Hierarchical Reasoning in Probabilistic CSP

Probabilistic CSP extends the language of CSP with an operator for probabilistic choice. However reasoning about an intricate combination of nondeterminism, communication and probabilistic behaviour can be complicated. In standard CSP, and in formal methods generally, such complication is overcome (when possible) by use of hierarchical reasoning. In this paper we provide a foundation for lifting such reasoning to the probabilistic setting. First we formalise the common observation that the standard models of CSP (traces, refusals and refusals/divergences) form a hierarchy, by showing that they are linked by embedding-projection pairs. Such structure underlies hierarchical reasoning in which complex process behaviour is reasoned about in terms of its simpler projections. Then we show how that hierarchy can be extended to a corresponding hierarchy between the probabilistic models, by using each of those three models of standard CSP as a basis for a probabilistic extension. Finally we show that there is a projection from the probabilistic models onto the standard models, which can be used to reason about nonprobabilistic properties of probabilistic processes.