Arithmetizing Classes Around $\textsf{NC}$ 1 and $\textsf{L}$

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.