NE is not NP Turing Reducible to Nonexpoentially Dense NP Sets

Abstract

A long standing open problem in the computational complexity theory is to separate NE from BPP, which is a subclass of $N{P}_T(NP\cap P/poly)$. In this paper, we show that $NE⊈N{P}_{(}NP\cap$ Nonexponentially-Dense-Class), where Nonexponentially-Dense-Class is the class of languages A without exponential density (for each constant c>0,$|A^{\le n}|\le 2^{n^c}$ for infinitely many integers n). Our result implies $NE⊈N{P}_T(pad(NP,g(n)))$ for every time constructible super-polynomial function g(n) such as $g(n)=n^{\ceiling{\log\ceiling{\log n}}}$, where Pad(NP, g(n)) is class of all languages $L_B=\{s10^{g(|s|)-|s|-1}:s\in B\}$ for $B\in NP$. We also show $NE⊈N{P}_T(P_{tt}(NP)\cap Tally)$.