A Probabilistic Calculus of Cyber-Physical Systems

We propose a hybrid probabilistic process calculus for modelling and reasoning on cyber-physical systems (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.

💡 Research Summary

The paper introduces pCCPS, a Probabilistic Calculus of Cyber‑Physical Systems, designed to model and reason about CPSs that combine discrete‑time physical dynamics with stochastic uncertainties and logical communication. A CPS in pCCPS consists of a physical component—captured by a state triple (state variables, sensor values, actuator values) and a physical environment that provides two probabilistic maps: an evolution map (giving a distribution over next states based on current state and actuator settings) and a measurement map (giving a distribution over sensor readings based on the current state). An invariant set defines admissible states; violation leads to deadlock.

The cyber component is a process language extending Hennessy‑Regev’s Timed Process Language (TPL) with three new constructs: read s(x).C for sensor access, write a⟨v⟩.C for actuator control, and guarded probabilistic choice (∑ p_i : P_i). Standard CCS‑style operators such as parallel composition, channel restriction, timed prefix, and conditional are retained. Communication occurs over named channels with timeout semantics, while sensor/actuator interactions are distinguished from ordinary message passing.

Operational semantics are given as a probabilistic labelled transition system (pLTS) in SOS style. Transition labels include τ, tick (time‑step), snd/rcv (channel), read, and write, each possibly leading to a probability distribution over successor configurations. The authors prove that the pLTS satisfies key timed properties: time determinism, patience, maximal progress, and well‑timedness.

A weak probabilistic bisimulation (≈) is defined on the pLTS, abstracting from internal τ‑steps. Crucially, ≈ is shown to be a congruence: it is preserved under parallel composition, channel restriction, and other context operators, enabling compositional reasoning about CPSs. However, ≈ only distinguishes exact behavioural equivalence, which is too strict for practical CPS comparison where small stochastic variations are acceptable.

To address this, two quantitative behavioural metrics are introduced. The first, weak bisimulation metric ≈_p (p ∈