On the size of depth-three boolean circuits for computing multilinear functions

This paper introduces and initiates a study of a new model of arithmetic circuits coupled with new complexity measures. The new model consists of multilinear circuits with arbitrary multilinear gates, rather than the standard multilinear circuits that use only addition and multiplication gates. In light of this generalization, the arity of gates becomes of crucial importance and is indeed one of our complexity measures. Our second complexity measure is the number of gates in the circuit, which (in our context) is significantly different from the number of wires in the circuit (which is typically used as a measure of size). Our main complexity measure, denoted AN(·), is the maximum of these two measures (i.e., the maximum between the arity of the gates and the number of gates in the circuit). We also consider the depth of such circuits, focusing on depth-two and unbounded depth. Our initial motivation for the study of this arithmetic model is the fact that its two main variants (i.e., depth-two and unbounded depth) yield natural classes of depth-three Boolean circuits for computing multilinear functions. The resulting circuits have size that is exponential in the new complexity measure. Hence, lower bounds on the new complexity measure yield size lower bounds on a restricted class of depth-three Boolean circuits (for computing multilinear functions). Such lower bounds are a sanity check for our conjecture that multilinear functions of relatively low degree over GF(2) are good candidates for obtaining exponential lower bounds on the size of constant-depth Boolean circuits (computing explicit functions). Specifically, we propose to move gradually from linear functions to multilinear ones, and conjecture that, for any t≥2, some explicit t-linear functions F:({0,1}n)t→{0,1} require depth-three circuits of size exp(Ω(tn^{t / ( t + 1 )})). Letting AN₂(·) denote the complexity measure AN(·), when minimized over all depth-two circuits of the above type, our main results are as follows. For every t-linear function F, it holds that AN(F)≤AN₂(F)=O((tn)^{t / ( t + 1 )}).For almost all t-linear function F, it holds that AN₂(F)≥AN(F)=Ω((tn)^{t / ( t + 1 )}).There exists a bilinear function F such that AN(F)=O(n) but AN₂(F)=Ω(n^2/3). The main open problem posed in this paper is proving that AN₂(F)≥AN(F)=Ω((tn)^{t / ( t + 1 )}) holds for an explicit t-linear function F, with t≥2. For starters, we seek lower bound of Ω((tn)^0.51) for an explicit t-linear function F, preferably for constant t. We outline an approach that reduces this challenge (for t=3) to a question regarding matrix rigidity.