Predicate transformers facilitate reasoning about imperative programs, including those exhibiting demonic non-deterministic choice. Probabilistic predicate transformers extend that facility to programs containing probabilistic choice, so that one can in principle determine not only whether a program is guaranteed to establish a certain result, but also its probability of doing so. We bring together independent work of Claire Jones and Jifeng He, showing how their constructions can be made to correspond � from that link between a predicate-based and a relation-based view of probabilistic execution we are able to propose `probabilistic healthiness conditions', generalising those of Dijkstra for ordinary predicate transformers. The associated calculus seems suitable for exploring further the rigorous derivation of imperative probabilistic programs.

Jones and Plotkin give a general construction for forming a probabilistic powerdomain over any directed-complete partial order [Jon90, JP89]. We apply their technique to the failures/divergences semantic model for Communicating Sequential Processes [Hoa85]. The resulting probabilistic model supports a new binary operator, probabilistic choice, and retains all operators of CSP including its two existing forms of choice. An advantage of using the general construction is that it is easy to see which CSP identities remain true in the probabilistic model. A surprising consequence however is that probabilistic choice distributes through all other operators; such algebraic mobility means that the syntactic position of the choice operator gives little information about when the choice actually must occur. That in turn leads to some interesting interaction between probability and nondeterminism. A simple communications protocol is used to illustrate the probabilistic algebra, and several sugg...

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 paper we present an algebraic language for the specification of probabilistic and nondeterministic processes, PNAL , which is a probabilistic extension of EPL that maintains nondeterminism.

In this paper we present an algebraic language for the specification of probabilistic and nondeterministic processes, PNAL, which is a probabilistic extension of EPL (Algebraic Theory of Processes, M. Hennessy) that maintains nondeterminism.

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-

Abstract. Randomization is of paramount importance in practical applications and randomized algorithms are used widely, for example in co-ordinating distributed computer networks, message routing and cache management. The appeal of randomized algorithms is their simplicity and elegance. However, this comes at a cost: the analysis of such systems become very complex, particularly in the context of distributed computation. This arises through the interplay between probability and nondeterminism. To prove a randomized distributed algorithm correct one usually involves two levels: classical, assertion-based reasoning, and a probabilistic analysis based on a suitable probability space on computations. In this paper we describe a number of approaches which allows us to verify the correctness of randomized distributed algorithms. 1

We present an algebraic language for the description of probabilistic and non-deterministic processes, which allows us to evaluate performance indexes as well as to check some temporal requirements: ROSA (Reasoning On Stochastic Algebras). As application, we analyse the Alternating Bit Protocol obtaining the average time to send a message, considering that channels may fail with a known probability.

Abstract. We propose a new approach to verification of probabilistic processes for which the model may not be available. We use a technique from Reinforcement Learning to approximate how far apart two processes are by solving a Markov Decision Process. If two processes are equivalent, the algorithm will return zero, otherwise it will provide a number and a test that witness the non equivalence. We suggest a new family of equivalences, called K-moment, for which it is possible to do so. The weakest, 1-moment equivalence, is trace-equivalence. The others are weaker than bisimulation but stronger than trace-equivalence. 1

Abstract. The combination of probabilistic and nondeterministic choice in program calculi is a notoriously tricky problem, and one with a long history. We present a simple functional programming approach to this challenge, based on algebraic theories of computational effects. We make use of the powerful abstraction facilities of modern functional languages, to introduce the choice operations as a little embedded domain-specific language rather than having to define a language extension; we rely on referential transparency, to justify straightforward equational reasoning about program behaviour. 1