削弱丘奇-图灵命题的理由
A case for weakening the Church-Turing Thesis
摘要：
我们从1931年哥德尔的开创性论文中的哥德尔定理VII得出结论，每个递归函数f（x_{1}，x_{2}）在一阶皮亚诺算术PA中可由公式[F（x_{1}，x_{2}，x_{3}）]，其在算法上可验证，但在算法上不可计算，如果我们假设一阶谓词演算的一个普遍量化公式的否定总是指示在PA的标准解释下存在一个反例。我们的结论是，如果我们定义一个数论公式是有效可计算的，当且仅当它是算法上可验证的，那么丘奇-图灵命题的标准假设就不成立;需要用一个较弱的命题作为等价假设来代替。
英文摘要：
We conclude from Goedel's Theorem VII of his seminal 1931 paper that every recursive function f(x_{1}, x_{2}) is representable in the first-order Peano Arithmetic PA by a formula [F(x_{1}, x_{2}, x_{3})] which is algorithmically verifiable, but not algorithmically computable, if we assume that the negation of a universally quantified formula of the first-order predicate calculus is always indicative of the existence of a counter-example under the standard interpretation of PA. We conclude that the standard postulation of the Church-Turing Thesis does not hold if we define a number-theoretic formula as effectively computable if, and only if, it is algorithmically verifiable; and needs to be replaced by a weaker postulation of the Thesis as an equivalence.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
