# A Duality Between Depth-Three Formulas and Approximation by Depth-Two

## TL;DR

This paper reveals a duality between depth-3 formulas and depth-2 approximation, leading to new bounds and characterizations for circuit complexity, including tight bounds for parity and majority functions.

## Contribution

It establishes a duality linking depth-3 formulas with depth-2 approximation, enabling new complexity bounds and insights into circuit size and function approximation.

## Key findings

- Depth-3 formula size is inversely related to depth-2 correlation bounds.
- Any function can be approximated by a small CNF with one-sided error, tight up to a constant.
- Depth-3 formulas computing parity require exponential size, showing a quadratic separation.

## Abstract

We establish an explicit link between depth-3 formulas and one-sided approximation by depth-2 formulas, which were previously studied independently. Specifically, we show that the minimum size of depth-3 formulas is (up to a factor of n) equal to the inverse of the maximum, over all depth-2 formulas, of one-sided-error correlation bound divided by the size of the depth-2 formula, on a certain hard distribution. We apply this duality to obtain several consequences: 1. Any function f can be approximated by a CNF formula of size $O(\epsilon 2^n / n)$ with one-sided error and advantage $\epsilon$ for some $\epsilon$, which is tight up to a constant factor. 2. There exists a monotone function f such that f can be approximated by some polynomial-size CNF formula, whereas any monotone CNF formula approximating f requires exponential size. 3. Any depth-3 formula computing the parity function requires $\Omega(2^{2 \sqrt{n}})$ gates, which is tight up to a factor of $\sqrt n$. This establishes a quadratic separation between depth-3 circuit size and depth-3 formula size. 4. We give a characterization of the depth-3 monotone circuit complexity of the majority function, in terms of a natural extremal problem on hypergraphs. In particular, we show that a known extension of Turan's theorem gives a tight (up to a polynomial factor) circuit size for computing the majority function by a monotone depth-3 circuit with bottom fan-in 2. 5. AC0[p] has exponentially small one-sided correlation with the parity function for odd prime p.