A Logical Characterization of Bisimulation for Labeled Markov Processes

This paper gives a logical characterization of probabilistic bisimulation for Markov processes introduced in BDEP97]. The thrust of that work was an extension of the notion of bisimulation to systems with continuous state spaces; for example for systems where the state space is the real numbers. In the present paper we study the logical characterization of probabilistic bisimulation for such general systems. This study revealed some unexpected results even for discrete probabilistic systems. Bisimulation can be characterized by a very weak modal logic. The most striking feature is that one has no negation or any kind of negative proposition. Bisimulation can be characterized by several inequivalent logics; we report ve in this paper. We do not need any nite branching assumption yet there is no need of innnitary conjunction. The proofs that we give are of an entirely diierent character than the typical proofs of these results. They use quite subtle facts about analytic spaces and appear at rst sight to be entirely nonconstructive. Yet one can derive an algorithm for deciding bisimilarity of nite state systems which constructs a formula that witnesses the failure of bisimulation.