Theory of Computing Systems

, Volume 31, Issue 3, pp 215–229 | Cite as

A Tight Upper Bound on Kolmogorov Complexity and Uniformly Optimal Prediction

  • L. Staiger


This paper links the concepts of Kolmogorov complexity (in complexity theory) and Hausdorff dimension (in fractal geometry) for a class of recursive (computable) ω -languages.

It is shown that the complexity of an infinite string contained in a Σ 2 -definable set of strings is upper bounded by the Hausdorff dimension of this set and that this upper bound is tight. Moreover, we show that there are computable gambling strategies guaranteeing a uniform prediction quality arbitrarily close to the optimal one estimated by Hausdorff dimension and Kolmogorov complexity provided the gambler's adversary plays according to a sequence chosen from a Σ 2 -definable set of strings.

We provide also examples which give evidence that our results do not extend further in the arithmetical hierarchy.


Complexity Theory Hausdorff Dimension Fractal Geometry Prediction Quality Kolmogorov Complexity 
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Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • L. Staiger
    • 1
  1. 1.Martin-Luther-Universität Halle-Wittenberg, Institut für Informatik, Kurt-Mothes-Strasse 1, D-06120 Halle, Germany staiger@cantor.informatik.uni-halle.deDE

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