The Spectrum of Strong Behavioral Equivalences for Nondeterministic and Probabilistic Processes

We present a spectrum of trace-based, testing, and bisimulation equivalences for nondeterministic and probabilistic processes whose activities are all observable. For every equivalence under study, 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 deterministic schedulers. We show that the first approach -- which compares two resolutions relatively to the probability distributions of all considered events -- results in a fragment of the spectrum compatible with the spectrum of behavioral equivalences for fully probabilistic processes. In contrast, the second approach -- which compares the probabilities of the events of a resolution with the probabilities of the same events in possibly different resolutions -- gives rise to another fragment composed of coarser equivalences that exhibits several analogies with the spectrum of behavioral equivalences for fully nondeterministic processes. Finally, the third approach -- which only compares the extremal probabilities of each event stemming from the different resolutions -- yields even coarser equivalences that, however, give rise to a hierarchy similar to that stemming from the second approach.