Undecidability of the transitive graded modal logic with converse

Abstract

We extend the language of the modal logic $\mathbf{K4}$ of transitive frames with two sorts of modalities. In addition to the usual possibility modality (which means that a formula holds in some successor of a given point), we consider graded modalities (a formula holds in at least $n$ successors) and converse graded modalities (a formula holds in at least $n$ predecessors). We show that the resulting logic, $\mathbf{GrIK4}$ , is both locally and globally undecidable. The same result is obtained for all logics between $\mathbf{GrIK4}$ and its reflexive companion $\mathbf{GrIS4}$ and for some other modal logics. As a consequence, for the ‘unrestricted version’ of the description logic $\mathcal{SIQ}$ , the problem of concept satisfiability (even with respect to the empty terminology) is undecidable. We also give a survey of results on the local and global decidability, complexity, and the finite model property for fragments of $\mathbf{GrIK4}$ .