Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity

TL;DR

This paper establishes exponential lower bounds for homogeneous depth-3 and depth-4 arithmetic circuits computing certain polynomials in VNP, using new complexity measures, with implications for boolean circuit complexity and separation of complexity classes.

Contribution

It introduces new exponential lower bounds for homogeneous depth-3 and depth-4 arithmetic circuits for polynomials in VNP, and develops the shifted evaluation dimension as a novel complexity measure.

Findings

Exponential lower bounds for depth-3 circuits in VNP.

Exponential lower bounds for depth-4 circuits with bounded degree in VNP.

Implications for boolean circuit complexity and class separations.

Abstract

We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-$3$ and depth-$4$ arithmetic circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-$3$ arithmetic circuits for a polynomial in $VNP$. 2. Exponential lower bounds for homogeneous depth-$4$ arithmetic circuits with bounded individual degree for a polynomial in $VNP$. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-$4$ arithmetic circuits for the Permanent imply a separation between ${\#}P$ and $ACC$. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean circuit complexity. Besides, it is known from a recent result of Kumar and Saptharishi [KS15] that over constant sized finite fields, strong enough average case functional lower bounds for homogeneous depth-$4$ circuits imply superpolynomial lower bounds for homogeneous depth-$5$ circuits. Our proofs are based on a family of new complexity measures called shifted evaluation dimension, and might be of independent interest.