Arithmetizing Classes Around NC1 and L

  • Nutan Limaye
  • Meena Mahajan
  • B. V. Raghavendra Rao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)


The parallel complexity class NC 1 has many equivalent models such as bounded width branching programs. Caussinus [10] considered arithmetizations of two of these classes, #NC 1 and #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 has the same power as #BWBP, while counting proof-trees in logarithmic width formulae has the same power as #NC 1. We also consider polynomial-degree restrictions of \({\sf SC}^{i}\), denoted \({\sf sSC}^{i}\), and show that the Boolean class \({\sf sSC}{^1}\) lies between NC 1 and L, whereas \({\sf sSC}^0\) equals \({\sf NC}^1\). On the other hand, \({\sf \#}{\sf sSC}^0\) contains #BWBP and is contained in FL, and #sSC 1 contains #NC 1 and is in \({\sf SC}^{2}\). We also investigate some closure properties of the newly defined arithmetic classes.


Polynomial Degree Closure Property Arithmetic Circuit Polynomial Size Boolean Circuit 
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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© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Nutan Limaye
    • 1
  • Meena Mahajan
    • 1
  • B. V. Raghavendra Rao
    • 1
  1. 1.The Institute of Mathematical Sciences, Chennai 600 113India

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