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.

网络物理系统(CPS) 是网络和分布式计算系统与物理过程的集成，其中反馈循环允许物理过程影响计算，反之亦然。尽管 CPS 可以在多个现实领域中找到，但它们的验证通常依赖于模拟测试系统而不是正式的方法。

我们提出了一种混合概率过程演算，用于对 CPS 进行建模和推理。微积分的动力学用Plotkin 的 SOS 风格的概率标记转换系统表示。这用于定义支持组合推理的基于互模拟的概率行为语义。为了在 CPS 之间进行更仔细的比较，我们提供了两个组合概率度量来形式化系统之间的行为距离的概念，在有界计算的情况下也是如此。最后，我们提供了一个来自工程应用程序的重要案例研究，并用它来说明我们的定义和我们的 CPS 组合行为理论。