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Conference on Computability in Europe

CiE 2012: How the World Computes pp 35–45Cite as

Turing’s Normal Numbers: Towards Randomness

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7318)

Abstract

In a manuscript entitled “A note on normal numbers” and written presumably in 1938 Alan Turing gave an algorithm that produces real numbers normal to every integer base. This proves, for the first time, the existence of computable normal numbers and it is the best solution to date to Borel’s problem on giving examples of normal numbers. Furthermore, Turing’s work is pioneering in the theory of randomness that emerged 30 years after. These achievements of Turing are largely unknown because his manuscript remained unpublished until its inclusion in his Collected Works in 1992. The present note highlights Turing’s ideas for the construction of normal numbers. Turing’s theorems are included with a reconstruction of the original proofs.

Keywords

  • Initial Segment
  • Unit Interval
  • Block Length
  • Computable Function
  • Output Sequence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Author information

Authors and Affiliations

  1. Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, (1428), Buenos Aires, Argentina

    Verónica Becher

Authors
  1. Verónica Becher
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Editor information

Editors and Affiliations

  1. School of Mathematics, University of Leeds, LS2 9JT, Leeds, UK

    S. Barry Cooper

  2. Computer Laboratory, University of Cambridge, Willialm Gates Building, J.J. Thomson Avenue, CB3 0FD, Cambridge, UK

    Anuj Dawar

  3. Institute for Logic, Language and Computation, University of Amsterdam, 1090 GE, Amsterdam, The Netherlands

    Benedikt Löwe

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Becher, V. (2012). Turing’s Normal Numbers: Towards Randomness. In: Cooper, S.B., Dawar, A., Löwe, B. (eds) How the World Computes. CiE 2012. Lecture Notes in Computer Science, vol 7318. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30870-3_5

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  • DOI: https://doi.org/10.1007/978-3-642-30870-3_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30869-7

  • Online ISBN: 978-3-642-30870-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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