M. D. Davis. A note on universal Turing machines. Automata studies, edited by C. E. Shannon and J. McCarthy, Annals of Mathematics studies no. 34, lithoprinted, Princeton University Press, Princeton1956, pp. 167–175. - Martin Davis. The definition of universal Turing machine. Proceedings of the American Mathematical Society, vol. 8 (1957), pp. 1125–1126.

1970 ◽  
Vol 35 (4) ◽  
pp. 590-590
Author(s):  
R. J. Nelson
Keyword(s):  
Turing Machine ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Turing Machines ◽  
Princeton University ◽  
Universal Turing Machine ◽  
Definition Of ◽  
Martin Davis

Algorithms and Turing Machines

1989 ◽  
Author(s):  
Roger Penrose ◽  
Martin Gardner
Keyword(s):  
Elementary School ◽  
Turing Machine ◽  
Mathematical Knowledge ◽  
Ninth Century ◽  
The Other ◽  
Turing Machines ◽  
Universal Turing Machine ◽  
Other Hand ◽  
Mathematical Expressions ◽  
Mathematical Textbook

What Precisely is an algorithm, or a Turing machine, or a universal Turing machine? Why should these concepts be so central to the modern view of what could constitute a ‘thinking device’? Are there any absolute limitations to what an algorithm could in principle achieve? In order to address these questions adequately, we shall need to examine the idea of an algorithm and of Turing machines in some detail. In the various discussions which follow, I shall sometimes need to refer to mathematical expressions. I appreciate that some readers may be put off by such things, or perhaps find them intimidating. If you are such a reader, I ask your indulgence, and recommend that you follow the advice I have given in my ‘Note to the reader’ on p. viii! The arguments given here do not require mathematical knowledge beyond that of elementary school, but to follow them in detail, some serious thought would be required. In fact, most of the descriptions are quite explicit, and a good understanding can be obtained by following the details. But much can also be gained even if one simply skims over the arguments in order to obtain merely their flavour. If, on the other hand, you are an expert, I again ask your indulgence. I suspect that it may still be worth your while to look through what I have to say, and there may indeed be a thing or two to catch your interest. The word ‘algorithm’ comes from the name of the ninth century Persian mathematician Abu Ja’far Mohammed ibn Mûsâ alKhowârizm who wrote an influential mathematical textbook, in about 825 AD, entitled ‘Kitab al-jabr wa’l-muqabala’. The way that the name ‘algorithm’ has now come to be spelt, rather than the earlier and more accurate ‘algorism’, seems to have been due to an association with the word ‘arithmetic’. (It is noteworthy, also, that the word ‘algebra’ comes from the Arabic ‘al-jabr’ appearing in the title of his book.) Instances of algorithms were, however, known very much earlier than al-Khowârizm’s book.

On Differentiation with respect to a Function

1925 ◽  
Vol 22 (6) ◽  
pp. 924-934 ◽  
Author(s):  
E. C. Francis
Keyword(s):  
Detailed Analysis ◽  
Continued Fractions ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Stieltjes Integral ◽  
Base Function ◽  
General Base ◽  
The Future ◽  
Definition Of ◽  
The Way

Thirty years ago, in a paper on continued fractions, Stieltjes published a definition of the integral which bears his name. His replacement of the variable of integration x by a more general “base function” φ(x)—a change which throws so much light upon other theories of integration—received at first little attention, but has later sprung into greater prominence; so much so that Professor Hildebrandt, in summarizing these various theories in a paper to the American Mathematical Society, makes the statement that “it [the Stieltjes Integral] seems destined to play the central rôle in the integrational and summational processes of the future.” Yet even now the integral and the allied theory of differentiation with respect to a function have been subjected to little detailed analysis, and the possibilities of extension have been only touched upon. It is the object of this present paper to establish certain results which are of some value in themselves and which prepare the way for an attack upon the integral.

scholarly journals An Informal Arithmetical Approach to Computability and Computation, II

1964 ◽  
Vol 7 (2) ◽  
pp. 183-200 ◽  
Author(s):  
Z.A. Melzak
Keyword(s):  
Turing Machine ◽  
Turing Machines ◽  
Symbol Space ◽  
Computing Device ◽  
Computing Power ◽  
Adequate Supply ◽  
Universal Turing Machine ◽  
At Will

In the first part of this paper [l] there was introduced a hypothetical computing device, the Q-machine. It was derived by abstracting from the process of calculating carried out by a man on his fingers, assuming an adequate supply of hands and the ability to grow fingers at will. The Q-machine was shown to be equal in computing power to a universal Turing machine. That is, the Q-machine could compute any number regarded as computable by any theory of computability developed so far. It may be recalled here that Turing machines were obtained by Turing [2] by abstracting from the process of calculating carried out by a man on some concrete 'symbol space' (tape, piece of paper, blackboard) by means of fixed but arbitrary symbols. Hence the contrast between the Q-machine and the Turing machines is that between arithmetical manipulation of counters and logical manipulation of symbols. In particular, one might say, loosely, that in a Turing machine, as in arithmetic, numbers are represented by signs whereas in the Q-machine, as on a counting frame, numbers represent themselves.

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