Improved bounds on the AN-complexity of $O(1)$-linear functions

Abstract

We consider arithmetic circuits with arbitrary gates for computing Boolean functions that are represented by low-degree polynomials over GF(2). An adequate complexity measure for such circuits is the maximum between the arity of the gates and their number. This model and the corresponding complexity measure, called AN-complexity, were introduced by Goldreich and Wigderson (ECCC, TR13-043, 2013), and it is meaningful only for low-degree polynomials (where the arity of a gate is not due to the degree of the polynomial that the gate computes but rather to the number of variables in it).

The AN-complexity of a function yields an upper bound on the size of depth-three Boolean circuits for computing the function. Specifically, the depth-three size of Boolean circuits is at most exponential in the AN-complexity of the function. Hence, proving linear lower bounds on the AN-complexity of explicit O(1)-linear functions is an essential step toward proving that depth-three Boolean circuits for these functions require exponential size.

In this work, we present explicit O(1)-linear functions that require depth-two arithmetic circuits of almost linear AN-complexity. Specifically, for every $\epsilon > 0$, we show an explicit poly $({1/ \epsilon})$-linear function $f:\{0,1\}^{\mathrm{poly} (1/\epsilon)\cdot n}\to \{0,1\}$ such that any depth-two arithmetic circuit that computes f must use gates of arity at least $n^{1-\epsilon}$. In particular, for every $\epsilon > 0$ and $t=O(1/\epsilon^2)$, the $\Omega(n^{1-\epsilon})$ lower bound holds also for the t-linear function

$$\begin{aligned}f(x^{(1)},x^{(2)},...,x^{(t)}) = \sum_{i_1,...,i_{t-1}\in[n]} \left(\prod_{j\in[t-1]} x^{(j)}_{i_{j}}\right) \cdot x^{(t)}_{{i_1+i_2+\cdots+i_{t-1}}} \end{aligned}$$

This improves over a corresponding lower bound of $\tilde \Omega(n^{2/3})$ that was known for an explicit trilinear function (Goldreich and Tal, Computational Complexity, 2018), but leaves open the problem of showing similar AN-complexity lower bounds for arithmetic circuits of larger depth.

A key aspect in our proof is considering many (extremely skewed) random restrictions, and contrasting the sum of the values of the original function and the circuit (which supposedly computes it) taken over a (carefully chosen) subset of these random restrictions. We show that if the original circuit has too low AN-complexity, then these two sums cannot be equal, which yields a contradiction.