The Average Sensitivity of Bounded-Depth Formulas

We show that unbounded fan-in Boolean formulas of depth d + 1 and size s have average sensitivity O(1/dlog s)^d . In particular, this gives a tight 2^Ω(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^Ω(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.