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

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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.

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  1. The Institute of Mathematical Sciences, Chennai, 600113, India

    Nutan Limaye, Meena Mahajan & B. V. Raghavendra Rao

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  1. Nutan Limaye
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  2. Meena Mahajan
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  3. B. V. Raghavendra Rao
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Correspondence to Meena Mahajan.

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Limaye, N., Mahajan, M. & Rao, B.V.R. Arithmetizing Classes Around \(\textsf{NC}\) 1 and \(\textsf{L}\) . Theory Comput Syst 46, 499–522 (2010). https://doi.org/10.1007/s00224-009-9233-3

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