Bisimulation for Communicating Piecewise Deterministic Markov Processes (CPDPs)

Abstract

CPDPs (Communicating Piecewise Deterministic Markov Processes) can be used for compositional specification of systems from the class of stochastic hybrid processes formed by PDPs (Piecewise Deterministic Markov Processes). We define CPDPs and the composition of CPDPs, and prove that the class of CPDPs is closed under composition. Then we introduce a notion of bisimulation for PDPs and CPDPs and we prove that bisimilar PDPs as well as bisimilar CPDPs have equal stochastic behavior. Finally, as main result, we prove the congruence property that, for a composite CPDP, substituting components by different but bisimilar components results in a CPDP that is bisimilar to the original composite CPDP (and therefore has equal stochastic behavior).

Algorithmic bisimulation for Communicating Piecewise Deterministic Markov Processes

“…These notions are formally defined in [14]. Briefly said, a measurable relation R ⊂ X × Y on measurable spaces (X, B(X)) and (Y, B(Y )) is a relation such that the R-projection of each Borel …”

“…In [12] and [14] it is shown how complex CPDPs can be modelled in a compositional way by first specifying all component-CPDPs of the complex system and second by interconnecting these component-CPDPs by using a parallel composition operator. In [13] it is shown that the behavior of a CPDP that is closed (i.e.…”

“…The notion of bisimulation was introduced by Milner [7] in the context of discrete state processes. Bisimulation has also been established in the context of probabilistic and stochastic automata [6], [1], continuous time interactive Markov chains (IMC) [4], continuous dynamical systems [8], [10], general (non-stochastic) hybrid systems [5], [11], and CPDPs [14].…”

Compositional Modeling and Minimization of Time-Inhomogeneous Markov Chains

“…In addition, a new characterization of transient probabilities is provided for rate functions that are piecewise uniform. In contrast to works on communicating PDPs [SvdS06,SJvdS03,SvdS05], this paper considers weak bisimulation, congruence results and axiomatization, and, more importantly a notion of bisimulation which respects maximal progress. Current work consists of investigating improvements to the quotienting algorithm akin to [DPP04], model-checking algorithms [Kat06], and simulation relations for ICTMCs.…”

“…Compositional modeling formalisms for hybrid systems have been considered by, e.g., [AGLS06,AGH + 00]. Strong bisimulation has been proposed for several classes of (stochastic) hybrid systems, see e.g., [BLB05,HTP05,SvdS05]. Our notion of bisimulation is closely related to that for CPDPs [SvdS05] but differs in the fact that the maximal progress assumption-a race between one or more rates and a transition that is not subject to interaction with the environment is resolved in favor of the internal transition-is not considered in [SvdS05].…”