Lower bounds for depth-three circuits with equals and mod-gates

Abstract

We say an integer polynomial p, on Boolean inputs, weakly m-represents a Boolean function f if p is non-constant and is zero (mod m) whenever f is zero. In this paper we prove that if a polynomial weakly m-represents the Mod_q function on n inputs, where q and m are relatively prime and m is otherwise arbitrary, then the degree of the polynomial is Ω(n). This generalizes previous results of Barrington, Beigel and Rudich [BBR] and Tsai [Tsai], which held only for constant (or slowly growing) m. The proof technique given here is quite different and somewhat simpler. We use a method in which the inputs are represented as complex q ^th roots of unity (following Barrington and Straubing [BS]). The representation is used to take advantage of a variant of the inverse Fourier transform and elementary properties of the algebraic integers. As a corollary of the main theorem and the proof of Toda's theorem, if q, p are distinct primes, any depth-three circuit which computes the Mod_q function, and consists of an equals gate at the output, Mod_p-gates at the next level, and AND-gates of small fan-in at the inputs, must be of exponential size. In terms of Turing machine complexity classes, there is an oracle A such that $Mod_q P^A \nsubseteq C_ = P^{Mod_p P^A }$.