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
Switching and Finite Automata Theory: Computer Science Series
Switching and Finite Automata Theory: Computer Science Series
Gales and the Constructive Dimension of Individual Sequences
ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
Dimension in Complexity Classes
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
SEQUENCES '97 Proceedings of the Compression and Complexity of Sequences 1997
COLT '02 Proceedings of the 15th Annual Conference on Computational Learning Theory
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Classical Hausdorff dimension was recently effectivized using gales (betting strategies that generalize martingales), thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary (infinite) sequences. In this paper we use gales computed by multi-account finite-state gamblers to develop the finite-state dimensions of sets of binary sequences and individual binary sequences. Every rational sequence (binary expansion of a rational number) has finite-state dimension 0, but every rational number in [0, 1] is the finite-state dimension of a sequence in the low-level complexity class AC0. Our main theorem shows that the finite-state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information-lossless finite-state compressors.