Resource-bounded 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 p-measure zero, meaning loosely that NP contains a non-negligible 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 p-measure 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 Kolmogorov-Loveland

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...

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This paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for ESPACE under nonuniform reductions that are computed by polynomial-size circuits (P/Poly-Turing reductions and P/Poly-many-one reductions). A tight, exponential lower bound on the space-bounded Kolmogorov complexities of weakly P/PolyTuring -complete problems is established. A Small Span Theorem for P/Poly-Turing reductions in ESPACE is proven and used to show that every P/Poly-Turing degree --- including the complete degree --- has measure 0 in ESPACE. (In contrast, it is known that almost every element of ESPACE is weakly P-many-one complete.) Every weakly P/Poly-many-one-complete problem is shown to have a dense, exponential, nonuniform complexity core. More importantly, the P/Poly-many-one-complete problems are shown to be unusually simple elements of ESPACE, in the sense that they obey nontrivial upper bounds on nonuniform complexity (size of nonuniform complexit...

Given a real number ff ! 1, every language that is weakly P n ff=2 \GammaT -hard for E or weakly P n ff \GammaT -hard for E 2 is shown to be exponentially dense. This simultaneously strengthens results of Lutz and Mayordomo(1994) and Fu(1995). 1 Introduction In the mid-1970's, Meyer[15] proved that every P m -complete language for exponential time---in fact, every P m -hard language for exponential time---is dense. That is, E 6` Pm(DENSE c ); (1) where E = DTIME(2 linear ), DENSE is the class of all dense languages, DENSE c is the complement of DENSE, and Pm(DENSE c ) is the class of all languages that are P m -reducible to non-dense languages. (A language A 2 f0; 1g is dense if there is a real number ffl ? 0 such that jA n j ? 2 n ffl for all sufficiently large n, where An = A " f0; 1g n .) Since that time, a major objective of computational complexity theory has been to extend Meyer's result from P m -reductions to P T -reductions, i.e., to prove that ...

Certain natural decision problems are known to be intractable because they are complete for E, the class of all problems decidable in exponential time. Lutz recently conjectured that many other seemingly intractable problems are not complete for E, but are intractable nonetheless because they are weakly complete for E. The main result of this paper shows that Lutz's intuition is at least partially correct; many more problems are weakly complete for E than are complete for E. The main result of this paper states that weakly complete problems are not rare in the sense that they form a non-measure 0 subset of E. This extends a recent result of Lutz that establishes the existence of problems that are weakly complete, but not complete, for E. The proof of Lutz's original result employs a sophisticated martingale diagonalization argument. Here we simplify and extend Lutz's argument to prove the main result. This simplified martingale diagonalization argument may be applicable to other quest...

Resource-bounded measure theory [7] is a study of complexity classes via an adaptation of the probabilistic method. The central hypothesis in this theory is the assertion that NP does not have measure zero in Exponential Time. This is a quantitative strengthening of NP 6= P.

We show that BPP has p-measure 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 undertt-reductions. The zero-one law for each of these classes C follows: Within EXP, C has either measure zero or else measure one.

We establish a relationship between the online mistake-bound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work

Introduction One of the important questions in computational complexity theory is whether every NP problem is solvable by polynomial time circuits, i.e., NP `?P=poly. Furthermore, it has been asked what the deterministic time complexity of NP is if NP ` P=poly. That is, if NP is easy in the nonuniform complexity measure, how easy is NP in the uniform complexity measure? Let P T (SPARSE) be the class of languages that are polynomial time Turing reducible to some sparse sets. Then it is well known that P T (SPARSE) = P=poly. Hence the above question is equivalent to the following question. NP `?PT (SPARSE): It has been shown by Wilson [18] that thi