Asymptotic behaviour in temporal logic

Asarin, Eugène; Blockelet, Michel; Degorre, Aldric; Dima, Cătălin; Mu, Chaoxu

doi:10.1145/2603088.2603158

Cited by 4 publications

(6 citation statements)

References 12 publications

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“…A key difference to our work is that the safety level does not give the probability of a model and does not distinguish between properties of the same class but with different density values. Also related is Asarin et al's investigation of the asymptotic behavior in temporal logic [1]. The authors use the notion of entropy to show the relation between formulas in parametric linear-time temporal logic and formulas in standard LTL as some bounds tend to infinity.…”

Section: Related Workmentioning

confidence: 99%

“…All lassos of length greater or equal to 2 belong to one of the sets of base models or base non-models, depending on whether the second position of the lasso is labeled with p or not, and thus, the density of p is determined by the rates of base models and base non-models. The number of base models of p is equal to 2 AP−1 • (2 AP ) n−1 • n for n > 1, which results in a density of 1 2 . The rates of base models and base non-models also determine the density of the safety property q R p over AP = {p, q}, which convergences to a value of 1 3 .…”

Section: Asymptotic Densitymentioning

confidence: 99%

“…The number of base models of p is equal to 2 AP−1 • (2 AP ) n−1 • n for n > 1, which results in a density of 1 2 . The rates of base models and base non-models also determine the density of the safety property q R p over AP = {p, q}, which convergences to a value of 1 3 . The property has no loop non-models and n loop models for each bound n, namely those where all positions are labeled with p and not labeled with q.…”

Section: Asymptotic Densitymentioning

confidence: 99%

“…It is clear that the rate of both the classes of models with no good-prefix and non-models with no bad-prefix converge to 0. This means, the upper and the lower bound of the density, determined by the classes of bad-prefix non-models and good-prefix models, meet in the limit and the density function converges to 1 2 . For the special case of ω-regular properties, the limit of the density function can be computed algorithmically.…”

Section: Introductionmentioning

confidence: 96%

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The Density of Linear-Time Properties

Finkbeiner

Torfah

2017

Automated Technology for Verification and Analysis

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Finding models for linear-time properties is a central problem in verification and planning. We study the distribution of lineartime models by investigating the density of linear-time properties over the space of ultimately periodic words. The density of a property over a bound n is the ratio of the number of lasso-shaped words of length n, that satisfy the property, to the total number of lasso-shaped words of length n. We investigate the problem of computing the density for both linear-time properties in general and for the important special case of ωregular properties. For general linear-time properties, the density is not necessarily convergent and can oscillates indefinitely for certain properties. However, we show that the oscillation is bounded by the growth of the sets of bad-and good-prefix of the property. For ω-regular properties, we show that the density is always convergent and provide a general algorithm for computing the density of ω-regular properties as well as more specialized algorithms for certain sub-classes and their combinations.

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Section: Related Workmentioning

confidence: 99%

Section: Asymptotic Densitymentioning

confidence: 99%

Section: Asymptotic Densitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

The Density of Linear-Time Properties

Finkbeiner

Torfah

2017

Automated Technology for Verification and Analysis

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“…In recent years, some of us have been working on a new non-probabilistic quantitative approach to classical models in computer science based on the notion of language entropy (growth rate). This approach has produced new insights about timed automata and languages [1] as well as temporal logics [2]. In this article, we apply it to game theory and obtain a new natural class of games that we call entropy games (EGs).…”

Section: Introductionmentioning

confidence: 99%

Entropy Games and Matrix Multiplication Games

Asarin,

Cervelle,

Degorre

et al. 2015

Preprint

Self Cite

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Two intimately related new classes of games are introduced and studied: entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on a finite arena by twoand-a-half players: Despot, Tribune and the non-deterministic People. Despot wants to make the set of possible People's behaviors as small as possible, while Tribune wants to make it as large as possible. An MMG is played by two players that alternately write matrices from some predefined finite sets. One wants to maximize the growth rate of the product, and the other to minimize it. We show that in general MMGs are undecidable in quite a strong sense. On the positive side, EGs correspond to a subclass of MMGs, and we prove that such MMGs and EGs are determined, and that the optimal strategies are simple. The complexity of solving such games is in NP ∩ coNP. * The support of Agence Nationale de la Recherche under the project EQINOCS (ANR-11-BS02-004) is gratefully acknowledged. The results of Section 4 were obtained at the Institute for Information Transmission Problems, Russian Academy of Science, by V. Kozyakin at the expense of the Russian Science Foundation (project 14-50-00150).1 Although this term is mostly used for stochastic games, it is also an appropriate description of EGs.

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A Weakness Measure for GR(1) Formulae

2018

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In spite of the theoretical and algorithmic developments for system synthesis in recent years, little effort has been dedicated to quantifying the quality of the specifications used for synthesis. When dealing with unrealizable specifications, finding the weakest environment assumptions that would ensure realizability is typically a desirable property; in such context the weakness of the assumptions is a major quality parameter. The question of whether one assumption is weaker than another is commonly interpreted using implication or, equivalently, language inclusion. However, this interpretation does not provide any further insight into the weakness of assumptions when implication does not hold. To our knowledge, the only measure that is capable of comparing two formulae in this case is entropy, but even it fails to provide a sufficiently refined notion of weakness in case of GR(1) formulae, a subset of linear temporal logic formulae which is of particular interest in controller synthesis. In this paper we propose a more refined measure of weakness based on the Hausdorff dimension, a concept that captures the notion of size of the omega-language satisfying a linear temporal logic formula. We identify the conditions under which this measure is guaranteed to distinguish between weaker and stronger GR(1) formulae. We evaluate our proposed weakness measure in the context of computing GR(1) assumptions refinements.

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Asymptotic behaviour in temporal logic

Cited by 4 publications

References 12 publications

The Density of Linear-Time Properties

The Density of Linear-Time Properties

Entropy Games and Matrix Multiplication Games

A Weakness Measure for GR(1) Formulae

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