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15
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 92 (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 78 (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
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
Entropy rates and finitestate dimension
 THEORETICAL COMPUTER SCIENCE
, 2005
"... The effective fractal dimensions at the polynomialspace level and above can all be equivalently defined as the Centropy rate where C is the class of languages corresponding to the level of effectivization. For example, pspacedimension is equivalent to the PSPACEentropy rate. At lower levels of c ..."
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Cited by 13 (0 self)
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The effective fractal dimensions at the polynomialspace level and above can all be equivalently defined as the Centropy rate where C is the class of languages corresponding to the level of effectivization. For example, pspacedimension is equivalent to the PSPACEentropy rate. At lower levels of complexity the equivalence proofs break down. In the polynomialtime case, the Pentropy rate is a lower bound on the pdimension. Equality seems unlikely, but separating the Pentropy rate from pdimension would require proving P != NP. We show that at the finitestate level, the opposite of the polynomialtime case happens: the REGentropy rate is an upper bound on the finitestate dimension. We also use the finitestate genericity of AmbosSpies and Busse (2003) to separate finitestate dimension from the REGentropy rate. However, we point out that a blockentropy rate characterization of finitestate dimension follows from the work of Ziv and Lempel (1978) on finitestate compressibility and the compressibility characterization of finitestate dimension by Dai, Lathrop, Lutz, and Mayordomo (2004). As applications of the REGentropy rate upper bound and the blockentropy rate characterization, we prove that every regular language has finitestate dimension 0 and that normality is equivalent to finitestate dimension 1.
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
Points on computable curves
 In Proceedings of the FortySeventh Annual IEEE Symposium on Foundations of Computer Science
, 1999
"... The “analyst’s traveling salesman theorem ” of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length. This result, proven for the plane by Jones (1990) and extended to higherdimensional Euclidean spaces by Okikiolu (1992), says that a ..."
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Cited by 5 (3 self)
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The “analyst’s traveling salesman theorem ” of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length. This result, proven for the plane by Jones (1990) and extended to higherdimensional Euclidean spaces by Okikiolu (1992), says that a bounded set K is contained in some curve of finite length if and only if a certain “square beta sum”, involving the “width of K ” in each element of an infinite system of overlapping “tiles” of descending size, is finite. In this paper we characterize those points of Euclidean space that lie on computable curves of finite length. We do this by formulating and proving a computable extension of the analyst’s traveling salesman theorem. Our extension, the computable analyst’s traveling salesman theorem, says that a point in Euclidean space lies on some computable curve of finite length if and only if it is “permitted ” by some computable “Jones constriction”. A Jones constriction here is an explicit assignment of a rational cylinder to each of the abovementioned tiles in such a way that, when the radius of the cylinder corresponding to a tile is used in place of the “width of K ” in each tile, the square beta sum is finite. A point is permitted by a Jones constriction if it is
Effective packing dimension of Π 0 1classes
, 2007
"... We construct a Π0 1class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relatio ..."
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Cited by 2 (1 self)
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We construct a Π0 1class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relationship between upper box dimension and packing A major theme of computability theory is the effectivization of classical mathematics. To do this one takes an existing (i.e. classical) mathematical notion and develops a new computabilitytheoretic analogue of that notion. Afterwards, one tries to determine the similarities and differences between the
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.