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An Informal Arithmetical Approach to Computability andComputation, II
Canadian mathematical bulletin, Tome 7 (1964) no. 2, pp. 183-200

Voir la notice de l'article provenant de la source Cambridge University Press

In the first part of this paper [l] there was introduced a hypotheticalcomputing device, the Q-machine. It was derived by abstracting from theprocess of calculating carried out by a man on his fingers, assuming anadequate supply of hands and the ability to grow fingers at will. TheQ-machine was shown to be equal in computing power to a universal Turingmachine. That is, the Q-machine could compute any number regarded ascomputable by any theory of computability developed so far. It may berecalled here that Turing machines were obtained by Turing [2] byabstracting from the process of calculating carried out by a man on someconcrete 'symbol space' (tape, piece of paper, blackboard) by means of fixedbut arbitrary symbols. Hence the contrast between the Q-machine and theTuring machines is that between arithmetical manipulation of counters andlogical manipulation of symbols. In particular, one might say, loosely, thatin a Turing machine, as in arithmetic, numbers are represented by signswhereas in the Q-machine, as on a counting frame, numbers representthemselves. 
Melzak, Z.A. An Informal Arithmetical Approach to Computability andComputation, II. Canadian mathematical bulletin, Tome 7 (1964) no. 2, pp. 183-200. doi: 10.4153/CMB-1964-017-6
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[1] 1. Melzak, Z.A., An Informal Arithmetical Introduction to Computability and Computation, Can. Math. Bull., vol.4, no. 3, Sept. 1961. Google Scholar

[2] 2. Turing, A.M., On Computable Numbers with an Application to the Entscheidungsproblem, J. Lond. Math. Soc., Spring 1936. Google Scholar

[3] 3. Shapiro, H.S., Rational Recurrence Formulae, Comm. Pure Appl. Math., vol.12, no. 3, Aug. 1959. Google Scholar

[4] 4. Cassels, J.W.S., An Introduction to Diophantine Approximation, Cambridge Univ. Press, 1957. Google Scholar

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