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The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... 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 ..."
Abstract

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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
Gales and the Dimensions of Individual Strings and Sequences
"... A constructive version of Hausdorff dimension is developed using constructive gales, which are betting strategies that generalize the constructive martingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence ..."
Abstract
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A constructive version of Hausdorff dimension is developed using constructive gales, which are betting strategies that generalize the constructive martingales 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 MartinLöf) have dimension 1, while sequences that are decidable, 0 1 , or 0 1 have dimension 0. It is shown that for every 0 2 computable real number in [0,1] there is a 0 2 sequence S such that dim(S) = : A discrete version of constructive dimension is also developed using termgales, which are galelike 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 in mum 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 in mum of the average Kolmogorov complexity of its first n bits. Every sequence that is random relative to any computable sequence of cointoss biases that converge to a real number in (0; 1) is shown to have dimension H( ), the binary entropy of .