Focus-style proofs for the two-way alternation-free μ\mu-calculus

We introduce a cyclic proof system for the two-way alternation-free modal $\mu$-calculus. The system manipulates one-sided Gentzen sequents and locally deals with the backwards modalities by allowing analytic applications of the cut rule. The global effect of backwards modalities on traces is handled by making the semantics relative to a specific strategy of the opponent in the evaluation game. This allows us to augment sequents by so-called trace atoms, describing traces that the proponent can construct against the opponent's strategy. The idea for trace atoms comes from Vardi's reduction of alternating two-way automata to deterministic one-way automata. Using the multi-focus annotations introduced earlier by Marti and Venema, we turn this trace-based system into a path-based system. We prove that our system is sound for all sequents and complete for sequents not containing trace atoms.Comment: To appear in proceedings of WoLLIC 202

An analytic proof system for common knowledge logic over S5

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