Choosing a Variable Ordering for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition

England, Matthew; Bradford, Russell J.; Davenport, James H.; Wilson, David J.

doi:10.1007/978-3-662-44199-2_68

Cited by 13 publications

(10 citation statements)

References 23 publications

(41 reference statements)

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“…The ordering can have a great effect on the time / memory use of CAD, the number of cells, and even the underlying complexity [5]. Human designed heuristics have been developed to make the choice [11], [4], [3], [14] and are used in most implementations.…”

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

Preprint

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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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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

Preprint

Self Cite

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show abstract

“…The ordering can have a great effect on the time/memory use of CAD, the number of cells, and even the underlying complexity [5]. Human designed heuristics have been developed to make the choice [3,4,11,14] and are used in most implementations.…”

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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“…There is a large set derived from university mathematics entrance exams [72], which but may be publicly available in the future. An obvious extension of Case Study A would be to test additional heuristics, such as the greedy sotd heuristic [41], one based on the number of full dimensional cells [98], or those developed for CAD by Regular Chains [45]. For Case Study B there are now further CAD optimisations for problems with multiple equalities under development [43,46,39] which may affect the role of GB preconditioning from CAD.…”

Section: Future Workmentioning

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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Choosing a Variable Ordering for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition

Cited by 13 publications

References 23 publications

A machine learning based software pipeline to pick the variable ordering for algorithms with polynomial inputs

A machine learning based software pipeline to pick the variable ordering for algorithms with polynomial inputs

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

Using Machine Learning to Improve Cylindrical Algebraic Decomposition

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