Bisimilar and logically equivalent programs in PDL with parallel operator

Abstract

In standard Propositional Dynamic Logic (PDL) literature, the semantics is given by Labeled Transition Systems, where for each program π we associate a binary relation Rπ. Process Algebras also give semantics to process (terms) by means of Labeled Transition Systems. In both formalisms, PDL and Process Algebra, the key notion to compare processes is bisimulation. In PDL, we also have the notion of logic equivalence, that can be used to prove that two programs π1 and π2 are logically equivalent ⊢〈π1〉φ↔〈π2〉φ. Unfortunately, logic equivalence and bisimulation do not match in PDL. Bisimilar programs are logic equivalent but the converse does not hold.This paper proposes a semantics and an axiomatization for PDL that makes logically equivalent programs also bisimilar. This allows for developing Dynamic Logics to reasoning about CCS specification. As in CCS the bisimulation is the main tool to establish equivalence of programs, it is very important that these two relations coincide. We propose a new Propositional Dynamic Logic with a new non-deterministic choice operator, PDL+. We prove its soundness, completeness, finite model property and EXPTIME-completeness for the satisfiability problem.We also add to PDL+ the parallel composition operator (PPDL+) and prove its soundness and completeness. We establish that the satisfiability problem for PPDL+ is in 2-EXPTIME. Finally, we define some fragments of PPDL+ and prove its EXPTIME-completeness.