Polynomial Termination Over ℕ Is Undecidable

Authors Fabian Mitterwallner , Aart Middeldorp

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Fabian Mitterwallner
  • Department of Computer Science, Universität Innsbruck, Austria
Aart Middeldorp
  • Department of Computer Science, Universiät Innsbruck, Austria
  • Future Value Creation Research Center, Nagoya University, Japan

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Fabian Mitterwallner and Aart Middeldorp. Polynomial Termination Over ℕ Is Undecidable. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


In this paper we prove via a reduction from Hilbert’s 10th problem that the problem whether the termination of a given rewrite system can be shown by a polynomial interpretation in the natural numbers is undecidable, even for rewrite systems that are incrementally polynomially terminating. We also prove that incremental polynomial termination is an undecidable property of terminating rewrite systems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Theory of computation → Rewrite systems
  • Theory of computation → Computability
  • Term Rewriting
  • Polynomial Termination
  • Undecidability


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