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24
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...
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.
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
When Worlds Collide: Derandomization, Lower Bounds, and Kolmogorov Complexity
 OF REDUCTIONS,IN“PROC.29THACM SYMPOSIUM ON THEORY OF COMPUTING
, 1997
"... This paper has the following goals:  To survey some of the recent developments in the field of derandomization.  To introduce a new notion of timebounded Kolmogorov complexity (KT), and show that it provides a useful tool for understanding advances in derandomization, and for putting vario ..."
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Cited by 18 (5 self)
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This paper has the following goals:  To survey some of the recent developments in the field of derandomization.  To introduce a new notion of timebounded Kolmogorov complexity (KT), and show that it provides a useful tool for understanding advances in derandomization, and for putting various results in context.  To illustrate the usefulness of KT, by answering a question that has been posed in the literature, and  To pose some promising directions for future research.
An Excursion to the Kolmogorov Random Strings
 In Proceedings of the 10th IEEE Structure in Complexity Theory Conference
, 1995
"... We study the set of resource bounded Kolmogorov random strings: R t = fx j K t (x) jxjg for t a time constructible function such that t(n) 2 n 2 and t(n) 2 2 n O(1) . We show that the class of sets that Turing reduce to R t has measure 0 in EXP with respect to the resourcebounded measure ..."
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Cited by 17 (8 self)
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We study the set of resource bounded Kolmogorov random strings: R t = fx j K t (x) jxjg for t a time constructible function such that t(n) 2 n 2 and t(n) 2 2 n O(1) . We show that the class of sets that Turing reduce to R t has measure 0 in EXP with respect to the resourcebounded measure introduced by [17]. From this we conclude that R t is not Turingcomplete for EXP . This contrasts the resource unbounded setting. There R is Turingcomplete for coRE . We show that the class of sets to which R t bounded truthtable reduces, has p 2 measure 0 (therefore, measure 0 in EXP ). This answers an open question of Lutz, giving a natural example of a language that is not weaklycomplete for EXP and that reduces to a measure 0 class in EXP . It follows that the sets that are p btt hard for EXP have p 2 measure 0. 1 Introduction One of the main questions in complexity theory is the relation between complexity classes, such as for example P ; NP , and EXP . It is well known that ...
Measure on P: Robustness of the Notion
, 1995
"... In #AS#, we de#ned a notion of measure on the complexity class P #in the spirit of the work of Lutz #L92# that provides a notion of measure on complexity classes at least as large as E, and the work of Mayordomo #M# that provides a measure on PSPACE#. In this paper, we show that several other ways ..."
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Cited by 15 (2 self)
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In #AS#, we de#ned a notion of measure on the complexity class P #in the spirit of the work of Lutz #L92# that provides a notion of measure on complexity classes at least as large as E, and the work of Mayordomo #M# that provides a measure on PSPACE#. In this paper, we show that several other ways of de#ning measure in terms of covers and martingales yield precisely the same notion as in #AS#. #Similar #robustness" results have been obtained previously for the notions of measure de#ned by #L92# and #M#, but # for reasons that will become apparent below # di#erent proofs are required in our setting.# To our surprise, and in contrast to the measures of Lutz #L92# and Mayordomo #M#, one obtains strictly more measurable sets if one considers #nonconservative" martingales that succeed merely in the lim sup rather than having a limit of in#nity.For example, it is shown in #AS# that the class of sparse sets does not have measure zero in P, whereas here we show that using the #nonconservative " measure, the class of sparse sets #and in fact the class of sets with density ##1=2# does have measure zero. We also show that our #nonconservative" measure on PSPACE is incomparable with that of #M#.
A generalization of resourcebounded measure, with application to the BPP vs. EXP problem
 SIAM J. Comput
, 2000
"... We introduce resourcebounded betting games, and propose a generalization of Lutz’s resourcebounded measure in which the choice of next string to bet on is fully adaptive. Lutz’s martingales are equivalent to betting games constrained to bet on strings in lexicographic order. We show that if strong ..."
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Cited by 13 (1 self)
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We introduce resourcebounded betting games, and propose a generalization of Lutz’s resourcebounded measure in which the choice of next string to bet on is fully adaptive. Lutz’s martingales are equivalent to betting games constrained to bet on strings in lexicographic order. We show that if strong pseudorandom number generators exist, then betting games are equivalent to martingales, for measure on E and EXP. However, we construct betting games that succeed on certain classes whose Lutz measures are important open problems: the class of polynomialtime Turingcomplete languages in EXP, and its superclass of polynomialtime Turingautoreducible languages. If an EXPmartingale succeeds on either of these classes, or if betting games have the “finite union property ” possessed
Measure on P: Strength of the Notion
 Inform. and Comp
, 1996
"... We give a notion of measure on P that overcomes some limitations of earlier formulations. In the process, we investigate the significance for resourcebounded measure of the choice of the lexicographic ordering of the words. 1 Introduction Resourcebounded measure was introduced by Lutz in [8]. Int ..."
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Cited by 9 (2 self)
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We give a notion of measure on P that overcomes some limitations of earlier formulations. In the process, we investigate the significance for resourcebounded measure of the choice of the lexicographic ordering of the words. 1 Introduction Resourcebounded measure was introduced by Lutz in [8]. Intuitively, this theory gives a notion of big and small to sets of languages. In recent years this tool has been used with many successes to illuminate the structure of complexity classes, notably E and E 2 = EXP [9]. The theory of resourcebounded measure is a parametrized tool. For many complexity classes C, one plugs C into the general theory, and one gets out a notion of measure at C, in which each singleton set of a C langauge is small, but C itself is not small. Unfortunately, Lutz's formulation only works directly for C ' E. Generalizing Lutz's notion, in [1] the authors introduced a notion of measure on P, PSPACE, and other subexponential classes. This notion satisfies many nice theor...
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.