Abstract.
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Author information
Authors and Affiliations
Additional information
Received February 1995, and in revised form February 1997, and in final form October 1997.
Rights and permissions
About this article
Cite this article
Staiger, L. A Tight Upper Bound on Kolmogorov Complexity and Uniformly Optimal Prediction. Theory Comput. Systems 31, 215–229 (1998). https://doi.org/10.1007/s002240000086
Issue Date:
DOI: https://doi.org/10.1007/s002240000086