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 μ-calculus, the variant of the polymodal μ-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 μ-calculus.

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 μ -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 μ -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 μ -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 μ -calculus.