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56
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 25 (11 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.
A lower cone in the wtt degrees of nonintegral effective dimension
 In Proceedings of IMS workshop on Computational Prospects of Infinity
, 2006
"... ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the e ..."
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Cited by 23 (2 self)
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ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the existence of a wttlower cone of effective dimension r. 1.
Correspondence Principles for Effective Dimensions
 In Proceedings of the 29th International Colloquium on Automata, Languages, and Programming
, 2001
"... We show that the classical Hausdorff and constructive dimensions of any union of 1  definable sets of binary sequences are equal. If the union is effective, that is, the set of sequences is 2 definable, then the computable dimension also equals the Hausdorff dimension. This second result is implic ..."
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Cited by 20 (6 self)
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We show that the classical Hausdorff and constructive dimensions of any union of 1  definable sets of binary sequences are equal. If the union is effective, that is, the set of sequences is 2 definable, then the computable dimension also equals the Hausdorff dimension. This second result is implicit in the work of Staiger (1998). Staiger also proved related results using entropy rates of decidable languages. We show that Staiger's computable entropy rate provides an equivalent definition of computable dimension. We also prove that a constructive version of Staiger's entropy rate coincides with constructive dimension.
Extracting Kolmogorov complexity with applications to dimension zeroone laws
 IN PROCEEDINGS OF THE 33RD INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING
, 2006
"... We apply recent results on extracting randomness from independent sources to "extract " Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), with K(y ..."
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Cited by 20 (4 self)
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We apply recent results on extracting randomness from independent sources to "extract " Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), with K(y) ? (1 \Gamma ffl)jyj. This result holds for both classical and spacebounded Kolmogorov complexity. We use the extraction procedure for spacebounded complexity to establish zeroone laws for polynomialspace strong dimension. Our results include: (i) If Dimpspace(E) ? 0, then Dimpspace(E=O(1)) = 1. (ii) Dim(E=O(1) j ESPACE) is either 0 or 1. (iii) Dim(E=poly j ESPACE) is either 0 or 1. In other words,
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
Constructive Dimension equals Kolmogorov Complexity
 INFORMATION PROCESSING LETTERS
, 2003
"... ..."
Natural halting probabilities, partial randomness, and zeta functions
 Inform. and Comput
, 2006
"... We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin’s Omega number, halting probability, and programsize complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent ..."
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Cited by 17 (8 self)
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We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin’s Omega number, halting probability, and programsize complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness—which cannot be naturally characterised in terms of plain complexity—asymptotic randomness admits such a characterisation. 1
Dimension is compression
 In Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science
, 2005
"... Effective fractal dimension was defined by Lutz (2003) in order to quantitatively analyze the structure of complexity classes. Interesting connections of effective dimension with information theory were also found, in fact the cases of polynomialspace and constructive dimension can be precisely cha ..."
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Cited by 16 (9 self)
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Effective fractal dimension was defined by Lutz (2003) in order to quantitatively analyze the structure of complexity classes. Interesting connections of effective dimension with information theory were also found, in fact the cases of polynomialspace and constructive dimension can be precisely characterized in terms of Kolmogorov complexity, while analogous results for polynomialtime dimension haven’t been found. In this paper we remedy the situation by using the natural concept of reversible timebounded compression for finite strings. We completely characterize polynomialtime dimension in terms of polynomialtime compressors. 1
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
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 14 (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.