Bisimulations that abstract from internal computation have proven to be useful for verification of compositionally defined transition systems. In the literature of probabilistic extensions of such transition systems, similar bisimulations are rare. In this paper, we introduce weak and branching bisimulation for fully probabilistic systems, transition systems where nondeterministic branching is replaced by probabilistic branching. In contrast to the nondeterministic case, both relations coincide. We give an algorithm to decide weak (and branching) bisimulation with a time complexity cubic in the number of states of the fully probabilistic system. This meets the worst case complexity for deciding branching bisimulation in the nondeterministic case. In addition, the relation is shown to be a congruence with respect to the operators of PLSCCS , a lazy synchronous probabilistic variant of CCS. We illustrate that due to these properties, weak bisimulation provides all the crucial ingredients...

This paper surveys the theoretical developments in the field of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems --- like large-scale computers, client-server architectures, networks --- can accurately be described using such stochastic specification formalisms.

We study various notions of probabilistic bisimulation from a coalgebraic point of view, accumulating in a hierarchy of probabilistic system types. In general, a natural transformation between two Set-functors straightforwardly gives rise to a transformation of coalgebras for the respective functors. This latter transformation preserves homomorphisms and thus bisimulations. For comparison of probabilistic system types we also need reflection of bisimulation. We build the hierarchy of probabilistic systems by exploiting the new result that the transformation also reflects bisimulation in case the natural transformation is componentwise injective and the first functor preserves weak pullbacks. Additionally, we illustrate the correspondence of concrete and coalgebraic bisimulation in the case of general Segala-type systems.

Published results show that various models may be obtained by combining parallel composition with probability and with or without non-determinism. In this paper we treat this problem in the setting of process algebra in the form of ACP. First, probabilities are introduced by an operator for the internal probabilistic choice. In this way we obtain the Basic Process Algebra with probabilistic choice prBPA.After- wards, prBPA is extended with parallel composition to ACP # .Wegive the axiom system for ACP # and a complete operational semantics that preserves the interleaving model for the dynamic concurrent processes. Considering the PAR protocol, a communication protocol that can be used in the case of unreliable channels, we investigate the applicability of ACP # . Using in addition only the priority operator and the preabstraction operator we obtain a recursive specification of the behaviour of the protocol that can be viewed as a Markov chain. 1

We survey various notions of probabilistic automata and probabilistic bisimulation, accumulating in an expressiveness hierarchy of probabilistic system types. The aim of this paper is twofold: On the one hand it provides an overview of existing types of probabilistic systems and, on the other hand, it explains the relationship between these models.

This paper deals with probabilistic and nondeterministic processes represented by a variant of labelled transition systems where any outgoing transition of a state s is augmented with probabilities for the possible successor states. Our main contribution are algorithms for computing the bisimulation equivalence classes as introduced by Larsen & Skou [44] and the simulation preorder `a la Segala & Lynch [57]. The algorithm for deciding bisimilarity is based on a variant of the traditional partitioning technique [43, 51] and runs in time O(mn(log m+ log n)) where m is the number of transitions and n the number of states. The main idea for computing the simulation preorder is the reduction to maximum flow problems in suitable networks. Using the method of Cheriyan, Hagerup & Mehlhorn [15] for computing the maximum flow, the algorithm runs in time O((mn 6 +m 2 n 3 )= log n). Moreover, we show that the network-based technique is also applicable to compute the simulation-like relation...

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Abstract. We study a process algebra which combines both nondeterministic and probabilistic behavior in the style of Segala and Lynch’s simple probabilistic automata. We consider strong bisimulation and observational equivalence, and provide complete axiomatizations for a language that includes parallel composition and (guarded) recursion. The presence of the parallel composition introduces various technical difficulties and some restrictions are necessary in order to achieve complete axiomatizations. 1

We are interested in describing timed systems that exhibit probabilistic behaviour. To this purpose, we consider a model of Probabilistic Timed Automata and introduce a concept of weak bisimulation for these automata, together with an algorithm to decide it. The weak bisimulation relation is shown to be preserved when either time, or probability are abstracted away. As an application, we use weak bisimulation for Probabilistic Timed Automata to model and analyze a timing attack on the dining cryptographers protocol.

Abstract. This paper is the extended technical report that corresponds to a published paper [14]. This paper introduces means to specify system randomness within UML statecharts, and to verify probabilistic temporal properties over such enhanced statecharts which we call probabilistic UML statecharts. To achieve this, we develop a general recipe to extend a statechart semantics with discrete probability distributions, resulting in Markov decision processes as semantic models. We apply this recipe to the requirementslevel UML semantics of [8]. Properties of interest for probabilistic statecharts are expressed in PCTL, a probabilistic variant of CTL for processes that exhibit both non-determinism and probabilities. Verification is performed using the model checker Prism. A model checking example shows the feasibility of the suggested approach.