Separation of AC^0[oplus] Formulas and Circuits

This paper gives the first separation between the power of formulas and circuits of equal depth in the AC^0[oplus] basis (unbounded fan-in AND, OR, NOT and MOD_2 gates). We show, for all d(n) <= O(log n/log log n), that there exist polynomial-size depth-d circuits that are not equivalent to depth-d formulas of size n^{o(d)} (moreover, this is optimal in that n^{o(d)} cannot be improved to n^{O(d)}). This result is obtained by a combination of new lower and upper bounds for 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. AC^0[oplus] formula lower bound. We show that every depth-d AC^0[oplus] formula of size s has a (1/8)-error polynomial approximation over F_2 of degree O((log s)/d)^{d-1}. This strengthens a classic $O(log s)^{d-1}$ degree approximation for circuits due to 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 AC^0[oplus] formula size of all Approximate Majority functions for all d(n) <= O(log n). Monotone AC^0 circuit upper bound. For all d(n) <= O(log n/log log n), we give a randomized construction of depth-d monotone 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 formulas of size exp(O(dn^{1/2(d-1)})) due to Amano