# Testing Labelled Markov Processes

## Abstract

Larsen and Skou introduced a notion of bisimulation for probabilistic transition systems. They characterized probabilistic bisimilarity in terms of a probabilistic modal logic and also in terms of ‘button pressing’ tests. Desharnais et al. extended the notion of probabilistic bisimulation and the logical characterization of probabilistic bisimilarity to labelled Markov processes. These processes generalize probabilistic transition systems in that they also allow continuous state spaces. We extend the characterization of probabilistic bisimilarity in terms of testing to labelled Markov processes. One of our main technical contributions is the construction of a final object in a category of labelled Markov processes and the identification of a natural metric on the state space of the final labelled Markov process. This metric provides us with another characterization of probabilistic bisimilarity: states are probabilistic bisimilar if and only if they have distance 0.

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