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Published in:
2012 | OriginalPaper | Chapter
NE Is Not NP Turing Reducible to Nonexponentially Dense NP Sets
Author : Bin Fu
Published in: LATIN 2012: Theoretical Informatics
Publisher: Springer Berlin Heidelberg
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A long standing open problem in the computational complexity theory is to separate NE from BPP, which is a subclass of NP
T
(NP) ∩ P/Poly. In this paper, we show that
${\rm NE}\not\subseteq {\rm NP}_{\rm T}({\rm NP}\cap$
$\mbox{\rm{Nonexponentially-Dense-Class}})$
, where
$\mbox{\rm{Nonexponentially-Dense-Class}}$
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
${\rm NE}\not\subseteq {\rm NP}_{\rm T}({{\rm padding}({\rm NP}, g(n))})$
for every time constructible super-polynomial function
g
(
n
) such as
$g(n)=n^{\left\lceil\log\left\lceil\log n\right\rceil \right\rceil }$
, where Padding(NP,
g
(
n
)) is class of all languages
L
B
= {
s
10
g
(|
s
|) − |
s
| − 1
:
s
∈
B
} for
B
∈ NP. We also show
${\rm NE}\not\subseteq {\rm NP}_{{\rm T}}({\rm P}_{tt}({\rm NP})\cap{\rm TALLY}).$