A Tight Upper Bound on Kolmogorov
نویسنده
چکیده
The present paper links the concepts of Kolmogorov complexity (in Complexity theory) and Hausdorr dimension (in Fractal geometry) for a class of recursive (computable) !-languages. It is shown that the complexity of an innnite string contained in a 2-deenable set of strings is upper bounded by the Hausdorr 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 Hausdorr dimension and Kolmogorov complexity provided the gambler's adversary plays according to a sequence chosen from a 2-deenable set of strings. We provide also examples which give evidence that our results do not extend further in the Arithmetical hierarchy.
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