Parity Games and Automata for Game Logic (Extended Version)

TL;DR

This paper explores the semantics of Parikh's game logic using parity games and automata, providing an automata-theoretic characterization of the logic's monotone μ-calculus fragment through innovative syntax graphs.

Contribution

It introduces a new automata-theoretic framework for game logic semantics using parity games and syntax graphs, linking it to the monotone μ-calculus.

Findings

Semantics of game logic over neighbourhood structures via parity games

Automata-theoretic characterization of the game logic fragment

Introduction of syntax graphs as a flexible automata representation

Abstract

Parikh's game logic is a PDL-like fixpoint logic interpreted on monotone neighbourhood frames that represent the strategic power of players in determined two-player games. Game logic translates into a fragment of the monotone $\mu$-calculus, which in turn is expressively equivalent to monotone modal automata. Parity games and automata are important tools for dealing with the combinatorial complexity of nested fixpoints in modal fixpoint logics, such as the modal $\mu$-calculus. In this paper, we (1) discuss the semantics a of game logic over neighbourhood structures in terms of parity games, and (2) use these games to obtain an automata-theoretic characterisation of the fragment of the monotone $\mu$-calculus that corresponds to game logic. Our proof makes extensive use of structures that we call syntax graphs that combine the ease-of-use of syntax trees of formulas with the flexibility and succinctness of automata. They are essentially a graph-based view of the alternating tree automata that were introduced by Wilke in the study of modal $\mu$-calculus.