Elements of information theory
Elements of information theory
Kolmogorov complexity and Hausdorff dimension
Information and Computation
The complexity and effectiveness of prediction algorithms
Journal of Complexity
Journal of the ACM (JACM)
A game of prediction with expert advice
Journal of Computer and System Sciences - Special issue on the eighth annual workshop on computational learning theory, July 5–8, 1995
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
Gales and the Constructive Dimension of Individual Sequences
ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
Dimension in Complexity Classes
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Structural Complexity I
Note: fractal dimension and logarithmic loss unpredictability
Theoretical Computer Science
Scaled dimension and nonuniform complexity
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
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Given a set X of sequences over a finite alphabet, we investigate the following three quantities. (i) The feasible predictability of X is the highest success ratio that a polynomial-time randomized predictor can achieve on all sequences in X. (ii) The deterministic feasible predictability of X is the highest success ratio that a polynomial-time deterministic predictor can achieve on all sequences in X. (iii) The feasible dimension of X is the polynomial-time effectivization of the classical Hausdorff dimension ("fractal dimension") of X.Predictability is known to be stable in the sense that the feasible predictability of X驴Y is always the minimum of the feasible predictabilities of X and Y. We showt hat deterministic predictability also has this property if X and Y are computably presentable. We show that deterministic predictability coincides with predictability on singleton sets. Our main theorem states that the feasible dimension of X is bounded above by the maximum entropy of the predictability of X and bounded below by the segmented self-information of the predictability of X, and that these bounds are tight.