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73
Almost Everywhere High Nonuniform Complexity
, 1992
"... . We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds ..."
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Cited by 176 (40 self)
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. We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds, and thereby strengthens, the Shannon 2 n n lower bound for almost every problem, with no computability constraint.) In exponential time complexity classes, we prove that the strongest relativizable lower bounds hold almost everywhere for almost all problems. Finally, we show that infinite pseudorandom sequences have high nonuniform complexity almost everywhere. The results are unified by a new, more powerful formulation of the underlying measure theory, based on uniform systems of density functions, and by the introduction of a new nonuniform complexity measure, the selective Kolmogorov complexity. This research was supported in part by NSF Grants CCR8809238 and CCR9157382 and in ...
The quantitative structure of exponential time
 Complexity Theory Retrospective II
, 1997
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Effective strong dimension in algorithmic information and computational complexity
 SIAM Journal on Computing
, 2004
"... The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded exten ..."
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Cited by 75 (25 self)
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The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomialspace, polynomialtime, and finitestate dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual
Almost Every Set in Exponential Time is PBiImmune
 Theoretical Computer Science
, 1994
"... . A set A is Pbiimmune if neither A nor its complement has an infinite subset in P. We investigate here the abundance of Pbiimmune languages in linearexponential time (E). We prove that the class of Pbiimmune sets has measure 1 in E. This implies that `almost' every language in E is Pbi ..."
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Cited by 57 (8 self)
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. A set A is Pbiimmune if neither A nor its complement has an infinite subset in P. We investigate here the abundance of Pbiimmune languages in linearexponential time (E). We prove that the class of Pbiimmune sets has measure 1 in E. This implies that `almost' every language in E is Pbiimmune, that is to say, almost every set recognizable in linear exponential time has no algorithm that recognizes it and works in polynomial time on an infinite number of instances. A bit further, we show that every prandom (pseudorandom) language is Ebiimmune. Regarding the existence of Pbiimmune sets in NP, we show that if NP does not have measure 0 in E, then NP contains a Pbiimmune set. Another consequence is that the class of p m complete languages for E has measure 0 in E. In contrast, it is shown that in E, and even in REC, the class of Pbiimmune languages lacks the property of Baire (the Baire category analogue of Lebesgue measurability). * This work was supported by a Spani...
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
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 45 (17 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 43 (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.
Resourcebounded measure
 In Proceedings of the 13th IEEE Conference on Computational Complexity
, 1998
"... ar ..."
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...