Decidability of Quasi-Dense Modal Logics

The decidability of axiomatic extensions of the modal logic K with modal reduction principles, i.e. axioms of the form $\Diamond^{k} p \rightarrow \Diamond^{n} p$, has remained a long-standing open problem. In this paper, we make significant progress toward solving this problem and show that decidability holds for a large subclass of these logics, namely, for 'quasi-dense logics.' Such logics are extensions of K with with modal reduction axioms such that $0 < k < n$ (dubbed 'quasi-density axioms'). To prove decidability, we define novel proof systems for quasi-dense logics consisting of disjunctive existential rules, which are first-order formulae typically used to specify ontologies in the context of database theory. We show that such proof systems can be used to generate proofs and models of modal formulae, and provide an intricate model-theoretic argument showing that such generated models can be encoded as finite objects called 'templates.' By enumerating templates of bound size, we obtain an EXPSPACE decision procedure as a consequence.