Multi-Structural Games and Number of Quantifiers

Fagin, Ronald; Lenchner, Jonathan; Regan, Kenneth W.; Vyas, Nikhil

doi:10.1109/lics52264.2021.9470756

Cited by 5 publications

(14 citation statements)

References 15 publications

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“…Next, we give an example of a setting in which one can completely nail down the number of quantifiers that are necessary and sufficient for expressing a property. Building on Fagin et al [10], which gives results on the number of quantifiers needed to distinguish linear orders of different sizes, we study the number of quantifiers needed to distinguish rooted trees of different depths.…”

Section: Theorem (Theorem 4 Section 2)mentioning

confidence: 99%

“…In this paper we make use of a variant of E-F games, which have come to be called multi-structural games [10]. Multi-structural games (henceforth M-S games) make Duplicator more powerful and can be used to characterize the number of quantifiers, rather than the quantifier rank.…”

Section: Multi-structural Gamesmentioning

confidence: 99%

“…In [10] the authors provide a simple example of a property P of a directed graph that requires 3 quantifiers but which can be expressed with a sentence of quantifier rank 2. P is the property of having a vertex with both an in-edge and an out-edge.…”

Section: Multi-structural Gamesmentioning

confidence: 99%

“…One of the reasons for the difficulty of directly analyzing the number of quantifiers is that the separability game is played on a pair (A, B) of sets of structures, rather than on a pair of structures as in a conventional Ehrenfeucht-Fraïssé game. However, last year Fagin, Lenchner, Regan and Vyas [10] rediscovered the games, provided some tools for analyzing them, and showed how to utilize them to characterize the number of quantifiers needed to express linear orders of different sizes. In this paper, we push forward in the study of number of quantifiers as a bona fide complexity measure by establishing several new results, using these rediscovered games as an important, though not exclusive, tool.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we push forward in the study of number of quantifiers as a bona fide complexity measure by establishing several new results, using these rediscovered games as an important, though not exclusive, tool. Although Immerman called his game the "separability game," we keep to the more evocative "multi-structural game," as coined in [10].…”

Section: Introductionmentioning

confidence: 99%

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On the Number of Quantifiers as a Complexity Measure

Fagin¹,

Lenchner²,

Vyas³

et al. 2022

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In 1981, Neil Immerman described a two-player game, which he called the "separability game" [14], that captures the number of quantifiers needed to describe a property in first-order logic. Immerman's paper laid the groundwork for studying the number of quantifiers needed to express properties in first-order logic, but the game seemed to be too complicated to study, and the arguments of the paper almost exclusively used quantifier rank as a lower bound on the total number of quantifiers. However, last year Fagin, Lenchner, Regan and Vyas [10] rediscovered the game, provided some tools for analyzing them, and showed how to utilize them to characterize the number of quantifiers needed to express linear orders of different sizes. In this paper, we push forward in the study of number of quantifiers as a bona fide complexity measure by establishing several new results. First we carefully distinguish minimum number of quantifiers from the more usual descriptive complexity measures, minimum quantifier rank and minimum number of variables. Then, for each positive integer k, we give an explicit example of a property of finite structures (in particular, of finite graphs) that can be expressed with a sentence of quantifier rank k, but where the same property needs 2 Ω(k 2 ) quantifiers to be expressed. We next give the precise number of quantifiers needed to distinguish two rooted trees of different depths. Finally, we give a new upper bound on the number of quantifiers needed to express s-t connectivity, improving the previous known bound by a constant factor.

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Section: Theorem (Theorem 4 Section 2)mentioning

confidence: 99%

Section: Multi-structural Gamesmentioning

confidence: 99%

Section: Multi-structural Gamesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Number of Quantifiers as a Complexity Measure

Fagin¹,

Lenchner²,

Vyas³

et al. 2022

Preprint

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Succinctness of Cosafety Fragments of LTL via Combinatorial Proof Systems

Geatti,

Mansutti,

Montanari

2024

Lecture Notes in Computer Science

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This paper focuses on succinctness results for fragments of Linear Temporal Logic with Past ($$\textsf{LTL}$$ LTL ) devoid of binary temporal operators like until, and provides methods to establish them. We prove that there is a family of cosafety languages $$(\mathcal {L}_n)_{n \ge 1}$$ ( L n ) n ≥ 1 such that $$\mathcal {L}_n$$ L n can be expressed with a pure future formula of size $$\mathcal {O}(n)$$ O ( n ) , but it requires formulae of size $$2^{\varOmega (n)}$$ 2 Ω ( n ) to be captured with past formulae. As a by-product, such a succinctness result shows the optimality of the pastification algorithm proposed in [Artale et al., KR, 2023].We show that, in the considered case, succinctness cannot be proven by relying on the classical automata-based method introduced in [Markey, Bull. EATCS, 2003]. In place of this method, we devise and apply a combinatorial proof system whose deduction trees represent $$\textsf{LTL}$$ LTL formulae. The system can be seen as a proof-centric (one-player) view on the games used by Adler and Immerman to study the succinctness of $$\textsf{CTL}$$ CTL .

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Game characterizations for the number of quantifiers

Hella,

Luosto

2024

Math. Struct. Comp. Sci.

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A game that characterizes equivalence of structures with respect to all first-order sentences containing a given number of quantifiers was introduced by Immerman in 1981. We define three other games and prove that they are all equivalent to the Immerman game, and hence also give a characterization for the number of quantifiers needed for separating structures. In the Immerman game, Duplicator has a canonical optimal strategy, and hence Duplicator can be completely removed from the game by replacing her moves with default moves given by this optimal strategy. On the other hand, in the last two of our games there is no such optimal strategy for Duplicator. Thus, the Immerman game can be regarded as a one-player game, but two of our games are genuine two-player games.

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Multi-Structural Games and Number of Quantifiers

Cited by 5 publications

References 15 publications

On the Number of Quantifiers as a Complexity Measure

On the Number of Quantifiers as a Complexity Measure

Succinctness of Cosafety Fragments of LTL via Combinatorial Proof Systems

Game characterizations for the number of quantifiers

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