Axiomatizing Probabilistic Processes: ACP with Generative Probabilities

This paper is concerned with finding complete axiomatizations of probabilistic processes. We examine this problem within the context of the process algebra ACP and obtain as our end-result the axiom system prACPi, a probabilistic version of ACP which can be used to reason algebraically about the reliability and performance of concurrent systems. Our goal was to introduce probability into ACP in as simple a fashion as possible. Optimally, ACP should be the homomorphic image of the probabilistic version in which the probabilities are forgotten. We begin by weakening slightly ACP to obtain the axiom system ACPi. The main difference between ACP and ACPi is that the axiom z + 6 = z, which does not yield a plausible interpretation in the generative model of probabilistic computation, is rejected in ACPi. We argue that this does not affect the usefulness of ACPi in practice, and show how ACP can be reconstructed from ACP; with a minimal amount of technical machinery. prACPi is obtained from ACPi through the introduction of probabilistic alternative and parallel composition operators, and a process graph model for prACPi based on probabilistic bisimulation is developed. We show that prACPi is a sound and complete axiomatization of probabilistic bisimulation for finite processes, and that prACPi can be homomorphically embedded in ACPi as desired. Our results for ACPi and prACPi are presented in a modular fashion by first considering several subsets of the signatures. We conclude with a discussion about the suitability of an internal probabilisti:: choice operaior in the context of p1'ACPr' • A preliminary version 0: t.b.is paper appeared in Proceetiing& o.z COSCCR '9! Thire International Conference on Concurrency Theory, Yo1. 630 of the Springer .. Verlag series Lecture SotC$ in Computer Science, pp. 472-485, Aug. 1992. The research ofthe first and second authors was supported by ESPRIT Basic Research Action 7166, CONCUR2. The second author was also supported by RACE project 1046, SPECS. This document does not necessarily reflect the views of the SPECS consortium. The research of the third author was supported by NSF grants CCR-8704309, CCR-9120995, and CCR-9208585.