Complexity results for modal logic with recursion via translations and tableaux

TL;DR

This paper investigates the complexity of classical modal logics extended with fixed-point operators, using translations and tableaux to transfer results and develop decision procedures, including a general 2EXP upper bound.

Contribution

It introduces translation-based methods to analyze complexity and develop tableau systems for modal logics with fixed points, connecting them to the $$-calculus.

Findings

Established complexity bounds for multi-agent modal logics via translations

Developed tableau systems based on $$-calculus and modal logic

Reduced satisfiability to $$-calculus with a 2EXP upper bound

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 $\mu$-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 $\mu$-calculus and the one of Fitting and Massacci for modal logic. Finally, we describe these tableaux with $\mu$-calculus formulas, thus reducing the satisfiability of each of these logics to the satisfiability of the $\mu$-calculus, resulting in a general 2EXP upper bound for satisfiability testing.