Separation of AC$^0[\oplus]$ Formulas and Circuits.

This paper gives the first separation between the power of {em formulas} and {em circuits} of equal depth in the basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) le O(frac{log n}{loglog n})$, that there exist {em polynomial-size depth-$d$ circuits} that are not equivalent {em depth-$d$ formulas of size $n^{o(d)}$} (moreover, this is optimal in that $n^{o(d)}$ cannot be improved $n^{O(d)}$). This result is obtained by a combination of new lower and upper bounds for {em Approximate Majorities}, the class of Boolean functions ${0,1}^n to {0,1}$ that agree with the Majority function on $3/4$ fraction of inputs. $mathrm{AC}^0[oplus]$ formula lower bound: We show that every depth-$d$ formula of size $s$ has a {em $1/8$-error polynomial approximation} over $mathbb{F}_2$ of degree $O(frac{1}{d}log s)^{d-1}$. This strengthens a classic $O(log s)^{d-1}$ degree approximation for underline{circuits} due Razborov. Since the Majority function has approximate degree $Theta(sqrt n)$, this result implies an $exp(Omega(dn^{1/2(d-1)}))$ lower bound on the depth-$d$ formula size of all Approximate Majority functions for all $d(n) le O(log n)$. Monotone $mathrm{AC}^0$ circuit upper bound: For all $d(n) le O(frac{log n}{loglog n})$, we give a randomized construction of depth-$d$ monotone $mathrm{AC}^0$ circuits (without NOT or MOD$_2$ gates) of size $exp(O(n^{1/2(d-1)}))$ that compute an Approximate Majority function. This strengthens a construction of underline{formulas} of size $exp(O(dn^{1/2(d-1)}))$ due Amano.