Prediction and Dimension
The feasible predictability of X is the highest success ratio that a polynomial-time randomized predictor can achieve on all sequences in X.
The deterministic feasible predictability of X is the highest success ratio that a polynomial-time deterministic predictor can achieve on all sequences in X.
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∪Y is always the minimum of the feasible predictabilities of X and Y. We show that 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.
KeywordsFractal Dimension Polynomial Time Binary Sequence Kolmogorov Complexity Bell System Technical Journal
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