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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&qu ..."
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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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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
FiniteState Dimension
, 2001
"... Classical Hausdorff dimension (sometimes called fractal 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 ..."
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Cited by 32 (13 self)
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Classical Hausdorff dimension (sometimes called fractal 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 multiaccount finitestate gamblers to develop the finitestate dimensions of sets of binary sequences and individual binary sequences. The theorem of Eggleston (1949) relating Hausdorff dimension to entropy is shown to hold for finitestate dimension, both in the space of all sequences and in the space of all rational sequences (binary expansions of rational numbers). Every rational sequence has finitestate dimension 0, but every rational number in [0; 1] is the finitestate dimension of a sequence in the lowlevel complexity class AC0 . Our main theorem shows that the finitestate dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by informationlossless finitestate compressors.
Scaled dimension and nonuniform complexity
 Journal of Computer and System Sciences
, 2004
"... Resourcebounded dimension is a complexitytheoretic extension of classical Hausdorff dimension introduced by Lutz (2000) in order to investigate the fractal structure of sets that have resourcebounded measure 0. For example, while it has long been known that the Boolean circuitsize complexity cla ..."
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Cited by 22 (9 self)
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Resourcebounded dimension is a complexitytheoretic extension of classical Hausdorff dimension introduced by Lutz (2000) in order to investigate the fractal structure of sets that have resourcebounded measure 0. For example, while it has long been known that the Boolean circuitsize complexity class SIZE � α 2n n has measure 0 in ESPACE for all 0 ≤ α ≤ 1, we now know that SIZE � α 2n n has dimension α in ESPACE for all 0 ≤ α ≤ 1. The present paper furthers this program by developing a natural hierarchy of “rescaled” resourcebounded dimensions. For each integer i and each set X of decision problems, we define the ithorder dimension of X in suitable complexity classes. The 0thorder dimension is precisely the dimension of Hausdorff (1919) and Lutz (2000). Higher and lower orders are useful for various sets X. For example, we prove the following for 0 ≤ α ≤ 1 and any polynomial q(n) ≥ n2. 1. The class SIZE(2 αn) and the time and spacebounded Kolmogorov complexity classes KT q (2 αn) and KS q (2 αn) have 1 storder dimension α in ESPACE. 2. The classes SIZE(2nα), KT q (2nα), and KS q (2nα) have 2ndorder dimension α in ESPACE.
Gales and the Dimensions of Individual Strings and Sequences
"... A constructive version of Hausdorff dimension is developed using constructive gales, which are betting strategies that generalize the constructive martingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence ..."
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A constructive version of Hausdorff dimension is developed using constructive gales, which are betting strategies that generalize the constructive martingales 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 are random (in the sense of MartinLöf) have dimension 1, while sequences that are decidable, 0 1 , or 0 1 have dimension 0. It is shown that for every 0 2 computable real number in [0,1] there is a 0 2 sequence S such that dim(S) = : A discrete version of constructive dimension is also developed using termgales, which are galelike functions that bet on the terminations of (finite, binary) strings as well as on their successive bits. This discrete dimension is used to assign each individual string w a dimension, which is a nonnegative real number dim(w). The dimension of a sequence is shown to be the limit in mum of the dimensions of its prefixes. The Kolmogorov complexity of a string is proven to be the product of its length and its dimension. This gives a new characterization of algorithmic information and a new proof of Mayordomo's recent theorem stating that the dimension of a sequence is the limit in mum of the average Kolmogorov complexity of its first n bits. Every sequence that is random relative to any computable sequence of cointoss biases that converge to a real number in (0; 1) is shown to have dimension H( ), the binary entropy of .
Dimension of measures: the probabilistic approach
, 2007
"... Various tools can be used to calculate or estimate the dimension of measures. Using a probabilistic interpretation, we propose very simple proofs for the main inequalities related to this notion. We also discuss the case of quasiBernoulli measures and point out the deep link existing between the ca ..."
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Various tools can be used to calculate or estimate the dimension of measures. Using a probabilistic interpretation, we propose very simple proofs for the main inequalities related to this notion. We also discuss the case of quasiBernoulli measures and point out the deep link existing between the calculation of the dimension of auxiliary measures and the multifractal analysis.
c World Scientic Publishing Company THE DIMENSION OF SETS DETERMINED BY THEIR CODE BEHAVIOR
, 2002
"... By prescribing their code run behavior, we consider some subsets of Moran fractals. Fractal dimensions of these subsets are exactly obtained. Meanwhile, an interesting decomposition of Moran fractals is given. ..."
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By prescribing their code run behavior, we consider some subsets of Moran fractals. Fractal dimensions of these subsets are exactly obtained. Meanwhile, an interesting decomposition of Moran fractals is given.
www.elsevier.com/locate/spa Individual behaviors of oriented walks
, 2000
"... Given an innite sequence t = (k)k of −1 and +1, we consider the oriented walk dened by Sn(t) = Pn k=1 12: : : k. The set of t’s whose behaviors satisfy Sn(t) bn is considered (b 2 R and 0<61 being xed) and its Hausdor dimension is calculated. A twodimensional model is also studied. A threed ..."
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Given an innite sequence t = (k)k of −1 and +1, we consider the oriented walk dened by Sn(t) = Pn k=1 12: : : k. The set of t’s whose behaviors satisfy Sn(t) bn is considered (b 2 R and 0<61 being xed) and its Hausdor dimension is calculated. A twodimensional model is also studied. A threedimensional model is described, but the problem remains open. c © 2000