Undecidability of Semi-Unification on a Napkin

Author Andrej Dudenhefner

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Andrej Dudenhefner
  • Saarland University, Saarbrücken, Germany

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Andrej Dudenhefner. Undecidability of Semi-Unification on a Napkin. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Semi-unification (unification combined with matching) has been proven undecidable by Kfoury, Tiuryn, and Urzyczyn in the 1990s. The original argument reduces Turing machine immortality via Turing machine boundedness to semi-unification. The latter part is technically most challenging, involving several intermediate models of computation. This work presents a novel, simpler reduction from Turing machine boundedness to semi-unification. In contrast to the original argument, we directly translate boundedness to solutions of semi-unification and vice versa. In addition, the reduction is mechanized in the Coq proof assistant, relying on a mechanization-friendly stack machine model that corresponds to space-bounded Turing machines. Taking advantage of the simpler proof, the mechanization is comparatively short and fully constructive.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
  • undecidability
  • semi-unification
  • mechanization


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