Filters








23 Hits in 8.5 sec

A Decision Procedure for Univariate Real Polynomials in Isabelle/HOL

Manuel Eberl
2015 Proceedings of the 2015 Conference on Certified Programs and Proofs - CPP '15  
Sturm sequences are a method for computing the number of real roots of a univariate real polynomial inside a given interval efficiently.  ...  Building upon this, an Isabelle/HOL proof method was then implemented to prove interesting statements about the number of real roots of a univariate real polynomial and related properties such as non-negativity  ...  We also thank César Muñoz for providing us with a preliminary version of his paper, allowing us to reference it as related work and compare our work to his.  ... 
doi:10.1145/2676724.2693166 dblp:conf/cpp/Eberl15 fatcat:ofuh7d52wfbr5bnoyj2steyuwa

Deciding Univariate Polynomial Problems Using Untrusted Certificates in Isabelle/HOL

Wenda Li, Grant Olney Passmore, Lawrence C. Paulson
2017 Journal of automated reasoning  
We present a proof procedure for univariate real polynomial problems in Isabelle/HOL. The core mathematics of our procedure is based on univariate cylindrical algebraic decomposition.  ...  We present experiments demonstrating the efficacy of this approach, in many cases yielding orders of magnitude improvements over previous methods.  ...  Acknowledgements We thank Florian Haftmann for helping with code generation for our procedure. We are also grateful to the anonymous referees for their constructive suggestions.  ... 
doi:10.1007/s10817-017-9424-6 fatcat:hmwbpdzomvf4vgctwnerib4tp4

From LCF to Isabelle/HOL

Lawrence C. Paulson, Tobias Nipkow, Makarius Wenzel
2019 Formal Aspects of Computing  
Linear arithmetic decision procedures were also widely available by then. Our title echoes Mike Gordon's paper "From LCF to HOL: a Short History" [Gor00].  ...  Here, we focus on Isabelle/HOL and its distinctive strengths.  ...  Acknowledgements We thank the referees, Jasmin Blanchette, Michael Norrish and Andrei Popescu for valuable comments on drafts of this paper.  ... 
doi:10.1007/s00165-019-00492-1 fatcat:rv5zmo22fjedvjxjz4lm22v6de

A Verified Decision Procedure for Univariate Real Arithmetic with the BKR Algorithm [article]

Katherine Cordwell and Yong Kiam Tan and André Platzer
2021 arXiv   pre-print
We formalize the univariate fragment of Ben-Or, Kozen, and Reif's (BKR) decision procedure for first-order real arithmetic in Isabelle/HOL.  ...  Its key insight is a clever recursive procedure that computes the set of all consistent sign assignments for an input set of univariate polynomials while carefully managing intermediate steps to avoid  ...  Acknowledgments We would very much like to thank Brandon Bohrer, Fabian Immler, and Wenda Li for useful discussions about Isabelle/HOL and its libraries.  ... 
arXiv:2102.03003v2 fatcat:s2f73nvp2bgaxbffmvlalejeju

A modular, efficient formalisation of real algebraic numbers

Wenda Li, Lawrence C. Paulson
2016 Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs - CPP 2016  
This paper presents a construction of the real algebraic numbers with executable arithmetic operations in Isabelle/HOL.  ...  This work can be the basis for decision procedures that rely on real algebraic numbers.  ...  Formalizing them in Isabelle/HOL [13] , an interactive theorem prover, opens the way to numerous decision procedures in computer algebra.  ... 
doi:10.1145/2854065.2854074 dblp:conf/cpp/LiP16 fatcat:df2xm27wc5dhpoikym3xdur5f4

Verified Quadratic Virtual Substitution for Real Arithmetic [article]

Matias Scharager, Katherine Cordwell, Stefan Mitsch, André Platzer
2021 arXiv   pre-print
The proofs necessitate various contributions to the existing multivariate polynomial libraries in Isabelle/HOL, including a method for re-indexing variables in a polynomial.  ...  This paper presents a formally verified quantifier elimination (QE) algorithm for first-order real arithmetic by linear and quadratic virtual substitution (VS) in Isabelle/HOL.  ...  We wish to thank Fabian Immler for his substantial contributions at CMU to the polynomial theories of Isabelle/HOL and regret that his current industry position precludes our ability to include him as  ... 
arXiv:2105.14183v1 fatcat:e2oldaaqozdl7luolm62svg7he

A Decision Procedure for Univariate Polynomial Systems Based on Root Counting and Interval Subdivision

Cesar Munoz, Anthony Joseph Narkawicz, Aaron M. Dutle
2018 Journal of Formalized Reasoning  
This paper presents a formally verified decision procedure for determinining the satisfiability of a system of univariate polynomial relations over the real line.  ...  In PVS, the decision procedure is specified as a computable Boolean function on a deep embedding of polynomial relations.  ...  Decision Procedure for Rational Polynomials on the Real Line This section presents a decision procedure called hutch that determines the satisfiability of a system of polynomials over the real line.  ... 
doi:10.6092/issn.1972-5787/8212 pmid:31534636 pmcid:PMC6749613 fatcat:hhpapzyxxvcobp4swpsessz73y

Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers [chapter]

Grant Olney Passmore
2015 Lecture Notes in Computer Science  
We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., "x n ∈ Q" and "x n ∈ Z."  ...  Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.  ...  We thank Jeremy Avigad, Wenda Li, Larry Paulson, András Salamon and the anonymous referees for their helpful comments.  ... 
doi:10.1007/978-3-319-21401-6_12 fatcat:4h7rzdni7zbu3jlfmmljhibrxm

Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers [article]

Grant Olney Passmore
2015 arXiv   pre-print
We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$.  ...  Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.  ...  We thank Jeremy Avigad, Wenda Li, Larry Paulson, András Salamon and the anonymous referees for their helpful comments.  ... 
arXiv:1506.04863v1 fatcat:iksfyqth45cjjbltxdshr7w62i

Formalization of real analysis: a survey of proof assistants and libraries

SYLVIE BOLDO, CATHERINE LELAY, GUILLAUME MELQUIOND
2015 Mathematical Structures in Computer Science  
We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS.  ...  We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library.  ...  We are also grateful to the anonymous reviewers for their constructive remarks.  ... 
doi:10.1017/s0960129514000437 fatcat:qjnpetpbcbb4rbvm26geem3jai

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm's and Tarski's Theorems

Anthony Narkawicz, César Muñoz, Aaron Dutle
2015 Journal of automated reasoning  
Sturm's Theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semiopen interval.  ...  The procedure and its correctness properties enable the implementation of a PVS strategy for automatically proving existential and universal univariate polynomial inequalities.  ...  Conclusion This paper presented a formalization Sturm's Theorem in PVS along with a decision procedure for deciding the sign of univariate rational polynomials where the polynomial variable ranges over  ... 
doi:10.1007/s10817-015-9320-x fatcat:l7jxvtfsxbhypbq3v2yje6dmbq

Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination

Assia Mahboubi, Cyril Cohen, Henk Barendregt
2012 Logical Methods in Computer Science  
The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic  ...  This paper describes a formalization of discrete real closed fields in the Coq proof assistant.  ...  Acknowledgments The authors wish to thank Georges Gonthier for his precious suggestion to use continuationpassing style in the last part of this work.  ... 
doi:10.2168/lmcs-8(1:2)2012 fatcat:kespb2lx6rh6dl3ykz65ufjfcu

Real World Verification [chapter]

André Platzer, Jan-David Quesel, Philipp Rümmer
2009 Lecture Notes in Computer Science  
Despite substantial advances in verification technology, complexity issues with classical decision procedures are still a major obstacle for formal verification of real-world applications, e.g., in automotive  ...  Finally, we introduce a new decision procedure combining Gröbner Bases and semidefinite programming for the real Nullstellensatz that outperforms the individual approaches on an interesting set of problems  ...  Related Work Nipkow [29] presented a formally verified implementations of quantifier elimination in an executable fragment of Isabelle/HOL, currently for linear real arithmetic only.  ... 
doi:10.1007/978-3-642-02959-2_35 fatcat:4dd6gqu5bbeopbuwikdprliieq

Coquelicot: A User-Friendly Library of Real Analysis for Coq

Sylvie Boldo, Catherine Lelay, Guillaume Melquiond
2014 Mathematics in Computer Science  
Real analysis is pervasive to many applications, if only because it is a suitable tool for modeling physical or socio-economical systems.  ...  The Coq system comes with an axiomatization of standard real numbers and a library of theorems on real analysis. Unfortunately, this standard library is lacking some widely used results.  ...  Acknowledgements The authors are grateful to Pierre Michalak andÉvelyne Roudneff for allowing us to take the Baccalaureate exam in real-life conditions in a high school in Massy, and for organizing the  ... 
doi:10.1007/s11786-014-0181-1 fatcat:eptv7v543bg63nlm6gzyakcctu

Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals [chapter]

Leonardo de Moura, Grant Olney Passmore
2013 Lecture Notes in Computer Science  
Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental  ...  Much research has gone into making RCF decision procedures practical, especially for restricted classes of formulas commonly arising in applications.  ...  Major research on reasoning with infinitesimals has been done both in the ACL2 [11] and Isabelle/HOL [9] proof assistants.  ... 
doi:10.1007/978-3-642-38574-2_12 fatcat:rjof3qq5tvhv7cv3n5sqregami
« Previous Showing results 1 — 15 out of 23 results