Hostname: page-component-cb9f654ff-r5d9c Total loading time: 0 Render date: 2025-08-11T10:20:43.591Z Has data issue: false hasContentIssue false
  • English
  • Français

An Informal Arithmetical Approach to Computability andComputation, II

Published online by Cambridge University Press:  20 November 2018

Z.A. Melzak
Z.A. Melzak*
Affiliation:
The University of British Columbia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 2 , June 1964 , pp. 183 - 200
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Melzak, Z.A., An Informal Arithmetical Introduction to Computability and Computation, Can. Math. Bull., vol.4, no. 3, Sept. 1961.Google Scholar
2. Turing, A.M., On Computable Numbers with an Application to the Entscheidungsproblem, J. Lond. Math. Soc., Spring 1936.Google Scholar
3. Shapiro, H.S., Rational Recurrence Formulae, Comm. Pure Appl. Math., vol.12, no. 3, Aug. 1959.Google Scholar
4. Cassels, J.W.S., An Introduction to Diophantine Approximation, Cambridge Univ. Press, 1957. Google Scholar