# Weak Bisimulation is Sound and Complete for PCTL^{*}

## Abstract

We investigate weak bisimulation of probabilistic systems in the presence of nondeterminism, i.e. labelled concurrent Markov chains (LCMC) with silent transitions. We build on the work of Philippou, Lee and Sokolsky [17] for finite state LCMCs. Their 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 estimates on the possible probabilities.

Our analysis leads to three new developments: - We identify an axiomatization 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 *linear combinations* of transitions. This idea is closely related to the use of randomized schedulers. - We study a minor variant of the probabilistic logic _{p}CTL✱— 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✱.

## Keywords

Label Transition System Weak Transition State Formula Silent Transition Nondeterministic Choice## Preview

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