Idempotent Transductions for Modal Logics

We investigate the extension of modal logics by bisimulation quantifiers and present a class of modal logics which is decidable when augmented with bisimulation quantifiers. These logics are refered to as the idempotent transduction logics and are defined using the programs of propositional dynamic logic including converse and tests. This is a nontrivial extension of the decidability of the positive idempotent transduction logics which do not use converse operators in the programs (French, 2006). This extension allows us to apply bisimulation quantifiers to, for example, logics of knowledge, logics of belief and tense logics. We show the idempotent transduction logics preserve the axioms of propositional quantification and are decidable. The definition of idempotent transduction logics allows us to apply these results to a number of combined modal logics with a variety of interactions between modalities.