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

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

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

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

Algorithmically generating new algebraic features of polynomial systems for machine learning

There are a variety of choices to be made in both computer algebra systems (CASs) and satisfiability modulo theory (SMT) solvers which can impact performance without affecting mathematical correctness. Such choices are candidates for machine learning (ML) approaches, however, there are difficulties in applying standard ML techniques, such as the efficient identification of ML features from input data which is typically a polynomial system. Our focus is selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm implemented in several CASs, and now also SMT-solvers. We created a framework to describe all the previously identified ML features for the problem and then enumerated all options in this framework to automatically generation many more features. We validate the usefulness of these with an experiment which shows that an ML choice for CAD variable ordering is superior to those made by human created heuristics, and further improved with these additional features. We expect that this technique of feature generation could be useful for other choices related to CAD, or even choices for other algorithms with polynomial systems for input.Comment: To appear in Proc SC-Square Workshop 2019. arXiv admin note: substantial text overlap with arXiv:1904.1106

Dataset supporting "Using Machine Learning to decide when to Precondition Cylindrical Algebraic Decomposition with Groebner Bases"

Dataset supporting the paper: Z. Huang, M. England, J.H. Davenport and L.C. Paulson Using Machine Learning to decide when to Precondition Cylindrical Algebraic Decomposition with Groebner Bases. Proceedings of the 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC '16), pp. 45--52. IEEE, 2016. Digital Object Identifier: 10.1109/SYNASC.2016.02

Dataset supporting "Using Machine Learning to decide when to Precondition Cylindrical Algebraic Decomposition with Groebner Bases"

<p>Dataset supporting the paper:</p> <p>Z. Huang, M. England, J.H. Davenport and L.C. Paulson<br> Using Machine Learning to decide when to Precondition Cylindrical Algebraic Decomposition with Groebner Bases.<br> Proceedings of the 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC '16), pp. 45--52. IEEE, 2016. Digital Object Identifier: 10.1109/SYNASC.2016.020 </p