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

On decidable extensions of Propositional Dynamic Logic with Converse