Quantifier Elimination and Cylindrical Algebraic Decomposition

Caviness, B. F.; Johnson, Jeremy

doi:10.1007/978-3-7091-9459-1

Cited by 165 publications

(21 citation statements)

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“…There have been a great many developments in the theory and implementation of CAD since Collins' original work which we do not describe here. The collection [12] summarises the work up to the mid-90s while the second author's journal articles [5] [21] attempt summaries of CAD progress since in their introduction and background sections. CAD is the backbone of all QE implementations as it is the only implemented complete procedure for the problem.…”

Section: Cylindrical Algebraic Decompositionmentioning

confidence: 99%

Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output Correctness

Florescu

England

2020

Lecture Notes in Computer Science

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Our topic is the use of machine learning to improve software by making choices which do not compromise the correctness of the output, but do affect the time taken to produce such output. We are particularly concerned with computer algebra systems (CASs), and in particular, our experiments are for selecting the variable ordering to use when performing a cylindrical algebraic decomposition of n-dimensional real space with respect to the signs of a set of polynomials.In our prior work we explored the different ML models that could be used, and how to identify suitable features of the input polynomials. In the present paper we both repeat our prior experiments on problems which have more variables (and thus exponentially more possible orderings), and examine the metric which our ML classifiers targets. The natural metric is computational runtime, with classifiers trained to pick the ordering which minimises this. However, this leads to the situation were models do not distinguish between any of the non-optimal orderings, whose runtimes may still vary dramatically. In this paper we investigate a modification to the cross-validation algorithms of the classifiers so that they do distinguish these cases, leading to improved results.

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Section: Cylindrical Algebraic Decompositionmentioning

confidence: 99%

Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output Correctness

Florescu

England

2020

Lecture Notes in Computer Science

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“…CAD was introduced in 1975 [10] and is still an active area of research. The collection [7] summarises the work up to the mid-90s while the background section of [13], for example, includes a summary of progress since. QE has numerous applications in science [2], engineering [25], and even the social sciences [23].…”

Section: Cylindrical Algebraic Decompositionmentioning

confidence: 99%

A Machine Learning Based Software Pipeline to Pick the Variable Ordering for Algorithms with Polynomial Inputs

Florescu

England

2020

Lecture Notes in Computer Science

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We are interested in the application of Machine Learning (ML) technology to improve mathematical software. It may seem that the probabilistic nature of ML tools would invalidate the exact results prized by such software, however, the algorithms which underpin the software often come with a range of choices which are good candidates for ML application. We refer to choices which have no effect on the mathematical correctness of the software, but do impact its performance. In the past we experimented with one such choice: the variable ordering to use when building a Cylindrical Algebraic Decomposition (CAD). We used the Python library Scikit-Learn (sklearn) to experiment with different ML models, and developed new techniques for feature generation and hyper-parameter selection. These techniques could easily be adapted for making decisions other than our immediate application of CAD variable ordering. Hence in this paper we present a software pipeline to use sklearn to pick the variable ordering for an algorithm that acts on a polynomial system. The code described is freely available online.

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“…Then by Bezout's theorem this point set is generally zero dimensional if f 1 and f 2 are coprime. This system can be solved using a technique similar to Cylindrical Algebraic Decomposition (Caviness and Johnson, 1998). Briefly, one can eliminate one variable by computing the resultant with respect to this variable, then find the real roots of this resultant, and finally lift the real roots to the solutions of the system.…”

Section: Curve Segmentsmentioning

confidence: 99%

Representing rational curve segments and surface patches using semi-algebraic sets

Shen

Pérez–Díaz

Goldman

et al. 2019

Computer Aided Geometric Design

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Quantifier Elimination and Cylindrical Algebraic Decomposition

Cited by 165 publications

References 0 publications

Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output Correctness

Improved Cross-Validation for Classifiers that Make Algorithmic Choices to Minimise Runtime Without Compromising Output Correctness

A Machine Learning Based Software Pipeline to Pick the Variable Ordering for Algorithms with Polynomial Inputs

Representing rational curve segments and surface patches using semi-algebraic sets

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