Approximating Bisimilarity for Markov Processes

In this paper we investigate bisimilarity for general Markov processes through the correspondence between sub-σ-algebras and equivalence relations. In particular, we study bisimulations from the perspective of fixed-point theory. Given a Markov process M=〈Ω,Σ,τ〉, we characterize its state bisimilarity as the greatest fixed point of a composition of two natural set operators between equivalence relations on Ω and sub-σ-algebras of Σ. Moreover, we employ a Smith-Volterra-Cantor-set-construction to obtain an example to show that state bisimilarity is beyond ω iterations of these two operators alternately from event bisimilarity and hence the composite operator is not continuous. This process of iteration illustrates the gap between event bisimilarity (or logical equivalence) and state bisimilarity, and hence provides insights about the Hennessy-Milner property for general Markov processes. At the end of this paper, we also study approximation of Markov processes related to filtration.