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A Generalised Branch-and-Bound Approach and Its Application in SAT Modulo Nonlinear Integer Arithmetic

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Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

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Abstract

The branch-and-bound framework has already been successfully applied in SAT-modulo-theories (SMT) solvers to check the satisfiability of linear integer arithmetic formulas. In this paper we study how it can be used in SMT solvers for non-linear integer arithmetic on top of two real-algebraic decision procedures: the virtual substitution and the cylindrical algebraic decomposition methods. We implemented this approach in our SMT solver SMT-RAT and compared it with the currently best performing SMT solvers for this logic, which are mostly based on bit-blasting. Furthermore, we implemented a combination of our approach with bit-blasting that outperforms the state-of-the-art SMT solvers for most instances.

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Notes

  1. 1.

    Z3 has a command-line option to solve, instead of QF_NIA problems, their QF_NRA relaxations. This way Z3 can detect unsatisfiability (no real solution is found) or sometimes even satisfiability (the found real solution happens to be integer), but otherwise it returns “unknown”.

  2. 2.

    These sub-problems are not necessarily constraints or conjunctions of constraints, but in general formulas with arbitrary Boolean structure.

  3. 3.

    In the experimental results we use final lemmas only.

  4. 4.

    Note that modern SAT solvers also allow to forget learnt clauses that did not contribute to conflicts recently. This applies also to branching clauses.

  5. 5.

    A projected polynomial can have several “parents”; in this case we can choose any of them. In practice, one could store the chronologically “oldest” parents, as backtracking removes input polynomials in chronologically reverse order.

  6. 6.

    This form of multiple-branch lemmas are handled analogously to the 2-branch-case.

  7. 7.

    Additionally, all of these strategies employ a common preprocessing.

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Authors and Affiliations

  1. RWTH Aachen University, Aachen, Germany

    Gereon Kremer, Florian Corzilius & Erika Ábrahám

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  1. Gereon Kremer
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  2. Florian Corzilius
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  3. Erika Ábrahám
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Correspondence to Gereon Kremer .

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Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Kremer, G., Corzilius, F., Ábrahám, E. (2016). A Generalised Branch-and-Bound Approach and Its Application in SAT Modulo Nonlinear Integer Arithmetic. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_21

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