An analytic proof system for common knowledge logic over S5

Abstract

In this paper we present an analytic proof system for multi-modal logic with common-knowledge over S5 (called S5-CKL). The system is an annotated cyclic calculus manipulating two-sided Gentzen sequents and extending a known system for multi-modalS5. First a direct argument is used to show that the system is sound. Using a canonical model construction, we then show that the system is analytically complete. In particular, the use of the cut-rule is restricted to analytic cuts. Exploiting this analyticity, we then reduce the provability problem of a given sequent to the problem of solving a certain parity game. As a consequence we obtain an optimal decision procedure for proof search and thereby for the validity problem of S5-CKL