NE Is Not NP Turing Reducible to Nonexponentially Dense NP Sets

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)


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 \({\rm NE}\not\subseteq {\rm NP}_{\rm T}({\rm NP}\cap\) \(\mbox{\rm{Nonexponentially-Dense-Class}})\), 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  = {s10 g(|s|) − |s| − 1:s ∈ B} for B ∈ NP. We also show \({\rm NE}\not\subseteq {\rm NP}_{{\rm T}}({\rm P}_{tt}({\rm NP})\cap{\rm TALLY}).\)


Polynomial Time Turing Machine Deterministic Turing Machine Computational Complexity Theory Nondeterministic Turing Machine 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bin Fu
    • 1
  1. 1.Department of Computer ScienceUniversity of Texas-Pan AmericanEdinburgUSA

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