Compositional Testing Preorders for Probabilistic Processes

Transition systems are well-established as a semantic model for distributed systems. There are widely accepted preorders that serve as criteria for reenement of a more abstract transition system to a more concrete one. To reason about probabilistic phenomena such as failure rates, we need to extend models and methods that have proven successful for non-probabilistic systems to a probabilistic setting. In this paper, we consider a model of probabilistic transition systems, containing probabilistic choice and nondeterministic choice as independent concepts. We present a notion of testing for these systems. Our main contributions are denotational characterizations of the testing pre-orders. The characterizations are given in terms of chains for may-testing and refusal chains for must-testing, that are analogous to traces and failures in denotational models of CSP. Reenement corresponds to inclusion between chains and refusal chains mod-ulo closure operations. The preorders are shown to be compositional. We also show that when restricted to non-probabilistic systems, these preorders collapse to the standard simulation and refusal-simulation.