Group-by-Group Probabilistic Bisimilarities and Their Logical Characterizations

Abstract

Abstract. We provide two interpretations, over nondeterministic and probabilistic processes, of PML, the probabilistic version of HennessyMilner logic used by Larsen and Skou to characterize bisimilarity of probabilistic processes without internal nondeterminism. We also exhibit two new bisimulation-based equivalences, which are in full agreement with the two different interpretations of PML. The new equivalences are coarser than the bisimilarity for nondeterministic and probabilistic processes proposed by Segala and Lynch, which instead is in agreement with a version of Hennessy-Milner logic extended with an additional probabilistic operator interpreted over state distributions rather than over individual states. The modal logic characterizations provided for the new equivalences thus offer a uniform framework for reasoning on purely nondeterministic processes, reactive probabilistic processes, and nondeterministic and probabilistic processes.

Relating strong behavioral equivalences for processes with nondeterminism and probabilities

“…2 be identified. This is indeed the case for equivalences defined according to the second approach, like the other probabilistic bisimilarity in [8].…”

“…3. Recently, in [12,38,37,4,8] a further approach has been proposed that compares resolutions on the basis of the probabilities of individual equivalence-specific events. 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 [12], [38,8], and [37] that, unlike the one in [34], 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 for fully nondeterministic models and fully probabilistic models, while in the case of trace equivalence it surprisingly results in a congruence with respect to parallel composition (see the full version of [4]).…”

“…1. In fact, these two processes are identified by equivalences defined according to the third approach, as ensured by one of the two probabilistic bisimilarities in [8]. Similarly, if what matters is the extremal -i.e., minimal and maximal -probabilities of winning (0.3 and 0.7), it is conceivable that the two processes in Fig.…”

“…While in [34] the challenger and the defender must stepwise behave the same along two matching resolutions of nondeterminism, one might offer the defender the possibility of choosing different resolutions in response to different directions taken by the challenger. In other words, instead of requiring, as in [34], that for each resolution of the challenger there is a fully matching resolution of the defender, one could consider bisimulation games with partially matching resolutions as in [12,38,37,8]. The new equivalences can then be obtained by comparing processes according to all sets of probabilities or only to the extremal probabilities.…”