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

Lynn Pickering, Tereso del Río, Matthew England, Kelly Cohen

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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.
Original languageEnglish
Article number102276
Number of pages24
JournalJournal of Symbolic Computation
Early online date15 Nov 2023
Publication statusE-pub ahead of print - 15 Nov 2023

Bibliographical note

© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Matthew England acknowledges the support of UKRI EPSRC Grant EP/T015748/1, “Pushing Back the Doubly-Exponential Wall of Cylindrical Algebraic Decomposition” (DEWCAD). Tereso del Río and Lynn Pickering acknowledge the Coventry University Research Excellence grant that allowed them to work together in person on this paper. Lynn Pickering acknowledges the support of the Rindsberg Fellowship from the University of Cincinnati, the Ohio Space Grant Consortium Research Fellowship, and a University of Cincinnati International Study Abroad Scholarship that allowed her to spend a semester at Coventry University.


  • Explainable AI
  • Computer Algebra
  • Heuristic Development
  • Cylindrical Algebraic
  • Decomposition
  • Variable Ordering


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