Abstract
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 Δ0 2-computable real number α in [0, 1] there is a Δ0 2 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.
This work was supported in part by National Science Foundation Grant 9610461.
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Lutz, J.H. (2000). Gales and the Constructive Dimension of Individual Sequences. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_76
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