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Using Machine Learning to Improve Cylindrical Algebraic Decomposition

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  • Published: 03 April 2019
  • Volume 13, pages 461–488, (2019)
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Using Machine Learning to Improve Cylindrical Algebraic Decomposition
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  • Zongyan Huang1,
  • Matthew England2,
  • David J. Wilson3,
  • James Bridge1,
  • James H. Davenport3 &
  • …
  • Lawrence C. Paulson1 

  • 1182 Accesses

  • 28 Citations

  • 3 Altmetric

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Abstract

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.

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Acknowledgements

This work was supported by EPSRC Grant EP/J003247/1; the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No 712689 (SC\(^2\)); and the China Scholarship Council (CSC). The authors acknowledge both the anonymous referees of the present paper, and those of [66, 67], whose comments also helped improve this article.

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  1. Computer Laboratory, University of Cambridge, Cambridge, CB3 0FD, UK

    Zongyan Huang, James Bridge & Lawrence C. Paulson

  2. Faculty of Engineering, Environment and Computing, Coventry University, Coventry, CV1 2JH, UK

    Matthew England

  3. Department of Computer Science, University of Bath, Bath, BA2 7AY, UK

    David J. Wilson & James H. Davenport

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  1. Zongyan Huang
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Huang, Z., England, M., Wilson, D.J. et al. Using Machine Learning to Improve Cylindrical Algebraic Decomposition. Math.Comput.Sci. 13, 461–488 (2019). https://doi.org/10.1007/s11786-019-00394-8

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  • Received: 31 January 2017

  • Revised: 07 May 2018

  • Accepted: 25 February 2019

  • Published: 03 April 2019

  • Issue Date: December 2019

  • DOI: https://doi.org/10.1007/s11786-019-00394-8

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Keywords

  • Symbolic Computation
  • Computer Algebra
  • Machine Learning
  • Support Vector Machine
  • Cylindrical Algebraic Decomposition
  • Gröbner Basis
  • Parameter Selection

Mathematics Subject Classification

  • 68W30 (Symbolic Computation and Algebraic Computation)
  • 68T05 (Learning and Adaptive Systems)

Profiles

  1. Matthew England View author profile
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Associated Content

Part of a collection:

Mathematics in Computational Logic

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