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34
The quantitative structure of exponential time
 Complexity theory retrospective II
, 1997
"... ABSTRACT Recent results on the internal, measuretheoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with biimmunity, complexity cores, polynomialtime reductions, completeness, circuit ..."
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Cited by 90 (13 self)
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ABSTRACT Recent results on the internal, measuretheoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with biimmunity, complexity cores, polynomialtime reductions, completeness, circuitsize complexity, Kolmogorov complexity, natural proofs, pseudorandom generators, the density of hard languages, randomized complexity, and lowness. Possible implications for the structure of NP are also discussed. 1
Equivalence of Measures of Complexity Classes
"... The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases ..."
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Cited by 70 (19 self)
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The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases) fi i 2 [ffi; 1 \Gamma ffi], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomialtime, truthtable reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the cointoss probability measure given by the sequence ~ fi. (2) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measu...
Cook versus KarpLevin: Separating Completeness Notions If NP Is Not Small
 Theoretical Computer Science
, 1992
"... Under the hypothesis that NP does not have pmeasure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is show n that there is a language that is P T complete ("Cook complete "), but not P m complete ("KarpLevin complete"), for NP. This conclusion, widely be ..."
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Cited by 56 (12 self)
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Under the hypothesis that NP does not have pmeasure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is show n that there is a language that is P T complete ("Cook complete "), but not P m complete ("KarpLevin complete"), for NP. This conclusion, widely believed to be true, is not known to follow from P 6= NP or other traditional complexitytheoretic hypotheses. Evidence is presented that "NP does not have pmeasure 0" is a reasonable hypothesis with many credible consequences. Additional such consequences proven here include the separation of many truthtable reducibilities in NP (e.g., k queries versus k+1 queries), the class separation E 6= NE, and the existence of NP search problems that are not reducible to the corresponding decision problems. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International. 1 Introduction The NPcompleteness of decision problems has...
Measure on small complexity classes, with applications for BPP
 In Proceedings of the 35th Symposium on Foundations of Computer Science
, 1994
"... We present a notion of resourcebounded measure for P and other subexponentialtime classes. This genemlization is based on Lutz’s notion of measure, but overcomes the limitations that cause Lptz’s definitions to apply only to classes at least as large as E. We present many of the basic properties ..."
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Cited by 48 (7 self)
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We present a notion of resourcebounded measure for P and other subexponentialtime classes. This genemlization is based on Lutz’s notion of measure, but overcomes the limitations that cause Lptz’s definitions to apply only to classes at least as large as E. We present many of the basic properties of this measure, and use it to ezplore the class of sets that are hard for BPP. Bennett and Gill showed that almost all sets are hard for BPP; Lutz improved this from Lebesgue measure to measure on ESPACE. We use OUT measure to improve this still further, showing that for all E> 0, almost every set in E, is hard for BPP, where E, = Us<rDTIME(2”6), which is the best that can be achieved without showing that BPP is properly contained in E. A number of related results are also obtained in this way. 1
The Complexity and Distribution of Hard Problems
 SIAM JOURNAL ON COMPUTING
, 1993
"... Measuretheoretic aspects of the P m reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in ..."
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Cited by 45 (16 self)
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Measuretheoretic aspects of the P m reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are P m  hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bound says that the P m hard languages for E are unusually simple, in the sense that they have smaller complexity cores than most languages in E. It follows that the P m complete languages for E form a measure 0 subset of E (and similarly in E 2 ). This latter fact is seen to be a special case of a more general theorem, namely, that every P m degree (e.g., the degree of all P m complete languages for NP) has measure 0 in E and in E 2 .
Resource Bounded Randomness and Weakly Complete Problems
 Theoretical Computer Science
, 1994
"... We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ([7, 8]). We concentrate on n c  randomness (c 2) which corresponds to the polynomial time bounded (p) measure of Lutz, and which is adequate for studying the internal and quantitative struct ..."
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Cited by 37 (6 self)
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We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ([7, 8]). We concentrate on n c  randomness (c 2) which corresponds to the polynomial time bounded (p) measure of Lutz, and which is adequate for studying the internal and quantitative structure of E = DTIME(2 lin ). However we will also comment on E2 = DTIME(2 pol ) and its corresponding (p2 ) measure. First we show that the class of n c random sets has pmeasure 1. This provides a new, simplified approach to pmeasure 1results. Next we compare randomness with genericity (in the sense of [2, 3]) and we show that n c+1  random sets are n c generic, whereas the converse fails. From the former we conclude that n c random sets are not pbttcomplete for E. Our technical main results describe the distribution of the n c random sets under pmreducibility. We show that every n c random set in E has n k random predecessors in E for any k 1, whereas the amou...
MAX3SAT Is Exponentially Hard to Approximate If NP Has Positive Dimension
 Theoretical Computer Science
, 2002
"... Under the hypothesis that NP has positive pdimension, we prove that any approximation for MAX3SAT must satisfy at least one of the following: time. ..."
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Cited by 29 (14 self)
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Under the hypothesis that NP has positive pdimension, we prove that any approximation for MAX3SAT must satisfy at least one of the following: time.
Pseudorandom Generators, Measure Theory, and Natural Proofs
, 1995
"... We prove that if strong pseudorandom number generators exist, then the class of languages that have polynomialsized circuits (P/poly) is not measurable within exponential time, in terms of the resourcebounded measure theory of Lutz. We prove our result by showing that if P/poly has measure zero in ..."
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Cited by 29 (4 self)
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We prove that if strong pseudorandom number generators exist, then the class of languages that have polynomialsized circuits (P/poly) is not measurable within exponential time, in terms of the resourcebounded measure theory of Lutz. We prove our result by showing that if P/poly has measure zero in exponential time, then there is a natural proof against P/poly, in the terminology of Razborov and Rudich [25]. We also provide a partial converse of this result.
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