Results 1  10
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14
Small Spans in Scaled Dimension
 SIAM Journal on Computing
, 2004
"... Juedes and Lutz (1995) proved a small span theorem for polynomialtime manyone reductions in exponential time. This result says that for language A decidable in exponential time, either the class of languages reducible to A (the lower span) or the class of problems to which A can be reduced (the up ..."
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Cited by 19 (5 self)
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Juedes and Lutz (1995) proved a small span theorem for polynomialtime manyone reductions in exponential time. This result says that for language A decidable in exponential time, either the class of languages reducible to A (the lower span) or the class of problems to which A can be reduced (the upper span) is small in the sense of resourcebounded measure and, in particular, that the degree of A is small. Small span theorems have been proven for increasingly stronger polynomialtime reductions, and a small span theorem for polynomialtime Turing reductions would imply BPP � = EXP. In contrast to the progress in resourcebounded measure, AmbosSpies, Merkle, Reimann, and Stephan (2001) showed that there is no small span theorem for the resourcebounded dimension of Lutz (2003), even for polynomialtime manyone reductions. Resourcebounded scaled dimension, recently introduced by Hitchcock, Lutz, and Mayordomo (2004), provides rescalings of resourcebounded dimension. We use scaled dimension to further understand the contrast between measure and dimension regarding polynomialtime spans and degrees. We strengthen prior results by showing that the small span theorem holds for polynomialtime manyone reductions in the −3 rdorder scaled dimension, but fails to hold in the −2 ndorder scaled dimension. Our results also hold in exponential space. As an application, we show that determining the −2 nd or −1 storder scaled dimension in ESPACE of the manyone complete languages for E would yield a proof of P = BPP or P � = PSPACE. On the other hand, it is shown unconditionally that the complete languages for E have −3 rdorder scaled dimension 0 in ESPACE and −2 nd and −1 storder scaled dimension
Hardness hypotheses, derandomization, and circuit complexity
 In Proceedings of the 24th Conference on Foundations of Software Technology and Theoretical Computer Science
, 2004
"... Abstract We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: * The measure hypothesis: NP does not have pmeasure 0.* The pseudoNP hypothesis: there is an NP language that can be distinguished from anyDT ..."
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Cited by 18 (5 self)
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Abstract We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: * The measure hypothesis: NP does not have pmeasure 0.* The pseudoNP hypothesis: there is an NP language that can be distinguished from anyDTIME(2 nffl) language by an NP refuter. * The NPmachine hypothesis: there is an NP machine accepting 0 * for which no 2n ffltime machine can find infinitely many accepting computations. We show that the NPmachine hypothesis is implied by each of the first two. Previously, norelationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuitsize lower bounds that are known to followfrom the first two hypotheses also follow from the NPmachine hypothesis. In particular, the NPmachine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizesAM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses. 1 Introduction The following uniform hardness hypotheses are known to imply full derandomization of ArthurMerlin games (NP = AM): * The measure hypothesis: NP does not have pmeasure 0 [24].
Partial Biimmunity, Scaled Dimension, and NPCompleteness
"... The Turing and manyone completeness notions for NP have been previously separated under measure, genericity, and biimmunity hypotheses on NP. The proofs of all these results rely on the existence of a language in NP with almost everywhere hardness. In this ..."
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Cited by 14 (6 self)
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The Turing and manyone completeness notions for NP have been previously separated under measure, genericity, and biimmunity hypotheses on NP. The proofs of all these results rely on the existence of a language in NP with almost everywhere hardness. In this
A note on dimensions of polynomial size circuits
 Electronic Colloquium on Computational Complexity
, 2004
"... In this paper, we use resourcebounded dimension theory to investigate polynomial size circuits. We show that for every i ≥ 0, P/poly has ith order scaled p 3strong dimension 0. We also show that P/poly i.o. has p ..."
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Cited by 10 (0 self)
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In this paper, we use resourcebounded dimension theory to investigate polynomial size circuits. We show that for every i ≥ 0, P/poly has ith order scaled p 3strong dimension 0. We also show that P/poly i.o. has p
Online learning and resourcebounded dimension: Winnow yields new lower bounds for hard sets
 SIAM Journal on Computing
, 2007
"... 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 ..."
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Cited by 8 (4 self)
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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
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
Hausdorff dimension and oracle constructions
 Theoretical Computer Science
, 2004
"... Bennett and Gill (1981) proved that P A � = NP A relative to a random oracle A, or in other words, that the set O [P=NP] = {A  P A = NP A} has Lebesgue measure 0. In contrast, we show that O [P=NP] has Hausdorff dimension 1. This follows from a much more general theorem: if there is a relativizabl ..."
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Cited by 4 (1 self)
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Bennett and Gill (1981) proved that P A � = NP A relative to a random oracle A, or in other words, that the set O [P=NP] = {A  P A = NP A} has Lebesgue measure 0. In contrast, we show that O [P=NP] has Hausdorff dimension 1. This follows from a much more general theorem: if there is a relativizable and paddable oracle construction for a complexitytheoretic statement Φ, then the set of oracles relative to which Φ holds has Hausdorff dimension 1. We give several other applications including proofs that the polynomialtime hierarchy is infinite relative to a Hausdorff dimension 1 set of oracles and that P A � = NP A ∩ coNP A relative to a Hausdorff dimension 1 set of oracles. 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.