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为什么“停机问题”归于阿兰·图灵?

已有 951 次阅读 2023-10-18 23:23 |个人分类:解读哥德尔不完全性定理|系统分类:科研笔记

Stackexchange论坛中有人提出这样的问题:为什么停机问题归于阿兰·图灵?


一,译文


为什么停机问题归于阿兰·图灵?


停机问题是可计算性理论中一个非常著名的例子,停机问题是不可判定的,人们常说其不可判定性的证明是由阿兰·图灵给出的,事实上维基百科也多次这么说。


然而,图灵的论文“On Computable Numbers, with an Application to the Entscheidungsproblem”根本没有处理停机问题。 正如 Charles Petzold 在《图灵注释》第 234 页中所写:

- 随后,停机问题与图灵机密切相关,但这个概念对于图灵的原始论文来说是陌生的。


图灵的论文确实展示了不可判定的问题,即确定图灵机是否是circle-free问题(这与停机不同,因为circle-freecircular的机器都可以永远保持打印),判定机器是否一直打印一些特定的符号的问题,即 Entscheidungsproblem


当然,判断图灵机是否 circle-free问题非常接近停机问题,但它仍然不是停机问题。 我知道停机问题这个术语是 Martin Davis 1958 创造的。


我的问题是,如果图灵从未讨论过停机问题,为什么这么多消息来源将其归于他?


二,原文(https://hsm.stackexchange.com/questions/2933/why-is-the-halting-problem-attributed-to-alan-turing


Why is the Halting problem attributed to Alan Turing?


The halting problem is a very famous example from computability theory of a problem that is undecidable. It is often said that the proof of its undecidability was given by Alan Turing, indeed wikipedia repeatedly says so.


However, Turing's paper "On Computable Numbers, with an Application to the Entscheidungsproblem" does not deal with the halting problem at all. As Charles Petzold writes in "The Annotated Turing", page 234:

- The halting problem has subsequently become closely identified with Turing Machines, but the concept is foreign to Turing's original paper.


The problems that Turing's paper does show are undecidable are the problem of deciding whether a Turing machine is circle-free (which is not the same as halting, since both circle-free and circular machines can keep printing forever), that of deciding whether a machine ever prints some specific symbol, and that of the Entscheidungsproblem. 


Of course, the problem of deciding whether a Turing machine is circle-free is very close to the halting problem, but it is nonetheless not the halting problem. I am aware that the term "halting problem" was coined by Martin Davis in 1958.


My question is, if Alan Turing never discussed the halting problem, why do so many sources attribute it to him?




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