The parallel complexity class
NC
1 has many equivalent models such as bounded width branching programs. Caussinus et.al [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 SCi{\sf SC}^{i}
, denoted
\sf sSCi{\sf sSC}^{i}
, and show that the Boolean class
\sf sSC1{\sf sSC}{^1}
lies between
NC
1 and
L, whereas
\sf sSC0{\sf sSC}^0
equals
\sf NC1{\sf NC}^1
. On the other hand,
\sf #\sf sSC0{\sf \#}{\sf sSC}^0
contains
#BWBP and is contained in
FL, and
#sSC
1 contains
#NC
1 and is in
\sf SC2{\sf SC}^{2}
. We also investigate some closure properties of the newly defined arithmetic classes.