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39
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...
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 .
Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Al ..."
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Cited by 30 (3 self)
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The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
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.
Separation of NPcompleteness notions
 SIAM Journal on Computing
, 2001
"... Abstract. We use hypotheses of structural complexity theory to separate various NPcompleteness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P Tcomplete but not ¡ P ttcomplete. We provide fairly thorough analyses of the hypotheses that we introduc ..."
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Cited by 26 (12 self)
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Abstract. We use hypotheses of structural complexity theory to separate various NPcompleteness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P Tcomplete but not ¡ P ttcomplete. We provide fairly thorough analyses of the hypotheses that we introduce. Key words. Turing completeness, truthtable completeness, manyone completeness, pselectivity, pgenericity AMS subject classifications. 1. Introduction. Ladner, Lynch, and Selman [LLS75] were the first to compare the strength of polyno), truth), that mialtime reducibilities. They showed, for the common polynomialtime reducibilities, ( ¢ Turing P T ( ¢ table P tt), bounded truthtable ( ¢ P btt), and manyone ( ¢ P m
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
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 18 (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 14 (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...