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重点是对一个固定的可迭代的算术运算序列进行编程。条件性迭代和条件性转移对于计算机的一般理论的根本重要性没有得到承认...- Gandy，第55页
重点是对一个固定的可迭代的算术运算序列进行编程。条件性迭代和条件性转移对于计算机的一般理论的根本重要性没有得到承认...- Gandy，第55页
===决策问题: 希尔伯特1900年提出的第10号问题===
===判定问题: 希尔伯特1900年提出的第10号问题===
With regard to [[Hilbert's problems]] posed by the famous mathematician [[David Hilbert]] in 1900, an aspect of problem #10 had been floating about for almost 30 years before it was framed precisely. Hilbert's original expression for #10 is as follows:
With regard to [[Hilbert's problems]] posed by the famous mathematician [[David Hilbert]] in 1900, an aspect of problem #10 had been floating about for almost 30 years before it was framed precisely. Hilbert's original expression for #10 is as follows:
关于著名数学家大卫•希尔伯特（David Hilbert）在1900年提出的希尔伯特问题中，第10号问题的一个方面浮动了近30年，才被准确地框定下来。希尔伯特对10号问题的原始表述如下：
关于著名数学家大卫•希尔伯特（David Hilbert）在1900年提出的希尔伯特问题中，第10号问题的一个方面浮动了近30年，才被准确地框定下来。希尔伯特对10号问题的原始表述如下：
10. Diophantine方程的可解性的确定。给出一个具有任意数量的未知量和有理积分系数的Diophantine方程。设计一个过程，根据这个过程可以在有限的操作中确定该方程是否可以用有理整数来解。当我们知道一个程序，允许任何给定的逻辑表达式通过有限的多次操作来决定其有效性或可满足性时，决策问题（一阶逻辑的决定问题）就解决了......决策问题必须被认为是数理逻辑的主要问题。--2008年，德尔肖维茨（Dershowitz）和古列维奇（Gurevich）引用了此译文和德文原文
10. Diophantine方程的可解性的确定。给出一个具有任意数量的未知量和有理积分系数的Diophantine方程。设计一个过程，根据这个过程可以在有限的操作中确定该方程是否可以用有理整数来解。当我们知道一个程序，允许任何给定的逻辑表达式通过有限的多次操作来决定其有效性或可满足性时，判定问题（一阶逻辑的决定问题）就解决了......判定问题必须被认为是数理逻辑的主要问题。--2008年，德尔肖维茨（Dershowitz）和古列维奇（Gurevich）引用了此译文和德文原文
By 1922, this notion of "[[Entscheidungsproblem]]" had developed a bit, and [[Heinrich Behmann|H. Behmann]] stated that
By 1922, this notion of "[[Entscheidungsproblem]]" had developed a bit, and [[Heinrich Behmann|H. Behmann]] stated that
到了1922年，“判定问题”的概念有了进一步的发展，Behmann指出：
判定问题的一般形式如下:需要一个相当明确的、普遍适用的处方，它将允许人们在有限的步骤中决定一个给定的纯逻辑论断的真假... ..
—Gandy，第57页，引用Behmann的话。
—Gandy，第57页，引用Behmann的话。
Behmann指出：一般问题相当于决定哪些数学命题是真的问题。如果能够解决判定问题，那么人们就会有一个"解决许多（甚至所有）数学问题的程序"。
By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise. First, was mathematics ''[[Completeness (logic)|complete]]'' ... Second, was mathematics ''[[Consistency proof|consistent]]'' ... And thirdly, was mathematics ''[[Decidability (logic)|decidable]]''?" (Hodges p. 91, Hawking p. 1121). The first two questions were answered in 1930 by [[Kurt Gödel]] at the very same meeting where Hilbert delivered his retirement speech (much to the chagrin of Hilbert); the third—the Entscheidungsproblem—had to wait until the mid-1930s.
By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise. First, was mathematics ''[[Completeness (logic)|complete]]'' ... Second, was mathematics ''[[Consistency proof|consistent]]'' ... And thirdly, was mathematics ''[[Decidability (logic)|decidable]]''?" (Hodges p. 91, Hawking p. 1121). The first two questions were answered in 1930 by [[Kurt Gödel]] at the very same meeting where Hilbert delivered his retirement speech (much to the chagrin of Hilbert); the third—the Entscheidungsproblem—had to wait until the mid-1930s.
1928年的国际数学家大会，希尔伯特把他的问题描述的非常的细致。第一，数学是完整的吗? 第二，数学是一致的吗? 第三，数学是可判定的吗? ”1930年，在希尔伯特发表退休演讲的同一次会议上，库尔特 · 哥德尔(Kurt Gödel)回答了前两个问题; 而第三个问题（决策问题）不得不等到上世纪30年代中期。
1928年的国际数学家大会，希尔伯特把他的问题描述的非常的细致。第一，数学是完整的吗? 第二，数学是一致的吗? 第三，数学是可判定的吗? ”1930年，在希尔伯特发表退休演讲的同一次会议上，库尔特 · 哥德尔(Kurt Gödel)回答了前两个问题; 而第三个问题（判定问题）不得不等到上世纪30年代中期。
The problem was that an answer first required a precise definition of "''definite general applicable prescription''", which Princeton professor [[Alonzo Church]] would come to call "[[effective calculability]]", and in 1928 no such definition existed. But over the next 6–7 years [[Emil Post]] developed his definition of a worker moving from room to room writing and erasing marks per a list of instructions (Post 1936), as did Church and his two students [[Stephen Kleene]] and [[J. B. Rosser]] by use of Church's [[lambda-calculus]] and Gödel's [[recursion theory]] (1934). Church's paper (published 15 April 1936) showed that the Entscheidungsproblem was indeed "undecidable" and beat Turing to the punch by almost a year (Turing's paper submitted 28 May 1936, published January 1937). In the meantime, Emil Post submitted a brief paper in the fall of 1936, so Turing at least had priority over Post. While Church refereed Turing's paper, Turing had time to study Church's paper and add an Appendix where he sketched a proof that Church's lambda-calculus and his machines would compute the same functions.
The problem was that an answer first required a precise definition of "''definite general applicable prescription''", which Princeton professor [[Alonzo Church]] would come to call "[[effective calculability]]", and in 1928 no such definition existed. But over the next 6–7 years [[Emil Post]] developed his definition of a worker moving from room to room writing and erasing marks per a list of instructions (Post 1936), as did Church and his two students [[Stephen Kleene]] and [[J. B. Rosser]] by use of Church's [[lambda-calculus]] and Gödel's [[recursion theory]] (1934). Church's paper (published 15 April 1936) showed that the Entscheidungsproblem was indeed "undecidable" and beat Turing to the punch by almost a year (Turing's paper submitted 28 May 1936, published January 1937). In the meantime, Emil Post submitted a brief paper in the fall of 1936, so Turing at least had priority over Post. While Church refereed Turing's paper, Turing had time to study Church's paper and add an Appendix where he sketched a proof that Church's lambda-calculus and his machines would compute the same functions.
问题在于，要回答这个问题，首先需要对“明确的通用规则”下一个精确定义。普林斯顿大学的教授阿朗佐•丘奇（Alonzo Church）将其称为“有效可计算性”，而在1928年并不存在这样的定义。但在接下来的6-7年里，埃米尔•波斯特（Emil Post）拓展了他的定义，即一个工人按照一张指令表从一个房间移动到另一个房间书写和擦除标记（1936），邱奇和他的两个学生斯蒂芬•克莱恩（Stephen Kleene）和（j.B. Rosser）利用邱奇的λ微积分和哥德尔的递归理论也是如此。丘奇的论文(发表于1936年4月15日)表明，决策问题确实是“无法决策的” ，比图灵早了将近一年(图灵的论文1936年5月28日提交，1937年1月发表)。与此同时，波斯特在1936年秋天提交了一篇短文，所以图灵至少比波斯特更有优先权。当丘奇评审图灵的论文时，图灵有时间研究丘奇的论文，并在附录中添加了一个草图，证明邱奇的λ微积分和他的机器可以计算同样的函数。
问题在于，要回答这个问题，首先需要对“明确的通用规则”下一个精确定义。普林斯顿大学的教授阿朗佐•丘奇（Alonzo Church）将其称为“有效可计算性”，而在1928年并不存在这样的定义。但在接下来的6-7年里，埃米尔•波斯特（Emil Post）拓展了他的定义，即一个工人按照一张指令表从一个房间移动到另一个房间书写和擦除标记（1936），邱奇和他的两个学生斯蒂芬•克莱恩（Stephen Kleene）和（j.B. Rosser）利用邱奇的λ微积分和哥德尔的递归理论也是如此。丘奇的论文(发表于1936年4月15日)表明，判定问题确实是“无法决策的” ，比图灵早了将近一年(图灵的论文1936年5月28日提交，1937年1月发表)。与此同时，波斯特在1936年秋天提交了一篇短文，所以图灵至少比波斯特更有优先权。当丘奇评审图灵的论文时，图灵有时间研究丘奇的论文，并在附录中添加了一个草图，证明邱奇的λ微积分和他的机器可以计算同样的函数。
1935年春天，图灵作为英国剑桥大学国王学院的一名年轻硕士生，接受了这个挑战；他受到逻辑学家纽曼（Newman）的讲座的鼓舞，并从他们那里了解了哥德尔的工作和判定问题。在图灵1955年的讣告中，纽曼写道:
I suppose, but do not know, that Turing, right from the start of his work, had as his goal a proof of the undecidability of the Entscheidungsproblem. He told me that the 'main idea' of the paper came to him when he was lying in Grantchester meadows in the summer of 1935. The 'main idea' might have either been his analysis of computation or his realization that there was a universal machine, and so a [[Cantor's diagonal argument|diagonal argument]] to prove unsolvability.|''ibid.'', p. 76
I suppose, but do not know, that Turing, right from the start of his work, had as his goal a proof of the undecidability of the Entscheidungsproblem. He told me that the 'main idea' of the paper came to him when he was lying in Grantchester meadows in the summer of 1935. The 'main idea' might have either been his analysis of computation or his realization that there was a universal machine, and so a [[Cantor's diagonal argument|diagonal argument]] to prove unsolvability.|''ibid.'', p. 76
我想，但我不知道，图灵从他工作的一开始，就把证明判定问题的不可解性作为他的目标。1935年夏天，他告诉我，当他躺在格兰切斯特草地的时候，他完成了这篇论文的构想。这个 "构想"可能是他对计算的分析，也可能是他意识到有一个普遍的机器，因此有一个对角线论证来证明不可解性。- 同上，第76页
While Gandy believed that Newman's statement above is "misleading", this opinion is not shared by all. Turing had a lifelong interest in machines: "Alan had dreamt of inventing typewriters as a boy; [his mother] Mrs. Turing had a typewriter; and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'" (Hodges p. 96). While at Princeton pursuing his PhD, Turing built a Boolean-logic multiplier (see below). His PhD thesis, titled "[[Systems of Logic Based on Ordinals]]", contains the following definition of "a computable function":
While Gandy believed that Newman's statement above is "misleading", this opinion is not shared by all. Turing had a lifelong interest in machines: "Alan had dreamt of inventing typewriters as a boy; [his mother] Mrs. Turing had a typewriter; and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'" (Hodges p. 96). While at Princeton pursuing his PhD, Turing built a Boolean-logic multiplier (see below). His PhD thesis, titled "[[Systems of Logic Based on Ordinals]]", contains the following definition of "a computable function":
当图灵回到英国后，他负责破解名为“英格玛”的加密机创造的德国密码；他还参与了ACE（自动计算引擎）的设计。“图灵的ACE建议实际上是自成一体的，其根源不在于EDVAC（美国的倡议），而在于他自己的通用机器"（Hodges p.318）。关于被Kleene（1952年）命名为 "图灵论文 "的起源和性质的争论仍在继续。但是，图灵用他的计算机模型所证明的东西出现在他的论文中"On Computable Numbers, with an Application to the Entscheidungsproblem"。
当图灵回到英国后，他负责破解名为“英格玛”的加密机创造的德国密码；他还参与了ACE（自动计算引擎）的设计。“图灵的ACE建议实际上是自成一体的，其根源不在于EDVAC（美国的倡议），而在于他自己的通用机器"（Hodges p.318）。关于被Kleene（1952年）命名为 "图灵论文 "的起源和性质的争论仍在继续。但是，图灵用他的计算机模型所证明的东西出现在他的论文中"On Computable Numbers, with an Application to the Entscheidungsproblem"。
希尔伯特-决策问题不可能有解......因此我建议，不可能有一个一般的过程来确定函数微积分K的一个给定公式U是否可证明，也就是说，不可能有一台机器在提供任何一个公式U的情况下，最终会说出U是否可证明。- 摘自图灵的论文，详见《The Undecidable》，第145页。
希尔伯特-判定问题不可能有解......因此我建议，不可能有一个一般的过程来确定函数微积分K的一个给定公式U是否可证明，也就是说，不可能有一台机器在提供任何一个公式U的情况下，最终会说出U是否可证明。- 摘自图灵的论文，详见《The Undecidable》，第145页。
Turing's example (his second proof): If one is to ask for a general procedure to tell us: "Does this machine ever print 0", the question is "undecidable".
Turing's example (his second proof): If one is to ask for a general procedure to tell us: "Does this machine ever print 0", the question is "undecidable".