Tableaux Methods for Propositional Dynamic Logics with Separating Parallel Composition

Abstract

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Abstract. PRSPDL is a propositional dynamic logic with an operator for parallel compositions of programs. We first give a complexity upper bound for this logic. Then we focus on the class of ⊳-deterministic frames and give tableaux methods for two fragments of PRSPDL over this class of frames.

Complexity Optimal Decision Procedure for a Propositional Dynamic Logic with Parallel Composition

“…Since PDL can be trivially embedded into PPDL det , this decision procedure is optimal and the satisfiability problem of PPDL det is EXPTIME-complete. This result extends a previous similar result for the iteration-free fragment of PPDL det [2]: adding a ⊳-deterministic separating parallel composition to PDL does not increase the theoretical complexity of the logic. This result contrasts with the 2EXPTIME complexity of the satisfiability problem of both PDL with intersection [17] and interleaving PDL [20].…”

“…For instance, for the class of all models, only a 2EXPTIME upper bound is currently known [2]. For the class of models where the separation of states is deterministic, even though PRSPDL has been proved to be undecidable [3], it may be possible that in the absence of the four store/recover programs of PRSPDL the logic is decidable.…”

“…Despite these interesting features, little is known about PRSPDL. To our knowledge, the only complexity results to date are that the satisfiability problem of PRSPDL is in 2EXPTIME [2] and that variants of PRSPDL interpreted in classes of models where the decomposition of states is deterministic are undecidable [3].…”

“…The adaptation of the elimination of Hintikka sets procedure to PPDL det is not straightforward. First, as it has been already outlined in [2,5], a comprehensive decomposition of formulas such as the Fischer-Ladner closure is not expressible in PPDL det . Second, whereas in Pratt [23] states can be considered independently, for PPDL det the decomposition path leading to each state must be remembered.…”