Gales and the Constructive Dimension of Individual Sequences

A constructive version of Hausdorff dimension is developed and used to assign to every individual infinite binary sequence A a constructive dimension, which is a real number cdim(A) in the interval [0, 1]. Sequences that are random (in the sense of Martin-Löf) have constructive dimension 1, while sequences that are decidable, r.e., or co-r.e. have constructive dimension 0. It is shown that for every Δ02-computable real number α in [0, 1] there is a Δ02 sequence A such that cdim(A) = α. Every sequence’s constructive dimension is shown to be bounded above and below by the limit supremum and limit infimum, respectively, of the average Kolmogorov complexity of the sequence’s first n bits. Every sequence that is random relative to a computable sequence of rational biases that converge to a real number β in (0,1) is shown to have constructive dimension H(β), the binary entropy of β. Constructive dimension is based on constructive gales, which are a natural generalization of the constructive martingales used in the theory of random sequences.