Separating NE from some nonuniform nondeterministic complexity classes

  • Authors:

  • Bin Fu;Angsheng Li;Liyu Zhang

  • Affiliations:

  • Department of Computer Science, University of Texas-Pan American, Edinburg, USA 78539;Institute of Software, Chinese Academy of Sciences, Beijing, China;Department of Computer and Information Sciences, University of Texas at Brownsville, Brownsville, USA 78520

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2011

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Abstract

We investigate the question whether NE can be separated from the reduction closures of tally sets, sparse sets and NP. We show that (1) $\mathrm{NE}\not\subseteq R^{\mathrm{NP}}_{n^{o(1)}-T}(\mathrm{TALLY})$ ; (2) $\mathrm{NE}\not\subseteq R^{SN}_{m}(\mathrm{SPARSE})$ ; (3) $\mathrm{NEXP}\not\subseteq \mathrm{P}^{\mathrm{NP}}_{n^{k}-T}/n^{k}$ for all k驴1; and (4) $\mathrm{NE}\not\subseteq \mathrm{P}_{btt}(\mathrm{NP}\oplus\mathrm{SPARSE})$ . Result (3) extends a previous result by Mocas to nonuniform reductions. We also investigate how different an NE-hard set is from an NP-set. We show that for any NP subset A of a many-one-hard set H for NE, there exists another NP subset A驴 of H such that A驴驴 A and A驴驴A is not of sub-exponential density.