Arithmetizing Classes Around NC 1 and L

The parallel complexity class NC has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. [CMTV98] considered arithmetizations of two of these classes, #NC 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 is in FLogDCFL, while counting proof-trees in logarithmic width formulae has the same power as #NC. We also consider polynomial-degree restrictions of SC, denoted sSC, and show that the Boolean class sSC is sandwiched between NC and L, whereas sSC equals NC. On the other hand, the arithmetic class #sSC contains #BWBP and is contained in FL, and #sSC contains #NC and is in SC. We also investigate some closure properties of the newly defined arithmetic classes.