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Decision Procedures for Theories of Sets with Measures

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Automated Deduction – CADE 26 (CADE 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10395))

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Abstract

In this paper we introduce a decision procedure for checking satisfiability of quantifier-free formulae in the combined theory of sets, measures and arithmetic. Such theories are important in mathematics (e.g. probability theory and measure theory) and in applications. We indicate how these ideas can be used for obtaining a decision procedure for a fragment of the duration calculus.

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Notes

  1. 1.

    For example, a model in which the set variables \({x}{}_{1}, {x}{}_{2}, {x}{}_{3}\) are interpreted as intervals \(I_1, I_2, I_3\) and the atomic sets \({\mathcal{{S}}{}_{001}}, {\mathcal{{S}}{}_{010}}, {\mathcal{{S}}{}_{100}}, {\mathcal{{S}}{}_{111}}\) are assigned non-empty values and the other atomic intervals are empty, is inconsistent: There can be no section in which the intervals overlap (\(\mathcal{{S}}{}_{111}\) not empty) that is joined by three sections where every set is disjoint from the other two (\({\mathcal{{S}}{}_{001}}, {\mathcal{{S}}{}_{010}},{\mathcal{{S}}{}_{100}}\) are not empty).

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Acknowledgments

We thank Ernst-Rüdiger Olderog, Martin Fränzle and Calogero Zarba for helpful discussions on the duration calculus. We also thank the anonymous reviewers for their constructive comments.

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  1. Universität Koblenz-Landau, Koblenz, Germany

    Markus Bender & Viorica Sofronie-Stokkermans

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  1. Markus Bender
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  2. Viorica Sofronie-Stokkermans
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Correspondence to Markus Bender .

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  1. Microsoft Research, Redmond, Washington, USA

    Leonardo de Moura

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Bender, M., Sofronie-Stokkermans, V. (2017). Decision Procedures for Theories of Sets with Measures. In: de Moura, L. (eds) Automated Deduction – CADE 26. CADE 2017. Lecture Notes in Computer Science(), vol 10395. Springer, Cham. https://doi.org/10.1007/978-3-319-63046-5_11

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  • DOI: https://doi.org/10.1007/978-3-319-63046-5_11

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