Weak Bisimulation Metrics in Models with Nondeterminism and Continuous State Spaces

Bisimulation metrics are used to estimate the behavioural distance between probabilistic systems. They have been defined in discrete and continuous state space models. However, the weak semantics approach, where non-observable actions are abstracted away, has been adopted only in the discrete case. We fill this gap and provide a weak bisimulation metric for models with continuous state spaces. A difficulty is to provide a notion of weak transition leaving from a continuous distribution over states. Our weak bisimulation metric allows for compositional reasoning. Systems at distance zero are equated by a notion of weak bisimulation. We apply our theory in a case study where continuous distributions derive by the evolution of the physical environment.