Algebraic Reasoning for Probabilistic Concurrent Systems

We extend Milner's SCCS to obtain a calculus, PCCS, for reasoning about communicating probabilistic processes. In particular, the nondeterministic process summation operator of SCCS is replaced with a probabilistic one, in which the probability of behaving like a particular summand is given explicitly. The operational semantics for PCCS is based on the notion of probabilistic derivation, and is given structurally as a set of inference rules. We then present an equational theory for PCCS based on probabilistic bisimulation, an extension of Milner's bisimulation proposed by Larsen and Skou. We provide the rst axiomatization of probabilistic bisimulation, a subset of which is relatively complete for nite-state probabilistic processes. In the probabilistic case, a notion of processes with almost identical behavior (i.e., with probability 1 ? , for suuciently small) appears to be more useful in practice than a notion of equivalence, since the latter is often too restrictive. We weaken probabilistic bisimulation to obtain a metric space for \deterministic" PCCS processes, and show that in most contexts the eeect of expression replacement on distance is isometric. Finally, we use PCCS to model and analyze the performance of the AUY communication protocol for deletion errors.