Powering requires threshold depth 3

We study the circuit complexity of the powering function, defined as POWm(Z) = Zm for an n-bit integer input Z and an integer exponent m poly(n). Let LTd denote the class of functions computable by a depth-d polynomial-size circuit of majority gates. We give a simple proof that POWm ∈ LT2 for any m 2. Specifically, we prove a 2Ω(n/logn) lower bound on the size of any depth-2 majority circuit that computes POWm. This work generalizes Wegener’s earlier result that the squaring function (i.e., POWm for the special case m = 2) is not in LT2. Our depth lower bound is optimal due to Siu and Roychowdhury’s matching upper bound: POWm ∈ LT3. The second part of this research note presents several counterintuitive findings about the membership of arithmetic functions in the circuit classes LT1 and LT2. For example,

• Type: Journal
• Year: 2007
• Where: IPL
• Authors: Alexander A. Sherstov
• Keywords: Depth-d Polynomial-size Circuit, IPL 2007, Lower Bound, N-bit Integer Input