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Towards Probabilistic Contracts for Intelligent Cyber-Physical Systems

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Leveraging Applications of Formal Methods, Verification and Validation. Specification and Verification (ISoLA 2024)

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Abstract

Cyber-physical systems are systems that exhibit both discrete computational and continuous physical behavior. They are often subject to different kinds of uncertainty, ranging from sensor noise over random component failures to inconfidences induced by sample-based statistical learning. Quantitative formal methods have proven to be especially useful for assessing the impact of uncertainty on the system evolution over time. However, they lack compositionality. Existing methods for compositional design and verification, such as contracts, traditionally abstract from or (over-)approximate probability distributions, and resort to purely qualitative safety assessments in worst-case scenarios. This paper proposes a first step towards the integration of probabilistic methods into contract-based verification schemes to enable compositional reasoning over uncertain system behavior. We discuss different sources of uncertainties, as well as the necessity of probabilistic contracts for cyber-physical systems. Our key idea for integrating probabilities into contracts is the identification of safe yet precise approximations for sets of distributions, for which we use subdistributions. With that, we hope to reconcile probabilistic with set-based reasoning.

The research reported herein originates from discussion at the Workshop on Contract Languages held at the Lorentz Center, Leiden, The Netherlands, March 4–8 2024. It has been partially funded by Germany’s Federal Ministry of Education and Research (BMBF) as part of AutoDevSafeOps (01IS22087Q) as well as by the Ministry of Science and Culture (MWK) of the State of Lower Saxony as part of Zukunftslabor Mobilität.

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Notes

  1. 1.

    In the qualitative setting, a contract \((A',G')\) is stronger than contract (A, G) iff \(A \subseteq A'\) and \(G' \subseteq G\), i.e. if it weakens assumptions and strengthens guarantees. In this case, any component satisfying \((A',G')\) can also safely be used in a context requiring contract (A, G).

  2. 2.

    “In terms of input and output variables” is to be understood broadly here. There is no need to express component behaviors directly in terms of the immediate (input and output) port variables of the component, as the mathematical framework applies equally well to images thereof obtained by coordinate transformations and projections or even non-linear images of the state-space spanned by the ports, as long as the semantics \([\![C]\!]\) of a component is well-defined over these images. In the example contract provided in Sect. 5, we will describe the input-output behavior by a one-dimensional (sub-)distribution on the difference \(d = a_{control} - \frac{1}{2} \frac{v^2}{s}\) between physical states and the value of a control output, rather than by a joint distribution on all these variables.

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  1. University of Münster, Münster, Germany

    Pauline Blohm, Paula Herber & Anne Remke

  2. Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany

    Martin Fränzle & Paul Kröger

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  1. University of Limerick, CSIS and Lero, Limerick, Ireland

    Tiziana Margaria

  2. TU Dortmund University, Dortmund, Germany

    Bernhard Steffen

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Blohm, P., Fränzle, M., Herber, P., Kröger, P., Remke, A. (2025). Towards Probabilistic Contracts for Intelligent Cyber-Physical Systems. In: Margaria, T., Steffen, B. (eds) Leveraging Applications of Formal Methods, Verification and Validation. Specification and Verification. ISoLA 2024. Lecture Notes in Computer Science, vol 15221. Springer, Cham. https://doi.org/10.1007/978-3-031-75380-0_3

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