A Quadratic Size-Hierarchy Theorem for Small-Depth Multilinear Formulas

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Suryajith Chillara, Nutan Limaye and Srikanth Srinivasan

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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.