A Quadratic Size-Hierarchy Theorem for Small-Depth Multilinear Formulas
Publisher Information
- Conference: ICALP 2018, Pages 36:1-36:13 (LIPIcs, Vol. 107)
Abstract
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\mathbb{F}[x_1,\ldots,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 \varepsilon-biased spaces.