Abstract. Cylindrical algebraic decomposition(CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. When using CAD, there is often a choice for the ordering placed on the variables. This can be important, with some problems infeasible with one variable ordering but easy with another. 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 use machine learning (specifically a support vector machine) to select between heuristics for choosing a variable ordering, outperforming each of the separate heuristics.

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.

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.

No abstract

Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples. BackgroundA cylindrical algebraic decomposition (CAD) dissects R n into cells, each described by polynomial relations and arranged cylindrically so that the projection of any two cells into lower coordinates is equal or disjoint. CAD is a key tool, both for its original motivation of quantifier elimination (QE) over real-closed fields and many other applications discovered since. For a more detailed introduction see Huang et al. [4] and the references within. When using CAD there may be a choice for the variable ordering and it is well known that this can have a great effect on the tractability of a problem. We recently tested the following three heuristics for picking this ordering.Brown: Suggested by Brown [2], this heuristic chooses a variable ordering according to three criteria on the input system. We start with the first and break ties with successive ones. The advice is to eliminate a variable first if: 1. it has lower overall degree in the input; 2. it has lower (maximum) total degree of those terms in the input in which it occurs; 3. there is a smaller number of terms in the input which contain the variable.sotd: Suggested by Dolzmann et al. [3], this heuristic constructs the full set of projection polynomials for each ordering and selects the ordering whose set has the lowest sum of total degree for each of the monomials in each of the polynomials.ndrr: Suggested by Bradford et al.[1], this heuristic also constructs the full projection set, and selects the one with the lowest number of distinct real roots of the univariate polynomials.1