## Abstract

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.

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.

Original language | English |
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Title of host publication | Mathematical Aspects of Computer and Information Sciences |

Subtitle of host publication | Proc. MACIS 2019 |

Editors | Daniel Slamanig , Elias Tsigaridas , Zafeirakis Zafeirakopoulos |

Publisher | Springer International Publishing |

Pages | 341-356 |

Number of pages | 16 |

Volume | 11989 |

ISBN (Electronic) | 978-3-030-43120-4 |

ISBN (Print) | 978-3-030-43119-8 |

DOIs | |

Publication status | Published - 18 Mar 2020 |

Event | Mathematical Aspects of Computer and Information Sciences 2019 - Istanbul, Turkey Duration: 13 Nov 2019 → 15 Nov 2019 Conference number: 8 http://macis2019.gtu.edu.tr/ |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11989 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | Mathematical Aspects of Computer and Information Sciences 2019 |
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Abbreviated title | MACIS 2019 |

Country/Territory | Turkey |

City | Istanbul |

Period | 13/11/19 → 15/11/19 |

Internet address |

### Bibliographical note

This work is funded by EPSRC Project EP/R019622/1: Embedding Machine Learning within Quantifier Elimination Procedures.

Publisher Copyright:

© 2020, Springer Nature Switzerland AG.

## Keywords

- Machine Learning
- cross-validation
- Computer Algebra
- Symbolic Computation
- Cylindrical Algebraic Decomposition
- Cylindrical algebraic decomposition
- Cross-validation
- Symbolic computation
- Computer algebra

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)