Prediction and dimension

  • Authors:

  • Lance Fortnow;Jack H. Lutz

  • Affiliations:

  • Department of Computer Science, University of Chicago, 1100 E 58th Street, Chicago, IL 60637, USA;Department of Computer Science, Iowa State University, Ames, IA 50011, USA

  • Venue:
  • Journal of Computer and System Sciences - Special issue on COLT 2002
  • Year:
  • 2005

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Abstract

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@?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.