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/d log s)^d. In particular, this gives a tight 2^Ω(d(n1/d-1)) lower bound on the size of depth d + 1 formulas computing the PARITY function. These results strengthen the corresponding O(log s)^d and 2^Ω(n1/d) bounds for circuits due to Boppana (1997) and Hastad (1986). Our proof technique studies a random process associated with formulas, in which the Switching Lemma is efficiently applied to subformulas.