Abstract

This paper gives the first separation between the power of {\em formulas} and {\em circuits} of equal depth in the $\mathrm{AC}^0[\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) \le O(\frac{\log n}{\log\log n})$, that there exist {\em polynomial-size depth-$d$ circuits} that are not equivalent to {\em 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 {\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-$d$ $\mathrm{AC}^0[\oplus]$ 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 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-$d$ $\mathrm{AC}^0[\oplus]$ formula size of all Approximate Majority functions for all $d(n) \le O(\log n)$. Monotone $\mathrm{AC}^0$ circuit upper bound: For all $d(n) \le O(\frac{\log n}{\log\log n})$, 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 to Amano.

Highlights

  • The relative power of formulas versus circuits is one of the great mysteries in complexity theory

  • With respect to either basis, there is a natural depth-d analogue of the NC1 vs. P/poly question (where d = d(n) is a parameter that may depend on n), namely whether every language decidable by polynomial-size depth-d circuits is decidable by depth-d formulas of size no(d) (i. e., better than the trivial nO(d) upper bound)

  • On the upper bound side, it is known from the work of O’Donnell and Wimmer [12] and Amano [1] that there exist Approximate Majorities that can be computed by monotone AC0 formulas of depth d and size exp(O(dn1/2(d−1))). (Note that the double exponent 1/(2(d − 1)) is the same in the upper and lower bounds.) In order to separate AC0[⊕] formulas and circuits, we show that both the upper and lower bounds from the previous paragraph are tight

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Summary

Introduction

The relative power of formulas versus circuits is one of the great mysteries in complexity theory. With respect to either basis, there is a natural depth-d analogue of the NC1 vs P/poly question (where d = d(n) is a parameter that may depend on n), namely whether every language decidable by polynomial-size depth-d circuits is decidable by depth-d formulas of size no(d) We show how to apply the Polynomial Method to obtain better parameters in the approximation of AC0[⊕] formulas by low-degree polynomials over F2 This leads to an exp(Ω(dn1/2(d−1))) lower bound on the AC0[⊕] formula size of all Approximate Majority functions. The other half of our formulas vs circuits separation comes from an exp(O(n1/2(d−1))) upper bound on the AC0[⊕] circuit size of some Approximate Majority function This upper bound is realized by a randomized construction of monotone AC0 circuits (AC0 circuits without NOT or MOD2 gates). For all 2 ≤ d ≤ O(log n), the depth-d AC0[⊕] formula size of any n-variable Approximate Majority function is exp(Ω(dn1/2(d−1)))

For all log n
Proof outline
Preliminaries
Lower bound
High-level idea
Formal proof
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