Arithmetizing classes around NC1 and L

The parallel complexity class NC1 has many equivalent models such as bounded width branching programs. Caussinus et.al[10] considered arithmetizations of two of these classes, #NC1 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 #NC1. We also consider polynomial-degree restrictions of SCi, denoted sSC(i), and show that the Boolean class sSC(1) lies between NC1 and L, whereas sSC(0) equals NC1. On the other hand, #sSC(0) contains #BWBP and is contained in FL, and #sSC(1) contains #NC1 and is in SC2. We also investigate some closure properties of the newly defined arithmetic classes.