Modal logics with hard diamond-free fragments

Abstract

We investigate the complexity of modal satisfiability for certain combinations of modal logics. In particular we examine four examples of multimodal logics with dependencies and demonstrate that even if we restrict our inputs to diamond-free formulas (in negation normal form), these logics still have a high complexity. This result illustrates that having D as one or more of the combined logics, as well as the interdependencies among logics can be important sources of complexity even in the absence of diamonds and even when at the same time in our formulas we allow only one propositional variable. We then further investigate and characterize the complexity of the diamond-free, 1-variable fragments of multimodal logics in a general setting.

Adventures in monitorability: from branching to linear time and back again

“…Satisfiability for HML (and therefore for HML as well) is known to be in PSPACE [Vardi 1988]. That satisfiability for HML is PSPACE-hard results from the observation that HML with at least two actions can encode the 1-variable, diamond-free fragment of D ⊕ ⊆ D4, which is PSPACE-complete [Achilleos 2016]. The reduction can be found in Appendix B.2.…”

“…B.2 The PSPACE-hardness of HML Here we prove that satisfiability for HML is PSPACE-hard. The reduction that we use is from the one-variable, diamond-free fragment of D ⊕ ⊆ K4, which is PSPACE-complete[Achilleos 2016]. D ⊕ ⊆ K4 is a modal logic with two modalities, [1] and [2], based on a serial transition relation…”