Bisimulations, logics, and trace distributions for stochastic systems with rewards

Stochastic systems with rewards yield a generic stochastic model where both the state and the action space might be uncountable and where every action is decorated by a real-valued reward. For every deterministic stochastic system with rewards we prove that the bisimulation relation and the trace-distribution relation collapse. As a second result, we also establish a characterisation of the bisimulation relation in terms of an expressive action-based probabilistic logic and show that this characterisation is still maintained by a small fragment of this logic.