The Average Sensitivity of Bounded-Depth Formulas

TL;DR

This paper establishes tight bounds on the average sensitivity of bounded-depth boolean formulas, leading to new lower bounds on formula size for computing parity, and introduces a novel proof technique involving a random process and the Switching Lemma.

Contribution

It provides the first tight bounds on average sensitivity for bounded-depth formulas and improves lower bounds for the size of formulas computing parity, using a new proof approach.

Findings

Average sensitivity of formulas is tightly bounded by $O((rac{1}{d} ext{log } s)^d)$.

Derived a lower bound of $2^{ ext{Omega}(d(n^{1/d}-1))}$ on formula size for parity.

Introduced a novel proof technique involving a random process and the Switching Lemma.

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 \textsc{parity} function. These results strengthen the corresponding $2^{\Omega(n^{1/d})}$ and $O(\log s)^d$ bounds for circuits due to H{\aa}stad (1986) and Boppana (1997). Our proof technique studies a random process where the Switching Lemma is applied to formulas in an efficient manner.