# Separation of AC0 [⊕] formulas and circuits

Conference Paper

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( log n/log log n ), 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 no(d) 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 F of degree O( 1/d log s) . This strengthens a classic O(log s) degree approximation for circuits due to Razborov [12]. Since the Majority function has approximate degree ⊖(√ n), this result implies an exp(ω(dn )) 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( log n/log log n ), we give a randomized construction of depth-d monotone AC circuits (without NOT or MOD gates) of size exp(O(n ))) that compute an Approximate Majority function. This strengthens a construction of formulas of size exp(O(dn )) due to Amano [1]. 0 o(d) O(d) n 0 0 d-1 d-1 1/2(d-1) 0 0 0 1/2(d-1 1/2(d-1) 2 2

### Cited Authors

• Rossman, B; Srinivasan, S

• July 1, 2017

• 80 /

• 1868-8969

### International Standard Book Number 13 (ISBN-13)

• 9783959770415

### Digital Object Identifier (DOI)

• 10.4230/LIPIcs.ICALP.2017.50

• Scopus