A Decidable Fragment of Strategy Logic

TL;DR

This paper introduces SL[1G], a syntactic fragment of Strategy Logic, which is more expressive than ATL* and has a decidable satisfiability problem, unlike the full logic.

Contribution

The paper defines SL[1G], a new fragment of Strategy Logic with a decidable satisfiability problem and bounded tree-model property, extending the applicability of strategic reasoning.

Findings

SL[1G] has an elementarily decidable model-checking problem.

SL[1G] satisfiability is decidable in 2ExpTime.

SL[1G] strictly subsumes ATL*.

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, $\Sigma_{1}^{1}$-Hard. In order to obtain a decidable sublogic, we introduce and study here One-Goal Strategy Logic (SL[1G], for short). This logic 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. SL[1G] is known to have an elementarily decidable model-checking problem. Here we prove that, unlike SL, it has the bounded tree-model property and its satisfiability problem is decidable in 2ExpTime, thus not harder than the one for ATL*.