A Reasoning About Strategies: On the Model-Checking Problem

In open systems veri cation, 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∗, 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 rst-order quanti cations over strategies and has been investigated in the very speci c 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 NonElementarySpacehard. This negative result has spurred us to investigate here syntactic fragments of Sl, strictly subsuming Atl∗, 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∗. In contrast, Sl[NG] turns out to be NonElementarySpace-hard, strengthening the corresponding result for Sl. Finally, we observe that Sl[BG] includes CHP-Sl, while its model-checking problem relies between NonElementaryTime and 2ExpTime.