The dimensions of individual strings and sequences

Abstract

A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that are random (in the sense of Martin-Löf) have dimension 1, while sequences that are decidable, Σ 0 1 , or Π 0 1 have dimension 0. It is shown that for every ∆ 0 2 -computable real number α in [0,1] there is a ∆ 0 2 sequence S such that dim(S) = α. A discrete version of constructive dimension is also developed using termgales, which are supergale-like functions that bet on the terminations of (finite, binary) strings as well as on their successive bits. This discrete dimension is used to assign each individual string w a dimension, which is a nonnegative real number dim(w). The dimension of a sequence is shown to be the limit infimum of the dimensions of its prefixes.The Kolmogorov complexity of a string is proven to be the product of its length and its dimension. This gives a new characterization of algorithmic information and a new proof of Mayordomo's recent theorem stating that the dimension of a sequence is the limit infimum of the average Kolmogorov complexity of its first n bits.Every sequence that is random relative to any computable sequence of coin-toss biases that converge to a real number β in (0, 1) is shown to have dimension H(β), the binary entropy of β.

Two Open Problems on Effective Dimension

“…A more detailed treatment, motivation, references and historical introduction can be found in [10], [15], and [14].…”

Scaled Dimension and Nonuniform Complexity

“…Martingales were later used extensively by Schnorr [18,19,20,21] in his investigations of random sequences and by Lutz [13,16] in the development of resource-bounded measure. Gales were introduced by Lutz [14,15] in the development of resource-bounded and constructive dimension. Scaled gales are introduced here in order to formulate scaled dimension.…”

A Divergence Formula for Randomness and Dimension

“…These two constructive dimensions, which were introduced in [13,1] where the logarithm is base-2 [16,1]. In these equations, K(w) is the Kolmogorov complexity of the prefix w of S, i.e., the length in bits of the shortest program that prints the string w. (See section * This research was supported in part by National Science Foundation Grants 9988483, 0344187, 0652569, and 0728806 and by the Spanish Ministry of Education and Science (MEC) and the European Regional Development Fund (ERDF) under project TIN2005-08832-C03-02.…”