A Resolution Calculus for Modal Logics

Abstract

A resolution calculus for the quantified versions of the modal logics K, T, K4, KB, S4, S5, B, D, D4 and DB is presented. It presupposes a syntax transformation, similar to the skolemization in predicate logic, that eliminates the modal operators from modal logic formulae and shifts the modal context information to the term level. The formulae in the transformed syntax can be translated into conjunctive normal form such that a clause based modal resolution calculus is definable without any additional inference rule, but with special modal unification algorithms. The method can be applied to first-order modal logics with the two operators □ and ◊ and with standard constant-domain possible worlds semantics with flexible constant and function symbols, where the accessibility relation may have any combination of the following properties: reflexivity, symmetry, transitivity, seriality or non-seriality. While extensions to other systems seem possible, they have not yet been investigated.