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arXiv · 2208.13428

Constructive Many-one Reduction from the Halting Problem to Semi-unification (Extended Version)

Andrej Dudenhefner
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Abstract

Semi-unification is the combination of first-order unification and first-order matching. The undecidability of semi-unification has been proven by Kfoury, Tiuryn, and Urzyczyn in the 1990s by Turing reduction from Turing machine immortality (existence of a diverging configuration). The particular Turing reduction is intricate, uses non-computational principles, and involves various intermediate models of computation. The present work gives a constructive many-one reduction from the Turing machine halting problem to semi-unification. This establishes RE-completeness of semi-unification under many-one reductions. 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. Arguably, this serves as comprehensive, precise, and surveyable evidence for the result at hand. The mechanization is incorporated into the existing, well-maintained Coq library of undecidability proofs. Notably, a variant of Hooper's argument for the undecidability of Turing machine immortality is part of the mechanization.

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BibTeXRIS

Andrej Dudenhefner. 2022-08-29. Constructive Many-one Reduction from the Halting Problem to Semi-unification (Extended Version). https://doi.org/10.46298/lmcs-19(4%3A22)2023

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