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The Matrix Reproved (Verification Pearl)

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Abstract

In this paper we describe a complete solution for the first challenge of the VerifyThis 2016 competition held at the 18th ETAPS Forum. We present the proof of two variants for the multiplication of matrices: a naive version using three nested loops and Strassen’s algorithm. The proofs are conducted using the Why3 platform for deductive program verification and automated theorem provers to discharge proof obligations. In order to specify and prove the two multiplication algorithms, we develop a new Why3 theory of matrices. In order to prove the matrix identities on which Strassen’s algorithm is based, we apply the proof by reflection methodology, which we implement using ghost state.To our knowledge, this is the first time such a methodology is used under an auto-active setting.

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Notes

  1. http://etaps2016.verifythis.org/.

  2. http://toccata.lri.fr/gallery/verifythis_2016_matrix_multiplication.en.html.

  3. For simplicity, the original task assumes that the matrices are square. Our implementation deals more generally with rectangular matrices.

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Acknowledgements

This work is partly supported by the Bware (ANR-12-INSE-0010, http://bware.lri.fr/), VOCAL (ANR-15-CE25-008, https://vocal.lri.fr/) Projects of the French National Research Organization (ANR), and by the Portuguese Foundation for the Sciences and Technology (Grant FCT-SFRH/BD/99432/2014). We thank Arthur Charguéraud, Jean-Christophe Filliâtre, and Claude Marché for their comments and remarks.

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Authors and Affiliations

  1. Lab. de Recherche en Informatique, CNRS, Univ. Paris-Sud, 91405, Orsay, France

    Martin Clochard, Léon Gondelman & Mário Pereira

  2. INRIA, Université Paris-Saclay, 91893, Palaiseau, France

    Martin Clochard, Léon Gondelman & Mário Pereira

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  1. Martin Clochard
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  2. Léon Gondelman
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  3. Mário Pereira
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Correspondence to Mário Pereira.

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Clochard, M., Gondelman, L. & Pereira, M. The Matrix Reproved (Verification Pearl). J Autom Reasoning 60, 365–383 (2018). https://doi.org/10.1007/s10817-017-9436-2

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  • DOI: https://doi.org/10.1007/s10817-017-9436-2

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