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Very Sparse Leaf Languages

  • Lance Fortnow
  • Mitsunori Ogihara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

Unger studied the balanced leaf languages defined via poly-logarithmically sparse leaf pattern sets. Unger shows that NP-complete sets are not polynomial-time many-one reducible to such balanced leaf language unless the polynomial hierarchy collapses to Θ\(^{p}_{\rm 2}\) and that Σ\(^{p}_{\rm 2}\)-complete sets are not polynomial-time bounded-truth-table reducible (respectively), polynomial-time Turing reducible) to any such balanced leaf language unless the polynomial hierarchy collapses to Δ\(^{p}_{\rm 2}\) (respectively, Σ\(^{p}_{\rm 4}\)).

This paper studies the complexity of the class of such balanced leaf languages, which will be denoted by VSLL. In particular, the following tight upper and lower bounds of VSLL are shown:

1. coNP ⊆ VSLL ⊆ coNP/poly (the former inclusion is already shown by Unger).

2. coNP/1 \(\not\subseteq\) VSLL unless PH = Θ\(^{p}_{\rm 2}\).

3. For all constant c>0, VSLL \(\not\subseteq\) coNP/n c .

4. P/(loglog(n) + O(1)) ⊆ VSLL.

5. For all h(n) = loglog(n) + ω(1), P\(/h \not\subseteq\) VSLL.

Keywords

Turing Machine Complexity Class Computation Tree Computation Path Kolmogorov Complexity 
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 2006

Authors and Affiliations

  • Lance Fortnow
    • 1
  • Mitsunori Ogihara
    • 2
  1. 1.University of Chicago 
  2. 2.University of Rochester 

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