NE Is Not NP Turing Reducible to Nonexponentially Dense NP Sets

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

Abstract

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 \({\rm NE}\not\subseteq {\rm NP}_{{\rm T}}({\rm P}_{tt}({\rm NP})\cap{\rm TALLY}).\)

Keywords

Polynomial Time Turing Machine Deterministic Turing Machine Computational Complexity Theory Nondeterministic Turing Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© 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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