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Constraint-Solving Techniques for the Analysis of Stochastic Hybrid Systems

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Provably Correct Systems

Abstract

The ProCoS project has been seminal in widening the perspective on verification of computer-based systems to a coverage of the detailed interaction and feedback dynamics between the embedded system and its environment. We have since then seen a steady increase both in expressiveness of the “hybrid” modeling paradigms adopting such an integrated perspective and in the power of automatic reasoning techniques addressing relevant fragments of logic and arithmetic. In this chapter we review definitions of stochastic hybrid automata and of parametric stochastic hybrid automata, both of which unify the hybrid view on system dynamics with stochastic modeling as pertinent to reliability evaluation, and we elaborate on automatic verification and synthesis methods based on arithmetic constraint solving. The procedures are able to solve step-bounded stochastic reachability problems and multi-objective parameter synthesis problems, respectively.

This research has partially been funded by the German Research Foundation through the Collaborative Research Action SFB-TR 14 “Automatic Verification and Analysis of Complex Systems” (AVACS, www.avacs.org) and the Research Training Group DFG-GRK 1765: “System Correctness under Adverse Conditions” (SCARE, scare.uni-oldenburg.de).

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Notes

  1. 1.

    The reader might expect to rather see finite sub-ranges of \({\mathbb Z}\) or other finite sets as domains. To avoid cluttering the notation, we abstained from this. It should be noted that this does not induce a loss of generality, as not all of \({\mathbb Z}\) need to be dynamically reachable.

  2. 2.

    As for discrete variables, this does not exclude the possibility that only a bounded sub-range may dynamically be reachable.

  3. 3.

    Defined as \(\chi _G(\sigma )= {\left\{ \begin{array}{ll} 1 &{} \text {if } \sigma \in G,\\ 0 &{} \text {if } \sigma \not \in G. \end{array}\right. }\)

  4. 4.

    In practice, we offer a selection from a set of predefined density functions over the reals. For discrete carriers, we offer the ability to write arbitrary distributions by means of enumeration.

  5. 5.

    In SSAT parlance, this is the body of the formula after rewriting it to prenex form and stripping all the quantifiers.

  6. 6.

    To this end please note that collapsing equivalent branches, as pursued in Fig. 3, can only be done after solving the instances of the matrix and thus only is an option in cases where continuity arguments (or similar) permit generalizations from samples to neighborhoods.

  7. 7.

    As usual in interval constraint solving, we call any product of intervals with computer-representable bounds a box.

  8. 8.

    Due to the generality of the PSHA model, defining rewards exclusively on the final state is as expressive as defining them via functions on the whole run.

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Fränzle, M., Gao, Y., Gerwinn, S. (2017). Constraint-Solving Techniques for the Analysis of Stochastic Hybrid Systems. In: Hinchey, M., Bowen, J., Olderog, ER. (eds) Provably Correct Systems. NASA Monographs in Systems and Software Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-48628-4_2

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