# Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition

arXiv: https://arxiv.org/abs/1404.6369 · HTML: https://mlsanity.com/paper/1404.6369/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2014.

## Built on by (most cited first, 10 of 10)

- [Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition](https://mlsanity.com/paper/2304.12154/) (2023)
- [Generating Elementary Integrable Expressions](https://mlsanity.com/paper/2306.15572/) (2023)
- [Symbolic Integration Algorithm Selection with Machine Learning: LSTMs vs Tree LSTMs](https://mlsanity.com/paper/2404.14973/) (2024)
- [Data Augmentation for Mathematical Objects](https://mlsanity.com/paper/2307.06984/) (2023)
- [Lessons on Datasets and Paradigms in Machine Learning for Symbolic Computation: A Case Study on CAD](https://mlsanity.com/paper/2401.13343/) (2024)
- [Learning a performance metric of Buchberger's algorithm](https://mlsanity.com/paper/2106.03676/) (2021)
- [Constrained Neural Networks for Interpretable Heuristic Creation to Optimise Computer Algebra Systems](https://mlsanity.com/paper/2404.17508/) (2024)
- [SC-Square: Future Progress with Machine Learning?](https://mlsanity.com/paper/2209.04361/) (2022)
- [Breaking the Data Barrier in Learning Symbolic Computation: A Case Study on Variable Ordering Suggestion for Cylindrical Algebraic Decomposition](https://mlsanity.com/paper/2601.13731/) (2026)
- [Learning Fast Monomial Orders for Gr"obner Basis Computations](https://mlsanity.com/paper/2602.02972/) (2026)

## Abstract (arXiv)

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.