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49
The quantitative structure of exponential time
 Complexity Theory Retrospective II
, 1997
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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 73 (22 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...
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 53 (9 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
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 c ..."
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Cited by 52 (14 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, stochasticity, and the density of hard languages
 Proceedings of the Tenth Symposium on Theoretical Aspects of Computer Science
, 1993
"... The main theorem of this paper is that, for every real number <1 (e.g., = 0:99), only a measure 0 subset of the languages decidable P in exponential time are n;ttreducible to languages that are not P exponentially dense. Thus every n;tthard language for E is exponentially dense. This strengthe ..."
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Cited by 44 (16 self)
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The main theorem of this paper is that, for every real number <1 (e.g., = 0:99), only a measure 0 subset of the languages decidable P in exponential time are n;ttreducible to languages that are not P exponentially dense. Thus every n;tthard language for E is exponentially dense. This strengthens Watanabe's 1987 result, that every P O(log n);tthard language for E is exponentially dense. The combinatorial technique used here, the sequentially most frequent query selection, also gives a new, simpler proof of Watanabe's result. The main theorem also has implications for the structure of NP under strong hypotheses. Ogiwara and Watanabe (1991) have shown P that the hypothesis P 6 = NP implies that every btthard language for NP is nonsparse (i.e., not polynomially sparse). Their technique does not appear to allow signi cant relaxation of either the query bound or the sparseness criterion. It is shown here that a stronger hypothesis namely, that NP does not have measure 0 in exponential timeimplies P the stronger conclusion that, for every real <1, every n;tthard language for NP is exponentially dense. Evidence is presented that this stronger hypothesis is reasonable. The proof of the main theorem uses a new, very general weak stochasticity theorem, ensuring that almost every language in E is statistically unpredictable by feasible deterministic algorithms, even How dense must a language A f0 � 1g be in order to be hard for a complexity class C? The ongoing investigation of this question, especially important
ResourceBounded Measure and Randomness
"... We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than ..."
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Cited by 42 (6 self)
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We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than timebounded randomness but which suffice for many of the applications in complexity theory.
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 s ..."
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Cited by 36 (7 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...
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 31 (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.