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 present a systematic way to generate (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of quantitative systems. Our quantitative systems include weighted versions of automata and transition systems, in which transitions are assigned a value in a monoid that represents cost, duration, probability, etc. Such systems are represented as coalgebras and (1) and (2) above are derived in a modular fashion from the underlying (functor) type of these coalgebras. In previous work, we applied a similar approach to a class of systems (without weights) that generalizes both the results of Kleene (on rational languages and DFA’s) and Milner (on regular behaviours and finite LTS’s), and includes many other systems such as Mealy and Moore machines. In the present paper, we extend this framework to deal with quantitative systems. As a consequence, our results now include languages and axiomatizations, both existing and new ones, for many different kinds of probabilistic systems.

The paper introduces symbolic bisimulations for a simple probabilistic π-calculus to overcome the infinite branching problem that still exists in checking ground bisimulations between probabilistic systems. Especially the definition of weak (symbolic) bisimulation does not rely on the random capability of adversaries and suggests a solution to the open problem on the axiomatization for weak bisimulation in the case of unguarded recursion. Furthermore, we present an efficient characterization of symbolic bisimulations for the calculus, which allows the ”on-the-fly ” instantiation of bound names and dynamic construction of equivalence relations for quantitative evaluation. This directly results in a local decision algorithm that can explore just a minimal portion of the state spaces of the probabilistic processes in question. 1

Abstract. This paper studies types and probabilistic bisimulations for a timed π-calculus as an effective tool for a compositional analysis of probabilistic distributed behaviour. The types clarify the role of timers as interface between nonterminating and terminating communication for guaranteeing distributed liveness. We add message-loss probabilities to the calculus, and introduce a notion of approximate bisimulation that discards transitions below a certain specified probability threshold. We prove this bisimulation to be a congruence, and use it for deriving quantitative bounds for practical protocols in distributed systems, including timer-driven message-loss recovery and the Two-Phase Commit protocol. 1

c t i v it y e p o r t

c t i v it y e p o r t 2008 Table of contents

Abstract. Coalgebras provide a uniform framework for the study of dynamical systems, including several types of automata. The coalgebraic view on systems has recently been proved relevant by the development of a number of expression calculi which generalize classical results by Kleene, on regular expressions, and by Kozen, on Kleene algebra. This note contains an overview of the motivation and results of the generic framework we developed – Kleene Coalgebra – to uniformly derive the aforementioned calculi. We present an historical overview of work on regular expressions and axiomatizations, as well a discussion of related work. We show applications of the framework to three types of probabilistic systems: simple Segala, stratified and Pnueli-Zuck. 1