , xn(y)) ∈ Rn and  φ−1  p (Ox1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' xn) ⊂ W s  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content=' Notice  that ρf is a topological invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' equivalent to the standard one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Bonatti and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='  For x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , xn) ∈ Rn denote ||x|| =  �  x2  1 + · · · + x2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  x2  1 + x2  2 + x2  3 ⩽ 4  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content='  x2  1 + x2  2 + x2  3 ⩽ 2  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content='  x2  1 + x2  2 + x2  3 ⩽ 2  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='  x2  1 + x2  2 + x2  3 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' 5), both being hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='  Now project the dynamics onto the 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  N  x  ��(x)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' 7 The stereographic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content='  N, x = N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Let γ be a curve from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' Consider this innermost curve γ ⊂ f k(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Denote this  point by w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , 3} critical points of index q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' Due to [5, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , α2n+1 numbered in counter-clockwise order (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='  Let ai, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , α2n+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' Let B, Ai ⊂ int K, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content='  12  Timur Medvedev, Olga Pochinka  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='  Now we show that gLn ⩾ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' Let γ be a curve from  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content=' Consider this innermost curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Denote the  resulting surface again by Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Let ˜Σ0 = ¯Σ0 \\ κ ∪ ˜κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content=' Denote this surface again by  Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content=' Due to [16, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  Notice (see, for instance, [4, Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , γ2n, each of which γi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' Then the  curves ˜γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content='  14  Timur Medvedev, Olga Pochinka  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' Due to [7, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' The disks d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' Moreover, the disks d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' , d2n cut P − making the  3-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+page_content=' According to [7, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+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'}
+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'}
+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'}
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+page_content='1421903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='  URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='1080/14689367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}
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+page_content=' 56–88 (1968)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQffgB5/content/2301.02405v1.pdf'}

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+Prepared for submission to JCAP
+Stochastic gravitational wave
+background from the collisions of dark
+matter halos
+Qiming Yana,b Xin Rena,b Yaqi Zhaoa,b Emmanuel N. Saridakisc,a,d
+aDeep Space Exploration Laboratory/School of Physical Sciences, University of Science and
+Technology of China, Hefei, Anhui 230026, China
+bCAS Key Laboratory for Researches in Galaxies and Cosmology/Department of Astronomy,
+School of Astronomy and Space Science, University of Science and Technology of China,
+Hefei, Anhui 230026, China
+cNational Observatory of Athens, Lofos Nymfon, 11852 Athens, Greece
+dDepartamento de Matem´aticas, Universidad Cat´olica del Norte, Avda.
+Angamos 0610,
+Casilla 1280 Antofagasta, Chile
+E-mail: asadoubi233@mail.ustc.edu.cn, rx76@mail.ustc.edu.cn,
+zxmyg86400@mail.ustc.edu.cn, msaridak@noa.gr
+Abstract.
+We investigate for the first time the effect of the dark matter (DM) halos colli-
+sions, namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung,
+on the stochastic gravitational wave background. We first calculate the gravitational wave
+signal of a single collision event, assuming point masses and linear perturbation theory. Then
+we proceed to the calculation of the energy spectrum of the collective effect of all dark matter
+collisions in the Universe. Concerning the DM halo collision rate we show that it is given
+by the product of the number density of DM halos, which is calculated by the extended
+Press-Schechter (EPS) theory, with the collision rate of a single DM halo, which is given by
+simulation results, with a function of the linear growth rate of matter density through cos-
+mological evolution. Hence, integrating over all mass and distance ranges, we finally extract
+the spectrum of the stochastic gravitational wave background created by DM halos collisions.
+As we show, the resulting contribution to the stochastic gravitational wave background is
+of the order of hc ≈ 10−30 in the pulsar timing array (PTA) band of f ≈ 10−9Hz, much
+smaller than other GW sources, such as super-massive black-hole mergers. However, in very
+low frequency band, it is larger. With current observational sensitivity it cannot be detected,
+nevertheless it may be accessible by PTA in the future, where techniques of distinguishing
+signal overlap should be used in order to isolate it and use it for cosmological studies.
+arXiv:2301.02414v1  [astro-ph.CO]  6 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Gravitational waves emitted during a single collision
+2
+3
+Effect on the stochastic gravitational wave background
+6
+3.1
+Energy spectrum of a single GW event
+7
+3.2
+Number density of GW sources
+8
+3.3
+The energy spectrum of the stochastic gravitational wave background
+10
+4
+Conclusions
+11
+1
+Introduction
+Recently, the gravitational wave (GW) detecting technology has been developing rapidly.
+In 2015, the detection of binary black holes merger GW150914 by the LIGO experimental
+cooperation signaled the first detection of gravitational waves [1], while in 2017, the joint
+detection of GW170817 [2] and GRB170817A [3] opened the new era of multi-messenger
+astronomy [4]. In general, with the increasing amount of detected gravitational wave events
+[5] one has improved statistics that allows to track the history of the universe [6, 7] and
+impose bounds on various cosmological parameters [8, 9], as well as constrain various theories
+of gravity [10–14]. Moreover, for different frequencies and types of gravitational wave sources,
+various detection means have been designed and implemented. Besides ground-based laser
+interferometers such as LIGO, Virgo and KAGRA, which probe high frequency bands (10 −
+104 Hz), space-based laser interferometers such as LISA [15, 16] for intermediate frequency
+gravitational waves (10−4 − 1 Hz), and the pulsar timing array (PTA) [17–20] for lower
+frequency bands (10−9 − 10−6 Hz), are also raised. These observational avenues allow us to
+acquire rich information from GWs of different types and sources, among which stochastic
+gravitational wave background is attracting increasing interest.
+Stochastic gravitational wave background (GWB) is a type of random background signal
+that exists in an analogous way to the cosmic microwave background.
+The contribution
+of GWB can be roughly divided into cosmological sources and astrophysical sources [21].
+Astrophysical originated GWB contains all types of unresolved GW emitting events, including
+binary black hole mergers [22–28]. These signals can provide information about astrophysical
+source populations and processes over the history of the universe [29–32]. On the other hand,
+cosmological originated GWB mainly involves primordial gravitational perturbations during
+the inflation epoch [33–35], or perturbations arising from primordial black holes fluctuations
+[36–39]. GW signals typically remain unaffected during their propagation, and thus they
+can provide valuable information about the very early stages of the universe. For instance,
+different inflationary models can lead to different predictions for the GWB spectrum [40–52],
+and thus GWB can be used as a probe of this primordial universe epoch. Since GWB can
+provide us with important astrophysical and cosmological probes, it is crucial to understand
+its composition and properties [53–63].
+On the other hand, according to observations, dark matter (DM) constitutes a significant
+fraction of the energy density of the universe [64–66]. Its microphysical nature and possible
+– 1 –
+
+interactions remain unknown [67–70], nevertheless we do know unambiguously that DM
+interacts gravitationally [71, 72].
+Current theory predicts that the main part of DM is
+concentrated in dark halos, which coincide in position with galaxy or galaxy clusters [73].
+These galaxies and galaxy clusters, and thus dark halos too, are typically accelerating and
+merging through their mutual attraction [74–76]. Such processes can in principle release GW
+signal through gravitational bremsstrahlung [77–87].
+In this work, we are interested in investigating for the first time the possible GW signals
+that could be emitted through bremsstrahlung during dark halo merger and collisions, and
+their contribution to the stochastic GWB. In particular, we will first consider a single event of
+two DM halos collision, and we will calculate the emitted GW signal. Then, we will calculate
+the energy spectrum contribution to the stochastic GWB, taking the DM halo collision rate
+into consideration. The structure of the article is as follows. In Section 2 we analyze the
+GW emitted during the collision of two galaxies or two galaxy clusters. In Section 3 we
+integrate over redshift and DM halos parameters to extract the contribution to stochastic
+GWB. Finally, in Section 4 we conclude and discuss our results.
+2
+Gravitational waves emitted during a single collision
+In this section, we aim at estimating the gravitational waves emitted during a single collision
+event. In particular, we calculate the GW radiated by the collision of two DM halos, which
+corresponds to the collision of two galaxies or two galaxy clusters.
+According to observations, such a collision typically has a huge duration, which in turn
+implies that the energy radiated through GWs per unit time is not very large, and thus we
+can safely use linear perturbation theory in the involved calculations. Specifically, we use
+[88]
+gµν = ηµν + hµν,
+|hµν| ≪ 1,
+(2.1)
+¯hij(t, x) = 2G
+rc4
+d2Iij (tr)
+dt2
+,
+tr = t − r
+c,
+(2.2)
+where G is the gravitational constant, c is the speed of light, and r is the distance from us
+to the center of mass of the two galaxies or galaxy clusters. Moreover, Iij is the quadruple
+moment
+Iij(t) =
+�
+yiyjT 00(t, y)d3y =
+�
+yiyjρ(t, y)d3y,
+(2.3)
+where T µν is energy-momentum tensor, ρ is energy density, and yi is the spatial coordinate.
+Since the goal of our calculation is to acquire an estimation of the order of the magnitude of
+the resulting signal, we can consider these two DM halos as mass points, with mass Ma and
+position y(a)(t) at time t. Hence, the density ρ can be written as
+ρ(t, y) =
+�
+a
+Maδ3(y − y(a)(t)),
+(2.4)
+while the quadruple moment Iij(t) becomes
+Iij(t) =
+�
+yiyjρ(t, y)d3y =
+�
+a
+Mayi
+(a)(t)yj
+(a)(t).
+(2.5)
+– 2 –
+
+Finally, since the relative speed of two galaxies or galaxy clusters is much smaller than the
+speed of light, we can use Newtonian mechanics to handle their dynamics.
+For simplicity we write the equations in the center-of-mass frame of these two mass
+points. By definition, we have
+MArA + MBrB = 0,
+(2.6)
+where MA, MB are the masses of the mass points A and B, with rA, rB their position vectors.
+From Newtonian mechanics we have
+¨rA = −
+GMB
+|rA − rB|2
+rA
+|rA|,
+(2.7)
+which using (2.6) gives
+¨rA = −µB
+rA
+|rA|3 ,
+(2.8)
+where we have defined µB ≡
+GMB
+(1+ MA
+MB )2 . Additionally, we assume that the two points are ini-
+tially at infinite distance, their relative speed is v∞ = vA∞ +vB∞, and the impact parameter
+is b = bA + bB. From Newtonian mechanics we know that the trajectory of each point is a
+hyperbola and the two points are moving in a plane (we set this plane as z = 0 plane, and
+thus rA = (xA, yA, 0)), while the total energy of the system is positive. Additionally, the
+mass center of these two DM halos will not follow a hyperbolic trajectory at all times, in
+order to acquire a collision. In Fig. 1 we depict an illustrative representation of the initial
+conditions of the collision.
+Figure 1.
+An illustrative representation of the initial conditions of the collision. The two galaxies
+or galaxy clusters are considered as points with masses MA and MB, where bA and bB are the impact
+parameters.
+Let us start with the beginning of the collision, when the two DM halos start moving
+towards each other. For point A we have
+(xA + aAeA)2 − (yA)2 = a2
+A,
+(2.9)
+– 3 –
+
+Y
+MB
+UB8
+bB
+O center of mass
+X
+L= 8
+b A
+VA8
+MAwhere
+aA =
+µB
+(vA∞)2 ,
+(2.10)
+eA =
+�
+1 + vA4∞b2
+A
+(µB)2 ,
+(2.11)
+a = aA + aB.
+(2.12)
+We proceed by defining λA through
+eA sinh(λA) − λA = vA∞t
+aA
+,
+(2.13)
+hence
+r1
+A = xA = aA [eA − cosh(λA)] ,
+(2.14)
+r2
+A = yA = aA
+��
+e2
+A − 1 sinh(λA)
+�
+.
+(2.15)
+Note that t = 0 corresponds to the time when the two mass points have the shortest distance.
+In order to obtain the GW amplitude hij, we proceed to the calculation of the quadrupole
+moment Iij(t) and its second time derivative. We have
+Iij = MAri
+Arj
+A + MBri
+Brj
+B,
+(2.16)
+d2Iij
+dt2
+= MA(¨ri
+Arj
+A + ri
+A¨rj
+A + 2 ˙ri
+A ˙rj
+A) + MB(¨ri
+Brj
+B + ri
+B¨rj
+B + 2 ˙ri
+B ˙rj
+B).
+(2.17)
+From (2.14), (2.15) we find
+˙xA = −
+aA sinh(λA)
+�
+a3
+A
+µB [eA cosh(λA) − 1]
+,
+(2.18)
+¨xA =
+µB(cosh(λA) − e)
+a2
+A[eA cosh(λA) − 1]3 ,
+(2.19)
+˙yA =
+aA
+�
+e2
+A − 1 cosh(λA)
+�
+a3
+A
+µB [eA cosh(λA) − 1]
+,
+(2.20)
+¨yA = −
+�
+e2
+A − 1µB sinh(λA)
+a2
+A[eA cosh(λA) − 1]3 ,
+(2.21)
+and thus inserting into (2.17) we extract all the second time derivatives of the quadrupole
+– 4 –
+
+moment Iij(t), namely
+d2I11
+dt2
+= µBMA{7eA cosh(λA) + eA[cosh(3λA) − 4eA] − 4 cosh(2λA)}
+2aA[eA cosh(λA) − 1]3
++µAMB{7eB cosh(λB) + eB[cosh(3λB) − 4eB] − 4 cosh(2λB)}
+2aB[eB cosh(λB) − 1]3
+,
+(2.22)
+d2I12
+dt2
+= −
+�
+e2
+A − 1µBMA sinh(λA){eA[cosh(2λA) + 3] − 4 cosh(λA)}
+aA[eA cosh(λA) − 1]3
+−
+�
+e2
+B − 1µBMA sinh(λB){eB[cosh(2λB) + 3] − 4 cosh(λB)}
+aB[eB cosh(λB) − 1]3
+,
+(2.23)
+d2I22
+dt2
+=
+�
+e2
+A − 1
+�
+µBMA[3eA cosh(λA) + eA cosh(3λA) − 4 cosh(2λA)]
+2aA[eA cosh(λA) − 1]3
++
+�
+e2
+B − 1
+�
+µAMB[3eB cosh(λB) + eB cosh(3λB) − 4 cosh(2λB)]
+2aB[eB cosh(λB) − 1]3
+.
+(2.24)
+Figure 2.
+The dimensionless components of the gravitational wave signal arising from a single
+event of the collision of two DM halos, i.e., the collision of two galaxies or clusters of galaxies. The
+left panel shows the ¯h11 component, the middle panel the ¯h12 component and the right panel the ¯h22
+component. The time t = 0 corresponds to the shortest distance between the two DM halos, that is
+the moment in which ¯h11 and ¯h22 reach their peaks and ¯h12 exhibits the largest variation. We have
+imposed the typical values MA = MB = 109M⊙, vA∞ = vB∞ = 300km/s, bA = bB = 104ly, and we
+have assumed that the distance from Earth is ∼ 109ly. Time t is measured in seconds.
+We can now use (2.2) in order to obtain the GW signal in the time domain. As typical
+values we set MA = MB = 109M⊙, namely the order of mass of a (dwarf) galaxy, where M⊙
+is the mass of the Sun, and we use vA∞ = vB∞ = 300km/s, bA = bB = 104ly, which are
+the typical values for galaxy collisions. Moreover, we assume that the collision happens at a
+distance of 109ly from the Earth, which is roughly the distance of the source of GW150914.
+Hence, we can estimate the magnitude of the GW signal. In Fig. 2 we present the obtained
+dimensionless GW signal ¯hij, as a function of time t. Since t = 0 corresponds to the time of
+shortest distance, the change rate of ¯hij is fastest at this time, as expected. As we observe,
+the variation of ¯hij is of the order of 5 × 10−22 during the collision. However, this variation
+– 5 –
+
+七
+-1.0×1017 -5.0×1016
+5.0 ×1016
+1.0×1017
+-1. × 10-22
+-3. × 10-22
+-5. × 10-22h12
+2.×10
+七
+-1.0×1017 -5.0×1016
+5.0 × 1016
+6 1.0×1017
+-1. × 10-22
+-2. × 10-22h22
+1.2538 × 10-18
+1.2534 × 10-18
+t
+-1.0×1017 -5.0×1016
+5.0×1016
+1.0 ×1017corresponds to a large time scale (about 1015s), which implies that a single signal of this kind
+of GW is extremely hard to be detected. Additionally, we can see that the evolution of ¯h12
+is faster than that of ¯h11, ¯h22, which implies that ¯h12 will be dominant in relatively higher
+frequency than that of ¯h11, ¯h22.
+We proceed by taking the Fourier transformation of ¯hij, in order to investigate its
+spectrum. In particular, we use
+˜¯hij(ω) =
+� t=+∞
+t=−∞
+dt eiωt ¯hij(t) ,
+(2.25)
+where ω = 2πf, with f the frequency. In Fig. 3 we present the dependence of ˜¯hij(f) on f.
+As we observe, all ˜¯hij(f) obey the power law in a very good approximation for a very wide
+frequency range. Besides, as ˜¯h11, ˜¯h22 ∝ 1/f2, while ˜¯h12 ∝ 1/f, we can infer that ˜¯h11, ˜¯h22
+will be dominant in the low frequency band while ˜¯h12 will be dominant in relatively high
+frequencies.
+Figure 3. The spectrum of the gravitational waves as a function of the frequency. The upper left panel
+shows the ˜¯h11(f) component, the upper right panel the ˜¯h12(f) component and the right panel the ˜¯h22(f)
+component. The blue dots represent the exact results at the time of shortest distance, while the red solid
+curves are power-law fits, specifically ˜¯h11(f) ≈ 1.7 × 10−53(Hz/f)2, ˜¯h12(f) ≈ 1.2 × 10−38(Hz/f)2,
+˜¯h22(f) ≈ 2.0 × 10−53(Hz/f)2.
+3
+Effect on the stochastic gravitational wave background
+In this section, we calculate the contribution of the DM halos collisions to the stochastic
+gravitational wave background. Specifically, we integrate the gravitational wave spectrum of
+– 6 –
+
+10-23
+10~24
+10~25
+10~26
+10~27
+10-28
+5.×10-15 10-14
+5. × 10-1.4 10-13
+5.× 10-13h11(f)f/H
+21. × 10-22
+5. × 10-23
+1.×10-23
+5.× 10-24
+1. × 10-24
+5. × 10-25
+5. × 10-15 10-14
+5. × 10-1.4 10-13
+5.×10-13h12(f)f/H
+210~23
+10~24
+10~25
+10-26
+10~27
+10-28
+5.× 10-15 10-14
+5. × 10-1.4 10-13
+5.× 10-13h22(f)f/H
+2a single collision event over the number density of GW sources.
+In principle, in order to compare a theoretical model with observations, one uses both
+the fractional energy density spectrum Ωgw(f), as well as the characteristic strain amplitude
+hc(f) [56]. They are related to the energy spectrum of GWB through the expression
+π
+4 f2h2
+c(f) = ρcΩgw(f) = dρgw(f)
+d ln f ,
+(3.1)
+where f is the frequency of GW detected on Earth, and ρc ≡ 3c2H2
+0/8πG is the critical
+energy density. The energy spectrum of the stochastic GWB, dρgw
+d ln f , can be written as
+dρgw(f)
+d ln f
+=
+� ∞
+0
+dz
+1
+1 + z
+�
+dξ dn
+dzdξ
+dE(ξ)gw
+d ln fr
+����
+fr=f(1+z)
+,
+(3.2)
+with z the redshift at the GW emission. Additionally, dE(ξ)gw
+d ln(fr) is the energy spectrum of a
+single GW event, which is calculated through the analysis of the previous subsection, and fr
+is the GW frequency in the rest frame of GW sources, and thus fr = (1 + z)f.
+We mention that we denote the parameters related to the number density of GW sources
+collectively by ξ = {ξ1, . . . , ξm}, and therefore
+dn
+dξ1...dξmdzdξ1 . . . dξmdz ≡
+dn
+dξdzdξdz is the
+number density of sources in the redshift interval [z, z + dz] and with source parameters
+in the interval [ξ, ξ + dξ]. Hence, in the simple single event of two DM halos collision of
+the previous section we have ξ = {M, x, v∞, b}, where M = MA + MB, x = MA/MB,
+v∞ = vA∞ + vB∞ and b = bA + bB.
+Let us now calculate the full distribution function
+dn
+dzdξ =
+dn
+dzdMdxdv∞db. As we have
+checked numerically, the variance of b, v∞ has a minor effect on the final result, not affecting
+the order of magnitude. Hence, it is a good approximation to omit the change of b, v∞, and
+consider that ξ = {M, x}. Hence, we have
+dρgw(f)
+d ln f
+=
+� ∞
+0
+dz
+1
+1 + z
+�
+dξ dn
+dzdξ
+dE(ξ)gw
+d ln fr
+����
+fr=f(1+z)
+≈
+� 10
+0
+dz
+1
+1 + z
+� Mmax=1015M⊙
+Mmin=109M⊙
+dM
+� xmax=105
+xmin=1
+dx
+dn
+dzdMdx
+dE(ξ)gw
+d ln, fr
+�����
+fr=f(1+z)
+, (3.3)
+where the varying range of M and x is taken from [89].
+In the following subsections we will separately calculate the energy spectrum of a single
+GW event dE(ξ)gw
+d ln fr , and the number density of GW sources
+dn
+dzdMdx.
+3.1
+Energy spectrum of a single GW event
+The energy density of a single GW event can be calculated from the (traceless) second time
+derivative of the quadrupole moment, namely [90]
+dE(ξ)gw
+d ln fr
+≈ fr
+2G
+5c5 (2πfr)2( ¨˜Qij(M, x; fr))( ¨˜Qij(M, x; fr)),
+(3.4)
+– 7 –
+
+where Qij is the traceless quadrupole moment and ¨˜Qij is the Fourier transformation of the
+second time derivative of Qij, which is related to Iij via
+Q11 = 2
+3I11 − 1
+3I22,
+(3.5)
+Q22 = −1
+3I11 + 2
+3I22,
+(3.6)
+Q33 = −1
+3I11 − 1
+3I22,
+(3.7)
+Q21 = Q12 = I12,
+(3.8)
+while all other Qij are equal to zero. Now, from Newtonian mechanics Iij can be written as
+¨˜Iij(M, x; fr) = 4
+�
+x
+(1 + x)3 +
+1/x
+(1 + 1/x)3
+� �
+M
+2 × 1012M⊙
+�2 ¨˜IG
+ij (fr),
+(3.9)
+where x is the mass ratio of the two masses, and IG
+ij is defined as Iij(M = 2×1012M⊙, x = 1).
+Therefore, from the calculation of Section 2, we can extract the values of ¨˜IG
+ij (fr) as
+¨˜IG
+11(fr) = 2.86 × 1021
+�Hz
+fr
+�2
+kg m2s−1,
+(3.10)
+¨˜IG
+22(fr) = 5.72 × 1020
+�Hz
+fr
+�2
+kg m2s−1,
+(3.11)
+¨˜IG
+12(fr) = ¨˜IG
+21(fr) = 1.29 × 1037
+�Hz
+fr
+�
+kg m2s−1.
+(3.12)
+Hence, inserting the above into (3.4) gives us the energy density of a single GW event.
+3.2
+Number density of GW sources
+Let us now calculate the number density of GW sources (per redshift, total mass and mass
+ratio interval)
+dn
+dzdMdx. This number density is equal to the DM matter halos mergers rate,
+which can be calculated by combining the extended Press-Schechter (EPS) theory [91] and
+numerical simulations [89]:
+dn
+dzdMdx = nhalo(M, z)dω
+dz
+�
+1
+nhalo
+dnmerger
+dωdx
+�
+,
+(3.13)
+where nhalo(M, z) is the number density of dark matter halos (per redshift per mass inter-
+val in the co-moving space), ω = ω(z) is a redshift-dependent function given below, and
+(
+1
+nhalo
+dnmerger
+dωdx
+) is the merger rate (at some ω) for a pair of DM halos with fixed total mass
+M and mass ratio x. In the following we handle these terms separately.
+We start with the definition of ω(z) [91]
+ω(z) = 1.69
+D(z),
+(3.14)
+where D(z) is the linear growth rate of matter density. D(z) can be written as
+D(z) =
+1
+g(z = 0)
+� g(z)
+1 + z
+�
+,
+(3.15)
+– 8 –
+
+where a good approximation of g(z) is
+g(z) ≈ 5
+2Ωm(z)
+�
+Ω4/7
+m (z) − ΩΛ(z) + [1 + Ωm(z)/2] [1 + ΩΛ(z)/70]
+�−1
+,
+(3.16)
+with ΩΛ(z), Ωm(z) the density parameters of dark energy and matter sectors given by
+ΩΛ(z) = ΩΛ,0
+E2(z);
+Ωm(z) = Ωm,0(1 + z)3
+E2(z)
+,
+(3.17)
+where the normalized Hubble function E(z) ≡ H(z)/H0 reads as
+E(z) ≈
+�
+ΩΛ,0 + Ωm,0(1 + z)3�1/2 ,
+(3.18)
+with the value of the Hubble function at present time given as [64]
+H0 ≈ 67.3 km s−1Mpc−1,
+(3.19)
+and with the values ΩΛ,0, Ωm,0 at present time taken as [64]
+ΩΛ,0 ≈ 0.685,
+(3.20)
+Ωm,0 ≈ 0.317.
+(3.21)
+Note that in the above we consider that the underlying cosmology is ΛCDM concordance
+scenario, i.e., the dark energy sector is the cosmological constant.
+We continue by using the EPS theory in order to write the formula of the number
+density of DM halos nhalo. We consider that the halos merge when the redshift is between z
+and z +dz , and that the emitted GW signals are detected at Earth at present. In co-moving
+space those halos are in the volume ∆V = 4πr2(z)d(r(z)). Now, the EPS theory provides
+the number density of DM halos nEPS(M, z) at some redshift z and mass M. Therefore, we
+have
+nhalo = 4πr2(z)dr(z)
+dz nEPS(M, z),
+(3.22)
+where the radius in the co-moving space r(z) is [91]
+r(z) =
+c
+H0
+� z
+0
+dz
+′
+1
+E(z
+′),
+(3.23)
+while the formula of nEPS(M, z) is [91]
+nEPS(M, z) =
+�
+2
+π
+¯ρ
+M2
+δc
+σ exp
+�
+− δ2
+c
+2σ2
+� ����
+d ln σ
+d ln M
+���� .
+(3.24)
+In the above expression ¯ρ = ρcΩm,0 is the mean density of the matter component, δc = ω =
+1.69
+D(z), while σ(M) is the variance of the matter density perturbation which can be estimated
+as [91]
+σ(M) ≈ σ8
+� R
+r8
+�−β
+,
+(3.25)
+– 9 –
+
+with M = 4π
+3 ¯ρR3 , σ8 ≈ 1, β ≈ 0.6 + 0.8(Ωm,0h), h = 0.673 , and r8 = 8 Mpc h−1, leading to
+����
+d ln σ
+d ln M
+���� = β
+3 .
+(3.26)
+Finally, the last term of (3.13), namely (
+1
+nhalo
+dnmerger
+dωdx
+) (dimensionless since both ω, x
+are dimensionless), can be found in [89] and it is given by
+�
+1
+nhalo
+dnmerger
+dωdx
+�
+= A
+�
+M
+1012M⊙
+�α
+xb exp [(˜x/x)γ] ,
+(3.27)
+where the best-fit parameters from simulations are A = 0.065, α = 0.15, b = −0.3, ˜x = 2.5,
+γ = 0.5 [89].
+In summary, inserting (3.14), (3.22) and (3.27) into (3.13), provides the value of the
+number density of GW sources
+dn
+dzdMdx.
+3.3
+The energy spectrum of the stochastic gravitational wave background
+We have now all the ingredients needed in order to calculate the energy spectrum of the
+stochastic gravitational wave background. This is given by (3.3), in which the energy spec-
+trum of a single GW event dE(ξ)gw
+d ln fr
+was calculated in subsection 3.1, while the number density
+of GW sources
+dn
+dzdMdx was calculated in subsection 3.2. Assembling everything, we finally
+obtain the stochastic gravitational wave background resulting from DM halos collisions in
+the universe, which is calculated numerically and it is shown in Fig. 4. Additionally, for com-
+parison we also depict the corresponding result of of the contribution of the super-massive
+black-hole mergers (SMBHM), which follows hc(f) ∝ f−2/3, as well as the current observa-
+tional sensitivity [92].
+10
+16
+10
+14
+10
+12
+10
+10
+10
+8
+10
+6
+10
+4
+10
+2
+Frequency f /Hz
+10
+34
+10
+30
+10
+26
+10
+22
+10
+18
+10
+14
+10
+10
+Characteristic Strain    hc
+DM halos collisions
+SMBHM
+current observation sensitivity
+Figure 4. The characteristic strain hc(f) as a function of the frequency of the stochastic gravita-
+tional wave background created by DM halos, namely galaxies and galaxy clusters, collisions (blue-solid
+curve). For comparison, with the green-dashed curve we depict the corresponding result of of the con-
+tribution of the super-massive black-hole mergers (SMBHM), which follows hc(f) ∝ f −2/3. Finally,
+with the red-dotted curve we present the current observational sensitivity [92].
+– 10 –
+
+As we can see, the contribution of GW radiated from the collisions of DM halos, namely
+galaxies and galaxy clusters, is quite small comparing to other sources. In the pulsar timing
+array (PTA) band, where f ≈ 10−9Hz, and where the current observational limit is hc ≈
+10−15 [93], we obtain an effect of the order of hc ≈ 10−30. Nevertheless, in very low frequency
+band hc will be larger. In general, with current observational sensitivity the effect of the DM
+halos collisions on the stochastic gravitational wave background cannot be detected, however
+it will be accessible in the future, in which case one could use the advanced techniques of
+distinguishing signal overlap in order to isolate it from other sources [94–97].
+Note that
+one could try to extend the analysis, by considering, instead of point masses, a group of
+mass points with Navarro, Frenk & White (NFW) density profile [91] to simulate DM halo
+collisions, nevertheless the results are expected to be at the same order of magnitude.
+4
+Conclusions
+In this work we investigated for the first time the effect of the dark matter halos collisions,
+namely collisions of galaxies and galaxy clusters, through gravitational bremsstrahlung, on
+the stochastic gravitational wave background.
+In order to achieve this goal, we first calculated the gravitational wave signal of a single
+DM halo collision event. As an estimation of the order of magnitude, we handled the two
+DM halos as mass points. Furthermore, since the strength of such GW signals is weak, we
+adopted linear perturbation theory of General Relativity, namely we extracted the GW signal
+using the second time derivative of the quadruple moment. Additionally, since the velocity
+of DM halos is small, we applied non-relativistic Newtonian Mechanics. Hence, we extracted
+the GW signal through bremsstrahlung from a single DM halo collision. As we showed, ¯hij is
+of the order of 10−22, and it becomes maximum at the time of shortest distance as expected.
+However, since such an event typically corresponds to durations of the order of 1015s, we
+deduce that a single signal of this kind of GW is extremely hard to be detected.
+As a next step we proceeded to the calculation of the energy spectrum of the collective
+effect of all DM halos collisions in the Universe. This can arise by the energy spectrum of
+a GW signal radiated by a single collision, multiplied by the DM halo collision rate, and
+integrating over the whole Universe.
+Firstly, knowing the signal of a single collision we
+calculated its energy spectrum. Secondly, concerning the DM halo collision rate we showed
+that it is given by the product of the number density of DM halos, which is calculated by the
+EPS theory, with the collision rate of a single DM halo, which is given by simulation results,
+with a function of the linear growth rate of matter density through cosmological evolution.
+Hence, integrating over all mass and distance ranges, we finally extracted the spectrum of
+the stochastic gravitational wave background created by DM halos collisions.
+As we saw, the resulting contribution to the stochastic gravitational wave background
+is of the order of hc ≈ 10−30 in the pulsar timing array (PTA) band of f ≈ 10−9Hz, much
+smaller than other GW sources, such as super-massive black-hole mergers. However, in very
+low frequency band, hc is larger. With current observational sensitivity it cannot be detected,
+nevertheless it may be accessible by PTA in the future, where techniques of distinguishing
+signal overlap should be used in order to isolate it and use it for cosmological studies.
+In summary, with the current and future significant advance in gravitational-wave as-
+tronomy, and in particular with the tremendous improvement on the sensitivity bounds that
+Collaborations like Laser Interferometer Space Antenna (LISA), Einstein Telescope (ET),
+Cosmic Explorer (CE), etc will bring, it is both interesting and necessary to investigate all
+– 11 –
+
+possibles contributions to the stochastic gravitational wave background. And the gravita-
+tional bremsstrahlung during galaxies and galaxy clusters collisions is one of them.
+Acknowledgments
+We are grateful to Yifu Cai, Jiewen Chen, Zihan Zhou and Jiarui Li for helpful discussions.
+This work is supported in part by the National Key R&D Program of China (2021YFC2203100),
+by the NSFC (11961131007, 11653002), by the Fundamental Research Funds for Central Uni-
+versities, by the CSC Innovation Talent Funds, by the CAS project for young scientists in
+basic research (YSBR-006), by the USTC Fellowship for International Cooperation, and by
+the USTC Research Funds of the Double First-Class Initiative. ENS acknowledges partici-
+pation in the COST Association Action CA18108 “Quantum Gravity Phenomenology in the
+Multimessenger Approach (QG-MM)”. All numerics were operated on the computer clusters
+LINDA & JUDY in the particle cosmology group at USTC.
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+UserSimCRS: A User Simulation Toolkit for Evaluating
+Conversational Recommender Systems
+Jafar Afzali
+University of Stavanger
+j.afzali@stud.uis.no
+Aleksander Mark Drzewiecki
+University of Stavanger
+am.drzewiecki@stud.uis.no
+Krisztian Balog
+University of Stavanger
+Stavanger, Norway
+krisztian.balog@uis.no
+Shuo Zhang
+Bloomberg
+London, United Kingdom
+szhang611@bloomberg.net
+ABSTRACT
+We present an extensible user simulation toolkit to facilitate auto-
+matic evaluation of conversational recommender systems. 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.
+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.
+CCS CONCEPTS
+• Information systems → Recommender systems.
+KEYWORDS
+Conversational recommender systems; user simulation
+ACM Reference Format:
+Jafar Afzali, Aleksander Mark Drzewiecki, Krisztian Balog, and Shuo Zhang.
+2023. UserSimCRS: A User Simulation Toolkit for Evaluating Conversational
+Recommender Systems. In Proceedings of the Sixteenth ACM International
+Conference on Web Search and Data Mining (WSDM ’23), February 27-March
+3, 2023, Singapore, Singapore. ACM, New York, NY, USA, 4 pages. https:
+//doi.org/10.1145/3539597.3573029
+1
+INTRODUCTION
+Conversational recommender systems (CRSs) elicit user preferences
+via multi-turn real-time interactions using natural language [6, 9].
+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]. A
+major challenges that remains, however, is evaluation [6]. 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. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from permissions@acm.org.
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+© 2023 Association for Computing Machinery.
+ACM ISBN 978-1-4503-9407-9/23/02...$15.00
+https://doi.org/10.1145/3539597.3573029
+the conversation level is not possible using offline test collections.
+While online evaluation with users of a live service is an option,
+it is expensive and does not scale. A promising solution to these
+issues is user simulation [1, 6]. The idea there is to build a simulated
+user that mimics how a real human would respond in a given
+dialogue situation [19, 25]. Simulation thus offers a repeatable and
+reproducible means of evaluation. (We note that it is not meant to
+replace, but rather to complement human evaluation.)
+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]. The current work aims to contribute to the de-
+velopment of novel CRSs by recognizing the need for better tooling
+for user simulation. In particular, we provide an extensible open-
+source toolkit that is designed specifically for evaluation. Our work
+is unique in at least three regards. 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. Second, it centers
+around evaluation as opposed to other uses of simulation (most
+commonly, synthetic data generation for reinforcement learning).
+Third, it is designed to work with existing CRSs, without needing ac-
+cess to source code or knowledge of their inner workings. It merely
+requires collecting and annotating a small sample of dialogues.
+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. Given its modular design, the
+toolkit can also be easily extended with other modeling options or
+additional components. 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.
+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. First, we focus on the task of conversational recom-
+mendations and hence place a strong emphasis on natural language
+understanding and generation. Thus, unlike others that operate in
+1https://github.com/iai-group/DialogueKit
+2https://github.com/iai-group/UserSimCRS
+arXiv:2301.05544v1  [cs.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. The parts in blue follow [25], while the yellow ones are novel additions.
+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).
+Our toolkit implements an agenda-based simulator [18], building
+on and extending the approach in [25]. Alternatively, model-based
+simulation could also be employed as it has been done recently for
+task-based dialogue systems. Shi et al. [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. 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. These simulators can be assembled by equipping a neu-
+ral network-based user policy with NLU and NLG components.
+Tseng et al. [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. 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. Importantly, such model-based approaches can also be
+incorporated into our framework in the future.
+3
+CONCEPTUAL OVERVIEW
+The goal of user simulation is to mimic how real users would re-
+spond in given dialogue situation [19, 25]. 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. Addi-
+tionally, there is a dedicated user modeling component; see Fig. 1.
+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. Our work builds on
+and extends the approach proposed in [25] as detailed below.
+Natural language understanding (NLU) is responsible for
+obtaining a structured representation of text utterances. Conven-
+tionally, it entails intent classification and entity recognition. Addi-
+tionally, motivated by recent research [17, 23], we also include a
+classifier for user satisfaction prediction.3
+Response generation is currently based on agenda-based simu-
+lation [18], however, it could be replaced with other approaches in
+the future. 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.
+updating it. 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; otherwise, it either repeats the
+same action as the previous turn or samples a new action.
+User modeling consists of three sub-components. The preference
+model captures users’ likes and dislikes. Following [25], it is modeled
+as a personal knowledge graph [2], where nodes correspond to items
+and attributes. Novel to our work is the modeling of persona, which
+can capture user-specific traits, e.g., patience or cooperativeness,
+and context, which can characterize the situation of the user, e.g.,
+temporal (time of the day and weekday vs. weekend), relational
+(alone vs. group setting), or conversational (user satisfaction). 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].
+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. Additionally, we extend the NLG such that it can
+be conditioned on context. 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.
+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. 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. See Fig. 2 for an overview of
+the main packages and their dependencies. Both libraries are made
+available in the Python Package Index (PyPI).
+4.1
+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. Utterances may be
+annotated with intents and slot-value pairs. DialogueKit currently
+supports two models for annotation, a cosine classifier for intents
+and a minimal pipeline DIET classifier [4] for slot-value pairs.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.
+
+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)
+<.
+Context model
+Conditional NLG
+Preference model
+PersonaUserSimCRS: A User Simulation Toolkit for Evaluating Conversational Recommender Systems
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+DialogueKit
+UserSimCRS
+Dialogue connector
+Platforms
+Utilities
+Evaluator
+NLU
+Entity extractor
+Intent classifier
+Satisfaction classifier
+NLG
+Template-based NLG
+Core components
+Annotation
+Intent
+Dialogue
+Utterance
+Domain
+…
+Agenda-based 
+simulator
+Interaction model
+User modeling
+Preference model
+Context model
+Persona
+Items
+Conditional NLG
+Participant
+Agent
+User
+Item
+Ratings
+Figure 2: Overview of the main packages (in yellow) with
+some of the core modules highlighted (in white). Arrows in-
+dicate intra-library dependencies (in blue) and inter-library
+dependencies (in black).
+machine-machine). Furthermore, the evaluation component pro-
+vides functionality required to evaluate a set of conversations with
+respect to standard metrics (such as AvgTurns and AvgSuccess).
+4.2
+UserSimCRS
+The UserSimCRS library implements the simulation-specific compo-
+nents in Fig. 1, specifically, response generation and user modeling.
+During a conversation, any time the user is asked to provide pref-
+erences, the preference model is consulted. Context is modeled in
+a generic way such that it can capture, among others, temporal,
+relational, and conversational factors. The generation of user utter-
+ances may be conditioned on the user’s context and persona. Next,
+we elaborate on how to use UserSimCRS for system evaluation.
+Note that the library may also be used for training agents, but that
+is outside the focus of the current paper.
+5
+SYSTEM EVALUATION USING SIMULATION
+This section discusses how to employ simulation for evaluating an
+existing CRS and illustrates this with a case study.
+5.1
+Methodology
+The main objective of simulation-based evaluation in this work
+is to establish a relative comparison between two systems. These
+may be different variants of the same CRS or two different systems.
+Importantly, the user simulator needs to target the differences that
+we care about. 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. Having a user simulator that asks only for genres but not
+for plot keywords will not capture the differences between these
+two systems. Therefore, as a general principle, the user simulator
+needs to be co-developed with the CRS and customized to mimic
+the targeted user behavior.
+5.2
+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. 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.
+Additionally, a file containing the item collection is required.
+(2) Provide preference data: Preference data is consumed in the
+form of item ratings (user ID, item ID, and rating triples).
+(3) Dialogue sample: A small sample of dialogues with the CRS
+needs to be collected. 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.
+(4) Define interaction model: A config file containing the space
+of user and agent intents (i.e., possible actions), as well as the set
+of expected agent responses for each user intent, is required for
+the interaction model. 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.
+(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. Note that
+the slots used for annotation should be the same as the ones
+defined in the domain file (cf. Step 1) and intents should follow
+the ones defined in the interaction model (cf. Step 4.).
+(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.g., weekday vs. weekend,
+alone vs. group setting) and personas (e.g., patient and impa-
+tient users). Further, the number of users to be generated is to
+be specified. Each user will have their own preference model,
+which may be instantiated by grounding it in actual preferences
+(i.e., the ratings dataset given in Step 2).
+(7) Train simulator: The NLU, NLG, and response generation
+components of the simulator are trained using the annotated
+dialogue sample.
+(8) Run simulation: Running the simulator will generate a set of
+simulated conversations for each user with the CRS and save
+those to files.
+(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.
+5.3
+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. This required creating a connector agent in
+DialogueKit, which can talk to IAI MovieBot via a RESTful API.
+We followed the steps listed above to prepare the user simulator.
+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. As it can be seen from the
+sample dialogue in Fig. 3, the simulator could successfully complete
+dialogues with the CRS, mimicking the behavior of users observed
+in the training data it was exposed to.
+5https://github.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).
+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.
+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.
+Future work is concerned with extending the components with ad-
+ditional modeling options, including alternatives to agenda-based
+simulation. We also plan to evaluate additional existing CRSs to
+ensure that our framework generalizes to diverse systems.
+Acknowledgment. We thank Nolwenn Bernard for her extensive
+contributions to the toolkit, made after the submission of this paper.
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+Recommender Systems. In Proc. of SIGIR ’20. 881–890.
+
+Hi there. I am IAl MovieBot, your movie recommending buddy. I
+can recommend you movies based on your preferences. I will ask
+you a few questions and based on your answers, I will try to find a
+movie for you.",
+Hello.
+DISCLOSE.NON-DISCLOSE
+Do you have any specific genres in mind?
+DISCLOSE.NON-DISCLOSE
+How about Crime?
+INQUIRE.ELICIT
+There are almost 5300 crime films. Please answer a few
+DISCLOSE
+more questions to help me find a good movie.Can you give
+me a few keywords? For example, 'organized crime' or
+'contact lens'",
+Pixar animation
+INQUIRE.ELICIT
+Do you like: Cat City 8.4 96 min
+DISCLOSE
+Tell me something about it
+REAVEAL
+What would you like to know about Cat City?
+INQUIRE.ITEMINFO
+藍
+INQUIRE.MORE
+Director name
+Its directed by Béla Ternovszky.
+INQUIRE.MORE
+I like this recommendation
+DISCLOSE.MORE
+NOTE.ACCEPT
+Please choose your next step.
+I would like to quite now.
+INQUIRE.NEXT
+I hope you had a good experience. Bye.
+COMPLETE
+END

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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='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='no  Shuo Zhang  Bloomberg  London, United Kingdom  szhang611@bloomberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  CCS CONCEPTS  Information systems → Recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  KEYWORDS  Conversational recommender systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='  2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+page_content=' ACM, New York, NY, USA, 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' https:  //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='org/10.' 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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='00  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='1145/3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Thus, unlike others that operate in  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='com/iai-group/DialogueKit  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='com/iai-group/UserSimCRS  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='05544v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  Tseng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  updating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+page_content='  User modeling consists of three sub-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' The preference  model captures users’ likes and dislikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=',  temporal (time of the day and weekday vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' weekend), relational  (alone vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' group setting), or conversational (user satisfaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' February 27-March 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Singapore  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='DialogueKit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='UserSimCRS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Dialogue connector  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Platforms  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Utilities  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Evaluator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='NLU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Entity extractor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Intent classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Satisfaction classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='NLG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Template-based NLG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Core components  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Annotation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Intent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Dialogue  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Utterance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Domain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='…  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Agenda-based  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='simulator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Interaction model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='User modeling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Preference model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Context model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Persona  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Items  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Conditional NLG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Participant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Agent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='User  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Item  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='Ratings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='some of the core modules highlighted (in white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='  machine-machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' 1, specifically, response generation and user modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=', weekday vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' weekend,  alone vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' group setting) and personas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=', patient and impa-  tient users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=', the ratings dataset given in Step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='  REFERENCES  [1] Krisztian Balog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Conversational AI from an Information Retrieval Perspec-  tive: Remaining Challenges and a Case for User Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+page_content='  [2] Krisztian Balog and Tom Kenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Personal Knowledge Graphs: A Research  Agenda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of ICTIR ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 217–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [3] Krisztian Balog, David Maxwell, Paul Thomas, and Shuo Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Report on  the 1st Simulation for Information Retrieval Workshop (Sim4IR 2021) at SIGIR  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' SIGIR Forum 55, 2, Article 10 (dec 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [4] Tanja Bunk, Daksh Varshneya, Vladimir Vlasov, and Alan Nichol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  DIET:  Lightweight  Language  Understanding  for  Dialogue  Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='09936 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='CL]  [5] Konstantina Christakopoulou, Filip Radlinski, and Katja Hofmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Towards  Conversational Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of KDD ’16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 815–824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [6] Chongming Gao, Wenqiang Lei, Xiangnan He, Maarten de Rijke, and Tat-Seng  Chua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Advances and Challenges in Conversational Recommender Systems:  A Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' AI Open 2 (2021), 100–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [7] Javeria Habib, Shuo Zhang, and Krisztian Balog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' IAI MovieBot: A Conver-  sational Movie Recommender System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of CIKM ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 3405–3408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [8] Eugene Ie, Chih wei Hsu, Martin Mladenov, Vihan Jain, Sanmit Narvekar, Jing  Wang, Rui Wu, and Craig Boutilier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' RecSim: A Configurable Simulation  Platform for Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='04847 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='LG]  [9] Dietmar Jannach, Ahtsham Manzoor, Wanling Cai, and Li Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' A Survey  on Conversational Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' ACM Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 54, 5 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [10] Ivica Kostric, Krisztian Balog, and Filip Radlinski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Soliciting User Preferences  in Conversational Recommender Systems via Usage-Related Questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  of RecSys ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+page_content='  [11] Karl Krauth, Sarah Dean, Alex Zhao, Wenshuo Guo, Mihaela Curmei, Benjamin  Recht, and Michael I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Do Offline Metrics Predict Online Performance  in Recommender Systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='07931 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='LG]  [12] Wenqiang Lei, Xiangnan He, Yisong Miao, Qingyun Wu, Richang Hong, Min-Yen  Kan, and Tat-Seng Chua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Estimation-Action-Reflection: Towards Deep  Interaction Between Conversational and Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of  WSDM ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+page_content='  [13] Raymond Li, Samira Kahou, Hannes Schulz, Vincent Michalski, Laurent Charlin,  and Chris Pal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
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+page_content=' Towards Deep Conversational Recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of  NIPS ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 9748–9758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [14] Martin Mladenov, Chih-Wei Hsu, Vihan Jain, Eugene Ie, Christopher Colby,  Nicolas Mayoraz, Hubert Pham, Dustin Tran, Ivan Vendrov, and Craig Boutilier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' RecSim NG: Toward Principled Uncertainty Modeling for Recommender  Ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='08057 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='LG]  [15] Namkee Park, Kyungeun Jang, Seonggyeol Cho, and Jinyoung Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Use of  Offensive Language in Human-Artificial Intelligence Chatbot Interaction: The  Effects of Ethical Ideology, Social Competence, and Perceived Humanlikeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Hum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Behav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 121 (2021), 106795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [16] David Rohde, Stephen Bonner, Travis Dunlop, Flavian Vasile, and Alexan-  dros Karatzoglou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  RecoGym: A Reinforcement Learning Environ-  ment for the problem of Product Recommendation in Online Advertising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='00720 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='IR]  [17] Alexandre Salle, Shervin Malmasi, Oleg Rokhlenko, and Eugene Agichtein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  Studying the Effectiveness of Conversational Search Refinement Through User  Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of ECIR ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 587–602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [18] Jost Schatzmann, Blaise Thomson, Karl Weilhammer, Hui Ye, and Steve Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of NAACL ’07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 149–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' Knowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 21, 2 (June 2006), 97–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [20] Ivan Sekulić, Mohammad Aliannejadi, and Fabio Crestani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of  WSDM ’22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 888–896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' Tragos,  James Geraci, and Aonghus Lawlor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' PyRecGym: A Reinforcement Learning  Gym for Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of RecSys ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 491–495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of EMNLP-IJCNLP ’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  1990–2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of SIGIR ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2499–2506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Transferable  Dialogue Systems and User Simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of ACL ’21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 152–166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [25] Shuo Zhang and Krisztian Balog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Evaluating Conversational Recommender  Systems via User Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of KDD ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 1512–1520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' In  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of SIGIR ’22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 133–143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' Bruce Croft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='  In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of CIKM ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 177–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='  In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of ACL ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 142–149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  [29] Jie Zou, Yifan Chen, and Evangelos Kanoulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Towards Question-Based  Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' of SIGIR ’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' 881–890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  Hi there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' I am IAl MovieBot, your movie recommending buddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+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'}
+page_content=' ",  Hello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  DISCLOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='  DISCLOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='NON-DISCLOSE  How about Crime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='ELICIT  There are almost 5300 crime films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='Can you give  me a few keywords?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='ELICIT  Do you like: Cat City 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+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'}
+page_content='  INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='ITEMINFO  藍  INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='MORE  Director name  Its directed by Béla Ternovszky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='MORE  I like this recommendation  DISCLOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='MORE  NOTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='ACCEPT  Please choose your next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  I would like to quite now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  INQUIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='NEXT  I hope you had a good experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content=' Bye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}
+page_content='  COMPLETE  END' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E5T4oBgHgl3EQfUg90/content/2301.05544v1.pdf'}

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+A Practical Runtime Security Policy
+Transformation Framework for Software Defined
+Networks
+Yunfei Menga, Changbo Keb, Zhiqiu Huangc, Guohua Shenc, ChunQiang Liua,
+Xiaojie Fenga
+aCollege of Information Engineering, Qingdao Binhai University, Qingdao 266555, China
+bSchool of Computer Science and Technology, Nanjing University of Posts and
+Telecommunications, Nanjing 210023, China
+cCollege of Computer Science and Technology, Nanjing University of Aeronautics and
+Astronautics, Nanjing 211106, China
+Abstract
+Software-defined networking (SDN) has been widely utilized to enforce the se-
+curity of traditional networks, thereby promoting the process of transforming
+traditional networks into SDN networks. However, SDN-based security enforce-
+ment mechanisms rely heavily on the security policies containing the underlying
+information of data plane, such as MAC address, IP address or switch ports.
+These security policies need to be specifically developed by the network opera-
+tors, and loaded into the control plane by manual inputting. With increasing the
+scale of underlying network, the current security policy management mechanism
+will confront more and more challenges. The security policy transformation for
+SDN networks is to research how to transform the high-level security policy
+without containing the underlying information of data plane into the practical
+flow entries used by the OpenFlow switches automatically, thereby implement-
+ing the automation of security policy management. Based on this insight, a
+practical runtime security policy transformation framework is proposed in this
+paper. First of all, we specify the security policies used by SDN networks as a
+system model of security policy (SPM). From the theoretical level, we establish
+the system model for SDN network and propose a formal method to transform
+SPM into the system model of flow entries automatically. From the practical
+level, we propose a runtime security policy transformation framework to solve
+the problem of how to find a connected path for each relationship of SPM in
+the data plane, as well as how to generate the practical flow entries according
+to the system model of flow entries. In order to validate the feasibility and ef-
+fectiveness of the framework, we set up an experimental system and implement
+the framework with POX controller and Mininet emulator. The experimental
+results illustrate the framework can synchronously perceive the changes caused
+by cutting down one edge or changing SPM, and keep the data plane holding
+the security properties defined by SPM continuously at runtime.
+Keywords:
+SDN, security policy, model transformation, data plane.
+Preprint submitted to Elsevier
+January 11, 2023
+arXiv:2301.03790v1  [cs.CR]  10 Jan 2023
+
+1. Introduction
+Software-defined networking (SDN) is a novel networking technique or ar-
+chitecture that changes the limitation of traditional network infrastructures by
+breaking the vertical integration, decoupling the control logics from the underly-
+ing forwarding devices, promoting the centralization of control and introducing
+the abilities to program the network directly[1]. In SDN networks, the control
+logics of network, such as routing, traffic engineering or security policy devel-
+oped in the application plane, are loaded into the control plane via the north-
+bound interfaces (NBI) and transformed into a set of forward entries used by
+the OpenFlow switches. After that, the control plane distributes the generated
+flow entries to the associated switches in the data plane via the southbound
+interfaces (SBI). Because of its programmable, centralized intelligent control
+as well as global traffic view, SDN has been widely utilized to enhance the
+security of tradition networks, thereby promoting the process of transforming
+traditional networks into SDN networks. For instances, Garay et al.[2] proposed
+a SDN-based network access control mechanism, flownac, which is a centralized
+EAP (extensible authentication protocol) for IEEE 802.1x wireless local area
+network (WLAN). Yakasai et al.[3] proposed a network access control mech-
+anism, flowidentity. This mechanism integrates EAP security authentication
+mechanism into the SDN controller. Hu et al.[4] proposed a dynamic firewall
+mechanism, flowguard, based on SDN. Koerner et al.[5] proposed a device se-
+curity authentication mechanism based on MAC address and SDN.
+However, SDN-based security enforcement mechanisms rely heavily on the
+security policies containing the underlying information of the data plane, such
+as MAC address, IP address or switch ports. These security policies need to
+be specifically developed by the network operators, and loaded into the control
+plane by means of the manual inputting. With increasing the scale of underlying
+network, the current security policy management mechanism will confront more
+and more challenges. First of all, it is nearly impossible for any operator to
+completely understand all the information of underlying network. In addition,
+with the emergence of multi-controller SDN[6], network operators need to man-
+age a variety of heterogeneous controllers at the same time. In this case, the
+same security policy often needs to be developed and deployed for the different
+types of controller, which inevitably increases the complexity and difficulty for
+network management. Therefore, a novel security policy management mecha-
+nism which can be completely transparent to the underlying information of data
+plane is urgently needed for SDN networks. That is, it can permit the operators
+only to define the high-level security policy without containing any underlying
+information, then by means of the security policy transformation, the high-level
+security policy can be automatically transformed into its corresponding flow
+entries used by the OpenFlow switches in the data plane, thereby implementing
+the automation of security policy management for SDN networks.
+2
+
+Based on these insights, we have proposed a security policy model transfor-
+mation and verification approach for SDN networks and published the approach
+in our previous paper[7]. In that paper, we proposed a security policy transfor-
+mation method to transform the high-level security policy model (SPM) without
+containing the underlying information into its corresponding low-level security
+policy model (LSPM) containing the underlying information.
+To verify the
+soundness of proposed security policy model transformation method, we further
+proposed a security policy verification method and proved that the problem of
+whether the data plane can satisfy the security properties defined by SPM is
+equivalent to the problem of searching the connected paths related with SPM in
+the data plane, that is, as long as each access control relationship Ri ∈ SPM can
+be transformed into a corresponding connected path Pi in the data plane, next
+transforms Pi into a set of flow entries used by the OpenFlow switches, then
+the data plane must can hold the security properties defined by SPM. However,
+that paper only proposed the method from the theoretical level, and did not
+specifically implement this method. Moreover, it did not solve the problem of
+how to find a connected path for each relationship of SPM in the data plane,
+and how to transform LSPM into the practical flow entries used by the switches.
+And based on the theoretical foundation of that paper, we propose a run-
+time security policy transformation framework for SDN networks in this paper.
+First of all, this paper further improves the system model of SDN networks and
+solves the problem of how to transform SPM into the flow entries used by the
+OpenFlow switches from the theoretical level. Moreover, this paper proposes
+a runtime security policy transformation framework from the practical level,
+thereby solving the problem of how to find a connected path Pi for each rela-
+tionship Ri ∈ SPM in the data plane, as well as how to transform the system
+model of flow entries into the practical flow entries used by the switches at run-
+time. In addition, this paper further implements the proposed framework with
+an experimental system. The experimental result illustrate the framework is
+completely effective at runtime.
+Hence, the contributions of this paper can be summarized as follows:
+• We specify the security policies used by SDN networks, such as access
+control policies or firewall policies, as a system model of security policy (SPM).
+SPM is of a high-level system model without containing any underlying infor-
+mation of data plane.
+• From the theoretical level, we establish the system model for SDN network,
+and propose a formal method to transform SPM into the system model of flow
+entries automatically. The system model of flow entry is of a low-level system
+model containing the underlying information of data plane.
+• From the practical level, we propose a runtime security policy transforma-
+tion framework which consists of the security policy module, topology discovery
+module, runtime monitoring module, path generation module and flow entry
+generation module. Leveraging these functional modules, the framework can
+solve the problem of how to find a connected path for each relationship of SPM
+in the data plane, how to transform the path into the system model of flow en-
+tries, as well we how to generate the practical flow entries by using the system
+3
+
+model of flow entries.
+• In order to validate the feasibility and effectiveness of the framework, we
+set up an experimental system and implement the framework by using POX con-
+troller and Mininet emulator. The experimental result illustrate the framework
+is completely effective at runtime.
+The remainder of this paper is structured as follows. Section 2 discusses
+some related works. Section 3 proposes the system model and elaborates on
+how to transform SPM into the flow entries from the theoretical level. Sec-
+tion 4 proposes the runtime security policy transformation framework from the
+practical level and introduces its functional modules. Section 5 implements the
+framework with an experimental system and elaborates on how to evaluate the
+effectiveness and performance of the framework. Finally, Section 6 concludes
+this paper and presents some future directions.
+2. Related Work
+In this section, we discuss some research works related with the policy model
+transformation and the security policy verification.
+2.1. Policy Model Transformation
+According to the definitions of model-driven architecture (MDA), the model
+transformation refers to the process of transforming the platform independent
+model (PIM) to its corresponding platform specific model (PSM)[8]. As far as
+the literatures we have read, the researches towards the policy model transforma-
+tion can be roughly divided into three categories, they are the template-based
+transformation, RBAC-oriented transformation as well as the transformation
+based on the system model and mapping rules[9]. Due to the limitation of tem-
+plate, the template-based model transformation has very limited transformation
+capability. Generally, RBAC-oriented model transformation[10] is only suitable
+for transforming RBAC (role-based access control) policies, and does not have
+enough capability to describe the complex system, so that these two methods
+are not suitable for SDN networks.
+At present, the model transformation based on the system model and map-
+ping rules has been widely used for transforming the policy models. The main
+idea of this method can be summarized as follows: (1) System Model: it de-
+fines the objects of system and the relationship between the system objects;
+(2) Policy Model: it defines the policy object and the relationship between the
+policy objects; (3) Mapping Rules: it establishes the mapping rules between the
+upper-level policy objects and the lower-level system objects[11][12]. The trans-
+formation based on the system model and mapping rules first establishes the
+policy model and the system model which can describe the underlying system,
+then establishes the mapping rules between the policy objects and the system
+objects, then transforms the upper-level policy model into its corresponding
+lower-level policy model by means of the established mapping rules. In partic-
+ular, Davy et al.[13] proposed a policy model transformation method based on
+4
+
+mapping rules, in which the policy model is defined as a tuple ( event, condi-
+tion, behavior, subject, object ) and used the ontology to establish the mapping
+rules between the different system layers. Luck et al.[14] proposed a method
+to transform RBAC model defined in service layer into the policy model used
+in the system layer. In this method, the system model is divided into three
+layers: roles and object (RO), subject and resources (SR) and processes and
+hosts (PH), and the mapping rules between the three layers have been estab-
+lished. Based on the Luck’s research, Porto et al.[15] further decomposes the
+PH layer into two sub layers, namely DAS (diagram abstract subsystem) layer
+and PH layer. DAS layer is mainly used to describe the network topology in
+the original PH layer, while PH layer is used to describe the specific network
+information in DAS layer. In addition, the authors also proposed a policy ver-
+ification framework, which can be used to verify the consistency problems in
+the process of policy transformation. In addition, Lampson et al.[16] proposed
+a network policy model transformation method for the distributed computing
+environment. Maullo et al.[17] proposed a policy transformation system based
+on the first-order predicate logics, which transforms the high-level policy model
+into the low-level network configuration policy through the network topology
+and other information. Nanxi et al.[18] proposed a SDN-oriented access control
+policy transformation framework. In this paper, In this paper, we also propose a
+security policy transformation framework based on the system model and map-
+ping rules. We first establish the system model of security policy (SPM) and
+data plane, then establish the transformation rules between the policy objects
+of SPM and the objects of the data plane, thereby transforming SPM into the
+system model of flow entries automatically.
+2.2. Security Policy Verification
+To assure the information systems running securely, security mechanisms
+of information system need to be validated whether it can satisfy the security
+properties defined by the security policy. The traditional validation methods
+based on the testing and simulation can only confirm the system can work prop-
+erly under the different testing scenarios, but it is difficult to find some hidden
+scenarios that occur with little probability. Formal verification methods have
+been applied to overcome the shortcomings existed in the traditional valida-
+tion methods. At present, the formal verification methods for validating the
+security policy mainly include theorem proving and model checking[19]. The-
+orem proving is unsuitable to validate the properties of complex systems due
+to its lower efficiency. Model checking[20] can be used to validate whether the
+system model can satisfy the expected dynamic behaviors and specific static
+properties.
+Model checking technique has been widely used for the security
+policy verification. For instances, Al-Shaer et al.[21] proposed a static policy
+inconsistency detection method for the firewall policies of network. Bandara et
+al.[22] proposed a security policy verification framework based on event calcu-
+lus (EC) and used the reasoning techniques for the policy conflict identification.
+May et al.[23] verified the privacy policies by means of an asynchronous model
+checker. Rubio-Loyola et al.[24] proposed a goal-oriented policy refinement and
+5
+
+conflict detection method by means of the model checking technique and linear
+temporal Logic (LTL). Graham et al.[25] proposed a policy conflict detection
+method with the model checking and an extended decision table. Baliosan and
+Serrat[26] proposed a specific finite automata based method for the policy con-
+flict detection.
+3. Problem Formalization
+The security policy transformation for SDN networks is to research how to
+transform the high-level security policy without containing the underlying infor-
+mation into the set of practical flow entries used by the OpenFlow switches in
+the data plane automatically, thereby implementing the automation of security
+policy management in SDN network. In the following of this section, we first
+establish the system model for SDN network, then propose a formal method to
+transform the security policy (SPM) into the system model of flow entries from
+the theoretical level.
+3.1.
+System Model
+Definition 1. (Security Policy):
+The high-level security policy is defined as
+a finite set of access control relationships: SPM = { R0, R1,...,Rn | ∀ Ri = (
+si, oj, a ) }, where si ∈ S represents the subject of the relationship, oj ∈ O
+represents the object of the relationship, a represents the access authorization,
+i.e., the subject can access the object.
+Definition 2. (Host):
+The host existed in the data plane is defined as a tuple:
+hi=( ipi, swi, portm
+i
+), where ipi represents the host’s IP address in the data
+plane, swi represents the OpenFlow switch connected with the host, portm
+swi
+represents the port connected with the host in swi.
+Definition 3. (OpenFlow Switch):
+The OpenFlow switch existed the data
+plane is defined as a finite set of flow entries: swi = { f0, f1, ..., fn }.
+Definition 4. (Flow Entry):
+The flow entry existed in the OpenFlow switch
+is defined as a tuple: fi = ( ipsrc, ipdst, portin
+swi =⇒ portout
+swi ), where ipsrc
+represents the traffic’s source IP address, ipdst represents the traffic’s destination
+IP address, portin
+swi =⇒ portout
+swi represents the traffic input from portin
+swi will be
+outputted from portout
+swi in the switch swi.
+Definition 5. (Edge):
+The edge existed in the data plane is defined as: ei
+= portout
+start �−→ portin
+end, where portout
+start represents the port connected with the
+edge in the start switch swstart, portin
+end represents the port connected with the
+edge in the end switch swend, so that the direction of the edge is from swstart
+to swend.
+Definition 6. (Topology):
+The topology of the data plane is defined as an
+graph: G = ( SW, E ), where SW represents a finite set of OpenFlow switches,
+E represents a finite set of edges.
+6
+
+Definition 7. (Connected Path):
+The connected path between the host hi
+and the host hj in the topology G is defined as: Pi = hi �−→ hj = { hi, e0, e1,...,
+en, hj | ∀ei ∈ E }, where the direction of the path is from hi to hj.
+3.2. Transforming SPM into the Flow Entries
+Based on the established system model, we propose a formal method to
+transform SPM into the system model of flow entries from the theoretical level.
+The method can be described as Figure 1 and summarized as follows: First of all,
+for ∀Ri ∈ SPM, the subject si ∈ Ri is transformed into a host h(si) in the data
+plane, the object oj ∈ Ri is transformed into a host h(oi) in the data plane;
+Next, the access authorization a ∈ Ri is transformed into a connected path
+Pi between h(si) and h(oj); After that, the connected path Pi is transformed
+into a set of flow entries used by the switches which are passed by Pi; Finally,
+SPM is transformed into the set of flow entries ∆ when all relationships of
+SPM have been transformed.
+As shown in Figure 1, the relationship R1 ∈
+SPM is transformed into the connected path P1 between h1 and h2, thus P1
+is transformed into the flow entries deployed in sw1, so as to implement the
+security policy transformation from the system model level. The soundness of
+the formal method has been proven in our previous paper[7]. That is, if the
+security properties defined by SPM is denoted as ϕ, the system model of data
+plane is denoted as D, the flow entries generated by the method is denoted as ∆,
+then the method can ensure the data plane D loaded with ∆ can synchronously
+and continuously hold the security properties ϕ at runtime, i.e., D(∆) |= ϕ.
+Specifically, the formal method to transform SPM into the system model of
+flow entries is defined as follows.
+• Transforming the Subject Given an access control relationship Ri ∈
+SPM and the subject si ∈ Ri, then si is transformed into a corresponding
+host existed in the data plane. The rule of transforming the subject of SPM is
+formally defined as follows:
+si ∈ Ri
+h(si) = (ipsrc, swsrc, portin
+src, )
+(1)
+where h(si) represents the host mapped with si in the data plane, ipsrc repre-
+sents the IP address of h(si) in the data plane, swsrc represents the OpenFlow
+switch connected with h(si) in the data plane, portin
+src represents the port con-
+nected with h(si) in swsrc.
+•
+Transforming the Object Given an access control relationship Ri ∈
+SPM and the object oj ∈ Ri, then oj is transformed into a corresponding host
+existed in the data plane. The rule of transforming the object of SPM is formally
+defined as follows:
+oj ∈ Ri
+h(oj) = (ipdst, swdst, portout
+dst, )
+(2)
+where h(oj) represents the host mapped with oj in the data plane, ipdst repre-
+sents the IP address of h(oj) in the data plane, swdst represents the OpenFlow
+7
+
+Figure 1:
+The method of transforming SPM into the system model of flow entries.
+8
+
+-
+Si
+R1
+S
+0
+Application PN
+h3
+P1
+sw1
+h1
+h2
+sw2switch connected with h(oj), portout
+dst represents the port connected with h(oj)
+in swdst.
+• Transforming the Authorization Given an access control relationship
+Ri ∈ SPM and the access authorization a ∈ Ri, if there existing a connected
+path Pi between h(si) and h(oj) in the topology, then a is transformed into the
+connected Pi. The rule of transforming the authorization is formally defined as
+follows:
+a ∈ Ri
+h(si) �−→ h(oj) ⊂ G
+(3)
+where h(si) �−→ h(oj) represents a directional connected path from h(si) to
+h(oj) in the topology.
+• Transforming the Path The connected path h(si) �−→ h(oj) is trans-
+formed into a set of flow entries deployed in the OpenFlow switches which are
+passed by the path. The rule of transforming the connected path is formally
+defined as follows:
+h(si) �−→ h(oj)
+dst
+�
+k=src
+fi(k)
+(4)
+where fi(k) =( ipsrc, ipdst, portin
+k =⇒ portout
+k
+) represents a flow entry deployed
+in the switch swk which is passed by h(si) �−→ h(oj). Leveraging the definitions
+of the system model, we can proof that the rule of transforming the connected
+path is sound.
+proof :
+h(si) �−→ h(oj)
+= { h(si), e0, e1, ..., en, h(oj) }
+= {(ipsrc, swsrc, portin
+src), (portout
+src �−→ portin
+0 )
+, ..., (portout
+n
+�−→ portin
+dst), (ipdst, swdst, portout
+dst)}
+= {(portin
+src =⇒ portout
+src), (portin
+0 =⇒ portout
+0
+),
+, ..., (portin
+n =⇒ portout
+n ), (portin
+dst =⇒ portout
+dst)
+(ipsrc, ipdst), (swsrc, swdst)}
+= {
+dst
+�
+k=src
+(portin
+k =⇒ portout
+k )} � (ipsrc, ipdst)
+=
+dst
+�
+k=src
+{(portin
+k =⇒ portout
+k ) � (ipsrc, ipdst)}
+=
+dst
+�
+k=src
+fi(k).
+Therefore, ∀Ri ∈ SPM, the subject si ∈ Ri, the object oj ∈ Ri and the
+autherization a ∈ Ri, if ∃Pi = h(si) �−→ h(oj) in the topology, then Ri can
+be transformed into a corresponding set of flow entries
+dst
+�
+k=src
+fi(k) by using the
+system model step by step, it can be formally defined as follows:
+Ri
+dst
+�
+k=src
+fi(k)
+(5)
+9
+
+•
+Transforming SPM SPM is transformed into a corresponding set of
+flow entries ∆ by using the equation (5). If || SPM || = m, then the rule of
+transforming SPM is formally defined as follows:
+∆ =
+m
+�
+i=1
+{
+dst
+�
+k=src
+fi(k)}
+(6)
+4. The Security Policy Transformation Framework
+The problem of how to transform SPM into the corresponding flow entries
+used by the OpenFlow switches has been solved from the theoretical level in
+Section 3. However, we cannot solve the problem of how to find a connected
+path in the data plane for each relationship Ri ∈ SPM. In addition, by means
+of the security policy transformation method, SPM can be transformed into the
+corresponding flow entries, but the flow entry transformed from SPM is of the
+system model, i.e. it is only the formal description of the real flow entry and
+cannot be used by the real OpenFlow switch directly, so that we need to further
+solve the problem of how to generate the practical flow entries by using the
+system model of flow entries. Based on this insight, a runtime security policy
+transformation framework for SDN networks is proposed from the practical level
+in this section. By means of the framework, we can solve the problem of how
+to find the connected path for each relationship defined by SPM, as well as
+the problem of how to generate the practical flow entries based on the system
+model at runtime. As shown in Figure 2, this framework consists of 5 functional
+modules, i.e., the security policy module, the topology discovery module, the
+runtime monitoring module, the path generation module and the flow entry
+generation module.
+4.1. Overview of the Framework
+• Security policy module is deployed in the application plane and responsible
+for maintaining the security policy (SPM). Each access control relationship Ri ∈
+SPM is designed as a 3-tuple: ( si, oj, fixed ) based on the system model, where
+si is the subject; oj is the object; fixed={ 0, 1 } is a tag bit, fixed=1 represents
+the relationship has been updated by user, fixed=0 represents it is unchanged.
+SPM is stored as a text document and can be updated by the user at runtime.
+• Topology discovery module is deployed in the control plane and responsible
+for creating a dynamic real-time topology of the entire data plane by capturing
+the link events transmitted from the OpenFlow switches. Based on the system
+model, each edge in the topology is designed as a tuple: ei = ( swsrc, portsrc,
+swdst, portdst, using, c ), where using={ True, False } is a tag bit and can
+be changed by the real-time link events at runtime, using=True represents the
+edge can be used now, using=False represents the edge is interrupted now. For
+the convenient of researching, the cost of each edge is set to 1, i.e., c =1. The
+generated topology is stored as a text document and can also be updated by
+the real-time link events at runtime.
+10
+
+Figure 2:
+The security policy transformation framework for SDN networks.
+• Runtime monitoring module is deployed in the control plane and respon-
+sible for monitoring all the traffics in the data plane by capturing the real-time
+packet-in events transmitted from the OpenFlow switches. When a new packet-
+in event arrives in the controller, the module first invokes the path generation
+module to transform the latest security policy (SPM) into a set of connected
+paths in the data plane, then invokes the flow entry generation module to trans-
+form all the connected paths into their corresponding flow entries deployed in
+the OpenFlow switches which are passed by these paths. As SPM and the topol-
+ogy of data plane will be evolved with the runtime environment, this module is
+designed to be triggered by the real-time packet-in events continuously, so that,
+when SPM is changed (i.e., occurring fixed=1) or the topology is changed (i.e.,
+occurring using=False) at runtime, the module will first delete all the current
+flow entries deployed in the OpenFlow switches, then update all the flow tables
+by using the latest generated flow entries, so as to ensure the security properties
+defined by SPM can be synchronously and continuously hold in the data plane
+at runtime.
+• Path generation module is deployed in the control plane and invoked by the
+runtime monitoring module at runtime. The module is responsible for trans-
+forming each Ri ∈ SPM input from the runtime monitoring module into a
+corresponding path Pi in the data plane by using the latest topology file and
+the path searching algorithm. Specifically, it first transforms the subject si ∈ Ri
+and the object oj ∈ Ri into the hosts h(si) and h(oj) in the data plane respec-
+11
+
+Security Policy
+Update
+SPM
+Module
+Application Plane
+SPM
+Path
+Topology
+-Topology
+Generation
+Module
+Control Plane
+Update
+Invoke
+Path
+Topology
+Flow Entry
+Runtime
+Discovery
+Generation
+Path
+Monitoring
+Module
+Module
+Module
+Link Events
+Flow Entries
+Packet-In Events
+OpenFlow Switches
+Data Planetively, then finds a shortest connected path Pi between h(si) and h(oj) by using
+the path searching algorithm, finally all the connected paths transformed from
+SPM are returned to the runtime monitoring module.
+• Flow entry generation module is also deployed in the control plane and in-
+voked by the runtime monitoring module at runtime. The module is responsible
+for transforming the connected path into a set of flow entries deployed in the
+OpenFlow switches which are passed by the path, then utilizing the instructions
+provided by the controller to generate the practical flow entries and distributing
+these flow entries to the corresponding OpenFlow switches at runtime.
+4.2. Runtime Monitoring Algorithm
+The runtime monitoring algorithm deployed in the runtime monitoring mod-
+ule plays the role of coordinator in the framework, and can be described as Al-
+gorithm 1 in pseudo code. First of all, the algorithm creates two dynamic lists
+S and T by reading the latest SPM and Topo file respectively. If there existing
+an access control relationship has been changed by user (Ri.fixed=1) or a edge
+has been shut down in the topology (ei.using=False) at runtime, it will clear
+all the current flow entries deployed in the OpenFlow switches for ready of the
+updating. In the following, for each access control relationship Ri ∈ S, it maps
+the subject si ∈ Ri and the object oj ∈ Ri with the switches swsrc and swdst
+in the data plane, then transforms Ri into a corresponding connected path Pi
+by invoking the path searching algorithm djk-route(swsrc, swdst, N). Based
+on the transformation rules, Pi can be further transformed into a set of flow
+entries. When all the relationships in the List S having been transformed, SPM
+has been transformed into a corresponding set of flow entries ∆, the algorithm
+invokes the flow entry generation module to update the data plane by using
+∆. Since the algorithm is designed to be triggered by the packet-in events at
+runtime, so that it ensures the framework can perceive any changes in time
+when the security policy or the topology of data plane has been evolved with
+the environment, and then update the data plane synchronously at runtime.
+4.3. Path Searching Algorithm
+Another important algorithm in the framework is the path searching algo-
+rithm deployed in the path generation module. The algorithm is improved from
+the classic Dijkstra algorithm and can be described as the Algorithm 2 in pseudo
+code. First of all, the algorithm creates a dynamic matrix djk[N][N] by using
+the sum of Openflow switches N and the topology file. In the following, it cal-
+culates a shortest connected path between swsrc and swdst in the data plane
+by using djk[N][N] and the created stacks. After multi-round calculating, the
+shortest path Pi between swsrc and swdst is found and returned to the runtime
+monitoring module. As the cost of each edge has been set to 1 and not consider-
+ing of the quality of services (QoS) of edges, so that the shortest path Pi found
+by the algorithm is generated by calculating the minimum number of hops in
+the topology. Moreover, since the matrix djk[N][N] is dynamically created by
+the topology file, so that the searched shortest path will be evolved with the
+changing of the topology at runtime.
+12
+
+5. Implementation and Evaluations
+In order to validate the feasibility and effectiveness of the framework pro-
+posed in Section 4, we set up an experimental system and implement the frame-
+work with POX controller[27] and Mininet emulator[28]. First of all, we im-
+plement a virtual SDN network by using the Mininet emulator. As shown in
+Figure 3, the topology of the network consists of 6 hosts (h1 ∼ h6) and 11 Open-
+Flow switches (sw1 ∼ sw11). We further implement the security policy module,
+topology discovery module, runtime monitoring module, path generation mod-
+ule and flow entry generation module with Python 3.6.1 and integrate these
+modules with the core of POX controller. The experimental system consists of
+a Lenovo workstation with Windows OS, Intel-i7 32Cores 2.60GHz CPU, 32GB
+RAM and a Raspberry platform with Linux OS, ARM-v7 CPU and 945MB
+RAM. The POX controller and the functional modules are deployed in Lenovo
+workstation, Mininet emulator is deployed in Raspberry platform, and Lenovo
+workstation is connected with Raspberry platform using coaxial cable directly.
+Table 1: The high-level security policy (SPM)
+R1
+( 1, 5, 1 )
+R2
+( 5, 1, 1 )
+R3
+( 2, 4, 1 )
+R4
+( 4, 2, 1 )
+5.1. Effectiveness Evaluation
+The security policy used for validating the effectiveness of the framework is
+shown in Table I. Since any effective interaction is bidirectional in SDN networks,
+i.e., the subject’s host and the object’s host must be ensured they can access
+each other in the data plane, so that we design the security policy as 4 access
+control relationships (R1 ∼ R4) to ensure h1 (1) and h5 (5) can access each
+other, h2 (2) and h4 (4) can access each other, and all the relationships of
+SPM are set as fixed=1, i.e., having been updated by user. In the following,
+the effectiveness evaluations towards the framework will be carried out under 4
+different scenarios at runtime, they are the effectiveness after loading the flow
+entries, the effectiveness after cutting down the path and the effectiveness after
+changing SPM.
+5.1.1. Effectiveness after Loading the Flow Entries
+The purpose of this experiment is to validate whether the data plane after
+loading the generated flow entries can hold the security properties defined by
+SPM. First of all, the subjects and objects of SPM shown in Table I, i,e., the
+1, 2, 4 and 5, are transformed into their corresponding hosts in the data plane
+by using security policy transformation. Specifically, the 1 is transformed into
+13
+
+Figure 3:
+The topology of the virtual SDN network.
+h1=( 10.0.0.1, sw1, 1 ), the 2 is transformed into h2=( 10.0.0.2, sw2, 1 ), the 4
+is transformed into h4=( 10.0.0.4, sw9, 1 ) and the 5 is transformed into h5=(
+10.0.0.5, sw10, 1 ) respectively. In the following, the path searching algorithm,
+i.e., Algorithm 2, searches the shortest path between the subject’s host and the
+object’s host for each Ri ∈ SPM based on the latest topology generated from
+the topology file.
+After that, the relationships shown in Table I have been transformed into 4
+corresponding shortest connected paths (P1 ∼ P4) in the data plane. As shown
+in Figure 4, the R1 is transformed into P1={ h1, e1, e2, e3, h5 }, the R2 is
+transformed into P2={ h5, e3, e2, e1, h1 }, the R3 is transformed into P3={ h2,
+e4, e5, h4 } and the R4 is transformed into P4={ h4, e5, e4, h2 }, where the
+P1 and P2 are depicted with the blue lines, the P3 and P4 are depicted with
+the orange lines. In the following, the flow entry generation module transforms
+each path into a set of flow entries deployed in the switches passed by the path.
+Specifically, P1 and P2 are transformed into the flow entries deployed in the
+switches { sw1, sw5, sw8, sw10 }, while P3 and P4 are transformed into the flow
+entries deployed in the switches { sw2, sw6, sw9 }.
+After that, we execute the pingall instruction in the Mininet CLI and observe
+the reachability of the entire data plane. As shown in Figure 5, h1 and h5 can
+access each other, h2 and h4 can access each other either, so that the data plane
+after loading the generated flow entries has been proven that it can hold all the
+security properties defined by SPM.
+14
+
+Figure 4:
+The shortest paths searched by Algorithm 2.
+Figure 5:
+The result of executing the pingall instruction.
+15
+
+1
+e
+es
+e3pi@raspberrypi:/mininet/custom
+口
+X
+s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11
+*** Adding links:
+(hl, sl) (h2, s2) (h3, s7) (h4, s9) (h5, s10) (h6, sl1) (sl, s2) (sl, s3) (sl, s
+5) (s2, s4) (s2, s6) (s3, s4) (s3, s7) (s4, s5) (s4, s7) (s4, s8) (s5, s6) (s5,
+s8) (s5. s9) (s6, s9) (s7. s8) (s7. s10) (s8. s9) (s8. s10) (s8. s11) (s9, s11)
+(s10, sl1)
+*** Configuring hosts
+hl h2 h3 h4 h5 h6
+*** starting controller
+c0
+*** starting ll switches
+s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11
+*** Starting CLI:
+mininet> pingall
+*** Ping: testing ping reachability
+hl 
+-> X X X h5 X
+h2
+-> x x h4 x x
+13
+XXXXX<-
+h4 -> x h2  X X
+n5 -> hl x x x X
+X X X X X <- 9u
+*** Results: 86% dropped (4/30 received)
+mininet>
+mininet>5.1.2. Effectiveness after Cutting down the Path
+The purpose of this experiment is to validate whether the security properties
+defined by SPM can be continuously hold in the data plane after the connected
+path between the subject’s host and the object’s host is shut down at runtime.
+If the framework can synchronously perceive this change from the topology
+and automatically find another new connected path to keep the data plane
+holding the security properties at runtime, then the framework will be proven
+to be effective under this scenario. First of all, we let h1 and h5 can access
+each other by loading the corresponding flow entries into the data plane, and
+the shortest path between h1 and h5 is initialed with { h1, e1, e2, e3, h5 }.
+In the following, we make a continuous TCP traffic sent from h1 to h5 by
+using the iperf instruction in the Mininet CLI, we set the duration time of the
+experiment equals 60 seconds and record the throughput of the traffics in h5.
+When the time reaches 23 seconds, we shut down the edge between sw8 and
+sw10 existed in the path P1 by using the instruction in the Mininet CLI and let
+the experiment going on. When the experiment is finished, we read the data
+recorded in h5 and plot them in Figure 6. As shown in Figure 6, the throughput
+of the traffic sent from h1 to h5 is sharply declined after the edge is shut down at
+23 seconds, and completely becomes zero from 25 seconds to 37 seconds. After
+38 seconds, the traffic quickly returns to normal until the end of the experiment.
+The experimental result has illustrated that the framework can synchronously
+perceive the change caused by cutting down one edge between sw8 and sw10,
+and automatically find another new shortest connected path between h1 and
+h5, i.e., { h1, e6, e7, e8, h5 }, so as to make the traffic returning to normal
+quickly and keep the data plane holding the security properties defined by SPM
+at runtime. The new shortest connected path searched by Algorithm 2 after
+cutting down the using path is shown in Figure 7.
+5.1.3. Effectiveness after Changing SPM
+The purpose of this experiment is to validate whether the security properties
+defined by SPM can be continuously hold in the data plane after SPM is changed
+by the user at runtime. If the framework can synchronously perceive this change
+from SPM and automatically update the flow entries deployed in the switches
+to keep the data plane holding the evolved security properties at runtime, then
+the framework will be proven to be effective under this scenario. First of all,
+we let h1 and h5 can access each other by loading a corresponding flow entries
+into the data plane, then make a continuous TCP traffic sent from h1 to h5 by
+using the iperf instruction in the Mininet CLI, we set the duration time of the
+experiment equals 60 seconds and record all the throughput data of the traffic
+in h5.
+In the following, we first validate the effectiveness of the framework under
+the scenario of changing SPM from h1 and h5 can access each other to h2 and h5
+can access each other at runtime, and the experimental result under this scenario
+is plotted in Figure 8. As shown in Figure 8, the throughput of the traffic sent
+from h1 to h5, which is depicted with the red line, is quickly declined when we
+load the new SPM into the controller at 29 seconds, and completely becomes
+16
+
+Figure 6:
+The blue line represents the throughput of the traffic sent from h1 to h5. The
+dotted line represents we shut down the connected path at 23 seconds.
+Figure 7:
+The new connected path searched by Algorithm 2 at runtime.
+17
+
+Throughput (Gbits/sec)
+the traffic from h1 to h5
+-shut down the path at runtime
+20
+60
+Time (sec)e6
+eA
+1
+e7
+es
+es
+eszero after 31 seconds. From 31 seconds until to the end of the experiment, h5
+can only receive the traffic sent from h2 which is depicted with the blue line.
+The experimental result illustrates the framework can synchronously perceive
+this change and keep the data plane holding the evolved security properties after
+changing SPM from h1 and h5 can access each other to h2 and h5 can access
+each other at runtime.
+Figure 8:
+The red line represents the throughput of the traffic sent from h1 to h5, the blue
+line represents the throughput of the traffic sent from h2 to h5, the dotted line represents we
+change SPM at 29 seconds.
+After that, we further validate the effectiveness of the framework under the
+scenario of adding a new relationship into SPM, i.e., adding h2 and h5 can access
+each other, at runtime. The experimental result under this scenario is plotted
+in Figure 9. As shown in Figure 9, the throughput of the traffic sent from h1 to
+h5, which is depicted with the red line, still keeps normal before we load the new
+SPM into the controller at 43 seconds. From 44 seconds until to the end of the
+experiment, h5 can receive the continuous traffic sent from h2 which is depicted
+with the blue line, and can also receive the traffic sent from h1 at the same time.
+Due to the crowding of the traffic sent from h2, the throughput of the traffic
+from h1 is declined from 3.71GB/s to 1.99 GB/s. The throughput of the traffic
+sent from h2 is still kept between 1.6GB/s and 2.2GB/s after 44 seconds. The
+experimental result illustrates the framework can synchronously perceive this
+change and keep the data plane holding the evolved security properties after
+adding a new relationship into SPM at runtime.
+18
+
+Throughput (Gbits/sec)
+the traffic from h1 to h5
+the traffic from h2 to h5
+-load the new SPM at runtime
+Time (sec)Figure 9:
+The red line represents the throughput of the traffic sent from h1 to h5, the blue
+line represents the throughput of the traffic sent from h2 to h5, the dotted line represents we
+change SPM at 43 seconds.
+5.2. Performance Evaluation
+As the critical algorithm used for implementing the security policy trans-
+formation, the performance of the path searching algorithm, i.e., Algorithm 2,
+needs to be further evaluated. First of all, the sum of access control relation-
+ships of the security policy (SPM) is denoted as M, and the sum of OpenFlow
+switches in the topology is denoted as N in this performance evaluation. Then
+by leveraging the Python programming, the execution time of Algorithm 2 have
+been recorded in milliseconds (ms) for calculating the shortest paths under set-
+ting the different value of M and N.
+The experimental result is plotted in
+Figure 10. As shown in Figure 10, with gradually amplifying the value of M
+from 2 to 10, and the value of N from 11 to 400 respectively, the execution time
+of Algorithm 2 shows an obvious exponential upward trend. Moreover, accord-
+ing to the description of Algorithm 2, the time complexity for calculating only
+one shortest path will reach O(N 2), because the Algorithm 2 needs to create
+a dynamic matrix djk[N][N] and further calculates the while loop, so that the
+time complexity for transforming all the access control relationships defined by
+SPM into their corresponding shortest paths will reach O(M × N 2).
+6. Conclusion
+In this paper, we propose a practical runtime security policy transforma-
+tion framework for SDN networks. First of all, we specify the security policies
+used by SDN networks, such as access control policies or firewall policies, as a
+19
+
+Throughput (Gbits/sec)
+the traffic from h1 to h5
+一the traffic from h2 to h5
+-load the new SPM at runtime
+Time (sec)Figure 10:
+The execution time of Algorithm 2 recorded in milliseconds (ms) for calculating
+the shortest paths under setting the different value of M and N, where M represents the sum
+of access control relationships of SPM, N represents the sum of OpenFlow switches in the
+topology.
+system model of security policy (SPM). SPM is of a high-level system model
+without containing any underlying information of data plane. From the theoret-
+ical level, we establish the system model for SDN network and propose a formal
+method to transform SPM into the corresponding flow entries automatically.
+The flow entry transformed from SPM is of a low-level system model containing
+the underlying information of data plane. From the practical level, we propose
+a runtime security policy transformation framework which consists of the se-
+curity policy module, topology discovery module, runtime monitoring module,
+path generation module, as well as flow entry generation module. Leveraging
+these functional modules, the framework can solve the problem of how to find
+a connected path for each relationship defined by SPM in the data plane, how
+to transform the path into the system model of flow entries, as well as how to
+generate the practical flow entries by using the system model of flow entries.
+In order to validate the feasibility and effectiveness of the framework, we set
+up an experimental system and implement the framework by using POX con-
+troller and Mininet emulator. The experimental result illustrate the framework
+is completely effective at runtime.
+However, there still exists some problems needed to be further researched
+in the future. The current path searching algorithm, i.e., Algorithm 2, used
+by the framework is improved from the classic Dijkstra algorithm and finds
+the shortest path by calculating the minimum number of hops in the topology.
+However, in the real SDN networks, the problem of searching a connected path
+between the two hosts need to consider the matters of quality of service (QoS),
+20
+
+350
+when M =2
+when M =4
+M
+300-
+when M =6
+when M =8
+when M =10
+250
+(sw)
+e 200
+Performance
+W
+150
+P
+100 -
+50 -
++0
+0
+50 
+100
+150
+200
+250
+300
+350
+400
+Sum of OpenFlow Switches (N)load balance, as well as some specific requirements about the traffic engineering
+at runtime, so that the framework needs to be further improved by employing
+some novel path searching algorithms based on multi-object optimization[29] or
+reinforcement learning[30] methods.
+Acknowledgment
+This paper has been sponsored and supported by National Key Research
+and Development Program of China (Grant No.2018YFB0803400), Doctoral
+Foundation of Qingdao Binhai University (Grant No. BS2022A10), partially
+supported by Key Program of National Natural Science Foundation of China
+(Grant No.61932013).
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+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. With this data and additional
+information on the person’s background, one can easily infer an individual’s private data. Many differentially private iterative
+algorithms have been proposed in interactive settings to protect an individual’s privacy from these inference attacks. 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. These DP mechanisms do not guarantee convergence of differentially
+private iterative algorithms and degrade the quality of the cluster. 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. 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. At every Lloyd’s step, the centroids are injected with the noise using the exponential DP mechanism.
+The results of the experiments indicate that our approach outperforms the current state-of-the-art method, i.e., the baseline
+algorithm, in terms of clustering quality while maintaining the same differential privacy requirements. The clustering quality
+significantly improved by 4.13 and 2.83 times than baseline for the Wine and Breast_Cancer dataset, respectively.
+Keywords
+differential privacy, 𝑘-means clustering, convergence guarantee
+1. Introduction
+Achieving extraordinary results is dependent on the data
+on which the machine learning models are trained. Data
+curators have a responsibility to provide datasets such
+that the privacy of data is not compromised. However,
+attackers use other public datasets to perform inference
+and adversarial attacks to get information about an indi-
+vidual in the dataset. Differential privacy is a potential
+technique for giving customers a mathematical guarantee
+of the privacy of their data[1]. 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; while in non-interactive setting the curator
+sanitized the data before publishing[2].
+Iterative clustering algorithms provide important in-
+sights about the dataset, which helps in a large number of
+applications. They are prone to privacy threats because
+they can reveal information about an individual with ad-
+ditional knowledge. 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.
+†These authors contributed equally.
+� devvrat.joshi@iitgn.ac.in (D. Joshi); janvi.thakkar@iitgn.ac.in
+(J. Thakkar)
+© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License
+Attribution 4.0 International (CC BY 4.0).
+CEUR
+Workshop
+Proceedings
+http://ceur-ws.org
+ISSN 1613-0073
+CEUR Workshop Proceedings (CEUR-WS.org)
+privacy [3]. In contrast to Lloyd’s K-means algorithm,
+which guarantees convergence, these algorithms do not
+provide any convergence guarantee. Getting differen-
+tially private centroids might not help in getting quality
+inferences because of this non-convergence. 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]. Their algorithm
+perturbs the centroids in a random direction from the
+center of the cluster. However, this lowers the quality of
+clustering, which is necessary for making inferences.
+In this work, we propose a variant of the existing ap-
+proach, which provides better clustering quality while
+using the same privacy budget. 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. Finally, we give the mathematical proof that our
+approach at any instance gives better clustering quality
+than the existing approaches. We have tested our ap-
+proach on breat_cancer, wine, iris, and digits datasets.
+We were able to get a significant improvement from the
+previous approach in terms of clustering quality.
+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.
+Our main contribution includes:
+1. We proposed SubClustering approach which has
+better clustering quality than the baseline (which
+is the current SOTA in terms of clustering qual-
+arXiv:2301.02896v1  [cs.LG]  7 Jan 2023
+
+ity). For the Wine and Breast_cancer dataset, the
+clustering quality improved by 4.13 and 2.83 times
+respectively.
+2. In addition to improving the clustering quality,
+our algorithm used same privacy budget as that
+of the existing work.
+2. 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. 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]. To achieve differential privacy, many im-
+plementations included infusing Laplace noise into each
+iteration of Lloyd’s algorithm. The proportion of noise
+added was based on a fixed privacy budget. Some of the
+strategies for allocating privacy budget included splitting
+the overall privacy budget uniformly to each iteration
+[12]. However, this requires us to calculate the number of
+iterations for the convergence, prior to the execution of
+algorithm, thus increasing the computational cost. 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. In another work, Mohan
+et al. [8] proposed GUPT, which uses Lloyd’s algorithm
+for local clustering of each bucket where the items were
+uniformly sampled to different buckets. The final result
+was the mean of locally sampled points in each bucket
+with added Laplace noise. But, the clustering quality of
+GUPT was unsatisfying because a large amount of noise
+was added in the aggregation stage.
+Based on the study of past literature on differentially
+private k-means clustering, Zhigang et al. [3] concluded
+that convergence of an iterative algorithm is important to
+the clustering quality. To solve this, they introduced the
+concept of the convergent zone and orientation controller.
+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. 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. The ap-
+proach maintains the same DP requirements as existing
+literature, with guaranteed convergence and improve-
+ment in clustering quality. However, their algorithm
+perturbs the centroids in a random direction from the
+center of the cluster, degrading the quality of clustering.
+Thus, in this work, we further build upon the approach
+and significantly improve the clustering quality with the
+same epsilon privacy.
+3. Preliminaries
+The definitions used in this work are briefly discussed
+in this section. The following is a formal definition of
+Differential Privacy:
+Definition 1 (𝜖-DP [9]). 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.
+Here, 𝑋 and 𝑋′ differ by only one item. Smaller val-
+ues of 𝜖 imply a better privacy guarantee. It is because
+the difference between the two neighboring datasets is
+reflected by the privacy budget. In this work, we use the
+ExpDP and LapDP. 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). Here 𝑋 is the dataset and 𝑥 is the response
+to the 𝑞(𝑋, 𝑥) on X.
+The formal definition of Exponential DP mechanism
+is defined as follow:
+Definition
+2
+(Exponential
+Mechanism
+[10]).
+Given a scoring function of a dataset 𝑋, 𝑞(𝑋, 𝑥),
+which reflects the quality of query respond x.
+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.
+Definition 3 (Convergent & Sampling Zones[3]).
+A region whose points satisfies the condition: { Node S:
+‖𝑆 − 𝑆𝑖
+(𝑡)‖ < ‖𝑆𝑖
+(𝑡−1) − 𝑆𝑖
+(𝑡)‖} is the convergent zone.
+𝑆𝑖
+(𝑡) is defined as the mean of 𝐶𝑖
+(𝑡). A sub-region inside
+convergent zone is defined as a sampling zone.
+Definition 4 (Orientation Controller[3]). 𝑋𝑖
+(𝑡) is
+a direction from the center of the convergent zone to a point
+on its circumference. This is the direction along which the
+center of the sampling zone will be sampled, defined as the
+orientation controller.
+4. Approach
+In this section, we explain our proposed approach and
+the baseline approach.
+4.1. 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.
+
+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. 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. The constraint guarantees that the
+perturbed centroid will eventually converge with the
+centroid of Lloyd’s algorithm.
+Their algorithm has four main steps to update the
+centroids at each Lloyd step t [3]. The overview of their
+approach can be seen in (Figure : 1).
+1. Let the differentially private centroid at iteration
+𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1). Using this centroid,
+run one iteration of Lloyd’s algorithm to get the
+current Lloyd’s centroid 𝑆𝑖
+(𝑡) for each cluster 𝑖.
+2. Using 𝑆𝑖
+(𝑡) and 𝑆𝑖
+(𝑡−1), generate a conver-
+gent zone for each cluster 𝑖 as described in
+𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3.
+3. 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.
+4. Sample a differentially private 𝑆𝑖ˆ (𝑡) with ExpDP
+in the sampling zone generated in step 3.
+The definition for the convergent zone (for convergence
+guarantee) and sampling zone (for centroid updating) is
+defined in Definition 3.
+4.2. Overview - SubCluster Guarantee
+We build upon the KMeans Guarantee algorithm to
+achieve better clustering quality. Our idea differs from
+the baseline in terms of creating a sampling zone. For
+each cluster, we execute Lloyd’s algorithm over its con-
+vergent zone to generate its sub-clustering. Further, we
+assign each sub-cluster with a probability linearly pro-
+portional to the number of points it contains. Finally, we
+sample the sub-cluster based on the assigned probability
+and define it as the sampling zone of the convergent zone.
+Drawing analogy from the KMeans Guarantee algorithm,
+our orientation controller is this sub-clustering and sam-
+pling technique. Intuitively, our algorithm ensures that
+Algorithm 1: Differentially Private 𝑘−Means
+SubClustering Algorithm
+Input: X = {𝑥1, 𝑥2, ...., 𝑥𝑁}: Dataset with N
+data points
+k: number of clusters
+𝜖𝑒𝑥𝑝: ExpDP privacy budget
+𝜖𝑙𝑎𝑝: Laplacian privacy budget for the converged
+centroids.
+𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾: number of sub-clusters per cluster
+Output: S: Final clustering centroids
+1 Select 𝑘 centroids S(0) = (𝑆(0)
+1 , 𝑆(0)
+2 , ..., 𝑆(0)
+𝑘 )
+uniformly from X.
+2 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝐹𝑜𝑟𝐿𝑙𝑜𝑦𝑑 = number of iterations to
+run the algorithm.
+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;
+6
+𝑆𝑖
+𝑡 ← centroid of 𝐶𝑖
+𝑡;
+7
+𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖
+(𝑡) ← List of data
+points inside the spherical region having
+𝑆𝑖
+𝑡 and 𝑆𝑖
+𝑡−1 as the endpoints of its
+radius.
+8
+𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖
+(𝑡) ← run Algorithm 2
+using 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒𝑖
+(𝑡) ,
+𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙𝐾;
+9
+𝑆𝑖ˆ (𝑡) ← sample from 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖
+(𝑡)
+using ExpDP with 𝑞 and 𝜖𝑒𝑥𝑝;
+10
+𝑆𝑖
+(𝑡) ← 𝑆𝑖ˆ (𝑡)
+11 Publish: 𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔𝑍𝑜𝑛𝑒𝑖
+(𝑡), 𝑞, 𝜖𝑒𝑥𝑝, 𝑆𝑖
+(𝑡)
+12 S ← add laplace noise with 𝜖𝑙𝑎𝑝 to S(𝑡);
+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.
+4 SamplingZonei
+(t) ← Sample a cluster from
+the ConvergentZoneClusters using
+ConvergentZoneProbability
+5 Return: SamplingZonei
+(t);
+
+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. 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. 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. Thus,
+we introduce the concept of sub-clustering in the conver-
+gent zone and consequently sample one sub-cluster as
+our sampling zone.
+We sample the centroid from the sampling zone using
+the ExpDP mechanism. Finally, we inject Laplace noise
+in the centroids of the clustering when our algorithm
+converges. It is because the differentially private cen-
+troids obtained are a subset of one of the local minima
+at which Lloyd’s algorithm converges. The overview of
+the proposed approach can be seen in (Figure : 2). We
+show that a randomized iterative algorithm satisfies an
+invariant (given in the claim of Lemma 1) and always
+converges (Proof: refer Lemma 1). Finally, we show
+that the SubCluster algorithm is a randomized iterative
+algorithm that satisfies the invariant(given in Lemma 1)
+(Proof: Refer Lemma 2).
+We have four main steps to update the centroids at
+each Lloyd step t.
+1. Let the differentially private centroid at iteration
+𝑡−1 for a cluster 𝑖 be 𝑆𝑖ˆ (𝑡−1). Using this centroid,
+run one iteration of Lloyd’s algorithm to get the
+current Lloyd’s centroid 𝑆𝑖
+(𝑡) for each cluster 𝑖.
+2. Using 𝑆𝑖
+(𝑡) and 𝑆𝑖
+(𝑡−1), generate a conver-
+gent zone for each cluster 𝑖 as described in
+𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 3.
+3. SubCluster the convergence zone and sample one
+of the sub-cluster as our sampling zone based on
+the probability assigned to each sub-cluster. The
+probability assignment is directly proportional to
+the number of points in each sub-cluster.
+4. Sample a differentially private 𝑆𝑖ˆ (𝑡) with EXpDP
+in the sampling zone generated in step 3.
+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). 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.
+The pseudo-code of our approach is shown in the Algo-
+rithm 1 and Algorithm 2.
+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.
+We reproduce this lemma from our baseline approach
+[3]. Lemma1 and Lemma 2 together provides the com-
+pleteness and proof for the convergence of our approach.
+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.
+Intuitively, the loss of 𝐶(𝑡)
+𝑖
+is minimum if the mean of
+𝐶(𝑡)
+𝑖
+is taken as centroid. But, if we slightly shift from
+the mean of 𝐶(𝑡)
+𝑖 , then the loss will increase. 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. For
+the mathematical proof, refer [3].
+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)||.
+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. It subclusters the points lying inside
+𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡)
+𝑖 . After subclustering, it samples
+one subcluster (sampling zone) with the assigned proba-
+bilities (linearly proportional to the number of data points
+in subcluster). 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;(t-1) > S;(t-1)
+S,(t+7)
+Srt+s
+convergent zone
+≤.(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. The algorithm was tested on four dataset, Digits (top-left), Wine (top-right),
+Breast Cancer (bottom-left), and Iris (bottom-right) datasets.
+troid of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡)
+𝑖 . Thus, our sampling zone
+always lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡)
+𝑖 . Therefore, the
+sampled point lies inside 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡)
+𝑖
+and it sat-
+isfies the invariant ||𝑆𝑖ˆ (𝑡) − 𝑆𝑖
+(𝑡)|| < ||𝑆𝑖
+(𝑡) − 𝑆𝑖
+(𝑡−1)||.
+5. Experimental Setup
+5.1. Dataset Used
+We used following four datasets to test our work Sub-
+Cluster Guarantee upon the baseline:
+1. Iris [13] dataset comprises total of 150 datapoints
+with four features and three classes.
+2. Wine[13] dataset comprises total of 178 data-
+points with 13 features and three classes.
+3. Breast Cancer[13] dataset comprises total of
+569 datapoints with 30 features and two classes.
+4. Digits[13] dataset comprises of 1797 datapoints
+with 64 dimensions and 10 classes.
+5.2. 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. In the experiments, we compare the clustering
+quality of SubCluster Guarantee with KMeans Guarantee.
+6. Results and Discussion
+We tested our algorithm on four datasets. All the datasets
+have different dimensions ranging from 4 to 64 dimen-
+sions and training sets ranging from 150 to 1800. As
+defined in metric smaller gap represents the better clus-
+tering quality. From the (Figure : 3) we can observe
+that, cost gap for all the dataset is smaller or equal to
+the baseline. Thus, it is evident that our algorithm has
+better clustering quality than the existing work for all the
+datasets experimented. We varied internalK (parameter
+for number of sub-clusters) from 2 to 5.
+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. Fi-
+nally, for each dataset, we took the average of all the
+experiments as our final result for plotting the graphs.
+
+KmeansGuaranteevsSubClusterGuaranteeDataset:Digits
+KmeansGuaranteevsSubClusterGuaranteeDataset:Wine
+0.200
+0.08
+KmeansGuarantee
+KmeansGuarantee
+0.175
+SubClusterGuarantee
+0.07
+SubClusterGuarantee
+0.150
+0.06
+0.125
+0.05
+costGap
+0.04
+0.075
+E00
+0.050
+0.02
+0.025
+0.01
+0.000
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+epsilon
+epsilon
+KmeansGuaranteevsSubClusterGuaranteeDataset:BreastCancer
+KmeansGuaranteevsSubClusterGuaranteeDataset:Iris
+0.035
+14
+KmeansGuarantee
+12
+SubClusterGuarantee
+0.030
+10
+0.025
+KmeansGuarantee
+8
+SubClusterGuarantee
+6
+0.015
+4
+0.010
+2
+0.005
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+epsilon
+epsilonFigure 4: Above figures plots the graph between costGap and epsilon budget for different internalK in SubClusterGuarantee
+Algorithm. 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). 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. In case of Iris dataset, the cost gap is 1.1 times
+smaller than baseline algorithm.
+2. In case of Wine dataset, the cost gap is 4.13 times
+smaller than baseline algorithm.
+3. In case of Breast_Cancer dataset, the cost gap
+is 2.83 times smaller than baseline algorithm.
+4. In case of Digits dataset, the cost gap is almost
+same as that of baseline algorithm.
+6.1. Detailed Analysis
+1. Iris: Iris dataset has four dimensions and a very
+small training set of 150 data points. Our al-
+gorithm achieves better clustering quality than
+the baseline algorithm for smaller epsilon values.
+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. 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. 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.
+2. Wine: The wine dataset has 13 dimensions and
+178 data points in the training set. Our algorithm
+performs significantly better than the baseline, as
+observed in (Figure : 3). 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). 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). From (Figure : 4),
+it is evident that internalK=4 for the wine dataset
+performs better than the rest of the internalK val-
+ues. Here, the number of dimensions is more than
+Iris. Therefore, the spatial arrangement will be in
+an n-sphere which allows better sub-clustering.
+3. Breast_Cancer: Breast_Cancer dataset has 569
+data points in its training set and 30 dimensions.
+Our algorithm performs exceptionally better than
+the baseline, with internalK equal to 4. From
+(Figure : 3), we can observe that there is no
+monotonous trend for the costGap. Trends are
+visible in other datasets due to the larger num-
+ber of classification classes, whereas this dataset
+has only two classes. Thus, adding Laplace noise
+does not have a relation to the clustering quality.
+Increasing the internalK improves the clustering
+
+VaryinginternalKforSubClusteringDataset:Digits
+VaryinginternalKforSubClusteringDataset:Wine
+K=2
+0.0035
+K=2
+0.20
+K=3
+K=3
+K=4
+K=4
+0.0030
+K=5
+K=5
+0.15
+0.0025
+0.10
+0.0020
+0.05
+0.0015
+0.00
+0.0010
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+epsilon
+epsilon
+VaryinginternalkforSubClusteringDataset:BreastCancer
+VaryinginternalKforSubClusteringDataset:Iris
+K=2
+10
+K=2
+0.025
+K=3
+K=3
+K=4
+K=4
+0.020
+8
+K=5
+h
+0.010
+0.005
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+epsilon
+epsilonquality, with internalK being 4 having the least
+loss. It is because this dataset has a high number
+of dimensions and a larger number of training
+points than other datasets.
+4. Digits: It has 64 dimensions and 1797 data points
+in the training dataset. Although it has a large
+number of dimensions, our algorithm has a very
+small improvement over the baseline algorithm as
+seen in (Figure : 3). Because of the higher time
+complexity of our algorithm, it is hard to tune
+the internalK parameter. 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. But, due to limited
+computational resources, we were not able to ex-
+periment with it further. We took internalK to
+be 5 for our experiments as it performed best in
+the range [2, 5] as in the (Figure : 4). 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.
+Our proposed algorithm significantly improves over the
+baseline in terms of clustering quality, especially for the
+wine and breast cancer dataset. In addition our algorithm
+maintains the same DP requirements as that of existing
+works.
+7. Conclusion
+This work presents a novel method for improving the
+clustering quality of differentially private k-means al-
+gorithms while ensuring convergence. 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. We proved that our work surpasses the current
+state-of-the-art algorithms in terms of clustering quality.
+Especially for the Wine and Breast_Cancer dataset, the
+clustering quality was significantly improved by 4.13 and
+2.83 times than the baseline. In addition, we maintain
+the same DP requirements as that of baseline and other
+existing approaches.
+8. Future Work
+• In this work, we proved our claim using empirical
+results. We further plan to validate the results
+by providing mathematical bounds for the con-
+vergence degree and rate of the SubClustering
+Lloyd’s algorithm. In terms of clustering qual-
+ity, the proposed algorithm in this work is com-
+pared with k-means guarantee clustering only;
+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].
+• 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.
+• We used Exponential and Laplace mechanisms
+of DP in the proposed approach; we further plan
+to explore the third mechanisms, i.e., sample and
+aggregate framework, by integrating it with the
+current algorithm.
+• In our algorithm, the number of data points inside
+a cluster is variable. Thus we plan to choose an
+internalK, custom to the size of the cluster to
+improve the clustering quality.
+Acknowledgement
+We would like to thank Prof. Anirban Dasgupta
+(IIT Gandhinagar) for his continuous support and
+guidance throughout the research.
+References
+[1] C. Dwork, Differential privacy: A survey
+of results, in: International conference on
+theory and applications of models of com-
+putation, Springer, 2008, pp. 1–19.
+[2] A. Narayanan, Data privacy: The non-
+interactive setting, The University of Texas
+at Austin, 2009.
+[3] Z. Lu, H. Shen,
+Differentially private k-
+means clustering with convergence guar-
+antee, IEEE Transactions on Dependable
+and Secure Computing (2020).
+[4] D. Su, J. Cao, N. Li, E. Bertino, H. Jin, Dif-
+ferentially private k-means clustering, in:
+Proceedings of the sixth ACM conference
+on data and application security and pri-
+vacy, 2016, pp. 26–37.
+[5] J. Lei, Differentially private m-estimators,
+Advances in Neural Information Processing
+Systems 24 (2011).
+[6] D. Su, J. Cao, N. Li, E. Bertino, M. Lyu, H. Jin,
+Differentially private k-means clustering
+and a hybrid approach to private optimiza-
+tion, ACM Transactions on Privacy and
+Security (TOPS) 20 (2017) 1–33.
+[7] C. Dwork, A firm foundation for private
+data analysis, Communications of the ACM
+54 (2011) 86–95.
+
+[8] P. Mohan, A. Thakurta, E. Shi, D. Song,
+D. Culler, Gupt: privacy preserving data
+analysis made easy, in: Proceedings of the
+2012 ACM SIGMOD International Confer-
+ence on Management of Data, 2012, pp. 349–
+360.
+[9] C. Dwork, F. McSherry, K. Nissim, A. Smith,
+Calibrating noise to sensitivity in private
+data analysis, in: Theory of cryptography
+conference, Springer, 2006, pp. 265–284.
+[10] F. McSherry, K. Talwar, Mechanism design
+via differential privacy, in: 48th Annual
+IEEE Symposium on Foundations of Com-
+puter Science (FOCS’07), IEEE, 2007, pp. 94–
+103.
+[11] K. Nissim, S. Raskhodnikova, A. Smith,
+Smooth sensitivity and sampling in private
+data analysis, in: Proceedings of the thirty-
+ninth annual ACM symposium on Theory
+of computing, 2007, pp. 75–84.
+[12] A. Blum, C. Dwork, F. McSherry, K. Nis-
+sim, Practical privacy: the sulq framework,
+in: Proceedings of the twenty-fourth ACM
+SIGMOD-SIGACT-SIGART symposium on
+Principles of database systems, 2005, pp.
+128–138.
+[13] A. Asuncion, Uci machine learning reposi-
+tory, university of california, irvine, school
+of information and computer sciences,
+http://www. ics. uci. edu/˜ mlearn/MLRepos-
+itory. html (2007).
+[14] J. Zhang, X. Xiao, Y. Yang, Z. Zhang,
+M. Winslett, Privgene: differentially pri-
+vate model fitting using genetic algorithms,
+in: Proceedings of the 2013 ACM SIGMOD
+International Conference on Management
+of Data, 2013, pp. 665–676.
+

F9E1T4oBgHgl3EQfFAMn/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf,len=432
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content=' The clustering quality  significantly improved by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  †These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='  � devvrat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='joshi@iitgn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' Joshi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' janvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='thakkar@iitgn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='in  (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content=' Use permitted under Creative Commons License  Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='0 International (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='  CEUR  Workshop  Proceedings  http://ceur-ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='org)  privacy [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  Our main contribution includes:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='02896v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='LG]  7 Jan 2023  ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='83 times  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' In another work, Mohan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  Definition 3 (Convergent & Sampling Zones[3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  Definition 4 (Orientation Controller[3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=', 𝑆(0)  𝑘 )  uniformly from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  6  𝑆𝑖  𝑡 ← centroid of 𝐶𝑖  𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' For  the mathematical proof, refer [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='(t-1) > S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  troid of 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡𝑍𝑜𝑛𝑒(𝑡)  𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' Experimental Setup  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  KmeansGuaranteevsSubClusterGuaranteeDataset:Digits  KmeansGuaranteevsSubClusterGuaranteeDataset:Wine  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='08  KmeansGuarantee  KmeansGuarantee  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='175  SubClusterGuarantee  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='07  SubClusterGuarantee  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  costGap  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='075  E00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='30  epsilon  epsilon  KmeansGuaranteevsSubClusterGuaranteeDataset:BreastCancer  KmeansGuaranteevsSubClusterGuaranteeDataset:Iris  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='035  14  KmeansGuarantee  12  SubClusterGuarantee  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='030  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='025  KmeansGuarantee  8  SubClusterGuarantee  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='015  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='010  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content='1 times  smaller than baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='13 times  smaller than baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='83 times smaller than baseline algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' Detailed Analysis  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='0035  K=2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='0030  K=5  K=5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='0025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='0020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='020  8  K=5  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content='13 and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='83 times than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
+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'}
+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'}
+page_content='  Acknowledgement  We would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+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'}
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+page_content=' ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
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+page_content=' edu/˜ mlearn/MLRepos-  itory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' html (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' Zhang,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' Winslett, Privgene: differentially pri-  vate model fitting using genetic algorithms,  in: Proceedings of the 2013 ACM SIGMOD  International Conference on Management  of Data, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}
+page_content=' 665–676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQfFAMn/content/2301.02896v1.pdf'}

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+ 
+ 
+ 
+Manuscript Template                                                                                           
+ Page 1 of 22 
+ 
+FRONT MATTER 
+ 
+Title  
+• Controlling Electromagnetic Surface Waves with Conformal Transformation Optics 
+ 
+Authors 
+Xiaoyu Zhao,1† Hong Deng,1† Xiaoke Gao,1† Xikui Ma,1 Tianyu Dong1* 
+1School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China. 
+ 
+†These authors contributed equally to this work. 
+*To whom correspondence should be addressed; E-mail: tydong@mail.xjtu.edu.cn. 
+ 
+Abstract 
+The application of transformation optics to the development of intriguing electromagnetic 
+devices can produce weakly anisotropic or isotropic media with the assistance of quasi-
+conformal and/or conformal mapping, as opposed to the strongly anisotropic media 
+produced by general mappings; however, it is typically limited to two-dimensional 
+applications. By addressing the conformal mapping between two manifolds embedded in 
+three-dimensional space, we demonstrate that electromagnetic surface waves can be 
+controlled without introducing singularity and anisotropy into the device parameters. 
+Using fruitful surface conformal parameterization methods, a near-perfect conformal 
+mapping between smooth manifolds with arbitrary boundaries can be obtained. 
+Illustrations of cloaking and illusions, including surface Luneburg and Eaton lenses and 
+black holes for surface waves, are provided. Our work brings the manipulation of surface 
+waves at microwave and optical wavelengths one step closer. 
+ 
+Teaser 
+Waves can be controlled at will on arbitrary open surfaces without holes, showing 
+fascinating applications such as invisible bumps for surface waves, reproducing 
+scatterings of one bump on other smooth surfaces, and controlling light beams on surfaces 
+to focus, to bend and/or to be absorbed akin to black holes without visible scatterings. 
+ 
+MAIN TEXT 
+ 
+Introduction 
+Since its inception in the design of electromagnetic cloaks (1, 2), transformation optics 
+(TO) has proven to be a powerful tool for understanding and customizing the physics in 
+acoustics (3), optics (4), mechanics (5), thermodynamics (6, 7), etc. Following the 
+groundbreaking work of cloaking, a number of other electromagnetic devices have been 
+reported within the theoretical framework of TO, such as electromagnetic concentrators 
+(8, 9), field rotators (10), optical lenses (11, 12) and optical illusion devices (13, 14). In 
+practice, however, traditional TO often yields significant anisotropy in a designed medium 
+(15). Thus, metamaterials are often used to infer spatial changes from coordinate 
+transformation geometry, which is based on the mathematical equivalence between 
+geometry and material (16). 
+
+Manuscript Template                                                                            
+Page 2 of 22 
+ 
+To reduce the anisotropy of the functional medium induced by TO, various approaches 
+have been developed. By constructing mapping in non-Euclidean space, for instance, it is 
+possible to remove singular points formed by traditional TO (17), hence minimizing 
+anisotropy in part. But for wavelengths comparable to the size of the transform region, 
+non-Euclidean TO may perform even worse (18); thus, several research projects focus on 
+conformal or quasi-conformal mappings to achieve isotropy (19). In ℝ2,  the concept of a 
+carpet cloak that resembles a flat ground plane is successfully realized with an isotropic 
+medium produced by minimizing the Modified-Liao functional under sliding boundary 
+conditions (20), or equivalently by constructing the quasi-conformal mapping via solving 
+inverse Laplace's equations (21). Although the concept of carpet cloak has been extended 
+to ℝ3 by the extrusion or revolution of a two-dimensional refractive index profile to 
+control the reflection of free-space waves, it is only applicable to surfaces with 
+translational or rotational symmetry (22). 
+Previous research has focused largely on controlling propagating waves by TO, whereas 
+less attention has been attached to the manipulation of surface waves (12, 23, 24). Perfect 
+surface wave cloaks have been proposed by equating the optical path length of a ray 
+traversing a flat plane with a homogeneous refractive index to the optical path on a curved 
+surface with an angle-dependent refractive index for two orthogonal paths (25, 26), which 
+have been experimentally validated (27). Although an electrically large object may be 
+hidden by such a cloak with an inhomogeneous isotropic medium, this approach is limited 
+to rotationally symmetric surfaces. By linking the governing eikonal equations on a virtual 
+flat plane and on a curved surface by transformation optics, the projection mapping yields 
+surface wave cloaks for non-rotationally symmetric geometries but with high anisotropy 
+(14, 28). Considerable effort has been devoted to reducing such anisotropy by employing 
+efficient numerical conformal algorithms such as boundary first flattening (29), yet only 
+non-rotationally symmetric surfaces with circular boundary are investigated (30). 
+In this work, we show how to manipulate surface waves on smooth manifolds embedded 
+in ℝ3 within the framework of conformal TO, requiring an effective isotropic material 
+under the regime of geometrical optics. Fig. 1 illustrates a conformal surface mapping 
+between two smooth manifolds in ℝ2 and ℝ3, i.e., 𝑓: ℳ′ → ℳ. The curved manifold ℳ 
+shown in Fig. 1A has been 𝑢𝑣-parameterized and the mesh grid can be regarded as the 
+mapping result of the Cartesian coordinate system {𝑥′, 𝑦′} in Fig. 1B. When the mapping 
+is conformal or quasi-conformal, the face element d𝑆 remains right-angled, indicating that 
+elements are just scaled up with little distortion. From the local coordinate systems on d𝑆 
+and d𝑆′ (Fig. S5), one can derive the Jacobian matrix 𝐉 of mapping 𝑓 with two singular 
+values 𝜎J1 = 𝜎J2 = 𝜎J that state equal scaling in two orthogonal directions (31). 
+Consequently, an isotropic cloaking medium distribution 𝑛 = 1/√det(𝐉) = 1/𝜎J may be 
+obtained based on the conformal TO (19), representing the ratio of line element d𝑙′ in 
+virtual space to the scaled element d𝑙 in physical space for compensating optical path 
+length (2). As a result, light propagating on curved ℳ behaves as propagating on flat ℳ′. 
+In practice, it is more convenient to describe mesh vertices in ℝ3 in a Cartesian coordinate 
+system {𝑥, 𝑦, 𝑧} and the Jacobian derived from the local coordinate system forms an 
+asymmetric rank-two matrix 𝐉3×2. In addition, the possible quasi-conformal mappings can 
+be measured by the conformality, i.e., the ratio 𝑄 = max(𝜎J1/𝜎J2, 𝜎J2/𝜎J1). A unity ratio 
+𝑄 allows an effective cloaking medium expressed as 𝑛cloak = 1/√𝜎J1𝜎J2 for every face 
+element (20). 
+
+Manuscript Template                                                                            
+Page 3 of 22 
+ 
+ 
+Results  
+Having obtained a conformal mapping between the manifolds ℳ ∈ ℝ3 and ℳ′ ∈ ℝ2, we 
+first design an isotropic surface wave cloak under the perspective of conformal TO and 
+compare its performance with the traditional surface wave cloak with anisotropic medium 
+(14). Simulations were conducted on a double-camelback bump with an elliptical base 
+profile embedded in ℝ3, as shown in Fig. 2. In comparison with the scattering when the 
+surface has no index profile (Fig. S1A), one can observe that the surface wave cloaking is 
+successfully achieved by two distinct approaches: one induced by the projection mapping 
+proposed in (14) (Fig. 2A) and the other originated from the proposed quasi-conformal 
+mapping (Fig. 2B). The corresponding material characteristics for the two types of cloaks 
+are displayed in Fig. 2C, indicating that the former is strongly anisotropic while the latter 
+is almost isotropic. In addition, the isotropic refractive index 𝑛c,double (the subscript "c" 
+denotes the cloak, and "double" denotes the double-camelback bump) ranges from 0.83 to 
+1, which decreases as the bump height rises because a longer geometrical distance need to 
+be compensated by a smaller refractive index in order to attain equal optical path length. 
+The proposed scheme based on conformal TO has achieved near-perfect surface wave 
+cloaking while eliminating the anisotropy in the transformation medium that the 
+traditional scheme presents. The distribution of 𝑛c,double in Fig. 2C outlines an 
+asymmetric geometric profile, manifesting that the effectiveness of this scheme is 
+independent from any symmetry. Such an achievement demands mappings with high 
+conformality rather than those bringing large distortion such as the projection mapping 
+(14). The numerical method we adopt here (29) can obtain a quasi-conformal mapping 
+with 𝑄 < 1.03, as shown in Fig. S1B, which is sufficient for designing an effective 
+isotropic cloaking medium distribution. 
+As the antithesis of cloaking, optical illusion devices can reproduce the scattering 
+characteristics of a specific object on other objects through a transformation medium (13, 
+14). Fig. 3A depicts the surface electromagnetic wave scattered by a single-camelback 
+bump ℳ filled with homogeneous material. Traditionally, if one wants to reproduce its 
+scattering on a plane region ℳ′, the quasi-conformal mapping for designing the illusion 
+device is 𝑓′: ℳ → ℳ′ with a Jacobian matrix 𝚲2×3. Fig. 3B shows the accurately 
+recurring scattering characteristics on plane region ℳ′ filled with 𝑛i,plane = 1/√𝜎Λ1𝜎Λ2 
+(the subscript "i" denotes the illusion, and "plane" denotes the plane region), where 𝜎Λ1 
+and 𝜎Λ2 are singular values of 𝚲2×3. Furthermore, Fig. 3C illustrates that the double-
+camelback bump filled with a carefully designed isotropic medium distribution can 
+reproduce the same scattering pattern as shown in Fig. 3A. Such an illusion is realized by 
+cascading two conformal mappings described in Fig. S3, i.e., 𝑓1 from ℝ3 (virtual space) to 
+ℝ2 (intermediate space), and 𝑓2 from ℝ2 to the ℝ3 (physical space). Thus, the illusion 
+medium for the double-camelback bump reads 𝑛i,double = 𝑛i,plane ∙ 𝑛c,double. Fig. 3D 
+displays the profiles of 𝑛i,plane (for Fig. 3B) and 𝑛i,double (for Fig. 3C), respectively, 
+which range from 1 to 1.25 (𝑛i,plane) and from 0.85 to 1.21 (𝑛i,double). 
+The scattering pattern of the single-camelback bump (Fig. 3A) has been successfully 
+reproduced on the plane region (Fig. 3B) and on the double-camelback bump (Fig. 3C), 
+which demonstrates that the proposed scheme is a general solution to illusion design on 
+smooth two-dimensional manifolds. The cascading method to construct mappings between 
+manifolds embedded in ℝ3 can even tackle surfaces with different base profiles, since a 
+
+Manuscript Template                                                                            
+Page 4 of 22 
+ 
+conformal mapping between simply-connected regions in ℝ2 exists according to the 
+Riemann mapping theorem (32). Moreover, the quasi-conformal ratios 𝑄 of the two 
+mappings for the double-camelback and single-camelback bump are smaller than 1.03 
+(Fig. S1B) and 1.012 (Fig. S2C), respectively, implicating that the cascaded mapping 
+meets the requirement for high conformality. The range of 𝑛i,single (1 to 1.25) is the 
+inverse of that of the cloaking refractive index 𝑛c,single (0.8 to 1) shown in Fig. S2B, 
+because the illusion can be regarded as the inverse design of the cloaking such that the 
+Jacobian matrices of their corresponding mappings are the Moore–Penrose pseudo-inverse 
+of each other (31). 
+Now that the wave behavior on the curved manifold can be manipulated flexibly, it is 
+natural to consider designing various complicated devices on it, such as surface wave 
+Luneburg lens, Eaton lens and black hole for surface waves (12, 23, 33, 34). Traditional 
+designs are usually based on spherical or circular profiles with a constant radius. While for 
+an elliptical profile without a constant radius, we adopt the distance from the point on the 
+ellipse to the center, also the coordinate origin, as the generalized radius, i.e., 𝑅(𝜃) =
+√(𝑎 cos 𝜃)2 + (𝑏 sin 𝜃)2 (35–37). Thus, the refractive index of the considered Luneburg 
+lens can be expressed as 
+ 
+𝑛L(𝑟, 𝜃) = √2 − (𝑟/𝑅(𝜃))2, 
+(1) 
+where 𝑟 = √𝑥2 + 𝑦2 and 𝜃 = arctan(𝑦/𝑥). Similar to the traditional circular Luneburg 
+lens, such a distribution retains 𝑛L = 1 on the boundary and 𝑛L = √2 at the center 𝑟 = 0 
+(38). Next, the medium distribution for a Luneburg lens on the double-camelback bump 
+can be expressed as 𝑛Luneburg = 𝑛c,double ⋅ 𝑛L. As illustrated in Fig. 4A, two Gaussian 
+beams with a free-space wavelength 𝜆G = 50 mm are incident along the 𝑥-direction at the 
+position ±0.8𝑏 on the 𝑦-direction and reflected by the Luneburg lens to interfere at the 
+focus point. The focal distance reads 20𝜆G that is identical to the unit circular Luneburg 
+lens. For the Eaton lens, the refractive index 𝑛E reads as 
+ 
+𝑛E(𝑟, 𝜃) = √2𝑅(𝜃)/𝑟 − 1, 
+(2) 
+which can approach infinity when 𝑟 = 0, leaving a singular point to be cared for. Fig. 4B 
+describes that a Gaussian beam going along the 𝑥-direction bends to the inverse 𝑥-
+direction after passing through the Eaton lens on the double-camelback bump. The 
+proposed surface wave Luneburg and Eaton lenses may be deployed in optical imaging, 
+signal acquisition and novel designs for surface wave microwave antennas. Another 
+functional device that can rotate beam propagation is the peripheral of the two-layer 
+optical black hole, where light is compelled to travel in a spiral path into the absorbing 
+medium at the core. The piece-wise refractive index distribution function 𝑛B can be 
+expressed as 
+ 
+𝑛B(𝑟, 𝜃) = {
+ 1,
+ 𝑟 > 𝑅(𝜃)
+ 𝑅(𝜃)/𝑟,
+ 𝑟𝑐 ⋅ 𝑅(𝜃) < 𝑟 < 𝑅(𝜃)
+ 1/𝑟𝑐 + i𝛾,
+ 𝑟 < 𝑟𝑐 ⋅ 𝑅(𝜃)
+,  
+(3) 
+where 𝑟𝑐 = 0.4 is the scaling factor of the internal ellipse core compared with the base 
+profile and 𝛾 = 0.1 is the loss factor. The refractive index distribution 𝑛Blackhole =
+𝑛c,double ⋅ 𝑛B on the double-camelback bump is depicted in Fig. 4D. The real part of 
+material parameters is matched on the inner boundary, and the imaginary part for 
+absorbing energy ranging from 0.083 to 0.097 only exists in the core. The same Gaussian 
+
+Manuscript Template                                                                            
+Page 5 of 22 
+ 
+beam that was used for the Eaton lens is employed, and the result in Fig. 4C shows that 
+the beam bends around 90∘ before it reaches the inner boundary and is absorbed by the 
+lossy core without reflection, showing potential application in interference reduction and 
+energy harvesting for electronic devices. Note that, the overall sizes of the simulation 
+models are larger than ten times the operating wavelength, demonstrating that the 
+proposed scheme is capable of managing surface wave behaviors on electrically large 
+objects. Moreover, the excellent performance of these functional surface wave devices 
+demonstrates that, based on the proposed scheme, a variety of novel devices may be 
+realized on smooth curved manifolds, which may facilitate the development of 
+miniaturized and integrated photonic devices. 
+ 
+Discussion  
+Our theory and method are based on geometrical optics. It requires small curvature and 
+little variation in wavelength (see (7) and (8) in Materials and Methods), which can be 
+expressed as 
+ 
+𝑤 = |∇𝜆| = |∇(𝜆0/𝑛)| = 𝜆0|∇𝑛|/𝑛2 ≪ 1,  
+(4) 
+ 
+𝜌 = |𝑅𝑖𝑗|𝜆2 = |𝐾𝑔𝑖𝑗|(𝜆0/𝑛)2 = det(𝑔𝑖𝑗) 𝐾2𝜆0
+2/𝑛2 = 𝐾2𝜆0
+2/𝑛6 ≪ 1,  
+(5) 
+where 𝑅𝑖𝑗 is the Ricci curvature tensor, 𝐾 is the Gaussian curvature, and 𝑔𝑖𝑗 is the metric 
+tensor. Both the wavelength index 𝑤 and the curvature index 𝜌 are inversely proportional 
+to powers of the refractive index 𝑛. In order to prevent 𝑤 and 𝜌 from increasing 
+drastically, a height lower than half of the base radius is favorable, and thereby the optical 
+path length can be compensated with a near-unity refractive index. On this basis, 
+requirements (4) and (5) demand shorter wavelength 𝜆0 and smoother geometric structure 
+to ease the changing rate |∇𝑛| and the Gaussian curvature 𝐾. As a negative example, a 
+hemisphere surface wave cloak is reviewed and results are displayed in Fig. S4, whose 
+refractive index 𝑛c,sphere is between 0.5 and 1 and the maximum of quasi-conformal ratio 
+𝑄 is smaller than 1.012. The visible scattering appearing in Fig. S4C implies the failure of 
+geometrical optics because of the high curvature index 𝜌 > 20 residing in the right-angle 
+connection between the hemisphere and the plane, as is depicted in Fig. S4F, and the 
+average curvature index 𝜌̅ = 1.57 is also larger than 1. The non-smooth connection causes 
+the phase distortion in the backward scattering, and the maximum of the forward 
+scattering  |𝐸𝑧 − 𝐸b𝑧|max = 0.75 V/m implies a phase difference arcsin (0.75) = 48.6∘ 
+resulted from the reconstruction of wave fronts. In comparison, Fig. S1C and Fig. S2D 
+display the average curvature index 𝜌̅ = 0.54 for double-camelback bump and 𝜌̅ = 0.39 
+for single-camelback bump, respectively, both satisfying the requirement (5) and leaving 
+near-zero 𝜌 on smooth boundaries. One may notice that the wavelength index 𝑤 for the 
+cloaks shown in Fig. S1D, Fig. S2E and Fig. S4E is smaller than unity everywhere 
+because it is related to lower powers of 𝜆0 and 𝑛; thus, it is much easier to meet the 
+requirement of (4) compared to (5). These selected curvature and wavelength 
+characteristics that validate the approximation of geometrical optics are indispensable for 
+the excellent performance of electromagnetic devices. 
+ 
+The isotropic case that determines the expression of requirements (4) and (5) is based on 
+the conformal or quasi-conformal mappings between two-dimensional manifolds. 
+Benefiting from the rapid development in conformal parameterization, a series of mapping 
+methods can be employed to design surface wave carpet cloak (29, 39, 40). The boundary 
+first flattening (BFF) method (29) adopted in our study can establish near-perfect 
+conformal mappings not only between smooth manifolds but also surfaces with cuspidal 
+
+Manuscript Template                                                                            
+Page 6 of 22 
+ 
+points, such as sharp corners and cone singularities, offering exhilarating promise for 
+wave manipulation on more complicated surfaces. In addition, there are algorithms aimed 
+at constructing quasi-conformal mappings between high-genus manifolds (41, 42), which 
+can be used to deal with phase regulation on surfaces with holes. One noteworthy idea is 
+to map a high-genus surface to a zero-genus plane region by transforming holes to slits 
+(43, 44) that implies the possibility for the scheme conducted in simply-connected regions 
+to manipulate wave behaviors on multiply-connected surfaces. By reasonably utilizing 
+advanced algorithms for a variety of particular cases, our method has the potential to be a 
+universal scheme for controlling surface electromagnetic waves on an arbitrary two-
+dimensional manifold. 
+ 
+In summary, we have proposed a general method to manipulate electromagnetic waves on 
+smooth two-dimensional manifolds without rotational symmetry by means of a certain 
+isotropic refractive index distribution derived from the quasi-conformal mapping. The 
+relationship between medium and mappings is induced from the wave equation on the 
+manifold under the geometrical optics approximation. Numerical quasi-conformal 
+algorithms are introduced to construct mappings between manifolds, and consequent 
+functional mediums are validated by cloaking surfaces and generating illusions on plane 
+regions. By cascading mappings between ℝ2 and ℝ3 to obtain a mapping between ℝ3, we 
+succeed in reproducing the scattering of a surface on another surface. In addition, 
+functional devices such as surface Luneburg lenses, surface Eaton lenses, and black holes 
+for surface waves are designed based on carpet cloaks. Finally, the indices required by 
+geometrical optics are reviewed to demonstrate the validity of the approximation on 
+simulation models. Our method paves the way for the regulation of surface 
+electromagnetic waves on any two-dimensional manifold, and can be utilized to control 
+surface waves in other fields, such as acoustics, mechanics, and thermodynamics. 
+ 
+Materials and Methods 
+Conformal transformation optics for surface waves 
+Wave equation on curved manifold. The concept of transformation medium stems from 
+the equivalence between geometry and media. Within the Einstein summation convention, 
+the Maxwell's wave equation for the electric field ∇ℳ × ∇ℳ × 𝐄 − 𝜇0𝜀0𝜕𝑡
+2𝐄 = 0 in free 
+space can be expressed as (16) 
+ 
+𝛻𝑗𝛻𝑗𝐸𝑖 − 𝑅𝑖𝑗𝐸𝑗 − 𝑐0
+−2𝜕𝑡
+2𝐸𝑖 = 0,  
+(6) 
+where 𝑐0 = 1/√𝜇0𝜀0 is the light velocity in free space; 𝑅𝑖𝑗 is the Ricci tensor of the 
+considered geometry ℳ. Supposing that the electromagnetic waves are confined nearby a 
+curved surface ℳ embedded in ℝ3 as surface waves, its local plane wave solution reads 
+as 𝐸𝑖 = ℰi𝑒i𝜑 with constant complex amplitudes ℰi, where the phase reads as 𝜑 = k ⋅ r −
+𝜔𝑡 with the wave vector 𝐤 = ∇ℳ𝜑 and angular frequency 𝜔 = −𝜕𝑡𝜑. For surface waves, 
+the wave vector k lies in the tangent space of the curved surface ℳ, i.e., 𝐤 ∈ 𝒯(ℳ). 
+Thus, (6) can be simplified and approximated in the regime of geometrical optics where 
+the wavelength 𝜆 = 2𝜋/𝑘 varies slowly with distance, i.e., 
+ 
+|∇ℳ𝜆| ≪ 1.  
+(7) 
+In addition, the effective curvature of the curved surface should be small enough 
+compared to the wavelength so that the assumption of locally plane waves is valid, i.e., 
+ 
+|𝑅𝑖𝑗|𝜆2 ≪ 1.  
+(8) 
+
+Manuscript Template                                                                            
+Page 7 of 22 
+ 
+As a result, inserting 𝐸𝑖 = ℰi𝑒i𝜑 into (6) and considering that the (spatial and temporal) 
+derivatives of ℰi vanish, one can obtain the dispersion relation for the surface wave 
+propagating on ℳ, which reads as 
+ 
+𝑘2 = 𝑘𝑗𝑘𝑗 = 𝑔𝑖𝑗𝑘𝑖𝑘𝑗 = 𝜔2/𝑐0
+2. 
+(9) 
+Here, 𝑔𝑖𝑗 is the induced metric tensor for the curved surface ℳ, which can be determined 
+from the transformation Jacobian matrix from the manifold ℳ′ in ℝ2 to ℳ (31),. 
+ 
+Wave equation on a flat plane. Alternatively, if ℳ is flat (i.e., 𝑅𝑖𝑗 = 0) and filled with 
+anisotropic medium denoted by relative permeability tensor 𝜇𝑖𝑗, (6) becomes 
+ 
+∇ × ∇ × 𝐄 − 𝜇0𝜀0𝝁 ⋅ 𝜕𝑡
+2𝐄 = 0. 
+(10) 
+Suppose that the electromagnetic waves are confined nearby ℳ and the electric field 𝐄 is 
+perpendicularly polarized. In a Cartesian coordinate system, if ℳ can be placed into 𝑥𝑦 
+plane, we focus on the case that the electric field vector 𝐄 lies in the normal space of the 
+flat plane ℳ, i.e., 𝐄 ∈ 𝒩(ℳ), and the global wave solution may read as 𝐸𝑧 = ℰ𝑧𝑒i𝜑. 
+Thus, the phase 𝜑 is independent of 𝑧 and the wave vector just lies on the plane as 𝐤 =
+(𝑘𝑥, 𝑘𝑦, 0), because a flat plane is coincident with its tangent space. Since the flat 
+manifold ℳ has a zero-curvature tensor, the condition (7) holds naturally. Once the other 
+condition (8) that wavelength varies slowly is satisfied, one may disregard the derivatives 
+of complex amplitude after inserting 𝐸𝑧 = ℰ𝑧𝑒i𝜑 into (10) and obtain the dispersion 
+relation for the surface wave propagating on ℳ, which reads as (𝜇𝑥𝑥𝑘𝑥2 + 2𝜇𝑥𝑦𝑘𝑥𝑘𝑦 +
+𝜇𝑦𝑦𝑘𝑦2)/det(𝝁) = 𝜔2/𝑐0
+2. By excluding consideration of the particular polarization, the 
+dispersion equation can be recast within the Einstein summation convention as 
+ 
+1
+det(𝝁) 𝜇𝑖𝑗𝑘𝑖𝑘𝑗 =
+𝜔2
+𝑐0
+2. 
+(11) 
+ 
+Transformation medium and geometry. For electromagnetic waves that behave 
+identically on two manifolds, one can obtain the equivalence between geometry and 
+material properties by comparing (9) and (11), which yields 
+ 
+𝜇𝑖𝑗
+det(𝝁) = 𝑔𝑖𝑗. 
+(12) 
+The relative permeability tensor 𝜇𝑖𝑗 actually creates an illusion on the flat plane because a 
+spatial point filled with medium 𝝁 is equivalent to be with a metric 𝒈 = det(𝝁)𝝁−1. If the 
+local Cartesian coordinate system at this point is aligned along the orthogonal 
+eigenvectors of 𝝁, the real and symmetric permeability tensor will reduce to 
+diag(𝜇𝑥, 𝜇𝑦, 𝜇𝑧) so that the square of the line element on 𝑥 direction is d𝑠2 = 𝑔𝑥𝑥d𝑥2 =
+𝜇𝑦𝜇𝑧d𝑥2, which is also the square of optical path length in curved free space. In 
+comparison to d𝑠2 = 𝑛𝑥2d𝑥2 on the flat manifold, one can derive 𝑛𝑥2 = 𝜇𝑦𝜇𝑧 and similar 
+results on 𝑦 and 𝑧 directions. Consequently, the relationship between the relative 
+permeability tensor 𝝁 and the refractive index tensor 𝒏 may be expressed as 𝒏2 =
+det(𝝁)𝝁−1 and one may further obtain 
+ 
+𝒏illustion
+2
+= 𝒈. 
+(13) 
+by referring to (12). 
+ 
+
+Manuscript Template                                                                            
+Page 8 of 22 
+ 
+Surface transformation and TO medium. The metric tensor in equation (13) is induced 
+from the mapping 𝑓: ℳ′ → ℳ and can be constructed by the Jacobian matrix 𝐉3×2 as 𝒈 =
+𝐉T𝐉 (31). Nevertheless, we prefer to associate 𝒏illusion with the Jacobian matrix 𝚲2×3 that 
+represents the transformation from ℝ3 (virtual space) to ℝ2 (physical space). Actually, the 
+asymmetric Jacobian matrices 𝐉3×2 and 𝚲2×3 can be denoted as the Moore–Penrose 
+pseudo-inverse of each other (31), i.e., 𝐉 = 𝚲†, where the superscript '†' denotes pseudo-
+inverse. Thus, one can rewrite the equivalence (13) as 
+ 
+𝒏illustion
+2
+= 𝒈 = 𝐉T𝐉 = (𝚲𝚲T)−1. 
+(14) 
+Similar relationship can be obtained for cloaking medium 𝒏cloak and corresponding 
+Jacobian matrix 𝐉3×2 from ℝ2 (virtual space) to ℝ3 (physical space) as 
+ 
+𝒏cloak
+2
+= (𝐉T𝐉)−1. 
+(15) 
+For the mapping between ℝ3 (Fig. S3), which is formed by cascading two transformations 
+between ℝ3 and ℝ2, the consequent medium for the illusion can be recast as the 
+combination of the cloaking and illusion refractive index tensors, i.e., 
+ 
+𝒏illustion
+2
+= (𝚲1𝚲1
+T)−1 ⋅ (𝐉2
+T𝐉2)−1. 
+(16) 
+where 𝚲1 and 𝐉2 are Jacobian matrices for mappings 𝑓1 and 𝑓2, as illustrated in Fig. S3, 
+respectively. In particular, when the mappings are conformal, the refractive index 
+becomes isotropic, and the corresponding Jacobian matrix has two identical singular 
+values. By taking the determinants of (14) and (15), the refractive indices can be denoted 
+by singular values of Jacobian matrices as 𝑛cloak = 1/𝜎J and 𝑛illusion = 1/𝜎Λ. 
+ 
+Discrete conformal mapping and transformation medium 
+Review on discrete conformal mapping. It has been demonstrated that an isotropic 
+refractive index distribution can be achieved by solving equations for equal optical path 
+length only on rotationally-symmetric surfaces (25). As to the non-rotationally symmetric 
+cloak, high anisotropy is introduced by the projection mapping that distorts the coordinate 
+grid (14). However, numerical algorithms for surface parameterization provide possible 
+conformal mappings for arbitrary surfaces. For example, the angle-based flattening (ABF) 
+method (45, 46) has been proposed to construct conformal parameterization by 
+minimizing a punishing functional to decrease angular distortion while its nonlinearity 
+reduces computational efficiency. Also, the so-called least-squares method (LSCM) (47) 
+and spectral method (SCP) (48) have been introduced to attain higher efficiency, 
+benefiting from their linearity. Their disadvantages are free target boundaries and non-
+bijectivity, whereas we expect a one-to-one mapping that includes every point on physical 
+and virtual space with controlled boundaries. Further research, like disk conformal 
+mapping (DCM) (40), has been reported as a linear and bijective conformal mapping 
+method but with a fixed disk boundary. Not until boundary first flattening (BFF) (29) 
+enabled editing boundary as demand were the drawbacks totally eliminated. To deal with 
+a certain electromagnetic circumstance, one could choose an appropriate algorithm among 
+the preceding techniques (49, 50). 
+ 
+Triangulation and Jacobian matrices. Supposing that the conformal mapping reads 
+𝑓1: ℳ2 → ℳ1 (or 𝑓2: ℳ1 → ℳ2) between manifolds ℳ1 ⊂ ℝ3 and ℳ2 ⊂ ℝ2, as shown in 
+Fig. S5A, one can find that a simplex 𝒮1 on meshed ℳ1 and its counterpart on meshed ℳ2 
+are a pair of similar triangles, which allows 𝒮1 and 𝒮2 to share a same barycentric 
+coordinate system. This local coordinate system, as shown in Fig. S5B, can represent any 
+
+Manuscript Template                                                                            
+Page 9 of 22 
+ 
+point inside the simplex as the linear combination of three vertices and helps quickly 
+induce the Jacobian matrix of numerical mappings based on triangular mesh 
+parameterization. For example, the location of the point 𝐪(𝑥′, 𝑦′) on 𝒮2 can be expressed 
+as 𝑥′ = ∑
+𝜆𝑖𝑥𝑖
+′
+3
+𝑖=1
+ and 𝑦′ = ∑
+𝜆𝑖𝑦𝑖
+′
+3
+𝑖=1
+ with 𝜆1 + 𝜆2 + 𝜆3 = 1, i.e., a linear combination of 
+vertices 𝐪1(𝑥1′,𝑦1′), 𝐪2(𝑥2′, 𝑦2′) and 𝐪3(𝑥3′,𝑦3′). For the triangulation mesh, we can 
+obtain the barycentric coordinates, which read as 
+ 
+𝜆1 = [(𝑦2
+′ − 𝑦3
+′)(𝑥′ − 𝑥3
+′) + (𝑥3
+′ − 𝑥2
+′)(𝑦′ − 𝑦3
+′)]/det (𝐌), 
+(17) 
+ 
+𝜆2 = [(𝑦3
+′ − 𝑦1
+′)(𝑥′ − 𝑥3
+′) + (𝑥1
+′ − 𝑥3
+′)(𝑦′ − 𝑦3
+′)]/det (𝐌), 
+(18) 
+ 
+𝜆3 = [(𝑦1
+′ − 𝑦2
+′)(𝑥′ − 𝑥2
+′) + (𝑥2
+′ − 𝑥1
+′)(𝑦′ − 𝑦2
+′)]/det (𝐌), 
+(19) 
+where det(𝐌) = |(𝐪1 − 𝐪3) × (𝐪2 − 𝐪3)|, with 𝐪𝑖(𝑥𝑖′, 𝑦𝑖′) being the 𝑖-th vertices (𝑖 =
+1, 2, 3). Here, (17), (18) and (19) show that the barycentric coordinate system (𝜆1, 𝜆2, 𝜆3) 
+can be expressed by the Cartesian coordinate system (𝑥′, 𝑦′). Regarding the point 
+𝐩(𝑥, 𝑦, 𝑧) on 𝒮1 ⊂ ℝ3, mapped from the point 𝐪 in ℝ2, we have 𝑥 = ∑
+𝜆𝑖𝑥𝑖
+3
+𝑖=1
+, 𝑦 =
+∑
+𝜆𝑖𝑦𝑖
+3
+𝑖=1
+ and 𝑧 = ∑
+𝜆𝑖𝑧𝑖
+3
+𝑖=1
+ as the linear combination of 𝐩1(𝑥1, 𝑦1, 𝑧1), 𝐩2(𝑥2, 𝑦2, 𝑧2) and 
+𝐩3(𝑥3, 𝑦3, 𝑧3), since 𝒮1 and 𝒮2 share the same barycentric coordinates 𝜆𝑖. As a result, the 
+Jacobian matrix 𝐉3×2 of the mapping from 𝒮2 ⊂ ℝ2 to 𝒮1 ⊂ ℝ3 can be derived according 
+to the derivatives of (𝜆1, 𝜆2, 𝜆3) with respect to (𝑥′, 𝑦′), which reads as 
+ 
+𝐉3×2 = (
+𝜕𝑥′𝑥
+𝜕𝑦′𝑥
+𝜕𝑥′𝑦
+𝜕𝑦′𝑦
+𝜕𝑥′𝑧
+𝜕𝑦′𝑧
+) =
+1
+det(𝐌) (
+𝑥1
+𝑥2
+𝑥3
+𝑦1
+𝑦2
+𝑦3
+𝑧1
+𝑧2
+𝑧3
+) (
+𝑦2
+′ − 𝑦3′
+𝑥3
+′ − 𝑥2
+′
+𝑦3
+′ − 𝑦1
+′
+𝑥1
+′ − 𝑥3
+′
+𝑦1
+′ − 𝑦2
+′
+𝑥2
+′ − 𝑥1
+′
+). 
+(20) 
+In a similar manner, one can derive the Jacobian matrix 𝚲2×3 of the numerical mapping 
+from 𝒮1 to 𝒮2; alternatively, one may calculate the Moore–Penrose pseudoinverse of 𝐉3×2 
+as 𝚲2×3 (31). By calculating the Jacobian matrices 𝐉3×2 or 𝚲2×3 on each simplex, the 
+information of mapping 𝑓1 or 𝑓2 can be fully described. 
+ 
+Simulation methods 
+FEM simulation. The wave behavior of electromagnetic devices is simulated using the 
+finite element method. The geometric model is an optical thin-film waveguide whose 
+thickness is less than one fifth of the wavelength. On the outer surfaces of the waveguide, 
+the perfect electric conductor (PEC) boundary condition is applied to emulate the 
+propagation of the surface wave on a two-dimensional manifold. Thus, the propagation of 
+the plane wave or Gaussian beam is restricted within the optical thin film. To mimic an 
+open and non-reflecting infinite domain, perfectly matched layers (PMLs) are applied on 
+the boundary of the propagating plane. The designed medium is configured to the 
+waveguide as a fitting function interpolated from the discrete data set calculated on extra 
+dense meshes. 
+ 
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+Manuscript Template                                                                            
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+Manuscript Template                                                                            
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+ 
+Acknowledgments 
+Funding: This work was supported by the National Natural Science Foundation of China 
+(NSFC) under grant no. 51977165. 
+ 
+Author contributions:  
+ 
+Conceptualization: X.Z., X.M., T.D. 
+
+Manuscript Template                                                                            
+Page 13 of 22 
+ 
+ 
+Methodology: X.Z., H.D., X.G., T.D. 
+ 
+Investigation: X.Z., H.D., X.G. 
+ 
+Visualization: X.Z., H.D., X.G. 
+ 
+Supervision: X.M., T.D. 
+ 
+Writing—original draft: All authors. 
+ 
+Writing—review & editing: All authors. 
+ 
+Competing interests: The authors declare that they have no competing interests. 
+ 
+Data and materials availability: All data needed to evaluate the conclusions in the paper 
+are present in the paper and/or the Supplementary Materials. Raw data and corresponding 
+simulation data are available upon request. 
+ 
+Figures and Tables 
+ 
+Fig. 1. The conformal mapping between manifolds. (A) A light beam crossing a curved 
+two-dimensional manifold ℳ embedded in ℝ3. (B) A light beam crossing a flat 
+two-dimensional manifold ℳ′ in ℝ2. The manifold ℳ is 𝑢𝑣-parameterized and 
+both manifolds are plotted with coordinate grid. One can obtain the manifold ℳ in 
+(A) from ℳ′ in (B) through a certain analytic or numerical mapping 𝑓: ℳ′ → ℳ. 
+ 
+
+A
+McR3
+u
+V
+f: M'-M
+B
+M' c R?
+XManuscript Template                                                                            
+Page 14 of 22 
+ 
+ 
+Fig. 2. The field and medium distribution for cloaks. Normalized electric field 
+distribution of surface electromagnetic wave cloaks achieved by (A) anisotropic 
+relative permeability and (B) isotropic refractive index. (C) Components of 
+anisotropic relative permeability, 𝜇𝑥𝑥, 𝜇𝑦𝑦 and 𝜇𝑥𝑦, applied in (A) and isotropic 
+refractive index 𝑛c,double applied in (B). The excitation is a 𝑧-polarized plane wave 
+with a magnitude of |𝐸𝑧| = 1 V/m; and the wavelength in free-space is 𝜆0 =
+20 mm. The bump with a height of 1.25𝜆0 is located in the center of the square 
+waveguide with a width of 12𝜆0. For the elliptical boundary, the semi-minor and 
+semi-major axis length are 𝑎 = 3.75𝜆0 and 𝑏 = 5𝜆0, respectively, along with 𝑥- 
+and 𝑦-axes. 
+ 
+ 
+
+A
+12^o
+Ez
+B
+c
+μxx
+Py
+μxy
+Nc.double
+1
+0.63
+0.63
+1
+0.19
+0.19
+0.83Manuscript Template                                                                            
+Page 15 of 22 
+ 
+ 
+Fig. 3. The field and medium distribution for illusions. Normalized electric field 
+distribution of surface electromagnetic wave scattering. (A) Scattering on the 
+single-camelback bump when filled with homogeneous medium. (B) Illusion of the 
+single-camelback bump appearing on the plane. (C) Illusion of the single-
+camelback bump appearing on the double-camelback bump. (D) Isotropic 
+refractive indices: 𝑛i,plane for the elliptic region in (B) and 𝑛i,double for the double-
+camelback bump in (C). The elliptical base profiles in (A), (B) and (C) are the 
+same. 
+
+A
+15^o
+B
+c
+D
+ni,plane
+ni.double
+1.25
+0.85
+1.21Manuscript Template                                                                            
+Page 16 of 22 
+ 
+ 
+ 
+Fig. 4. The field and medium distribution for devices. Normalized electric field 
+distribution on surface electromagnetic wave devices. (A) Luneburg lens; (B) 
+Eaton lens; and (C) Black hole. Gaussian beam is applied to demonstrate their 
+functions. (D) Isotropic refractive indices; 𝑛Luneburg for Luneburg lens in (A), 
+decimal logarithm of 𝑛Eaton for Eaton lens in (B), real and imaginary part of 
+𝑛Blackhole for black hole in (C). 
+ 
+ 
+ 
+
+12Ao
+A
+focus
+[EI2
+B
+c
+D
+nLuneburg
+log1o(nEaton)
+real(nglaothole)
+imag(nglaochole)
+1.37
+0
+8
+2.44
+0.083
+0.097Manuscript Template                                                                            
+Page 17 of 22 
+ 
+Supplementary Materials for 
+ 
+• 
+Controlling Electromagnetic Surface Waves with Conformal 
+Transformation Optics 
+ 
+Xiaoyu Zhao et al. 
+ 
+*Corresponding author. Email: tydong@mail.xjtu.edu.cn. 
+ 
+ 
+ 
+ 
+ 
+This PDF file includes: 
+ 
+Figs. S1 to S5 
+ 
+ 
+
+Manuscript Template                                                                            
+Page 18 of 22 
+ 
+ 
+Fig. S1. (A) Normalized electric field distribution of surface electromagnetic wave scattering on 
+double-camelback bump when filled with homogeneous medium. (B) Quasi-conformal ratio 𝑸 of 
+the mapping applied to design the cloak shown in Fig. 2A. (C) Curvature index 𝝆. (D) 
+Wavelength index 𝒘. 
+ 
+ 
+
+A
+Ez
+B
+C
+D
+p = 0.54
+Q
+d
+W
+1.028
+0
+10.2
+0
+0.24Manuscript Template                                                                            
+Page 19 of 22 
+ 
+ 
+Fig. S2. (A) Normalized electric field distribution of the surface electromagnetic wave cloak on 
+single-camelback bump achieved by (B) isotropic refractive index 𝒏𝐜,𝐬𝐢𝐧𝐠𝐥𝐞. (C) Quasi-conformal 
+ratio 𝑸 of the mapping applied to design the cloak shown in (A). (D) Curvature index 𝝆. (E) 
+Wavelength index 𝒘. 
+ 
+ 
+
+A
+12^o
+B
+C
+D
+E
+nc.single
+p=0.39
+Q
+d
+W
+0.8
+1.011
+0
+14.5
+0
+0.21Manuscript Template                                                                            
+Page 20 of 22 
+ 
+ 
+Fig. S3. A quasi-conformal mapping between two manifolds embedded in ℝ𝟑 constructed by 
+cascading two mappings between ℝ𝟑 and ℝ𝟐. (A) A single-camelback manifold 𝓜𝟏 embedded in 
+ℝ𝟑. (B) The plane region 𝓜𝟐 in ℝ𝟐 mapped from 𝓜𝟏 through mapping 𝒇𝟏. (C) The double-
+camelback manifold 𝓜𝟑 embedded in ℝ𝟑 mapped from 𝓜𝟐 through mapping 𝒇𝟐. 
+ 
+ 
+
+A
+M1 CR3
+fi : M1 → M2
+B
+M2 C IR2
+fz : M2 → M3
+c
+M3 CR3Manuscript Template                                                                            
+Page 21 of 22 
+ 
+ 
+Fig. S4. (A) Normalized electric field 𝑬𝒛, (B) background field 𝑬𝐛𝒛 and (C) scattering field 𝑬𝒛 −
+𝑬𝐛𝒛 of the hemisphere surface wave cloak achieved by (D) isotropic refractive index 𝒏𝐜,𝐬𝐩𝐡𝐞𝐫𝐞. (E) 
+Quasi-conformal ratio 𝑸 of the mapping applied to design the cloak shown in (A). (F) Curvature 
+index 𝝆. (G) Wavelength index 𝒘. The radius of the hemisphere is 𝟓𝝀𝟎. 
+ 
+ 
+
+A
+Ez
+1
+15^o
+B
+Epz
+Ez- Ebz
+c
+0.75
+0.75
+D
+E
+F
+G
+p = 1.57
+nc.sphere
+W
+0.5
+1.012
+0
+35
+0
+0.2Manuscript Template                                                                            
+Page 22 of 22 
+ 
+ 
+Fig. S5. (A) Simplices 𝓢𝟏 and 𝓢𝟐 as triangle elements in the mesh of double-camelback manifold 
+𝓜𝟏 embedded in ℝ𝟑 and the region 𝓜𝟐 in ℝ𝟐, related by quasi-conformal mappings 𝒇𝟏 and 𝒇𝟐. 
+(B) The same barycentric coordinates on simplices 𝓢𝟏 and 𝓢𝟐. 
+ 
+
+A
+B
+Pi(X1,Y1,Z1)
+Si
+MICR3
+P=入1P1+22P2+2P3
+p(x,y,z)
+Si
+P3(X3,Y3,Z3)
+P2(X2,Y2,Z2)
+f2:Mi→M2
+f:M2-→M1
+Barycentric
+Coordinates
+=A,(x,y-x(x.y,z)
+q1(xi,yi)
+S2
+M2CR2
+q=191+22+3
+q(x,y)
+S2
+q3(x3,y3)
+q2 (x2,y2)

NtAyT4oBgHgl3EQftPkw/content/tmp_files/2301.00590v1.pdf.txt

@@ -0,0 +1,539 @@
+arXiv:2301.00590v1  [astro-ph.IM]  2 Jan 2023
+End-to-end simulations of a near-infrared pyramid sensor on
+Keck II
+C. Planteta, G. Agapitoa, C. Giordanoa, S. Espositoa, P. Wizinowichb, and C. Bondc
+aINAF - Osservatorio di Arcetri, 50125 Firenze, Italy
+bW. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743, USA
+cInstitute for Astronomy, University of Hawaii, 640 N. 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). 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). 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. In this case, the pyramid would be used as a low-order sensor only. We report on a study
+of the pyramid sensor’s performance via end-to-end simulations, applied to Keck’s specific case. We present the
+expected Strehl ratio with optimized configurations in NGS mode, and the expected residual on low orders in
+LGS mode. 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.
+Keywords: Adaptive optics, Wavefront sensing, Infrared, Keck, Pyramid, LIFT
+1. 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).3 The main goal of this upgrade is to perform direct imaging and
+slit spectroscopy of exoplanets around M dwarfs. 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. 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.4 In addition to the NGS mode, the
+system will also provide a LGS mode. In this case, the pyramid would be used as a low-order sensor only. We
+report on a study of the pyramid sensor’s performance via end-to-end simulations made with PASSATA.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). 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. 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.
+2. SIMULATIONS PARAMETERS
+We list in Table 1 the simulation parameters used for the different cases of this study. The chosen values for
+the parameters that are optimized (modulation, frequency, control gain. . . ) are given in each specific case, and
+we only state here the explored ranges of values. The wavefront modes (turbulent Karhunen-Lo`eve and Zernike)
+are considered to be perfectly reproduced by the deformable mirror (DM). 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.P.: E-mail: plantet@arcetri.astro.it
+
+• 333 Hz ă f ď 666 Hz : 2 frames delay
+• f ď 333 Hz : 1 frame delay
+Table 1. Simulation parameters.
+Parameter
+NGS mode
+LGS mode
+20ˆ20
+32ˆ32
+20ˆ20
+32ˆ32
+Sensing band
+1.5 µm - 1.8 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)
+0.3
+Sky background in H
+14 mag/arcsec2
+Seeing
+0.63”
+Layers’ altitudes (km)
+0, 500, 1000, 2000, 4000, 8000, 16000
+C2
+n profile (normalized in energy)
+0.517, 0.119, 0.063, 0.061, 0.105, 0.081, 0.054
+Mean wind speed
+9.5 m/s
+Zenith angle
+300
+Subaperture size
+0.5625 m
+0.35 m
+0.5625 m
+0.35 m
+APD gain
+30
+Excess noise factor
+1.4
+Read-out noise
+0.1 or 1 e´
+1 e´
+0.8 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
+0.1-0.6
+LIFT: 0.1-0.6
+LIFT: 0.1-0.6
+Pyramid: 0.15-10
+Pyramid:0.25-5
+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.
+– LGS = high flux point source at finite distance.
+– Tip/tilt filtered and replaced by a residual jitter of 106 mas rms + turbulent tip/tilt.
+– Control gain: 0.3.
+• Focus loop (only in 32ˆ32 case):
+– Correction frequency: 10 Hz.
+– Input: focus residual from high-order control + sinusoid of period 5 seconds and amplitude 100 nm
+(80 nm rms).
+– Control gain range: 0.1-1 for LIFT, 0.1-4 for the pyramid.
+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.
+
+3. NGS MODE
+In this section, we study the performance of the pyramid, in terms of Strehl ratio, for different pupil samplings.
+We first considered a pupil sampling of 20ˆ20 subapertures, in agreement with the current DM’s number of
+actuators. However, the DM should be upgraded to a MEMS 32ˆ32. We thus study in a second step the impact
+of increasing the pupil sampling to 32ˆ32, or to 40ˆ40 for robustness reasons.
+3.1 Pyramid 20ˆ20
+We present here the results of the simulations with a pyramid 20ˆ20 in NGS mode (Fig. 3.1). The parameters,
+listed in Table 2, are optimized in the ranges described previously to get the highest Strehl ratio. This optimiza-
+tion is simply made by running simulations going through the whole set of parameters and selecting the best
+one.
+As we lacked information on the detector’s noise, we considered two cases: low noise (no dark current, read-
+out noise = 0.1 e´) and high noise (dark current = 100 e´/s, read-out noise = 1 e´). The difference between
+those two cases is not very significant (0.5 magnitude at faint end).
+Table 2. Optimized parameters (high noise/low noise) for the pyramid 20ˆ20 in NGS mode.
+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
+0.3/0.3
+0.2/0.2
+0.15/0.25
+0.15/0.3
+0.3/0.55
+0.3/0.6
+Modulation radius (λ{D)
+1.5
+1.5
+1.5
+1.5
+1.5
+2
+8
+9
+10
+11
+12
+13
+14
+15
+H-magnitude
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+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
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+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
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+SR(@K)
+Dark = 0 -- RON = 0.1     
+Dark = 100 -- RON = 1
+(c) Comparison
+Figure 1. Strehl ratio as a function of H magnitude with a pyramid 20ˆ20 in NGS mode. Left: Low noise case. Middle:
+High noise case. Right: Comparison of low and high noise cases in K band.
+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.
+• Increase the gain: we cannot remove the background in pyramid images, it is thus taken into account in
+the normalization when computing the slopes. In the end, the slopes are proportional to the ratio star
+flux/background, which decreases with respect to the magnitude. A higher gain is needed to compensate
+that effect. The increase in gain is also needed at lower frequencies, as the correction is done less often.
+• Increase in modulation: at low flux, the noise error makes the pyramid work in non-linear regime. The
+modulation reduces the non-linearity error, at the price of a lower sensitivity (hence greater noise error).
+A trade-off is made between those two errors to reach the lowest overall error. At high flux, using a high
+modulation lowers the non-linearity error.
+
+• Decrease the number of modes: estimating less modes improves the noise propagation behavior at low
+orders.
+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.
+3.2 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. We consider here only the impact at low flux, as it corresponds to more practical cases and
+is more critical for the system design.
+We simulated two different pupil samplings: 32ˆ32 and 40ˆ40 (in that case, only the subaperture size from
+Table 1 is changed). The first one matches the MEMS mirror sampling, while the second would help calibrate
+misregistration errors and thus gain in robustness.
+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´. The performance for the high noise case of the pyramid 20ˆ20 is recalled for
+reference. It should be noted that the dark current does not have a significant impact here, the results can thus
+be fairly compared.
+Table 3. Optimized parameters and Strehl ratios for the pyramid 32ˆ32 and 40ˆ40 in NGS mode.
+Frequency (Hz)
+Number of modes
+Gain
+Modulation radius (λ{D)
+Strehl ratio (K)
+20ˆ20
+600
+65
+0.3
+1.5
+29.8%
+32ˆ32
+200
+65
+0.75
+1.5
+26.7%
+40ˆ40
+200
+65
+0.75
+1.5
+25.1%
+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. 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.
+4. 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. We compare the pyramid to LIFT, in order to evaluate the gain of having a focal-plane
+sensor for this low-order estimation. 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.
+As in the previous section, we first considered a pyramid with 20ˆ20, and then checked the impact of a finer
+sampling. For LIFT, the only design parameter that will have an impact on the performance is the pixel scale.
+We consider here a pixel of 15 or 30 mas, corresponding respectively to a Nyquist and a Nyquist/2 sampling in
+H band.
+4.1 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.
+The focus estimation will be included in the next section. For practical reasons, the number of reconstructed
+modes for the pyramid is either 2 (lowest noise error) or 250 (lowest aliasing error).
+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. The residuals obtained with LIFT and the pyramid are compared in Fig. 4.1. We find that
+LIFT provides a gain of up to 2 magnitudes over the pyramid, either on axis or off axis.
+The behavior of the optimized parameters for the pyramid is as described in section 3.1. 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. The flux is sufficiently high in that case to use
+a strong modulation without a significant impact on noise error.
+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. On the contrary, it is less
+efficient off axis, or at high flux in general. 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. 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. The overall aliasing + noise error is in the end always better with the 15 mas pixel for the considered
+magnitudes.
+Table 4. Optimized parameters (on axis/off axis) for the pyramid 20ˆ20 in LGS mode.
+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
+0.5/0.25
+0.5/0.65
+0.75/0.65
+1/1
+3.5/3.5
+5/5
+Modulation radius (λ{D)
+0/2
+0/0
+0/0
+0/0
+0/0
+0/1
+TT residual (nm rms)
+21.4/48
+32.3/59.4
+44.6/73.3
+57.8/89.4
+88.6/123
+153.7/195.6
+Table 5. Optimized parameters (on axis/off axis) for LIFT with a 15 mas pixel in LGS mode.
+Magnitude
+10
+12
+13
+14
+15
+16
+Frequency (Hz)
+1000/1000
+1000/1000
+1000/333
+333/333
+333/200
+200/200
+Gain
+0.3/0.3
+0.3/0.3
+0.2/0.5
+0.4/0.5
+0.4/0.4
+0.4/0.3
+TT residual (nm rms)
+18.9/42.3
+23.2/45.1
+27.3/48.5
+33.1/56.4
+42.9/67.5
+63.5/93.2
+Table 6. Optimized parameters (on axis/off axis) for LIFT with a 30 mas pixel in LGS mode.
+Magnitude
+10
+12
+13
+14
+15
+16
+Frequency (Hz)
+1000/1000
+1000/1000
+1000/1000
+333/1000
+333/500
+200/333
+Gain
+0.4/0.3
+0.3/0.2
+0.2/0.2
+0.5/0.2
+0.4/0.2
+0.4/0.3
+TT residual (nm rms)
+25.7/62.5
+28.5/63.7
+31.5/66.5
+37.6/68.3
+45.6/81.1
+64.5/101.2
+(a) On axis
+(b) 15” off axis
+Figure 2. Comparison of the tip/tilt residual obtained with LIFT or the pyramid 20ˆ20 in LGS mode. Left: NGS on
+axis. Right: NGS at 15” off axis.
+
+4.2 Finer pupil sampling
+We now check the impact of having a finer-sampled pyramid on the performance at magnitude 14. 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).
+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).
+On axis, there is a clear advantage using LIFT for tip/tilt estimation, with a factor 2 in rms residual. 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 1.4).
+Off axis, we still have a better estimation of tip/tilt with LIFT, but with less difference (factor 1.4 at best).
+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. The focus correction is similar in all cases, and the
+tip/tilt estimation is always affected. 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). 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. For LIFT with 30 mas pixels, this error is lower, around 20 nm rms. This might
+be due to the fact that the tip/tilt estimation alone was already affected by non-linear effects.
+Table 7. Optimized parameters and residuals (on axis/off axis) for the pyramid 32ˆ32 and LIFT in LGS mode, with the
+estimation of focus.
+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
+0.4/0.1
+0.4/0.5
+0.4/0.3
+4/4
+TT residual (nm rms)
+39.3/80.5
+34.3/67.5
+34.7/69.8
+68.9/95.9
+TT residual (with focus)
+36.2/82.7
+33.6/75.9
+34.8/77.7
+69.3/100
+Gain on focus
+0.6/0.4
+0.5/0.4
+0.7/0.3
+2/1.5
+Focus residual (nm rms)
+37/52.9
+38.4/53.1
+35.9/53.2
+50.5/50.8
+5. 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. 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). The latter can
+be increased with a finer pupil sampling (32ˆ32, or 40ˆ40) and a higher degree of correction (i. e. 32ˆ32
+DM), without degrading significantly the performance at low flux. The 40ˆ40 sampling would also provide more
+robustness to errors such as misregistration. 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. 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. It was also shown that LIFT gives best results with images sampled at Nyquist
+(15 mas pixels). In future works, we will explore more off-axis distances and seeing conditions to confirm the
+advantage of using LIFT. We will also study the impact of the atmosphere dispersion on both sensors.
+ACKNOWLEDGMENTS
+This work was partly funded by INAF (Research Grant DD 27). The Keck II pyramid wavefront sensor is funded
+by the National Science Foundation under Grant No. AST-1611623.
+
+REFERENCES
+[1] Wizinowich, P., Le Mignant, D., Bouchez, A. H., Campbell, R. D., Chin, J. C., Contos, A. R., van Dam,
+M. A., Hartman, S. K., Johansson, E. M., Lafon, R. E., et al., “The WM Keck Observatory laser guide star
+adaptive optics system: overview,” Publications of the Astronomical Society of the Pacific 118(840), 297
+(2006).
+[2] Ragazzoni, R., “Pupil plane wavefront sensing with an oscillating prism,” Journal of modern optics 43(2),
+289–293 (1996).
+[3] Wizinowich, P., Chun, M., Mawet, D., Agapito, G., Dekany, R., Esposito, S., Fusco, T., Guyon, O., Hall, D.,
+Plantet, C., and Rigaut, F., “Near-infrared wavefront sensing,” Proc.SPIE 9909, 9909 – 9909 – 13 (2016).
+[4] Feautrier, P., Gach, J.-L., and Wizinowich, P., “State of the art IR cameras for wavefront sensing using
+e-APD MCT arrays,” in [AO4ELT4 Proceedings], (2015).
+[5] G. Agapito, A. Puglisi, S. E., “Passata: object oriented numerical simulation software for adaptive optics,”
+Proc.SPIE 9909, 9909 – 9909 – 9 (2016).
+[6] Meimon, S., Fusco, T., and Mugnier, L. M., “LIFT: a focal-plane wavefront sensor for real-time low-order
+sensing on faint sources,” Optics letters 35(18), 3036–3038 (2010).
+[7] Quir´os-Pacheco, F., Agapito, G., Riccardi, A., Esposito, S., Louarn, M. L., and Marchetti, E., “Performance
+simulation of the eris pyramid wavefront sensor module in the vlt adaptive optics facility,” Proc.SPIE 8447,
+8447 – 8447 – 12 (2012).
+

NtAyT4oBgHgl3EQftPkw/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='00590v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='IM]  2 Jan 2023  End-to-end simulations of a near-infrared pyramid sensor on  Keck II  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' Planteta, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' Agapitoa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' Giordanoa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' Espositoa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' Wizinowichb, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' Bondc  aINAF - Osservatorio di Arcetri, 50125 Firenze, Italy  bW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' Keck Observatory, 65-1120 Mamalahoa Hwy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' : E-mail: plantet@arcetri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content=' Simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='5 µm - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+page_content='517, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='119, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='063, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='061, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='105, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='081, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='054  Mean wind speed  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='5625 m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='35 m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5625 m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='4  Read-out noise  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1 or 1 e´  1 e´  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6  LIFT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6  LIFT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6  Pyramid: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='15-10  Pyramid:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='  – LGS = high flux point source at finite distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='  – Control gain: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content='  – Control gain range: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1-1 for LIFT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1-4 for the pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='5 magnitude at faint end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='15/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='15/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6  Modulation radius (λ{D)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='0  SR(@K)  Dark = 0 -- RON = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content=' Left: Low noise case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' Middle:  High noise case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='8%  32ˆ32  200  65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7%  40ˆ40  200  65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='75/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='65  1/1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  5/5  Modulation radius (λ{D)  0/2  0/0  0/0  0/0  0/0  0/1  TT residual (nm rms)  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4/48  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3/59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='8/89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6/123  153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7/195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  TT residual (nm rms)  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9/42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2/45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3/48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1/56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9/67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5/93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  TT residual (nm rms)  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7/62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5/63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5/66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6/68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6/81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5/101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content=' Left: NGS on  axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content=' Right: NGS at 15” off axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+page_content='4 at best).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content='  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  4/4  TT residual (nm rms)  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3/80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3/67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7/69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='8  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9/95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9  TT residual (with focus)  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2/82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6/75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='8/77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3/100  Gain on focus  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='6/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='7/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='3  2/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5  Focus residual (nm rms)  37/52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='4/53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='9/53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='2  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='5/50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+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'}
+page_content=' AST-1611623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAyT4oBgHgl3EQftPkw/content/2301.00590v1.pdf'}
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