License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2022.18
URN: urn:nbn:de:0030-drops-157380
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Dudenhefner, Andrej

Constructive Many-One Reduction from the Halting Problem to Semi-Unification

LIPIcs-CSL-2022-18.pdf (0.8 MB)


The undecidability of semi-unification (unification combined with matching) has been proven by Kfoury, Tiuryn, and Urzyczyn in the 1990s. The original argument is by Turing reduction from Turing machine immortality (existence of a diverging configuration).
There are several aspects of the existing work which can be improved upon. First, many-one completeness of semi-unification is not established due to the use of Turing reductions. Second, existing mechanizations do not cover a comprehensive reduction from Turing machine halting to semi-unification. Third, reliance on principles such as König’s lemma or the fan theorem does not support constructivity of the arguments.
Improving upon the above aspects, the present work gives a constructive many-one reduction from the Turing machine halting problem to semi-unification. This establishes many-one completeness of semi-unification. Computability of the reduction function, constructivity of the argument, and correctness of the argument is witnessed by an axiom-free mechanization in the Coq proof assistant. The mechanization is incorporated into the existing Coq library of undecidability proofs. Notably, the mechanization relies on a technique invented by Hooper in the 1960s for Turing machine immortality.
An immediate consequence of the present work is an alternative approach to the constructive many-one equivalence of System F typability and System F type checking, compared to the argument established in the 1990s by Wells.

BibTeX - Entry

  author =	{Dudenhefner, Andrej},
  title =	{{Constructive Many-One Reduction from the Halting Problem to Semi-Unification}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{18:1--18:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-157380},
  doi =		{10.4230/LIPIcs.CSL.2022.18},
  annote =	{Keywords: constructive mathematics, undecidability, mechanization, semi-unification}

Keywords: constructive mathematics, undecidability, mechanization, semi-unification
Collection: 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)
Issue Date: 2022
Date of publication: 27.01.2022
Supplementary Material: Software (Source Code):

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