Separation of AC[⊕] Formulas and Circuits

This paper gives the first separation between the power of formulas and circuits of equal depth in the AC[⊕] basis (unbounded fan-in AND, OR, NOT and MOD2 gates). We show, for all d(n) ≤ O( logn log logn ), that there exist polynomial-size depth-d circuits that are not equivalent to depth-d formulas of size n (moreover, this is optimal in that n cannot be improved to n). This result is obtained by a combination of new lower and upper bounds for Approximate Majorities, the class of Boolean functions {0, 1} → {0, 1} that agree with the Majority function on 3/4 fraction of inputs. AC[⊕] formula lower bound. We show that every depth-d AC[⊕] formula of size s has a 1/8-error polynomial approximation over F2 of degree O( 1 d log s) d−1. This strengthens a classic O(log s)d−1 degree approximation for circuits due to Razborov [12]. Since the Majority function has approximate degree Θ( √ n), this result implies an exp(Ω(dn1/2(d−1))) lower bound on the depth-d AC[⊕] formula size of all Approximate Majority functions for all d(n) ≤ O(log n). Monotone AC circuit upper bound. For all d(n) ≤ O( logn log logn ), we give a randomized construction of depth-d monotone AC circuits (without NOT or MOD2 gates) of size exp(O(n1/2(d−1))) that compute an Approximate Majority function. This strengthens a construction of formulas of size exp(O(dn1/2(d−1))) due to Amano [1].