Labeled transition systems are typically used as behavioral models of concurrent processes. Their labeled transitions define a one-step state-to-state reachability relation. This model can be generalized by modifying the transition relation to associate a state reachability distribution with any pair consisting of a source state and a transition label. The state reachability distribution is a function mapping each possible target state to a value that expresses the degree of one-step reachability of that state. Values are taken from a preordered set equipped with a minimum that denotes unreachability. By selecting suitable preordered sets, the resulting model, called ULTraS from Uniform Labeled Transition System, can be specialized to capture well-known models of fully nondeterministic processes (LTS), fully probabilistic processes (ADTMC), fully stochastic processes (ACTMC), and nondeterministic and probabilistic (MDP) or nondeterministic and stochastic (CTMDP) processes. This uniform treatment of different behavioral models extends to behavioral equivalences. They can be defined on ULTraS by relying on appropriate measure functions that express the degree of reachability of a set of states when performing multi-step computations. It is shown that the specializations of bisimulation, trace, and testing equivalences for the different classes of ULTraS coincide with the behavioral equivalences defined in the literature over traditional models except when nondeterminism and probability/stochasticity coexist; then new equivalences pop up.

We present a comparison of behavioral equivalences for nondeterministic and probabilistic processes whose activities are all observable. In particular, we consider trace-based, testing, and bisimulation-based equivalences. For each of them, we examine the discriminating power of three variants stemming from three approaches that differ for the way probabilities of events are compared when nondeterministic choices are resolved via schedulers. The first approach compares two resolutions with respect to the probability distributions of all considered events. The second approach requires that the probabilities of the set of events of a resolution be individually matched by the probabilities of the same events in possibly different resolutions. The third approach only compares the extremal probabilities of each event stemming from the different resolutions. The three approaches have very reasonable motivations and, when applied to fully nondeterministic processes or fully probabilistic processes, give rise to the classical well studied relations. We shall see that, for processes with nondeterminism and probability, they instead give rise to a much wider variety of behavioral relations, whose discriminating power is thoroughly investigated here in the case of deterministic schedulers.

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Vol. 10(1:16)2014, pp. 1–42 www.lmcs-online.org

Abstract. Testing equivalences have been deeply investigated on fully nondeterministic processes, as well as on processes featuring probabil-ities and internal nondeterminism. This is not the case with reactive probabilistic processes, for which it is only known that the discrimi-nating power of probabilistic bisimilarity is achieved when admitting a copying capability within tests. In this paper, we introduce for reactive probabilistic processes three testing equivalences without copying, which are respectively based on reactive probabilistic tests, fully nondetermin-istic tests, and nondeterministic and probabilistic tests. We show that the three testing equivalences are strictly finer than probabilistic failure-trace equivalence, and that the one based on nondeterministic and prob-abilistic tests is strictly finer than the other two, which are incomparable with each other. Moreover, we provide a number of facts that lead us to conjecture that (i) may testing and must testing coincide on reactive probabilistic processes and (ii) nondeterministic and probabilistic tests reach the same discriminating power as probabilistic bisimilarity. 1