An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for quantifier elimination

Iwane, Hidenao; Yanami, Hitoshi; Anai, Hirokazu; Yokoyama, Keitaro

doi:10.1145/1577190.1577203

Cited by 20 publications

(14 citation statements)

References 32 publications

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“…ACM 978-1-4503-3435-8/15/07. DOI: http://dx.doi.org /10.1145/2755996.2756678. may be due to the many extensions and optimisations of CAD since Collins including: partial CAD (to lift only when necessary for quantifier elimination); symbolic-numeric lifting schemes [29,22]; local projection approaches [8,30]; and decompositions via complex space [11,3]. Collins original algorithm is described in [1] while a more detailed summary of recent developments can be found, for example, in [5].…”

Section: Introductionmentioning

confidence: 99%

Improving the Use of Equational Constraints in Cylindrical Algebraic Decomposition

England

Bradford

Davenport

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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When building a cylindrical algebraic decomposition (CAD) savings can be made in the presence of an equational constraint (EC): an equation logically implied by a formula.The present paper is concerned with how to use multiple ECs, propagating those in the input throughout the projection set. We improve on the approach of McCallum in ISSAC 2001 by using the reduced projection theory to make savings in the lifting phase (both to the polynomials we lift with and the cells lifted over). We demonstrate the benefits with worked examples and a complexity analysis.

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Section: Introductionmentioning

confidence: 99%

Improving the Use of Equational Constraints in Cylindrical Algebraic Decomposition

England

Bradford

Davenport

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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“…• only constructing cells when necessary (e.g. partial CAD [36] and sub-CAD [96]); • symbolic-numeric lifting schemes [92,68].…”

Section: Cylindrical Algebraic Decompositionmentioning

confidence: 99%

Using Machine Learning to Improve Cylindrical Algebraic Decomposition

Huang¹,

England²,

Wilson³

et al. 2019

Math.Comput.Sci.

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Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, best known as a procedure to enable Quantifier Elimination over real-closed fields. However, it has a worst case complexity doubly exponential in the size of the input, which is often encountered in practice. It has been observed that for many problems a change in algorithm settings or problem formulation can cause huge differences in runtime costs, changing problem instances from intractable to easy. A number of heuristics have been developed to help with such choices, but the complicated nature of the geometric relationships involved means these are imperfect and can sometimes make poor choices. We investigate the use of machine learning (specifically support vector machines) to make such choices instead.Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we apply it in two case studies: the first to select between heuristics for choosing a CAD variable ordering; the second to identify when a CAD problem instance would benefit from Gröbner Basis preconditioning. These appear to be the first such applications of machine learning to Symbolic Computation. We demonstrate in both cases that the machine learned choice outperforms human developed heuristics.1 principally {= 0, > 0, < 0}, but the Boolean combinations allow { = 0, ≥ 0, ≤ 0} as well 2 The feature set they used for their support vector machine was seeded by Table 1.

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“…For some applications there exist algorithms with better complexity [3], but CAD implementations still remain the best general purpose approach for many. This may be due to the numerous approaches used to improve the efficiency of CAD since Collins' original work including: improvements to the projection operator [39], [47], [10], [38]: partial CAD (lift only when necessary for QE) [21]; and symbolic-numeric lifting schemes [54], [42]. Some recent advances include making use of any Boolean structure in the input [6], [7], [27]; local projection approaches [11], [55]; and decompositions via complex space [18], [5].…”

Section: A Cylindrical Algebraic Decompositionmentioning

confidence: 99%

Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition with Groebner Bases

Huang

England

Davenport

et al. 2016

2016 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)

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Publisher: Elsevier © 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. This document is the author's post-print version, incorporating any revisions agreed during the peer-review process. Some differences between the published version and this version may remain and you are advised to consult the published version if you wish to cite from it. Abstract-Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials using Groebner Basis (GB) theory. Previous experiments have shown that while this can often be very beneficial to the CAD algorithm, for some problems it can significantly worsen the CAD performance.In the present paper we investigate whether machine learning, specifically a support vector machine (SVM), may be used to identify those CAD problems which benefit from GB preconditioning. We run experiments with over 1000 problems (many times larger than previous studies) and find that the machine learned choice does better than the human-made heuristic.

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An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for quantifier elimination

Cited by 20 publications

References 32 publications

Improving the Use of Equational Constraints in Cylindrical Algebraic Decomposition

Improving the Use of Equational Constraints in Cylindrical Algebraic Decomposition

Using Machine Learning to Improve Cylindrical Algebraic Decomposition

Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition with Groebner Bases

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