Uniform strategies, rational relations and jumping automata

Abstract

A general concept of uniform strategies has recently been proposed as a relevant notion in game theory for computer science, which subsumes various notions from the literature. It relies on properties involving sets of plays in two-player turn-based arenas equipped with arbitrary binary relations between plays; these properties are expressed in a language based on CTL⁎ with a quantifier over related plays. There are two semantics for our quantifier, a strict one and a full one, that we study separately. Regarding the strict semantics, the existence of a uniform strategy is undecidable for rational binary relations, but introducing jumping tree automata and restricting attention to recognizable relations allows us to establish a 2-Exptime-complete complexity – and still capture a class of two-player imperfect-information games with epistemic temporal objectives. Regarding the full semantics, relying on information set automata we establish that the existence of a uniform strategy is decidable for rational relations and we provide a nonelementary synthesis procedure. We also exhibit an essentially optimal subclass of rational relations for which the problem becomes 2-Exptime-complete. Considering rich classes of relations makes the theory of uniform strategies powerful: it directly entails various results in logics of knowledge and time, some of them already known, and others new.