Reasoning About Strategies

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Abstract

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 multiagent games, such as A tl , A tl *, and the like. Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic , which we denote here by CHP-S l , with the aim of getting a powerful framework for reasoning explicitly about strategies. CHP-S l 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 nonelementary model-checking algorithm has been provided. While CHP-S l is a very expressive logic, we claim that it does not fully capture the strategic aspects of multiagent systems. In this article, we introduce and study a more general strategy logic, denoted S l , for reasoning about strategies in multiagent concurrent games. As a key aspect, strategies in S l are not intrinsically glued to a specific agent, but an explicit binding operator allows an agent to bind to a strategy variable. This allows agents to share strategies or reuse one previously adopted. We prove that S l strictly includes CHP-S l , while maintaining a decidable model-checking problem. In particular, the algorithm we propose is computationally not harder than the best one known for CHP-S l . Moreover, we prove that such a problem for S l is N on E lementary . This negative result has spurred us to investigate syntactic fragments of S l , strictly subsuming A tl *, with the hope of obtaining an elementary model-checking problem. Among others, we introduce and study the sublogics S l [ ng ], S l [ bg ], and S l [1 g ]. 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. Intuitively, for a goal, we mean a sequence of bindings, one for each agent, followed by an L tl formula. We prove that the model-checking problem for S l [1 g ] is 2E xp T ime - complete , thus not harder than the one for A tl *. In contrast, S l [ ng ] turns out to be N on E lementary -hard, strengthening the corresponding result for S l . Regarding S l [ bg ], we show that it includes CHP-S l and its model-checking is decidable with a 2E xp T ime lower-bound. It is worth enlightening that to achieve the positive results about S l [1 g ], we introduce a fundamental property of the semantics of this logic, called behavioral , which allows to strongly simplify the reasoning about strategies. Indeed, in a nonbehavioral logic such as S l [ bg ] and the subsuming ones, to satisfy a formula, one has to take into account that a move of an agent, at a given moment of a play, may depend on the moves taken by any agent in another counterfactual play.

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