A Quadratic Size-Hierarchy Theorem for Small-Depth Multilinear Formulas
Suryajith Chillara, Nutan Limaye, Srikanth Srinivasan · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2018
We show explicit separations between the expressive powers of multilinear formulas of small-depth and all polynomial sizes. Formally, for any s = s(n) = n^{O(1)} and any delta>0, we construct explicit families of multilinear polynomials P_n in F[x_1,...,x_n] that have multilinear formulas of size s and depth three but no multilinear formulas of size s^{1/2-delta} and depth o(log n/log log n). As far as we know, this is the first such result for an algebraic model of computation. Our proof can be viewed as a derandomization of a lower bound technique of Raz (JACM 2009) using epsilon-biased spaces.