In this paper we describe how to build semantic models that support both nondeterministic choice and probabilistic choice. Several models exist that support both of these constructs, but none that we know of satisfies all the laws one would like. Using domain-theoretic techniques, we show how models can be devised using the "standard model" for probabilistic choice, and then applying modified domain-theoretic models for nondeterministic choice. These models are distinguished by the fact that the expected laws for nondeterministic choice and probabilistic choice remain valid. We also describe some potential applications of our model to aspects of security.

Abstract. We consider a simple divergence-free language RP for reactive processes which includes prefixing, deterministic choice, actionguarded probabilistic choice, synchronous parallel and recursion. We show that the probabilistic bisimulation of Larsen & Skou is a congruence for this language. Following the methodology introduced by de Bakker & Zucker we give denotational semantics to this language by means of a complete metric space of (deterministic) probabilistic trees defined in terms of the powerdomain of closed sets. This new metric, although not an ultra-metric, nevertheless specialises to the metric of de Bakker & Zucker. Our semantic domain admits a full abstraction result with respect to probabilistic bisimulation. 1

Metric-Space Denotational Semantics for Reactive Probabilistic Processes M.Z. Kwiatkowska and G.J. Norman School of Computer Science, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Abstract We consider the calculus of Communicating Sequential Processes (CSP) [8] extended with action-guarded probabilistic choice and provide it with an operational semantics in terms of a suitable extension of Larsen and Skou's [14] reactive probabilistic transition systems. We show that a testing equivalence which identi es two processes if they pass all tests with the same probability is a congruence for a subcalculus of CSP including external and internal choice and the synchronous parallel. Using the methodology of de Bakker and Zucker [3] introduced for classical process calculi, we derive a metric-space semantic model for the calculus and show it is fully abstract.

In this thesis we present three mathematical frameworks for the modelling of reac-tive probabilistic communicating processes. We first introduce generalised labelled transition systems as a model of such processes and introduce an equivalence, coarser than probabilistic bisimulation, over these systems. Two processes are identified with respect to this equivalence if, for all experiments, the probabilities of the respective processes passing a given experiment are equal. We next consider a probabilistic pro-cess calculus including external choice, internal choice, action-guarded probabilistic choice, synchronous parallel and recursion. We give operational semantics for this calculus be means of our generalised labelled transition systems and show that our equivalence is a congruence for this language. Following the methodology introduced by de Bakker & Zucker, we then give deno-tational semantics to the calculus by means of a complete metric space of probabilistic processes. The derived metric, although not an ultra-metric, satisfies the intuitive property that the distance between two processes tends to 0 if a measure of the dif-

. It has already been noticed by C. Elgot and his collaborators that the algebra of (nite and innite) trees is completely iterative, i.e., every system of ideal recursive equations has a unique solution. We prove that this is a special case of a very general coalgebraic phenomenon: suppose that an endofunctor

This report summarises research undertaken by the author as part of his DPhil in computation. A draft table of contents of the thesis, a proposed timetable for submission, and a list of publications are also included.