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Extended Graded Modalities in Strategy Logic

Profile image of Aniello MuranoAniello Murano

2016, Electronic Proceedings in Theoretical Computer Science

https://doi.org/10.4204/EPTCS.218.1
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Abstract

Strategy Logic (SL) is a logical formalism for strategic reasoning in multi-agent systems. Its main feature is that it has variables for strategies that are associated to specific agents with a binding operator. We introduce Graded Strategy Logic (GRADEDSL), an extension of SL by graded quantifiers over tuples of strategy variables, i.e., "there exist at least g different tuples (x 1 , ..., x n) of strategies" where g is a cardinal from the set N ∪ {ℵ 0 , ℵ 1 , 2 ℵ 0 }. We prove that the model-checking problem of GRADEDSL is decidable. We then turn to the complexity of fragments of GRADEDSL. When the g's are restricted to finite cardinals, written GRADED N SL, the complexity of model-checking is no harder than for SL, i.e., it is non-elementary in the quantifier rank. We illustrate our formalism by showing how to count the number of different strategy profiles that are Nash equilibria (NE), or subgame-perfect equilibria (SPE). By analyzing the structure of the specific formulas involved, we conclude that the important problems of checking for the existence of a unique NE or SPE can both be solved in 2EXPTIME, which is not harder than merely checking for the existence of such equilibria.

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Reasoning about graded strategy quantifiers
Aniello Murano

Information and Computation, 2018

In this paper we introduce and study Graded Strategy Logic (GSL), an extension of Strategy Logic (SL) with graded quantifiers. SL is a powerful formalism that allows to describe useful game concepts in multi-agent settings by explicitly quantifying over strategies treated as first-order citizens. In GSL, by means of the existential construct x ≥ g ϕ, one can enforce that there exist at least g strategies x satisfying ϕ. Dually, via the universal construct [[x < g]]ϕ, one can ensure that all but less than g strategies x satisfy ϕ. Strategies in GSL are counted semantically. This means that strategies inducing the same outcome, even though looking different, are counted as one. While this interpretation is natural, it requires a suitable machinery to allow for such a counting, as we do. Precisely, we introduce a non-trivial equivalence relation over strategy profiles based on the strategic behavior they induce. To give an evidence of GSL usability, we investigate some basic questions about the Vanilla GSL[1g] fragment, that is the vanilla restriction of the wellstudied One-Goal Strategy Logic fragment of SL augmented with graded strategy quantifiers. We show that the model-checking problem for this logic is PTimecomplete. We also report on some positive results about the determinacy.

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Satisfiability in Strategy Logic Can Be Easier than Model Checking
Erman Acar

Proceedings of the AAAI Conference on Artificial Intelligence, 2019

In the design of complex systems, model-checking and satisfiability arise as two prominent decision problems. While model-checking requires the designed system to be provided in advance, satisfiability allows to check if such a system even exists. With very few exceptions, the second problem turns out to be harder than the first one from a complexity-theoretic standpoint. In this paper, we investigate the connection between the two problems for a non-trivial fragment of Strategy Logic (SL, for short). SL extends LTL with first-order quantifications over strategies, thus allowing to explicitly reason about the strategic abilities of agents in a multi-agent system. Satisfiability for the full logic is known to be highly undecidable, while model-checking is non-elementary.The SL fragment we consider is obtained by preventing strategic quantifications within the scope of temporal operators. The resulting logic is quite powerful, still allowing to express important game-theoretic propert...

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What Makes Atl* Decidable? A Decidable Fragment of Strategy Logic
Aniello Murano

Lecture Notes in Computer Science, 2012

Strategy Logic (SL, for short) has been recently introduced by Mogavero, Murano, and Vardi as a formalism for reasoning explicitly about strategies, as first-order objects, in multi-agent concurrent games. This logic turns out to be very powerful, strictly subsuming all major previously studied modal logics for strategic reasoning, including ATL, ATL * , and the like. The price that one has to pay for the expressiveness of SL is the lack of important model-theoretic properties and an increased complexity of decision problems. In particular, SL does not have the bounded-tree model property and the related satisfiability problem is highly undecidable while for ATL * it is 2EXPTIME-COMPLETE. An obvious question that arises is then what makes ATL * decidable. Understanding this should enable us to identify decidable fragments of SL. We focus, in this work, on the limitation of ATL * to allow only one temporal goal for each strategic assertion and study the fragment of SL with the same restriction. Specifically, we introduce and study the syntactic fragment One-Goal Strategy Logic (SL[1G], for short), which consists of formulas in prenex normal form having a single temporal goal at a time for every strategy quantification of agents. We show that SL[1G] is strictly more expressive than ATL * . Our main result is that SL[1G] has the bounded tree-model property and its satisfiability problem is 2EXPTIME-COMPLETE, as it is for ATL * .

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Reasoning About Strategies: On the Model-Checking Problem
Moshe Vardi

2011

In open systems verification, to formally check for reliability, one needs an appropriate formalism to model the interaction between agents and express the correctness of the system no matter how the environment behaves. An important contribution in this context is given by modal logics for strategic ability, in the setting of multi-agent games, such as ATL, ATL\star, and the like. Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic, which we denote here by CHP-SL, with the aim of getting a powerful framework for reasoning explicitly about strategies. CHP-SL is obtained by using first-order quantifications over strategies and has been investigated in the very specific setting of two-agents turned-based games, where a non-elementary model-checking algorithm has been provided. While CHP-SL is a very expressive logic, we claim that it does not fully capture the strategic aspects of multi-agent systems. In this paper, we introduce and study a more general strategy logic, denoted SL, for reasoning about strategies in multi-agent concurrent games. We prove that SL includes CHP-SL, while maintaining a decidable model-checking problem. In particular, the algorithm we propose is computationally not harder than the best one known for CHP-SL. Moreover, we prove that such a problem for SL is NonElementarySpace-hard. This negative result has spurred us to investigate here syntactic fragments of SL, strictly subsuming ATL\star, with the hope of obtaining an elementary model-checking problem. Among the others, we study the sublogics SL[NG], SL[BG], and SL[1G]. They encompass formulas in a special prenex normal form having, respectively, nested temporal goals, Boolean combinations of goals and, a single goal at a time. About these logics, we prove that the model-checking problem for SL[1G] is 2ExpTime-complete, thus not harder than the one for ATL\star.

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Strategy Logic with Imperfect Information
Moshe Vardi

ACM Transactions on Computational Logic, 2021

We introduce an extension of Strategy Logic for the imperfect-information setting, called SL ii and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, this problem is undecidable; but we introduce a syntactical class of “hierarchical instances” for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model, and we prove that model-checking SL ii restricted to hierarchical instances is decidable. This result, because it allows for complex patterns of existential and universal quantification on strategies, greatly generalises the decidability of distributed synthesis for systems with hierarchical information. It allows us to easily derive new decidability results concerning strategic problems under imperfect information such as the existence of Nash equilibria or rational synthesis. To establish this result, we go through an intermedia...

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Strategy Logic with Simple Goals: Tractable Reasoning about Strategies
Aniello Murano

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, 2019

In this paper we introduce Strategy Logic with simple goals (SL[SG]), a fragment of Strategy Logic that strictly extends the well-known Alternating-time Temporal Logic ATL by introducing arbitrary quantification over the agents&#39; strategies. Our motivation comes from game-theoretic applications, such as expressing Stackelberg equilibria in games, coercion in voting protocols, as well as module checking for simple goals. Most importantly, we prove that the model checking problem for SL[SG] is PTIME-complete, the same as ATL. Thus, the extra expressive power comes at no computational cost as far as verification is concerned.

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Model-checking games for logics of imperfect information
Erich Grädel

Theoretical Computer Science, 2013

Logics of dependence and independence have semantics that, unlike Tarski semantics, are not based on single assignments (mapping variables to elements of a structure) but on sets of assignments. Sets of assignments are called teams and the semantics is called team semantics. We design model-checking games for logics with team semantics in a general and systematic way. The construction works for any extension of first-order logic by atomic formulae on teams, as long as certain natural conditions are observed which are satisfied by all team properties considered so far in the literature, including dependence, independence, constancy, inclusion, and exclusion.

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Reasoning about strategies
Aniello Murano

Leibniz International Proceedings in Informatics, LIPIcs, 2010

In open systems verification, to formally check for reliability, one needs an appropriate formalism to model the interaction between open entities and express that the system is correct no matter how the environment behaves. An important contribution in this context is given by modal logics for strategic ability, in the setting of multi-agent games, such as ATL, ATL*, and the like. Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic, which we denote here by SL CHP , with the aim of getting a powerful framework for reasoning explicitly about strategies. SL CHP is obtained by using first-order quantifications over strategies and it has been investigated in the specific setting of two-agents turned-based game structures where a non-elementary model-checking algorithm has been provided. While SL CHP is a very expressive logic, we claim that it does not fully capture the strategic aspects of multi-agent systems.

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Representing strategic games and their equilibria in many-valued logics
Chris Fermueller

Logic Journal of IGPL, 2016

We introduce the notion of logical A-games for a fairly general class of algebras A of real truth-values. This concept generalizes the Boolean games of Harrenstein et al. as well as the recently defined Lukasiewicz games of Marchioni and Wooldridge. We demonstrate that a wide range of strategic n-player games can be represented as logical A-games. Moreover we show how to construct, under rather general conditions, propositional formulas in the language of A that correspond to pure and mixed Nash equilibria of logical A-games.

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On the complexity of constrained Nash equilibria in graphical games
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Modal transition systems (MTS) is a formalism which extends the classical notion of labelled transition systems by introducing transitions of two types: must transitions that have to be present in any implementation of the MTS and may transitions that are allowed but not required.

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Graded modalities in Strategy Logic
Aniello Murano

Information and Computation, 2018

Strategy Logic (SL) is a logical formalism for strategic reasoning in multi-agent systems. Its main feature is that it has variables for strategies that are associated to specific agents using a binding operator. In this paper we introduce Graded Strategy Logic (GradedSL), an extension of SL by graded quantifiers over tuples of strategy variables, i.e., "there exist at least g different tuples (x 1 , ..., x n) of strategies" where g is a cardinal from the set N ∪ {ℵ 0 , ℵ 1 , 2 ℵ0 }. We prove that the model-checking problem of GradedSL is decidable. We then turn to the complexity of fragments of GradedSL. When the g's are restricted to finite cardinals, written Graded N SL, the complexity of model-checking is no harder than for SL, i.e., it is non-elementary in the quantifier-block rank. We illustrate our formalism by showing how to count the number of different strategy profiles that are Nash equilibria (NE). By analysing the structure of the specific formulas involved, we conclude that the important problem of checking for the existence of a unique NE can be solved in 2ExpTime, which is not harder than merely checking for the existence of such an equilibrium.

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Graded Strategy Logic: Reasoning about Uniqueness of Nash Equilibria
Aniello Murano

adaptive agents and multi-agents systems, 2016

Strategy Logic (SL) is a well established formalism for strategic reasoning in multi-agent systems. In a nutshell, SL is built over LTL and treats strategies as first-order objects that can be associated with agents by means of a binding operator. In this work we introduce Graded Strategy Logic (Graded-SL), an extension of SL by graded quantifiers over tuples of strategy variables such as "there exist at least g different tuples (x1, ..., xn) of strategies". We study the model-checking problem of Graded-SL and prove that it is no harder than for SL, i.e., it is non-elementary in the quantifier rank. We show that Graded-SL allows one to count the number of different strategy profiles that are Nash equilibria (NE), or subgame-perfect equilibria (SPE). By analyzing the structure of the specific formulas involved, we conclude that the important problems of checking for the existence of a unique NE or SPE can both be solved in 2ExpTime, which is not harder than merely checking for the existence of such equilibria.

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A Decidable Fragment of Strategy Logic
Aniello Murano

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Abstract: Strategy Logic (SL, for short) has been recently introduced by Mogavero, Murano, and Vardi as a useful formalism for reasoning explicitly about strategies, as first-order objects, in multi-agent concurrent games. This logic turns to be very powerful, subsuming all major previously studied modal logics for strategic reasoning, including ATL, ATL*, and the like. Unfortunately, due to its expressiveness, SL has a non-elementarily decidable model-checking problem and a highly undecidable satisfiability problem, specifically, $\ Sigma_ { ...

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On the Counting of Strategies
Aniello Murano

2015 22nd International Symposium on Temporal Representation and Reasoning (TIME), 2015

In game theory, a classic qualitative question is to check whether a designated set of players has a winning strategy. In several safety-critical applications, however, it is important to ensure that some redundant strategies also exist, to be possibly used in case of some fault. In this paper, we introduce Graded Strategy Logic (GSL), an extension of Strategy Logic (SL) with graded quantifiers. SL is a powerful formalism that allows to describe useful game concepts in multi-agent settings by explicitly quantifying over strategies treated as first-order citizens. In GSL, by means of the existential construct x ≥ g ϕ one can enforce that there exist at least g strategies satisfying ϕ. Dually, via the universal construct [[x < g]]ϕ one can ensure that all but less than g strategies satisfy ϕ. As different strategies may induce the same outcome, although looking different, they need to be counted as one. While this interpretation is natural, it heavily complicates the definition and thus the reasoning about GSL. In order to accomplish this specific way of counting, we formally introduce a suitable equivalence relation over profiles based on the strategic behavior they induce. To give evidence of GSL usability, we investigate basic questions of one of its vanilla fragment, namely GSL[1G]. In particular, we report on positive results about the determinacy of games and the related model-checking problem, which we show to be PTIME-COMPLETE.

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Reasoning about Strategies: on the Satisfiability Problem
Moshe Vardi

Log. Methods Comput. Sci., 2017

Strategy Logic (SL, for short) has been introduced by Mogavero, Murano, and Vardi as a useful formalism for reasoning explicitly about strategies, as first-order objects, in multi-agent concurrent games. This logic turns out to be very powerful, subsuming all major previously studied modal logics for strategic reasoning, including ATL, ATL*, and the like. Unfortunately, due to its high expressiveness, SL has a non-elementarily decidable model-checking problem and the satisfiability question is undecidable, specifically Sigma_1^1. In order to obtain a decidable sublogic, we introduce and study here One-Goal Strategy Logic (SL[1G], for short). This is a syntactic fragment of SL, strictly subsuming ATL*, which encompasses formulas in prenex normal form having a single temporal goal at a time, for every strategy quantification of agents. We prove that, unlike SL, SL[1G] has the bounded tree-model property and its satisfiability problem is decidable in 2ExpTime, thus not harder than the ...

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