Semi-unification is the combination of first-order unification and first-order matching. It does arise naturally in the context of polymorphically recursive functional and logic programming. The undecidability of semi-unification has been proven by Kfoury, Tiuryn, and Urzyczyn in the 1990s. The original argument starts with the undecidability result by Hooper from the 1960s for Turing machine immortality (existence of a diverging configuration). The undecidability of semi-unification is established by Turing reduction from Turing machine immortality. The particular Turing reduction is intricate, uses non-computational principles, and involves various intermediate models of computation. There are several aspects of the existing work which can be improved upon in the setting of constructive mathematics. 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 K\"onig'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. 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 Turing machine immortality is part of the mechanization.
翻译:半统一是一阶统一和一阶匹配的结合。 在多变递归性功能和逻辑编程的背景下, 半统一性自然产生。 Kfoury、 Tiururyn 和 Urzyqzynen 在1990年代证明了半统一性。 最初的论据是从1960年代Hooper在图灵机不朽( 存在差异的配置) 中产生的不可降解性结果开始的。 半统一性由图灵机不朽的缩影减少而建立。 特别的图灵减少是复杂的, 使用非对流原则, 使用非对流原则, 涉及各种中间的计算模型。 在建设性数学的设置中, 现有半统一性有许多方面的完整性没有被确定。 第二, 现有的机械化并不包括从图灵机床停止到半异化的全面减少。 第三, 以K\\' sonma 或 Fangeroducation 等原则为基础调查, 将不精确性变现变现的精确性变现, 的精确性演算不支持了目前一个机械化, 机变现的精确性演算结果。