Group-by-group probabilistic bisimilarities and their logical characterizations

We provide two interpretations, over nondeterministic and probabilistic processes, of PML, the probabilistic version of Hennessy-Milner 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.