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Undecidability of the transitive graded modal logic with converse

We extend the language of the modal logic 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 introduce graded modalities (a formula holds in at least n successors) and graded inverse modalities (a formula holds in at least n predecessors). We show that the resulting logic, called GrIK4, is undecidable. The same result is obtained for all logics between GrIK4 and GrIS4. As a consequence, for the “unrestricted version” of the description logic SIQ, the problem of concept satisfiability (even with respect to the empty terminology) is undecidable. We also give a survey of complexity results for the local and global satisfiability problems for fragments of the logic GrIK4.