Continuity Properties of Distances for Markov Processes (With Proof Appendix)

In this paper we investigate distance functions on finite state Markov processes that measure the behavioural similarity of non-bisimilar processes. We consider both probabilistic bisimilarity metrics, and tracebased distances derived from standard Lp and Kullback-Leibler distances. Two desirable continuity properties for such distances are identified. We then establish a number of results that show that these two properties are in conflict, and not simultaneously fulfilled by any of our candidate natural distance functions. An impossibility result is derived that explains to some extent the fundamental difficulty we encounter.