Local Reductions for the Modal Cube

Abstract

AbstractThe modal logic $${\mathsf {K}}$$ K is commonly used to represent and reason about necessity and possibility and its extensions with combinations of additional axioms are used to represent knowledge, belief, desires and intentions. Here we present local reductions of all propositional modal logics in the so-called modal cube, that is, extensions of $${\mathsf {K}}$$ K with arbitrary combinations of the axioms $${\mathsf {B}}$$ B , $${\mathsf {D}}$$ D , $${\mathsf {T}}$$ T , $${\mathsf {4}}$$ 4 and $${\mathsf {5}}$$ 5 to a normal form comprising a formula and the set of modal levels it occurs at. Using these reductions we can carry out reasoning for all these logics with the theorem prover "Image missing" . We define benchmarks for these logics and experiment with the reduction approach as compared to an existing resolution calculus with specialised inference rules for the various logics.