The Kolmogorov complexity of real numbers

  • Ludwig Staiger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1684)


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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Ludwig Staiger
    • 1
  1. 1.Institut für InformatikMartin-Luther-Universität Halle-WittenbergHalleGermany

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