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Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

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NASA Formal Methods (NFM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 7871))

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Abstract

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Alexey Solovyev & Thomas C. Hales

Authors
  1. Alexey Solovyev
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  2. Thomas C. Hales
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Editor information

Editors and Affiliations

  1. NASA Ames Research Center, M/S 269-2, Moffett Field, 94035-0001, CA, USA

    Guillaume Brat

  2. Stinger Ghaffarian Technologies Inc., NASA Ames Research Center, M/S 269-2, Moffett Field, 94035-0001, CA, USA

    Neha Rungta

  3. NASA Ames Research Center, Carnegie Mellon University, M/S 269-2, Moffett Field, 94035-0001, CA, USA

    Arnaud Venet

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Solovyev, A., Hales, T.C. (2013). Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations. In: Brat, G., Rungta, N., Venet, A. (eds) NASA Formal Methods. NFM 2013. Lecture Notes in Computer Science, vol 7871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38088-4_26

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  • DOI: https://doi.org/10.1007/978-3-642-38088-4_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38087-7

  • Online ISBN: 978-3-642-38088-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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