Fully Abstract Characterizations of Testing Preorders for Probabilistic Processes

We present alternative characterizations of the testing preorders for probabilistic processes proposed in [CSZ92]. For a given probabilistic process, the characterization takes the form of a mapping from probabilistic traces to the interval [0, 1], where a probabilistic trace is an alternating sequence of actions and probability distributions over actions. Our results, like those of [CSZ92], pertain to divergence-free probabilistic processes, and are presented in two stages: probabilistic tests without internal τ-transitions are considered first, followed by probabilistic tests with τ-transitions. In each case, we show that our alternative characterization is fully abstract with respect to the corresponding testing pre-order, thereby resolving an open problem in [CSZ92]. In the second case, we use the alternative characterization to show that the testing preorder is actually an equivalence relation.