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41
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 18 (4 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
Relative to a random oracle, NP is not small
 In Proc. 9th Structures
, 1994
"... Resourcebounded measure as originated by Lutz is an extension of classical measure theory which provides a probabilistic means of describing the relative sizes of complexity classes. Lutz has proposed the hypothesis that NP does not have pmeasure zero, meaning loosely that NP contains a nonneglig ..."
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Cited by 17 (1 self)
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Resourcebounded measure as originated by Lutz is an extension of classical measure theory which provides a probabilistic means of describing the relative sizes of complexity classes. Lutz has proposed the hypothesis that NP does not have pmeasure zero, meaning loosely that NP contains a nonnegligible subset of exponential time. This hypothesis implies a strong separation of P from NP and is supported by a growing body of plausible consequences which are not known to follow from the weaker assertion P ̸ = NP. It is shown in this paper that relative to a random oracle, NP does not have pmeasure zero. The proof exploits the following independence property of algorithmically random sequences: if A is an algorithmically random sequence and a subsequence A0 is chosen by means of a bounded KolmogorovLoveland
Weakly Hard Problems
, 1994
"... A weak completeness phenomenon is investigated in the complexity class E = DTIME(2 linear ). According to standard terminology, a language H is P m hard for E if the set Pm (H), consisting of all languages A P m H , contains the entire class E. A language C is P m complete for E if it ..."
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Cited by 15 (6 self)
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A weak completeness phenomenon is investigated in the complexity class E = DTIME(2 linear ). According to standard terminology, a language H is P m hard for E if the set Pm (H), consisting of all languages A P m H , contains the entire class E. A language C is P m complete for E if it is P m hard for E and is also an element of E. Generalizing this, a language H is weakly P m hard for E if the set Pm (H) does not have measure 0 in E. A language C is weakly P m complete for E if it is weakly P m hard for E and is also an element of E. The main result of this paper is the construction of a language that is weakly P m complete, but not P m complete, for E. The existence of such languages implies that previously known strong lower bounds on the complexity of weakly P m hard problems for E (given by work of Lutz, Mayordomo, and Juedes) are indeed more general than the corresponding bounds for P m hard problems for E. The proof of this result in...
Weakly Complete Problems are Not Rare
 COMPUTATIONAL COMPLEXITY
, 1995
"... Certain natural decision problems are known to be intractable because they are complete for E, the class of all problems decidable in exponential time. Lutz recently conjectured that many other seemingly intractable problems are not complete for E, but are intractable nonetheless because they are we ..."
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Cited by 8 (2 self)
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Certain natural decision problems are known to be intractable because they are complete for E, the class of all problems decidable in exponential time. Lutz recently conjectured that many other seemingly intractable problems are not complete for E, but are intractable nonetheless because they are weakly complete for E. The main result of this paper shows that Lutz's intuition is at least partially correct; many more problems are weakly complete for E than are complete for E. The main result of this paper states that weakly complete problems are not rare in the sense that they form a nonmeasure 0 subset of E. This extends a recent result of Lutz that establishes the existence of problems that are weakly complete, but not complete, for E. The proof of Lutz's original result employs a sophisticated martingale diagonalization argument. Here we simplify and extend Lutz's argument to prove the main result. This simplified martingale diagonalization argument may be applicable to other quest...
Completeness and Weak Completeness under PolynomialSize Circuits
 Information and Computation
, 1996
"... This paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for ESPACE under nonuniform reductions that are computed by polynomialsize circuits (P/PolyTuring reductions and P/Polymanyone reductions). A tight, exponential lower bound on the ..."
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Cited by 8 (4 self)
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This paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for ESPACE under nonuniform reductions that are computed by polynomialsize circuits (P/PolyTuring reductions and P/Polymanyone reductions). A tight, exponential lower bound on the spacebounded Kolmogorov complexities of weakly P/PolyTuring complete problems is established. A Small Span Theorem for P/PolyTuring reductions in ESPACE is proven and used to show that every P/PolyTuring degree  including the complete degree  has measure 0 in ESPACE. (In contrast, it is known that almost every element of ESPACE is weakly Pmanyone complete.) Every weakly P/Polymanyonecomplete problem is shown to have a dense, exponential, nonuniform complexity core. More importantly, the P/Polymanyonecomplete problems are shown to be unusually simple elements of ESPACE, in the sense that they obey nontrivial upper bounds on nonuniform complexity (size of nonuniform complexit...
The zeroone law holds for BPP
"... We show that BPP has pmeasure zero if and only if BPP differs from EXP. The same holds when we replace BPP by any complexity class C that is contained in BPP and is closed underttreductions. The zeroone law for each of these classes C follows: Within EXP, C has either measure zero or else measur ..."
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Cited by 8 (0 self)
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We show that BPP has pmeasure zero if and only if BPP differs from EXP. The same holds when we replace BPP by any complexity class C that is contained in BPP and is closed underttreductions. The zeroone law for each of these classes C follows: Within EXP, C has either measure zero or else measure one.
Constant Depth Circuits and the Lutz Hypothesis
"... Resourcebounded measure theory [7] is a study of complexity classes via an adaptation of the probabilistic method. The central hypothesis in this theory is the assertion that NP does not have measure zero in Exponential Time. This is a quantitative strengthening of NP 6= P. ..."
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Cited by 8 (2 self)
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Resourcebounded measure theory [7] is a study of complexity classes via an adaptation of the probabilistic method. The central hypothesis in this theory is the assertion that NP does not have measure zero in Exponential Time. This is a quantitative strengthening of NP 6= P.
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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Cited by 7 (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 of Fu (1995) and Lutz and Zhao (2000), and solves one of Lutz and Mayordomo's "Twelve Problems in ResourceBounded Measure" (1999).
Nonuniform Lower Bounds for Exponential Time Classes
 Mathematical Foundations of Computer Science 1995, 20th International Symposium, volume 969 of lncs, pages 159168, Prague, Czech Republic, 1 September 28
, 1993
"... this paper we are interested in absolute results and consider advice classes slightly smaller than P=poly and circuit classes smaller than polynomialsize. And we establish several new lower bounds for exponential time problems with respect to these classes. This is not a new tack to explore. In la ..."
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Cited by 6 (0 self)
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this paper we are interested in absolute results and consider advice classes slightly smaller than P=poly and circuit classes smaller than polynomialsize. And we establish several new lower bounds for exponential time problems with respect to these classes. This is not a new tack to explore. In last years' Structures Bin Fu [Fu93] considered lower bounds for polynomial time reductions to sparse sets, where limits are placed on the number of queries to the sparse set. The main result of his paper was that there are sets in EXP which are not polynomial time Turing reducible to a sparse set when the reduction is restricted to querying the sparse set no more than n