On the undecidability of description and dynamic logics with recursion and counting

The evolution of Description Logics (DLs) and Propositional Dynamic Logics produced a hierarchy of decidable logics with multiple maximal elements. It would be desirable to combine different maximal logics into one super-logic, but then inference may turn out to be undecidable. Then it is important to characterize the decidability threshold for these logics. In this perspective, an interesting open question pointed out by Sattler and Vardi [Sattler and Vardi, 1999] is whether inference in a hybrid μ-calculus with restricted forms of graded modalities is decidable, and which complexity class it belongs to. In this paper we prove that this calculus and the corresponding are undecidable. Second, we prove undecidability results for logics that support both a transitive closure operator over roles and number restrictions.