Counting Classes and the Fine Structure between NC1 and L

  • Samir Datta
  • Meena Mahajan
  • B. V. Raghavendra Rao
  • Michael Thomas
  • Heribert Vollmer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)

Abstract

The class NC1of problems solvable by bounded fan-in circuit families of logarithmic depth is known to be contained in logarithmic space L, but not much about the converse is known. In this paper we examine the structure of classes in between NC1 and L based on counting functions or, equivalently, based on arithmetic circuits. The classes PNC1 and C=NC1, defined by a test for positivity and a test for zero, respectively, of arithmetic circuit families of logarithmic depth, sit in this complexity interval. We study the landscape of Boolean hierarchies, constant-depth oracle hierarchies, and logarithmic-depth oracle hierarchies over PNC1 and C=NC1. We provide complete problems, obtain the upper bound L for all these hierarchies, and prove partial hierarchy collapses—in particular, the constant-depth oracle hierarchy over PNC1 collapses to its first level PNC1, and the constant-depth oracle hierarchy over C=NC1 collapses to its second level.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Samir Datta
    • 1
  • Meena Mahajan
    • 2
  • B. V. Raghavendra Rao
    • 3
  • Michael Thomas
    • 4
  • Heribert Vollmer
    • 4
  1. 1.Chennai Mathematical InstituteIndia
  2. 2.The Institute of Mathematical SciencesChennaiIndia
  3. 3.Universität des SaarlandesSaarbrückenGermany
  4. 4.Leibniz UniversitätHannoverGermany

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