The Average Sensitivity of Bounded-Depth Formulas

Abstract

We show that unbounded fan-in Boolean formulas of depth d + 1 and size s have average sensitivity ${O(\frac{1}{d} \log s)^d}$. In particular, this gives a tight ${2^{\Omega(d(n^{1/d}-1))}}$ lower bound on the size of depth d + 1 formulas computing the parity function. These results strengthen the corresponding ${2^{\Omega(n^{1/d})}}$ and ${O(\log s)^d}$ bounds for circuits due to Håstad (Proceedings of the 18th annual ACM symposium on theory of computing, ACM, New York, 1986) and Boppana (Inf Process Lett 63(5): 257–261, 1997). Our proof technique studies a random process where the switching lemma is applied to formulas in an efficient manner.