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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.
The Density of Weakly Complete Problems under Adaptive Reductions
 SIAM Journal on Computing
, 2000
"... 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 mid1970's, Meyer[15] prov ..."
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Cited by 8 (1 self)
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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 mid1970's, Meyer[15] proved that every P m complete language for exponential timein fact, every P m hard language for exponential timeis 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 nondense 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 ...