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Utilisation et certification de l'arithmétique d'intervalles dans un assistant de preuves

Abstract : Nowadays, more and more calculations, including monitoring and control, are done by software. Our goal is to formally prove the correctness of some numerical calculations by using on computers some tools that have long been guaranteed by human beings like interval calculations, and particularly Taylor models calculations. Yet such theoretical tools are totally unknown to formal proof checkers.

This thesis presents the construction of a library of Taylor models for PVS proof checker. We have developed addition, subtraction, multiplication by a scalar, multiplication, square, power and square root operations on Taylor models. We have also developed the exponential, sine, arctangent and hyperbolic sine and cosine on Taylor models. We have proven in PVS that the operations and functions defined in our library preserve the containment property. To the best of our knowledge, such formal proofs have never been done for the implementations of Taylor models.

We have developed a PVS strategy called containment to prove the containment property of Taylor models built from the operations and functions defined. We have also developed another strategy called taylors to automatically certify inequalities and bound expressions. When a proof checker is used to establish an inequality, it can be necessary to guide the proof checker step by step during the proof. For this reason, users frequently don't do the proof. Then, our work, that simplifies the proof of inequalities and expressions boundaries, is incentive to the use of formal proof checkers.

This work can be used to construct Taylor models of expressions, derive more or less accurate bounds for arithmetic expressions, certify inequalities and bound expressions. We show with two applications that our method for verifying expressions in a proof checker allows us to formally prove some expressions that appear in life critical pieces of software.
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https://tel.archives-ouvertes.fr/tel-00177109
Contributor : Francisco José Chaves <>
Submitted on : Friday, October 5, 2007 - 2:36:13 PM
Last modification on : Wednesday, November 20, 2019 - 2:41:50 AM
Long-term archiving on: : Sunday, April 11, 2010 - 10:13:38 PM

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  • HAL Id : tel-00177109, version 1

Citation

Francisco Cháves. Utilisation et certification de l'arithmétique d'intervalles dans un assistant de preuves. Modélisation et simulation. Ecole normale supérieure de lyon - ENS LYON, 2007. Français. ⟨tel-00177109⟩

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