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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 ..."
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Cited by 93 (10 self)
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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 suffice for constructive dimension
 Information Processing Letters
, 2003
"... Supergales, generalizations of supermartingales, have been used by Lutz (2002) to define the constructive dimensions of individual binary sequences. Here it is shown that gales, the corresponding generalizations of martingales, can be equivalently used to define constructive dimension. 1 ..."
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Cited by 19 (4 self)
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Supergales, generalizations of supermartingales, have been used by Lutz (2002) to define the constructive dimensions of individual binary sequences. Here it is shown that gales, the corresponding generalizations of martingales, can be equivalently used to define constructive dimension. 1
Effective Fractal Dimension Bibliography
"... z. The dimensions of individual strings and sequences. Information and Computation, 187(1):4979, 2003. [23] J. H. Lutz. E#ective fractal dimensions. In Computability and Complexity in Analysis, volume 302 of Informatik Berichte, pages 8197. FernUniversitat in Hagen, August 2003. To appear in Ma ..."
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Cited by 6 (0 self)
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z. The dimensions of individual strings and sequences. Information and Computation, 187(1):4979, 2003. [23] J. H. Lutz. E#ective fractal dimensions. In Computability and Complexity in Analysis, volume 302 of Informatik Berichte, pages 8197. FernUniversitat in Hagen, August 2003. To appear in Mathematical Logic Quarterly. [24] E. Mayordomo. E#ective Hausdor# dimension. In Proceedings of Foundations of the Formal Sciences III. Kluwer Academic Press. To appear. [25] E. Mayordomo. A Kolmogorov complexity characterization of constructive Hausdor# dimension. Information Processing Letters, 84(1):13, 2002. [26] P. Moser. BPP has e#ective dimension at most 1/2 unless BPP = EXP. Technical Report TR03029, Electronic Colloquium on Computational Complexity, 2003. [27] S. Reid. The classes of algorithmically random reals. Master's thesis, Victoria University of Wellington, 2003. [28] L. Staiger. Constructive dimension equals Kolmogorov complexity. Technical Report CDMTCS210, Universi