Revisiting bisimilarity and its modal logic for nondeterministic and probabilistic processes

The logic PML is a probabilistic version of Hennessy–Milner logic introduced by Larsen and Skou to characterize bisimilarity over probabilistic processes without internal nondeterminism. In this paper, two alternative interpretations of PML over nondeterministic and probabilistic processes as models are considered, and two new bisimulation-based equivalences that are in full agreement with those interpretations are provided. The new equivalences include as coarsest congruences the two bisimilarities for nondeterministic and probabilistic processes proposed by Segala and Lynch. The latter equivalences are instead known to agree with two versions of Hennessy–Milner logic extended with an additional probabilistic operator interpreted over state distributions in place of individual states. The new interpretations of PML and the corresponding new bisimilarities are thus the first ones to offer a uniform framework for reasoning on processes that are purely nondeterministic or reactive probabilistic or that mix nondeterminism and probability in an alternating/nonalternating way.