Reasoning About Strategies: On the Model-Checking Problem

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 be- haves. 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 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 nonelementary 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 article, we introduce and study a more general strategy logic, denoted SL , for reasoning about strategies in multi-agent 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-SL. Moreover, we prove that such a problem for SL is NONELEMENTARY. This negative result has spurred us to investigate syntactic fragments of SL , strictly subsuming ATL*, with the hope of obtaining an elementary model-checking problem. Among others, we introduce and 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. Intuitively, for a goal, we mean a sequence of bindings, one for each agent, followed by an LTL formula. We prove that the model-checking problem for SL[1G] is 2EXPTIME-COMPLETE, thus not harder than the one for ATL* . In contrast, SL[NG] turns out to be NONELEMENTARY-HARD, strengthening the corresponding result for SL. Regarding SL[BG], we show that it includes CHP-SL and its model-checking is decidable with a 2EXPTIME lower-bound. It is worth enlightening that to achieve the positive results about SL[1G], 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 SL[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 counter-factual play.