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A reflexive tactic for polynomial positivity using numerical solvers and floating-point computations

Authors:

Érik Martin-Dorel and

Pierre RouxAuthors Info & Claims

CPP 2017: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs

January 2017

Pages 90 - 99

https://doi.org/10.1145/3018610.3018622

Published: 16 January 2017 Publication History

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Abstract

Polynomial positivity over the real field is known to be decidable but even the best algorithms remain costly. An incomplete but often efficient alternative consists in looking for positivity witnesses as sum of squares decompositions. Such decompositions can in practice be obtained through convex optimization. Unfortunately, these methods only yield approximate solutions. Hence the need for formal verification of such witnesses. State of the art methods rely on heuristic roundings to exact solutions in the rational field. These solutions are then easy to verify in a proof assistant. However, this verification often turns out to be very costly, as rational coefficients may blow up during computations.

Nevertheless, overapproximations with floating-point arithmetic can be enough to obtain proofs at a much lower cost. Such overapproximations being non trivial, it is mandatory to formally prove that rounding errors are correctly taken into account. We develop a reflexive tactic for the Coq proof assistant allowing one to automatically discharge polynomial positivity proofs. The tactic relies on heavy computation involving multivariate polynomials, matrices and floating-point arithmetic. Benchmarks indicate that we are able to formally address positivity problems that would otherwise be untractable with other state of the art methods.

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Cited By

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Martin-Dorel ÉMelquiond GRoux P(2023)Enabling Floating-Point Arithmetic in the Coq Proof AssistantJournal of Automated Reasoning10.1007/s10817-023-09679-x67:4Online publication date: 16-Sep-2023
https://dl.acm.org/doi/10.1007/s10817-023-09679-x
Khalife EGaroche PFarhood M(2023)Code-Level Formal Verification of Ellipsoidal Invariant Sets for Linear Parameter-Varying SystemsNASA Formal Methods10.1007/978-3-031-33170-1_10(157-173)Online publication date: 3-Jun-2023
https://doi.org/10.1007/978-3-031-33170-1_10
Bentkamp AMir RAvigad J(2023)Verified reductions for optimizationTools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-031-30820-8_8(74-92)Online publication date: 20-Apr-2023
https://doi.org/10.1007/978-3-031-30820-8_8
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Published In

CPP 2017: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs

January 2017

234 pages

ISBN:9781450347051

DOI:10.1145/3018610

Program Chairs:
Yves Bertot
Inria, France / Université Cote d'Azur, France
,
Viktor Vafeiadis
MPI-SWS, Germany

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 16 January 2017

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CPP '17

CPP '17: Certified Proofs and Programs

January 16 - 17, 2017

Paris, France

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Overall Acceptance Rate 18 of 26 submissions, 69%

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Martin-Dorel ÉMelquiond GRoux P(2023)Enabling Floating-Point Arithmetic in the Coq Proof AssistantJournal of Automated Reasoning10.1007/s10817-023-09679-x67:4Online publication date: 16-Sep-2023
https://dl.acm.org/doi/10.1007/s10817-023-09679-x
Khalife EGaroche PFarhood M(2023)Code-Level Formal Verification of Ellipsoidal Invariant Sets for Linear Parameter-Varying SystemsNASA Formal Methods10.1007/978-3-031-33170-1_10(157-173)Online publication date: 3-Jun-2023
https://doi.org/10.1007/978-3-031-33170-1_10
Bentkamp AMir RAvigad J(2023)Verified reductions for optimizationTools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-031-30820-8_8(74-92)Online publication date: 20-Apr-2023
https://doi.org/10.1007/978-3-031-30820-8_8
Sogokon AJackson PJohnson T(2018)Verifying Safety and Persistence in Hybrid Systems Using Flowpipes and Continuous InvariantsJournal of Automated Reasoning10.1007/s10817-018-9497-xOnline publication date: 24-Nov-2018
https://doi.org/10.1007/s10817-018-9497-x
Roux PVoronin YSankaranarayanan S(2018)Validating numerical semidefinite programming solvers for polynomial invariantsFormal Methods in System Design10.1007/s10703-017-0302-y53:2(286-312)Online publication date: 1-Oct-2018
https://dl.acm.org/doi/10.1007/s10703-017-0302-y
Roux PIguernlala MConchon S(2018)A Non-linear Arithmetic Procedure for Control-Command Software VerificationTools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-319-89963-3_8(132-151)Online publication date: 14-Apr-2018
https://doi.org/10.1007/978-3-319-89963-3_8

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Polytope-based computation of polynomial ranges

Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields

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