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The Kolmogorov complexity of real numbers

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Fundamentals of Computation Theory (FCT 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1684))

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Abstract

We consider for a real number α the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way which depends only on the relative information of one base with respect to the other.

More precisely, we show that the complexity of the length l · logr b prefix of the base r expansion of α is the same (up to an additive constant) as the logr b-fold complexity of the length l prefix of the base b expansion of α.

Then we use this fact to derive complexity theoretic proofs for the base independence of the randomness of real numbers and for some properties of Liouville numbers.

Article

The paper was prepared during my stay at the Centre for Discrete Mathematics and Theoretical Computer Science, University of Auckland, New Zealand, August 1998

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Author information

Authors and Affiliations

  1. Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, Kurt-Mothes-Str. 1, D-06099, Halle, Germany

    Ludwig Staiger

Authors
  1. Ludwig Staiger
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Editor information

Editors and Affiliations

  1. Faculty of Computer Science, “A.I.Cuza” University, 6600, Iaşi, Romania

    Gabriel Ciobanu

  2. Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 70700, Bucharest, Romania

    Gheorghe Păun

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© 1999 Springer-Verlag Berlin Heidelberg

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Staiger, L. (1999). The Kolmogorov complexity of real numbers. In: Ciobanu, G., Păun, G. (eds) Fundamentals of Computation Theory. FCT 1999. Lecture Notes in Computer Science, vol 1684. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48321-7_45

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  • DOI: https://doi.org/10.1007/3-540-48321-7_45

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66412-3

  • Online ISBN: 978-3-540-48321-2

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