The Spectrum of Strong Behavioral Equivalences for Nondeterministic and Probabilistic Processes

Abstract

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.

Quantitative Aspects of Programming Languages and Systems over the past 2^4 years and beyond

“…As far as quantitative notions of equivalence relations are concerned, in the last 20 years several different semantics have been considered, as also witnessed in the context of QAPL, which in such a sense proposes interesting representatives of the lines of research in this field. Such studies include linear-time equivalences, like trace equivalence for continuous-time Markov chains [145] and for interactive Markov chains [146]; probabilistic barbed congruence [57], which coincides with observational equivalence for a version of CCS including a probabilistic guarded choice operator, branching bisimulation congruence for probabilistic transition systems obeying a general alternating model of probabilistic and nondeterministic states [11] and for a more general probabilistic transition system specification format [104], weak bisimulation for continuous-time Markov chains [23] and for Markov automata [6]; testing equivalence for reactive probabilistic processes [71] and for nondeterministic, probabilistic, and Markovian processes [22], reward-based testing preorders for probabilistic labeled transition systems [59]; finally, a spectrum of different probabilistic equivalences, including trace, bisimulation, and testing semantics, in the setting of nondeterministic and probabilistic processes [24], a generalized notion of bisimulation for state-to-function transition systems that is comparable to many other quantitative notions of bisimulation [103], and undecidability results of bisimulation on Petri nets under durational semantics [100].…”

Trace and Testing Metrics on Nondeterministic Probabilistic Processes

“…We take inspiration from the extremal probabilities approach proposed in [5], which bases on the comparison, for each trace α, of both suprema and infima execution probabilities, wrt. resolutions, of α: two processes are equated if they assign the same extremal probabilities to all traces.…”

“…Several approaches to probabilistic trace equivalence are discussed in [7]: (i) The trace distribution [24] approach, comparing entire resolutions created by schedulers by checking if they assign the same probability to the same traces; (ii) The trace-by-trace [4] approach, in which firstly we take a trace and then we check if there are resolutions for processes assigning the same probability to it; (iii) The extremal probabilities [5] approach, considering for each trace only the infima and suprema of the probabilities assigned to it over all resolutions for the processes. We will argue that considering only supremal probabilities instead of both extremal probabilities is more tailored to reason on the verification problem.…”

“…We proceed to recall some notions, mostly from [5][6][7], necessary to reason on trace and testing semantics. A computation is a weighted sequence of process-to-process transitions.…”

Relating strong behavioral equivalences for processes with nondeterminism and probabilities

“…The rest of the paper, which is a revised and extended version of [6], is organized as follows. In Sect.…”