# Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition

arXiv: https://arxiv.org/abs/2304.12154 · HTML: https://mlsanity.com/paper/2304.12154/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 60 references that are in the map.
- First posted on arXiv 24 Apr 2023.

## Builds on (most cited first, 15 of 15)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Techniques for Interpretable Machine Learning](https://mlsanity.com/paper/1808.00033/) (2018)
- [Machine-Learning Mathematical Structures](https://mlsanity.com/paper/2101.06317/) (2021)
- [Stop ordering machine learning algorithms by their explainability! A user-centered investigation of performance and explainability](https://mlsanity.com/paper/2206.10610/) (2022)
- [Learning selection strategies in Buchberger's algorithm](https://mlsanity.com/paper/2005.01917/) (2020)
- [Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition](https://mlsanity.com/paper/1404.6369/) (2014)
- [Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition](https://mlsanity.com/paper/1904.11061/) (2019)
- [Using Machine Learning to Improve Cylindrical Algebraic Decomposition](https://mlsanity.com/paper/1804.10520/) (2018)
- [Algorithmically generating new algebraic features of polynomial systems for machine learning](https://mlsanity.com/paper/1906.01455/) (2019)
- [Improved cross-validation for classifiers that make algorithmic choices to minimise runtime without compromising output correctness](https://mlsanity.com/paper/1911.12672/) (2019)
- [Machine learning the real discriminant locus](https://mlsanity.com/paper/2006.14078/) (2020)
- [Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases](https://mlsanity.com/paper/1608.04219/) (2016)
- [A machine learning based software pipeline to pick the variable ordering for algorithms with polynomial inputs](https://mlsanity.com/paper/2005.11251/) (2020)
- [TheoryGuru: A Mathematica Package to apply Quantifier Elimination](https://mlsanity.com/paper/1806.10925/) (2018)

## Built on by (most cited first, 7 of 7)

- [Symbolic Integration Algorithm Selection with Machine Learning: LSTMs vs Tree LSTMs](https://mlsanity.com/paper/2404.14973/) (2024)
- [Lessons on Datasets and Paradigms in Machine Learning for Symbolic Computation: A Case Study on CAD](https://mlsanity.com/paper/2401.13343/) (2024)
- [Transformers to Predict the Applicability of Symbolic Integration Routines](https://mlsanity.com/paper/2410.23948/) (2024)
- [Constrained Neural Networks for Interpretable Heuristic Creation to Optimise Computer Algebra Systems](https://mlsanity.com/paper/2404.17508/) (2024)
- [Algebraic Representations for Faster Predictions in Convolutional Neural Networks](https://mlsanity.com/paper/2408.07815/) (2024)
- [Tree-Based Deep Learning for Ranking Symbolic Integration Algorithms](https://mlsanity.com/paper/2508.06383/) (2025)
- [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)

## Abstract (arXiv)

In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not directly call upon AI tools. We present a case study on the use of ML to select the variable ordering for cylindrical algebraic decomposition. It has already been demonstrated that ML can make the choice well, but here we show how the SHAP tool for explainability can be used to inform new heuristics of a size and complexity similar to those human-designed heuristics currently commonly used in symbolic computation.