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Constructive Dimension equals Kolmogorov Complexity
 INFORMATION PROCESSING LETTERS
, 2003
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Gales and supergales are equivalent for defining constructive Hausdorff dimension
, 2002
"... We show that for a wide range of probability measures, constructive gales are interchangable with constructive supergales for defining constructive Hausdorff dimension, thus generalizing a previous independent result of Hitchcock [2] and partially answering an open question of Lutz [5]. 1 ..."
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Cited by 14 (0 self)
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We show that for a wide range of probability measures, constructive gales are interchangable with constructive supergales for defining constructive Hausdorff dimension, thus generalizing a previous independent result of Hitchcock [2] and partially answering an open question of Lutz [5]. 1
Every sequence is decompressible from a random one
 In Logical Approaches to Computational Barriers, Proceedings of the Second Conference on Computability in Europe, Springer Lecture Notes in Computer Science, volume 3988 of Computability in Europe
, 2006
"... ddoty at iastate dot edu Kučera and Gács independently showed that every infinite sequence is Turing reducible to a MartinLöf random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a MartinLöf random sequence R such that the asymptotic number of b ..."
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Cited by 10 (6 self)
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ddoty at iastate dot edu Kučera and Gács independently showed that every infinite sequence is Turing reducible to a MartinLöf random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a MartinLöf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. It is shown that this is the optimal ratio of query bits to computed bits achievable with Turing reductions. As an application of this result, a new characterization of constructive dimension is given in terms of Turing reduction compression ratios.
Scaled dimension and the Kolmogorov complexity of Turinghard sets
 In Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science
, 2004
"... We study constructive and resourcebounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension ..."
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Cited by 6 (2 self)
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We study constructive and resourcebounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension of an infinite sequence is characterized by the scaled dimensions of its prefixes. We obtain an exact Kolmogorov complexity characterization of scaled dimension. Juedes and Lutz (1996) established a small span theorem for P/polyTuring reductions which asserts that for any problem A in ESPACE, either the class of problems reducible to A (the lower span) or the class of problems to which A is reducible (the upper span) has measure 0 in ESPACE. We apply our Kolmogorov complexity characterization to improve this to (−3) rdorder scaled dimension 0 in ESPACE. As a consequence we obtain a new upper bound on the Kolmogorov complexity of Turinghard sets for ESPACE. 1
Dimension, Entropy Rates, and Compression
"... Abstract This paper develops new relationships between resourcebounded dimension, entropy rates,and compression. New tools for calculating dimensions are given and used to improve previous results about circuitsize complexity classes.Approximate counting of SpanP functions is used to prove that th ..."
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Abstract This paper develops new relationships between resourcebounded dimension, entropy rates,and compression. New tools for calculating dimensions are given and used to improve previous results about circuitsize complexity classes.Approximate counting of SpanP functions is used to prove that the NPentropy rate is an upper bound for dimension in \Delta E3, the third level of the exponentialtime hierarchy. This generalresult is applied to simultaneously improve the results of Mayordomo (1994) on the measure on P/poly in \Delta E3 and of Lutz (2000) on the dimension of exponentialsize circuit complexity classesin ESPACE. Entropy rates of efficiently rankable sets, sets that are optimally compressible, are studied inconjunction with timebounded dimension. It is shown that rankable entropy rates give upper bounds for timebounded dimensions. We use this to improve results of Lutz (1992) aboutpolynomialsize circuit complexity classes from resourcebounded measure to dimension.
Effective Strong Dimension with Applications to Information and Complexity
, 2002
"... The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed by Tricot (1982). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geome ..."
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The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed by Tricot (1982). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomialspace, polynomialtime, and finitestate dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual of – and every bit as simple as – the gale characterization of Hausdorff dimension. Effectivizing our