We introduce a new notion of bisimulation, called event bisimulation on labelled Markov processes and compare it with the, now standard, notion of probabilistic bisimulation, originally due to Larsen and Skou. Event bisimulation uses a sub σ-algebra as the basic carrier of information rather than an equivalence relation. The resultingnotion is thus based on measurable subsets rather than on points: hence the name. Event bisimulation applies smoothly for general measure spaces; bisimulation, on the other hand, is known only to work satisfactorily for analytic spaces. We prove the logical characterization theorem for event bisimulation without havingto invoke any of the subtle aspects of analytic spaces that feature prominently in the correspondingproof for ordinary bisimulation. These complexities only arise when we show that on analytic spaces the two concepts coincide. We show that the concept of event bisimulation arises naturally from taking the cocongruence point of view for probabilistic systems. We show that the theory can be given a pleasing categorical treatment in line with general coalgebraic principles. As an easy application of these ideas we develop a notion of "almost sure" bisimulation; the theory comes almost "for free" once we modify Giry's monad appropriately. © 2005 Elsevier Inc. All rights reserved.
Danos, V., Desharnais, J., Laviolette, F., & Panangaden, P. (2006). Bisimulation and cocongruence for probabilistic systems. Information and Computation, 204(4), 503–523. https://doi.org/10.1016/j.ic.2005.02.004