NE Is Not NP Turing Reducible to Nonexponentially Dense NP Sets

A long standing open problem in the computational complexity theory is to separate NE from BPP, which is a subclass of NPT(NP)∩P/Poly. In this paper, we show that NE ⊈ NPT (NP∩ Nonexponentially-Dense-Class, where Nonexponentially-Dense-Class is the class of languages A without exponential density (for each constant c>0, |A≤n}| ≤ 2nc for infinitely many integers n). Our result implies NE ⊈ NPT (padding(NP, g(n))) for every time constructible super-polynomial function g(n) such as g(n)=n⌈log⌈log n⌉⌉, where Padding(NP, g(n)) is class of all languages LB={s10g(|s|)−|s|−1:s∈B} for B∈NP. We also show NE ⊈ NPT(Ptt(NP) ∩ TALLY).