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On the Generation of Positivstellensatz Witnesses in Degenerate Cases

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Book cover Interactive Theorem Proving (ITP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6898))

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Abstract

One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq.

The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty.

We implemented our method and illustrate its use with examples, including extractions of proofs to Coq.

This work was partially supported by ANR project “ASOPT”.

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Authors and Affiliations

  1. CNRS, VERIMAG, France

    David Monniaux

  2. Université Joseph Fourier, VERIMAG, France

    Pierre Corbineau

Authors
  1. David Monniaux
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  2. Pierre Corbineau
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Editor information

Editors and Affiliations

  1. Faculteit Informatica, Open Universiteit, Postbus 2960, Heerlen, 6401 DL, The Netherlands

    Marko van Eekelen

  2. FNWI/ICIS/IS, Radboud Universiteit Nijmegen, Postbus 9010, 6500 GL, Nijmegen, The Netherlands

    Herman Geuvers  & Freek Wiedijk  & 

  3. Faculteit Informatica, Open Universiteit, Postbus 2960, 6401 DL, Heerlen, The Netherlands

    Julien Schmaltz

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Monniaux, D., Corbineau, P. (2011). On the Generation of Positivstellensatz Witnesses in Degenerate Cases. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds) Interactive Theorem Proving. ITP 2011. Lecture Notes in Computer Science, vol 6898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22863-6_19

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  • DOI: https://doi.org/10.1007/978-3-642-22863-6_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22862-9

  • Online ISBN: 978-3-642-22863-6

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