In this paper we give a direct proof of $N_0=N_0^\prime$, i.e., the equivalence of
uniform $NC^1$ based on different recursion principles: one is OR-AND complete
binary tree (in depth $\log n$) and the other is the recursion on notation with value
bounded in $[0,k]$ and $|x|(=n)$ many ... more >>>

In this paper we first give a uniform $AC^0$ algorithm which uses
partial sums to compute multiple addition. Then we use it to show
that multiple addition is computable in uniform $TC^0$ by using
$count$ only once sequentially. By constructing bit matrix for
multiple addition, ... more >>>