NE is not NP Turing Reducible to Nonexpoentially Dense NP Sets

TL;DR

This paper proves that NE cannot be reduced to certain classes of NP sets with limited density or specific structural properties, advancing understanding of complexity class separations.

Contribution

It establishes new separations between NE and classes of NP sets with nonexponential density or particular structural restrictions.

Findings

NE is not contained in NP_T(NP ∩ Nonexponentially-Dense-Class)

NE is not reducible to NP_T(Pad(NP, g(n))) for super-polynomial g(n)

NE is not reducible to NP_T(P_{tt}(NP) ∩ Tally)

Abstract

A long standing open problem in the computational complexity theory is to separate NE from BPP, which is a subclass of $NP_T(NP\cap P/poly)$. In this paper, we show that $NE\not\subseteq NP_(NP \cap$ 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 $NE\not\subseteq NP_T({pad(NP, g(n))})$ 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 $NE\not\subseteq NP_T(P_{tt}(NP)\cap Tally)$.