Revisiting bisimilarity and its modal logic for nondeterministic and probabilistic processes

Abstract

We consider PML, the probabilistic version of Hennessy-Milner logic introduced by Larsen and Skou to characterize bisimilarity over probabilistic processes without internal nondeterminism. We provide two different interpretations for PML by considering nondeterministic and probabilistic processes as models, and we exhibit two new bisimulation-based equivalences that are in full agreement with those interpretations. Our new equivalences include as coarsest congruences the two bisimilarities for nondeterministic and probabilistic processes proposed by Segala and Lynch. The latter equivalences are instead in agreement with two versions of Hennessy-Milner logic extended with an additional probabilistic operator interpreted over state distributions rather than over individual states. Thus, our new interpretations of PML and the corresponding new bisimilarities offer a uniform framework for reasoning on processes that are purely nondeterministic or reactive probabilistic or are mixing nondeterminism and probability in an alternating/non-alternating way.

The Spectrum of Strong Behavioral Equivalences for Nondeterministic and Probabilistic Processes

“…In contrast, continuous rounded boxes contain equivalences assigning a weaker role to schedulers that have been recently introduced in [8,32,31,3,5], which compare separately the probabilities of individual equivalence-specific events. Continuous rectangular boxes instead contain old equivalences [35,19,27,11] and new equivalences [5] based on extremal probabilities. The only hybrid box is the one containing ∼ PTe-∀∃ , as this equivalence does not follow any of the three definitional approaches.…”

“…Recently, in [8,32,31,3,5] a further approach has appeared that compares resolutions on the basis of the probabilities of individual equivalence-specific events. Thus, a resolution of any of the two processes can be matched, with respect to different equivalence-specific events, by different resolutions of the other process (partially matching resolutions).…”

“…For the behavioral relations resulting from this approach, which weakens the impact of schedulers, we have that probabilistic bisimilarity implies probabilistic failure equivalence, which in turn implies probabilistic testing equivalence, which finally implies probabilistic trace equivalence. This approach has contributed to the development of new probabilistic bisimilarities in [8], [32,5], and [31] that, unlike the one in [28], are characterized by standard probabilistic logics such as quantitative µ-calculus, PML, and PCTL/PCTL*, respectively. Moreover, in the case of testing equivalence this approach has the advantage of being conservative also for fully nonde-terministic models [3], while in the case of trace equivalence it surprisingly results in a congruence with respect to parallel composition (full version of [3]).…”

“…To the best of our knowledge, this is the first comparative study of different kinds of behavioral equivalences over models featuring both nondeterministic and probabilistic aspects. Such a study is even more on demand after the recent introduction of new equivalences, like the ones in [8,32,31,3,5], that have interesting properties.…”

Probabilistic Bisimulation: Naturally on Distributions

“…They were considered for finite [11,34] and uncountable [8] state spaces. There are other bisimulations that identify more states than the classical [43] such as [54] and [4] designed to match a specific logic. Further, weak bisimulations coarser than usual state based analogues were given in [26,25,19], which also inspires our work, especially their approach to internal transitions.…”