Continuous stochastic logic characterizes bisimulation of continuous-time Markov processes

In a recent paper Baier, Haverkort, Hermanns and Katoen [BHHK00], analyzed a new way of model-checking formulas of a logic for continuoustime processes called Continuous Stochastic Logic (henceforth CSL) – against continuous-time Markov chains – henceforth CTMCs. One of the important results of that paper was the proof that if two CTMCs were bisimilar then they would satisfy exactly the same formulas of CSL. This raises the converse question – does satisfaction of the same collection of CSL formulas imply bisimilarity? In other words, given two CTMCs which are known to satisfy exactly the same formulas of CSL does it have to be the case that they are bisimilar? We prove that the answer to the question just raised is “yes”. In fact we prove a significant extension, namely that a subset of CSL suffices even for systems where the state-space may be a continuum. Along the way we prove a result to the effect that the set of Zeno paths has measure zero provided that the transition rates are bounded. ∗Research supported in part by NSERC and MITACS.