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19
Dimension in Complexity Classes
 SIAM Journal on Computing
, 2000
"... A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Othe ..."
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Cited by 113 (17 self)
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A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Other choices of the parameter yield internal dimension theories in E, E 2 , ESPACE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X j C) 2 [0; 1]. Along with the elements of this theory, two preliminary applications are presented: 1. For every real number 0 1 2 , the set FREQ( ), consisting of all languages that asymptotically contain at most of all strings, has dimension H()  the binary entropy of  in E and in E 2 . 2. For every real number 0 1, the set SIZE( 2 n n ), consisting of all languages decidable by Boolean circuits of at most 2 n n gates, has dimension in ESPACE.
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
Effective strong dimension in algorithmic information and computational complexity
 SIAM Journal on Computing
, 2004
"... The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded exten ..."
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Cited by 79 (29 self)
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The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). 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
Minimum Description Length Induction, Bayesianism, and Kolmogorov Complexity
 IEEE Transactions on Information Theory
, 1998
"... The relationship between the Bayesian approach and the minimum description length approach is established. We sharpen and clarify the general modeling principles MDL and MML, abstracted as the ideal MDL principle and defined from Bayes's rule by means of Kolmogorov complexity. The basic condition un ..."
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Cited by 67 (7 self)
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The relationship between the Bayesian approach and the minimum description length approach is established. We sharpen and clarify the general modeling principles MDL and MML, abstracted as the ideal MDL principle and defined from Bayes's rule by means of Kolmogorov complexity. The basic condition under which the ideal principle should be applied is encapsulated as the Fundamental Inequality, which in broad terms states that the principle is valid when the data are random, relative to every contemplated hypothesis and also these hypotheses are random relative to the (universal) prior. Basically, the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability, and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized. If we restrict the model class to the finite sets then application of the ideal principle turns into Kolmogorov's mi...
Small Spans in Scaled Dimension
 SIAM Journal on Computing
, 2004
"... Juedes and Lutz (1995) proved a small span theorem for polynomialtime manyone reductions in exponential time. This result says that for language A decidable in exponential time, either the class of languages reducible to A (the lower span) or the class of problems to which A can be reduced (the up ..."
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Cited by 19 (5 self)
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Juedes and Lutz (1995) proved a small span theorem for polynomialtime manyone reductions in exponential time. This result says that for language A decidable in exponential time, either the class of languages reducible to A (the lower span) or the class of problems to which A can be reduced (the upper span) is small in the sense of resourcebounded measure and, in particular, that the degree of A is small. Small span theorems have been proven for increasingly stronger polynomialtime reductions, and a small span theorem for polynomialtime Turing reductions would imply BPP � = EXP. In contrast to the progress in resourcebounded measure, AmbosSpies, Merkle, Reimann, and Stephan (2001) showed that there is no small span theorem for the resourcebounded dimension of Lutz (2003), even for polynomialtime manyone reductions. Resourcebounded scaled dimension, recently introduced by Hitchcock, Lutz, and Mayordomo (2004), provides rescalings of resourcebounded dimension. We use scaled dimension to further understand the contrast between measure and dimension regarding polynomialtime spans and degrees. We strengthen prior results by showing that the small span theorem holds for polynomialtime manyone reductions in the −3 rdorder scaled dimension, but fails to hold in the −2 ndorder scaled dimension. Our results also hold in exponential space. As an application, we show that determining the −2 nd or −1 storder scaled dimension in ESPACE of the manyone complete languages for E would yield a proof of P = BPP or P � = PSPACE. On the other hand, it is shown unconditionally that the complete languages for E have −3 rdorder scaled dimension 0 in ESPACE and −2 nd and −1 storder scaled dimension
Relations between varieties of Kolmogorov complexity
 Mathematical Systems Theory
, 1996
"... Abstract. There are several sorts of Kolmogorov complexity, better to say several Kolmogorov complexities: decision complexity, simple complexity, prefix complexity, monotonic complexity, a priori complexity. The last three can and the first two cannot be used for defining randomness of an infinite ..."
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Cited by 7 (3 self)
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Abstract. There are several sorts of Kolmogorov complexity, better to say several Kolmogorov complexities: decision complexity, simple complexity, prefix complexity, monotonic complexity, a priori complexity. The last three can and the first two cannot be used for defining randomness of an infinite binary sequence. All those five versions of Kolmogorov complexity were considered, from a unified point of view, in a paper by the first author which appeared in Watanabe’s book [23]. Upper and lower bounds for those complexities and also for their differences were announced in that paper without proofs. (Some of those bounds are mentioned in Section 4.4.5 of [16].) The purpose of this paper (which can be read independently of [23]) is to give proofs for the bounds from [23]. The terminology used in this paper is somehow nonstandard: we call “Kolmogorov entropy ” what is usually called “Kolmogorov complexity. ” This is a Moscow tradition suggested by Kolmogorov himself. By this tradition the term “complexity ” relates to any mode of description and “entropy ” is the complexity related to an optimal mode (i.e., to a mode that, roughly speaking, gives the shortest descriptions).
Effective Fractal Dimension in Algorithmic Information Theory
, 2006
"... Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that ..."
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Cited by 7 (7 self)
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Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that