Depth reduction for noncommutative arithmetic circuits

Abstract

We show that for every family of arithmetic circuits of polynomial size and degree over the algebra (Z*, max, concat ), there is an equivalent family of arithmetic circuits of depth log2 n. (The depth can be reduced to log n if unbounded fan-in is allowed.) This is the first depth-reduction result for arithmetic circuits Olrer a nonco~utative semiring, and it complements the lower bounds of [Ni91, K090] showing that depth reduction cannot be done in the general noncommutative setting. The (max,concat) semiring is of interest, because it characterizes certain classes of optimization problems [AJ92, Vi91]. In particular, our results show that OptSACi is contained in AC1. We also prove other results relating Boolean and arithmetic circuit complexity. We show that ACl has no more power than arithmetic circuits of polynomial size and degree n“(log log’) (improving the trivial bound of nOIIOg‘)). Connections are drawn between TCl and arithmetic circuits of polynomial size and degree.