The geometry of fractal sets
Kolmogorov complexity and Hausdorff dimension
Information and Computation
The complexity and effectiveness of prediction algorithms
Journal of Complexity
Proceedings of the 30th IEEE symposium on Foundations of computer science
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
On the Length of Programs for Computing Finite Binary Sequences
Journal of the ACM (JACM)
On the Length of Programs for Computing Finite Binary Sequences: statistical considerations
Journal of the ACM (JACM)
A Theory of Program Size Formally Identical to Information Theory
Journal of the ACM (JACM)
Dimension in Complexity Classes
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Process complexity and effective random tests
Journal of Computer and System Sciences
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
COLT '02 Proceedings of the 15th Annual Conference on Computational Learning Theory
Journal of Computer and System Sciences - Special issue on COLT 2002
Effective Dimensions and Relative Frequencies
CiE '08 Proceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Constructive dimension and Hausdorff dimension: the case of exact dimension
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Kolmogorov-Loveland randomness and stochasticity
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
A correspondence principle for exact constructive dimension
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
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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 Δ20-computable real number α in [0; 1] there is a Δ20 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.