1936 OnComputableNumberswithAnApplic

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Subject Headings: Turing Machine, Entscheidungsproblem.


Cited By


  • (Vardi, 2012b) ⇒ Moshe Y. Vardi. (2012). “What is An Algorithm?.” In: Communications of the ACM Journal, 55(3). doi:10.1145/2093548.2093549
    • QUOTE: But conflating algorithms with Turing machines is a misreading of Turing's 1936 paper "On Computable Numbers, with an Application to the Entscheidungsproblem." Turing's aim was to define computability, not algorithms. His paper argued that every function on natural numbers that can be computed by a human computer (until the mid-1940s a computer was a person who computes) can also be computed by a Turing machine. There is no claim in the paper that Turing machines offer a general model for algorithms. (See S. B. Cooper's article on page 74 for a perspective on Turing's model of computation.)



The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the relations of the computable numbers, functions, and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine.

In [9,10] I give some arguments with the intention of showing that the computable numbers include all numbers which could naturally be regarded as computable. In particular, I show that certain large classes of numbers are computable. They include, for instance, the real parts of all algebraic numbers, the real parts of the zeros of the Bessel functions, the numbers�X, e, etc. The computable numbers do not, however, include all definable numbers, and an example is given of a definable number which is not computable.

Although the class of computable numbers is so great, and in many ways similar to the class of real numbers, it is nevertheless enumerable. In [8] I examine certain arguments which would seem to prove the contrary. By the correct application of one of these arguments, conclusions are reached which are superficially similar to those of Gödel [1] . These results {231} have valuable applications. In particular, it is shown [11] that the Hilbertian Entscheidungsproblem can have no solution.

In a recent paper Alonzo Church[2] has introduced an idea of “effective calculability”, which is equivalent to my “computability”, but is very differently defined. Church also reaches similar conclusions about the Entscheidungsproblem.[3] The proof of equivalence between “computability” and “effective calculability” is outlined in an appendix to the present paper.return to the index

1.Computing machines.

We have said that the computable numbers are those whose decimals are calculable by finite means. This requires rather more explicit definition. No real attempt will be made to justify the definitions given until we reach [9]. For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited.




 AuthorvolumeDate ValuetitletypejournaltitleUrldoinoteyear
1936 OnComputableNumberswithAnApplicAlan Turing (1912-1954)
Moshe Y. Vardi
On Computable Numbers, with An Application to the Entscheidungsproblem1936
AuthorAlan Mathison Turing + and Moshe Y. Vardi +
titleOn Computable Numbers, with An Application to the Entscheidungsproblem +
year1936 +