In industrial-sized cyber-physical systems, ensuring fulfillment of requirements gets increasingly more costly as the number of components increases. To make the task feasible, compositional verification has been suggested as a scalable solution. Such techniques allow verification by divide-and-conquer, often using assume/guarantee contracts. Although previous research has focused mostly on the non-probabilistic setting, in the real world, probabilities often arise due to random hardware failures, communication delays, sensor ghost objects, machine learning, rounding errors, human behavior, and probabilistic algorithms. Therefore, for contract theories to be practically relevant to cyber-physical systems, there is a need to support probabilistic reasoning, for instance regarding safety and reliability. To this end, we first propose a contract metatheory for general input-output systems, allowing both probabilistic and non-probabilistic instantiations. Then, we instantiate the metatheory with probabilistic behaviors, introducing a new, fully trace-based probabilistic contract theory that supports general probability measures, continuous time, and continuous state spaces. To verify decompositions of such contracts, we also present a deductive system, which is illustrated by an industrially inspired automatic emergency braking example.

Abstract According to the principles of compositional verification, verifying that lower-level components satisfy their specification ensures that the whole system satisfies its top-level specification. The key step is to ensure that the lower-level specifications constitute a correct decomposition of the top-level specification. In a non-stochastic context, such decomposition can be analyzed using techniques of theorem proving. In industrial applications, especially in safety-critical systems, specifications are often of stochastic nature, for example, giving a bound on the probability that a system failure will occur before a given time. A decomposition of such a specification requires techniques beyond traditional theorem proving. The first contribution of the paper is a theoretical framework that allows the representation of, and reasoning about, stochastic and timed behavior of systems as well as specifications for such behavior. The framework is based on traces that describe the continuous-time evolution of a system, and specifications are formulated using timed automata combined with probabilistic acceptance conditions. The second contribution is a novel approach to verifying decompositions of such specifications by reducing the problem to checking emptiness of the solution space for a system of linear inequalities.

Although commonly used in industry, a major drawback of Stateflow is that it lacks support for stochastic properties; properties that are often needed to build accurate models of real-world systems. In order to solve this problem, as the first contribution, Stochastic Stateflow (SSF) is presented as a stochastic extension of a subset of Stateflow models. As the second contribution, the tool SMP-tool is updated with support for SSF models specified in Stateflow. Finally, as the third contribution, an industrial case study is presented.