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.

Input/Output Stochastic Automata with Urgency: Confluence and Weak Determinism

“…The semantics of IOSA is defined in terms of non-deterministic labeled Markov processes (NLMP) [14,26] which extends LMP [15] with internal non-determinism. The foundations of NLMP is strongly rooted in measure theory, hence we recall first some basic definitions.…”

“…This is only guaranteed by the following proposition. Larsen and Skou's probabilistic bisimulation [21] has been extended to NLMPs in [14]. It can be shown that the bisimulation equivalence is a congruence for parallel composition of IOSA.…”

“…Urgent actions are also partitioned in input and output actions and, though inputs behave reactively and passively as before, urgent outputs are executed instantaneously as soon as the enabling state is reached. We also give semantics to IOSA with urgent actions (from now on, we simply call it IOSA) in terms of NLMP [14,26] (Sec. 3), and define its parallel composition (Sec.…”

Distribution-Based Bisimulation for Labelled Markov Processes

“…Another direction is to deal with systems that are more complex than LMPs. For example, we can add non-determinism choices for the same action, as the model in [11]. In addition, we can add the internal action τ to the set of actions and investigate weak bisimulations for LMPs, and investigate the metric definition for continuous-time models [20].…”

Bisimulation and Hennessy-Milner Logic for Generalized Synchronization Trees

“…the identification of Hennessy-Milner classes. The few papers that do consider the problem of identifying HM classes seem to be concerned with probabilistic systems: see [5,6,7,17,8,3,1] for example. However, in most of these papers, the following quote is emblematic of the source of these results: "... the probabilistic systems we are considering, without explicit nondeterminism, resemble deterministic systems quite closely, rather than nondeterministic systems" [5].…”