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Online Learning and ResourceBounded Dimension: Winnow Yields New Lower Bounds for Hard Sets
"... We establish a relationship between the online mistakebound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work of Fu (1995) and Lutz and Z ..."
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We establish a relationship between the online mistakebound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work of Fu (1995) and Lutz and Zhao (2000), and solves one of Lutz and Mayordomo's "Twelve Problems in ResourceBounded Measure" (1999).
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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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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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
SIGACT News Complexity Theory Column 48
"... Here is a real gift to the field from David Johnson: After a thirteen year intermission, David is restarting his NPcompleteness column. His column will now appear about twice yearly in ACM Transactions on Algorithms. Welcome back David, and thanks! ..."
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Here is a real gift to the field from David Johnson: After a thirteen year intermission, David is restarting his NPcompleteness column. His column will now appear about twice yearly in ACM Transactions on Algorithms. Welcome back David, and thanks!
Dimension, Entropy Rates, and Compression
"... 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 NPen ..."
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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.