Bisimulations for non-deterministic labelled Markov processes

Abstract

We extend the theory of labelled Markov processes to include internal non-determinism, which is a fundamental concept for the further development of a process theory with abstraction on non-deterministic continuous probabilistic systems. We define non-deterministic labelled Markov processes (NLMP) and provide three definitions of bisimulations: a bisimulation following a traditional characterisation; a state-based bisimulation tailored to our 'measurable' non-determinism; and an event-based bisimulation. We show the relations between them, including the fact that the largest state bisimulation is also an event bisimulation. We also introduce a variation of the Hennessy-Milner logic that characterises event bisimulation and is sound with respect to the other bisimulations for an arbitrary NLMP. This logic, however, is infinitary as it contains a denumerable. We then introduce a finitary sublogic that characterises all bisimulations for an image finite NLMP whose underlying measure space is also analytic. Hence, in this setting, all the notions of bisimulation we consider turn out to be equal. Finally, we show that all these bisimulation notions are different in the general case. The counterexamples that separate them turn out to be non-probabilistic NLMPs.