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
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15738/
Dudenhefner, Andrej
Constructive Many-One Reduction from the Halting Problem to Semi-Unification
Abstract
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
@InProceedings{dudenhefner:LIPIcs.CSL.2022.18,
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 = {https://drops.dagstuhl.de/opus/volltexte/2022/15738},
URN = {urn:nbn:de:0030-drops-157380},
doi = {10.4230/LIPIcs.CSL.2022.18},
annote = {Keywords: constructive mathematics, undecidability, mechanization, semi-unification}
}
