Probabilistic Bisimulation: Naturally on Distributions

In contrast to the usual understanding of probabilistic systems as stochastic pro-cesses, recently these systems have also been regarded as transformers of probabili-ties. In this paper, we give a natural definition of strong bisimulation for probabilisticsystems corresponding to this view that treats probability distributions as first-classcitizens. Our definition applies in the same way to discrete systems as well as tosystems with uncountable state and action spaces. Several examples demonstratethat our definition refines the understanding of behavioural equivalences of prob-abilistic systems. In particular, it solves a longstanding open problem concerningthe representation of memoryless continuous time by memoryfull continuous time.Finally, we give algorithms for computing this bisimulation not only for finite butalso for classes of uncountably infinite systems.