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Kleene Algebra with Commutativity Conditions Is Undecidable

Authors Arthur Azevedo de Amorim , Cheng Zhang , Marco Gaboardi

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LIPIcs.CSL.2025.36.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Automated reasoning
  • Theory of computation → Regular languages
Keywords
  • Kleene Algebra
  • Hypotheses
  • Complexity

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Abstract

We prove that the equational theory of Kleene algebra with commutativity conditions on primitives (or atomic terms) is undecidable, thereby settling a longstanding open question in the theory of Kleene algebra. While this question has also been recently solved independently by Kuznetsov, our results hold even for weaker theories that do not support the induction axioms of Kleene algebra.

Cite As Get BibTex

Arthur Azevedo de Amorim, Cheng Zhang, and Marco Gaboardi. Kleene Algebra with Commutativity Conditions Is Undecidable. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 36:1-36:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.36

Author Details

Arthur Azevedo de Amorim
  • Rochester Institute of Technology, NY, USA
Cheng Zhang
  • University College London, UK
Marco Gaboardi
  • Boston University, MA, USA

Funding

  • Azevedo de Amorim, Arthur: National Science Foundation Grant No. 2314323.
  • Zhang, Cheng: National Science Foundation Award No. 1845803 and No. 2040249.
  • Gaboardi, Marco: National Science Foundation Award No. 1845803 and No. 2040249.

Acknowledgements

We want to thank Todd Schmid and Alexandra Silva for the valuable discussion during this work; and also acknowledge the useful feedback provided by the anonymous reviewers of CSL2025. Finally, we want to thank Stepan Kuznetsov for his detailed and valuable feedback on this work.

References

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