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

2007

The Game Description Language (GDL) is a special purpose declarative language for defining games. GDL is used in the AAAI General Game Playing Competition, which tests the ability of computer programs to play games in general, rather than just to play a specific game. Participants in the competition are provided with a game specified in GDL, and then required to play this game. Recently, there has been much interest in the use of strategic cooperation logics for reasoning about game-like scenarios -Alternatingtime Temporal Logic (ATL) is perhaps the best known example. The aim of this paper is to make a link between ATL and GDL. We show that a GDL specification can be viewed as a specification of an ATL model, and that ATL can thus be interpreted over GDL specifications. Our main result is that it is possible to translate a propositional GDL specification into an "equivalent" ATL specification, which is only polynomially larger than the original GDL specification. As a corollary, we are able to characterise the complexity of reasoning about GDL-specified games using ATL: it is EXPTIME-complete.

Lecture Notes in Computer Science, 2012

This paper is a study of Brihaye et al.'s ATL with strategy contexts. We focus on memory-less strategies and establish that the resulting logic is undecidable. An immediate corollary follows that the problem of satisfiability checking of every variant of ATL with strategy context introduced by Brihaye et al. is undecidable. We also relate ATLsc with memory-less strategies with ATL with explicit strategies, providing a decidable fragment.

This paper presents a logical framework that extends the Game Description Language with coalition operators from Alternatingtime Temporal Logic and prioritised strategy connectives. Our semantics is built upon the standard state transition model. The new framework allows us to formalise van Benthem’s game-oriented principles in multiplayer games, and formally deriveWeak Determinacy and Zermelo’s Theorem for two-player games. We demonstrate with a real-world game how to use our language to specify a game and design a strategy, and how to use our framework to verify a winning/no-losing strategy. Finally, we show that the model-checking problem of our logic is in 2EXPTIME with respect to the size of game structure and the length of formula, which is no worse than the model-checking problem in ATL*.

Electronic Proceedings in Theoretical Computer Science, 2016

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.

2013

Propositional Dynamic Logic or PDL was invented as a logic for reasoning about regular programming constructs. We propose a new perspective on PDL as a multi-agent strategic logic (MASL). This logic for strategic reasoning has group strategies as rst class citizens, and brings game logic closer to standard modal logic. We demonstrate that MASL can express key notions of game theory, social choice theory and voting theory in a natural way, we give a sound and complete proof system for MASL, and we show that MASL encodes coalition logic. Next, we extend the language to epistemic multi-agent strategic logic (EMASL), we give examples of what it can express, we propose to use it for posing new questions in epistemic social choice theory, and we give a calculus for reasoning about a natural class of epistemic game models. We end by listing avenues for future research and by tracing connections to a number of other logics for reasoning about strategies.

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...

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.

Synthese, 2008

We make a proposal for formalizing simultaneous games at the abstraction level of player's powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of 'concurrent game logic' CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic. ∀x∃y J J J J Rxyzu ∀z∃u s s s s

Synthese, 2008

We make a proposal for formalizing simultaneous games at the abstraction level of player's powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of 'concurrent game logic' CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.

Lecture Notes in Computer Science, 2014

Starting from the seminal work introducing Alternating Temporal Logic, formalisms for strategic reasoning have assumed a prominent role in multi-agent systems verification. Among the others, Strategy Logic (SL) allows to represent sophisticated solution concepts, by treating agent strategies as first-order objects. A drawback from the high power of SL is to admit non-behavioral strategies: a choice of an agent, at a given point of a play, may depend on choices other agents can make in the future or in counterfactual plays. As the latter moves are unpredictable, such strategies cannot be easily implemented, making the use of the logic problematic in practice. In this paper, we describe a hierarchy of SL fragments as syntactic restrictions of the recently defined Boolean-Goal Strategy Logic (SL[bg]). Specifically, we introduce Alternating-Goal Strategy Logic (SL[ag]) that, by imposing a suitable alternation over the nesting of the Boolean connectives in SL[bg], induces two dual chains of sets of formulas, the conjunctive and disjunctive ones. A formula belongs to the level i of the conjunctive chain if it just contains conjunctions of atomic goals together with a unique formula belonging to the disjunctive chain of level i − 1. The disjunctive chain is defined similarly. We formally prove that classic and behavioral semantics for SL [ag] coincide. Additionally, we study the related model-checking problem showing that it is 2ExpTime-complete.