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Satisfiability of Non-linear Transcendental Arithmetic as a Certificate Search Problem

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NASA Formal Methods (NFM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13903))

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Abstract

For typical first-order logical theories, satisfying assignments have a straightforward finite representation that can directly serve as a certificate that a given assignment satisfies the given formula. For non-linear real arithmetic with transcendental functions, however, no general finite representation of satisfying assignments is available. Hence, in this paper, we introduce a different form of satisfiability certificate for this theory, formulate the satisfiability verification problem as the problem of searching for such a certificate, and show how to perform this search in a systematic fashion. This does not only ease the independent verification of results, but also allows the systematic design of new, efficient search techniques. Computational experiments document that the resulting method is able to prove satisfiability of a substantially higher number of benchmark problems than existing methods.

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Notes

  1. 1.

    For example, for \(f(x)=x^2-1\), \(deg(f, [-10,10],0)=0\), while \(deg(f,[-10,0],0)=-1\), and \(deg(f,[0,10],0)=1\).

  2. 2.

    Available at https://www.cs.cas.cz/~ratschan/topdeg/topdeg.html.

  3. 3.

    For a description of the two tactics: https://microsoft.github.io/z3guide/docs/strategies/summary. The version of z3 used is 4.5.1.0.

  4. 4.

    The results of the experiments are available at https://doi.org/10.5281/zenodo.7774117.

  5. 5.

    For the results of such experiments, see [21].

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Acknowledgments

The authors thank Alessandro Cimatti, Alberto Griggio, and Roberto Sebastiani for helpful discussions on the topic of the paper. The work of Stefan Ratschan was supported by the project GA21-09458S of the Czech Science Foundation GA ČR and institutional support RVO:67985807.

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Authors and Affiliations

  1. DIBRIS, University of Genoa, Genoa, Italy

    Enrico Lipparini

  2. Institute of Computer Science of the Czech Academy of Sciences, Prague, Czechia

    Stefan Ratschan

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  1. Enrico Lipparini
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Correspondence to Enrico Lipparini .

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Editors and Affiliations

  1. Iowa State University, Ames, IA, USA

    Kristin Yvonne Rozier

  2. University of Texas at Austin, Austin, TX, USA

    Swarat Chaudhuri

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Lipparini, E., Ratschan, S. (2023). Satisfiability of Non-linear Transcendental Arithmetic as a Certificate Search Problem. In: Rozier, K.Y., Chaudhuri, S. (eds) NASA Formal Methods. NFM 2023. Lecture Notes in Computer Science, vol 13903. Springer, Cham. https://doi.org/10.1007/978-3-031-33170-1_29

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  • DOI: https://doi.org/10.1007/978-3-031-33170-1_29

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