Behavioural Pseudometrics for Nondeterministic Probabilistic Systems

For the model of probabilistic labelled transition systems that allow for the co-existence of nondeterminism and probabilities, we present two notions of bisimulation metrics: one is state-based and the other is distribution-based. We provide a sound and complete modal characterisation for each of them, using real-valued modal logics based on Hennessy-Milner logic. The logic for characterising the state-based metric is much simpler than an earlier logic by Desharnais et al. as it uses only two non-expansive operators rather than the general class of non-expansive operators. For the kernels of the two metrics, which correspond to two notions of bisimilarity, we give a comprehensive comparison with some typical distribution-based bisimilarities in the literature.