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Theory of Computing Systems

, Volume 46, Issue 3, pp 499–522 | Cite as

Arithmetizing Classes Around \(\textsf{NC}\) 1 and \(\textsf{L}\)

  • Nutan Limaye
  • Meena Mahajan
  • B. V. Raghavendra Rao
Article

Abstract

The parallel complexity class \(\textsf{NC}\) 1 has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. (J. Comput. Syst. Sci. 57:200–212, 1992) considered arithmetizations of two of these classes, \(\textsf{\#NC}\) 1 and \(\textsf{\#BWBP}\) . We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in \(\textsf{FLogDCFL}\) , while counting proof-trees in logarithmic width formulae has the same power as \(\textsf{\#NC}\) 1. We also consider polynomial-degree restrictions of \(\textsf{SC}\) i , denoted \(\textsf{sSC}\) i , and show that the Boolean class \(\textsf{sSC}\) 1 is sandwiched between \(\textsf{NC}\) 1 and \(\textsf{L}\) , whereas \(\textsf{sSC}\) 0 equals \(\textsf{NC}\) 1. On the other hand, the arithmetic class \(\textsf{\#sSC}\) 0 contains \(\textsf{\#BWBP}\) and is contained in \(\textsf{FL}\) , and \(\textsf{\#sSC}\) 1 contains \(\textsf{\#NC}\) 1 and is in \(\textsf{SC}\) 2. We also investigate some closure properties of the newly defined arithmetic classes.

Keywords

Bounded-width circuits Branching programs Circuit degree Arithmetization 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Nutan Limaye
    • 1
  • Meena Mahajan
    • 1
  • B. V. Raghavendra Rao
    • 1
  1. 1.The Institute of Mathematical SciencesChennaiIndia

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