Taming Strategy Logic: Non-Recurrent Fragments

Abstract

Strategy Logic ( SL for short) is one of the prominent languages for reasoning about the strategic abilities of agents in a multi-agent setting. This logic extends LTL with first-order quantifiers over the agent strategies and encompasses other formalisms, such as ATL* and CTL* . The model-checking problem for SL and several of its fragments have been extensively studied. On the other hand, the picture is much less clear on the satisfiability front, where the problem is undecidable for the full logic. In this work, we study two fragments of One-Goal SL , where the nesting of sentences within temporal operators is constrained. We show that the satisfiability problem for these logics, and for the corresponding fragments of ATL* and CTL* , is ExpSpace and PSpace-complete , respectively. Classification Theory of computation → Modal and temporal logics; Theory of computation → Logic and verification; Theory of computation → Automata over infinite objects Bounded-Fork Tree Automata. Bounded-fork automata are a restriction of the standard tree automata tailored to accept only trees having a bounded number of fork nodes along each path starting from the root (recall that bounded-fork trees are suitable models for SL ̸ ⟳ [1g] ). If at most k forks in a path are allowed, the ability to count the number of occurring forks is obtained by partitioning the set of states Q into k + 1 subsets Q 0 , . . . , Q k . Intuitively, a state q ∈ Q i can observe at most i additional forks along a path. Naturally, the initial states belong to Q k and only states in Q 0 , from where no more forks are admitted, can be involved in the Büchi acceptance condition.