Separation of AC 0 [⊕] Formulas and Circuits release_uicebiktbzbc5ipfgyasuqdd2y

Abstract

This paper gives the first separation between the power of formulas and circuits of equal depth in the AC 0 [⊕] 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 → {0, 1} that agree with the Majority function on 3/4 fraction of inputs. AC 0 [⊕ ⊕ ⊕] formula lower bound. We show that every depth-d AC 0 [⊕] formula of size s has a 1/8-error polynomial approximation over F 2 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(dn 1/2(d−1))) lower bound on the depth-d AC 0 [⊕] 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 [1].
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