On decidable extensions of Propositional Dynamic Logic with Converse

We describe a family of decidable propositional dynamic logics, where atomic modalities satisfy some extra conditions (for example, given by axioms of the logics K5, S5, or K45 for different atomic modalities). It follows from recent results (Kikot, Shapirovsky, Zolin, 2014; 2020) that if a modal logic $L$ admits a special type of filtration (so-called definable filtration), then its enrichments with modalities for the transitive closure and converse relations also admit definable filtration. We use these results to show that if logics $L_1,\dots ,L_n$ admit definable filtration, then the propositional dynamic logic with converse extended by the fusion $L_1\ast \dots \ast L_n$ has the finite model property.