Logics for Bisimulation and Divergence

Abstract

The study of modal logics and various bisimulation equivalences so far shows the following progression: 1. weak bisimilarity is characterized by Hennessy-Milner logic (HML), a simple propositional modal logic with a weak possibility modality, and 2. extending HML by refining the weak possibility modality one obtains a logic which characterizes branching bisimilarity, a refinement of weak bisimilarity, and 3. further extending the logic with a divergence modality one obtains a logic which characterizes branching bisimilarity with explicit divergence, a refinement of branching bisimilarity. In this paper, we explore the development by exchanging the above 2 and 3, i.e. by first extending HML with a divergence modality and then refining the weak possibility modality in the extended logic. We have the following findings: A. extending HML with a new divergence modality one obtains a new logic which characterizes complete weak bisimilarity, an equivalence relation with distinguishing power in between weak bisimilarity and branching bisimilarity with explicit divergence; B. further extending the obtained logic by refining the weak possibility modality in it one obtains another logic which characterizes branching bisimilarity with explicit divergence. As main results of the paper, the logic in A. provides a modal characterization for complete weak bisimilarity, and moreover the two new logics in A. and B. are both sub-logics of the known logic obtained in above 3.