A probabilistic calculus of cyber-physical systems

Cyber-Physical Systems (CPSs) are integrations of networking and distributed computing systems with physical processes, where feedback loops allow physical processes to affect computations and vice versa. Although CPSs can be found in several real-world domains, their verification often relies on simulation test systems rather than formal methodologies. We propose a hybrid probabilistic process calculus for modelling and reasoning on CPSs. The dynamics of the calculus is expressed in terms of a probabilistic labelled transition system in the SOS style of Plotkin. This is used to define a bisimulation-based probabilistic behavioural semantics which supports compositional reasonings. For a more careful comparison between CPSs, we provide two compositional probabilistic metrics to formalise the notion of behavioural distance between systems, also in the case of bounded computations. Finally, we provide a non-trivial case study, taken from an engineering application, and use it to illustrate our definitions and our compositional behavioural theory for CPSs.