Weak bisimulation is sound and complete for pCTL∗

We investigate weak bisimulation of probabilistic systems in the presence of nondeterminism, i.e. labelled concurrent Markov chains (LCMC) with silent transitions. We develop an approach based on allowing convex combinations of computations, similar to Segala and Lynch’s use of randomized schedulers.The definition of weak bisimulation destroys the additivity property of the probability distributions, yielding instead capacities. The mathematics behind capacities naturally captures the intuition that when we deal with nondeterminism we must work with bounds on the possible probabilities rather than with their exact values.Our analysis leads to three new developments:
• We identify a characterization of “image finiteness” for countable-state systems and present a new definition of weak bisimulation for these LCMCs. We prove that our definition coincides with that of Philippou, Lee and Sokolsky for finite state systems.
• We show that bisimilar states have matching computations. The notion of matching involves convex combinations of transitions.
• We study a minor variant of the probabilistic logic pCTL∗ – the variation arises from an extra path formula to address action labels. We show that bisimulation is sound and complete for this variant of pCTL∗.This is an extended complete version of a paper that was presented at CONCUR 2002.