Completeness for Game Logic

TL;DR

This paper develops sound and complete sequent calculi for game logic, extending Parikh's axiomatization and connecting it to the monotone mu-calculus through proof transformations and translations.

Contribution

It introduces new cut-free sequent calculi for game logic and the monotone mu-calculus, proving their completeness and thus establishing the completeness of Parikh's original axiomatization.

Findings

Established soundness and completeness of the new calculi.

Proved completeness of Parikh's original axiomatization.

Developed a translation from game logic to the monotone mu-calculus.

Abstract

Game logic was introduced by Rohit Parikh in the 1980s as a generalisation of propositional dynamic logic (PDL) for reasoning about outcomes that players can force in determined 2-player games. Semantically, the generalisation from programs to games is mirrored by moving from Kripke models to monotone neighbourhood models. Parikh proposed a natural PDL-style Hilbert system which was easily proved to be sound, but its completeness has thus far remained an open problem. In this paper, we introduce a cut-free sequent calculus for game logic, and two cut-free sequent calculi that manipulate annotated formulas, one for game logic and one for the monotone mu-calculus, the variant of the polymodal mu-calculus where the semantics is given by monotone neighbourhood models instead of Kripke structures. We show these systems are sound and complete, and that completeness of Parikh's axiomatization follows. Our approach builds on recent ideas and results by Afshari & Leigh (LICS 2017) in that we obtain completeness via a sequence of proof transformations between the systems. A crucial ingredient is a validity-preserving translation from game logic to the monotone mu-calculus.