Complexity results for modal logic with recursion via translations and tableaux

Abstract

This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the μ-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce terminating and non-terminating tableau systems for the logics we study, based on Kozen’s tableau for the μ-calculus and the one of Fitting and Massacci for modal logic. Finally, we describe these tableaux with μ-calculus formulas, thus reducing the satisfiability of each of these logics to the satisfiability of the μ-calculus, resulting in a general 2EXP upper bound for satisfiability testing.