Probabilistic communicating processes

We explore the suitability of two semantic spaces as a basis for a probabilistic variant of the language of Communicating Sequential Processes (CSP), so as to provide a formalism for the specification and proof of correctness of probabilistic algorithms. The two spaces give rise to two sublanguages, each of which is characterised by an algebraic axiomatisation which is shown to be sound and complete for finite processes.In the first semantics, processes are defined as probability measures on the space of infinite traces and operators are defined as functions (mostly transformations) of probability measures. The advantage of this semantics is that it is simple and good for reasoning about probabilistic properties such as self-stabilisation or fairness of random algorithms. The disadvantage is that neither external choice nor parallel composition other than fully synchronised parallel composition can be defined in this semantics.This problem is solved in the second model which is based on the space of conditional probability measures of infinite traces. This model leads to a set of proof rules for the deterministic properties of probabilistic algorithms, but it is more difficult to use in the analysis of probabilistic properties.The last part of the paper shows how the two models are related and how this can be exploited to combine their advantages and get around their disadvantages. This is illustrated by the example of a self-stabilising tokenring.