research-article

Variant real quantifier elimination: algorithm and application

Authors:

Mohab Safey El DinAuthors Info & Claims

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

Pages 183 - 190

https://doi.org/10.1145/1576702.1576729

Published: 28 July 2009 Publication History

Abstract

We study a variant of the real quantifier elimination problem (QE). The variant problem requires the input to satisfy a certain extra condition, and allows the ouput to be almost equivalent to the input. In a sense, we are strengthening the pre-condition and weakening the post-condition of the standard QE problem.

The motivation/rationale for studying such a variant QE problem is that many quantified formulas arising in applications do satisfy the extra conditions. Furthermore, in most applications, it is sufficient that the ouput formula is almost equivalent to the input formula. Thus, we propose to solve a variant of the initial quantifier elimination problem.

We present an algorithm (VQE), that exploits the strengthened pre-condition and the weakened post-condition. The main idea underlying the algorithm is to substitute the repeated projection step of CAD by a single projection without carrying out a parametric existential decision over the reals.

We find that the algorithm VQE can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormack's scheme. The problem has been practically out of reach for standard QE algorithms in spite of many attempts to tackle it. However the current implementation of VQE can solve it in about 1 day.

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Cited By

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https://doi.org/10.1016/j.jalgebra.2025.01.014
Berthomieu JFerguson ADin M(2021)Towards fast one-block quantifier elimination through generalised critical valuesACM Communications in Computer Algebra10.1145/3457341.345734854:3(109-113)Online publication date: 15-Mar-2021
https://dl.acm.org/doi/10.1145/3457341.3457348
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https://doi.org/10.1007/s11424-019-8351-5
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Variant real quantifier elimination: algorithm and application

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cover image ACM Conferences

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

July 2009

402 pages

ISBN:9781605586090

DOI:10.1145/1576702

General Chairs:
Jeremy Johnson
Drexel University, USA
,
Hyungju Park
Korea Institute for Advanced Study, Korea
,
Program Chair:
Erich Kaltofen
North Carolina State University, USA

Copyright © 2009 ACM.

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 ACM 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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Published: 28 July 2009

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ISSAC '09: International Symposium on Symbolic and Algebraic Computation

July 28 - 31, 2009

Seoul, Republic of Korea

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Cited By

Gopalakrishnan S(2025)On the arithmetic complexity of computing Gröbner bases of comaximal determinantal idealsJournal of Algebra10.1016/j.jalgebra.2025.01.014Online publication date: Jan-2025
https://doi.org/10.1016/j.jalgebra.2025.01.014
Berthomieu JFerguson ADin M(2021)Towards fast one-block quantifier elimination through generalised critical valuesACM Communications in Computer Algebra10.1145/3457341.345734854:3(109-113)Online publication date: 15-Mar-2021
https://dl.acm.org/doi/10.1145/3457341.3457348
Wang CYang ZZhi L(2019)Global Optimization of Polynomials over Real Algebraic SetsJournal of Systems Science and Complexity10.1007/s11424-019-8351-532:1(158-184)Online publication date: 14-Feb-2019
https://doi.org/10.1007/s11424-019-8351-5
Tiwari ALincoln P(2016)A search-based procedure for nonlinear real arithmeticFormal Methods in System Design10.1007/s10703-016-0245-848:3(257-273)Online publication date: 1-Jun-2016
https://dl.acm.org/doi/10.1007/s10703-016-0245-8
Bannwarth ISafey El Din MRobertz DLinton SYokoyama K(2015)Probabilistic Algorithm for Computing the Dimension of Real Algebraic SetsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756670(37-44)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756670
Xu MLi ZYang L(2015)Quantifier elimination for a class of exponential polynomial formulasJournal of Symbolic Computation10.1016/j.jsc.2014.09.01568:P1(146-168)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1016/j.jsc.2014.09.015
Faugère JEl Din MSpaenlehauer Pvan der Hoeven Jvan Hoeij M(2012)Critical points and Gröbner basesProceedings of the 37th International Symposium on Symbolic and Algebraic Computation10.1145/2442829.2442855(162-169)Online publication date: 22-Jul-2012
https://dl.acm.org/doi/10.1145/2442829.2442855
Sturm TTiwari ASchost ÉEmiris I(2011)Verification and synthesis using real quantifier eliminationProceedings of the 36th international symposium on Symbolic and algebraic computation10.1145/1993886.1993935(329-336)Online publication date: 8-Jun-2011
https://dl.acm.org/doi/10.1145/1993886.1993935
Kaltofen E(2011)The “Seven Dwarfs” of Symbolic ComputationNumerical and Symbolic Scientific Computing10.1007/978-3-7091-0794-2_5(95-104)Online publication date: 12-Oct-2011
https://doi.org/10.1007/978-3-7091-0794-2_5

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