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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
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
Pushdown dimension
 Theoretical Computer Science
, 2007
"... Abstract Resourcebounded dimension is a notion of computational information density of infinite sequences based on computationally bounded gamblers. This paper develops the theory of pushdown dimension and explores its relationship with finitestate dimension.The pushdown dimension of any sequence ..."
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Cited by 2 (0 self)
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Abstract Resourcebounded dimension is a notion of computational information density of infinite sequences based on computationally bounded gamblers. This paper develops the theory of pushdown dimension and explores its relationship with finitestate dimension.The pushdown dimension of any sequence is trivially bounded above by its finitestate dimension, since a pushdown gambler can simulate any finitestate gambler. We show thatfor every rational 0 < d < 1, there exists a sequence with finitestate dimension d whosepushdown dimension is at most d/2. This provides a stronger quantitative analogue of thewellknown fact that pushdown automata decide strictly more languages than finitestate
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.
www.stacsconf.org PUSHDOWN COMPRESSION
"... Abstract. The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation [6, 9], and in particular calls for a formulation of informationlossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitio ..."
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Abstract. The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation [6, 9], and in particular calls for a formulation of informationlossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. The celebrated LempelZiv algorithm LZ78 [10] was introduced as a general purpose compression algorithm that outperforms finitestate compressors on all sequences. We compare the performance of the LempelZiv algorithm with that of the pushdown compressors, or compression algorithms that can be implemented with a pushdown transducer. This comparison is made without any a priori assumption on the data’s source and considering the asymptotic compression ratio for infinite sequences. We prove that LempelZiv is incomparable with pushdown compressors. 1.
Polylog space compression, pushdown compression, and LempelZiv are incomparable
, 2009
"... The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation [11, 17], and in particular calls for a formulation of informationlossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use o ..."
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The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation [11, 17], and in particular calls for a formulation of informationlossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. We also consider online compression algorithms that use at most polylogarithmic space (plogon). These algorithms correspond to compressors in the data stream model. We compare the performance of these two families of compressors with each other and with the general purpose LempelZiv algorithm. This comparison is made without any a priori assumption on the data’s source and considering the asymptotic compression ratio for infinite sequences. We prove that in all cases they are incomparable.
Base Invariance of Feasible Dimension
, 2013
"... Effective fractal dimensions were introduced by Lutz (2003) in order to study the dimensions of individual sequences and quantitatively analyze the structure of complexity classes. Interesting connections of effective dimensions with information theory were also found, implying that constructive dim ..."
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Effective fractal dimensions were introduced by Lutz (2003) in order to study the dimensions of individual sequences and quantitatively analyze the structure of complexity classes. Interesting connections of effective dimensions with information theory were also found, implying that constructive dimension as well as polynomialspace dimension are invariant under basechange while finitestate dimension is not. We consider the intermediate case, polynomialtime dimension, and prove that it is indeed invariant under basechange by a nontrivial argument which is quite different from the Kolmogorov complexity ones used in the other cases. Polynomialtime dimension can be characterized in terms of predictionlossrate, entropy, and compression algorithms. Our result implies that in an asymptotic way each of these concepts is invariant under basechange. A corollary of the main theorem is any polynomialtime dimension 1 number (which may be established in any base) is an absolutely normal number, providing an interesting source of absolute normality. 1