Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

Abstract

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models. For depth-3 multilinear formulas, of size $\exp(n^\delta)$, we give a hitting set of size $\exp(\tilde{O}(n^{2/3 + 2\delta/3}))$. This implies a lower bound of $\exp(\tilde{\Omega}(n^{1/2}))$ for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size $\exp(n^\delta)$, we give a hitting set of size $\exp(\tilde{O}(n^{2/3 + 4\delta/3}))$. This implies a lower bound of $\exp(\tilde{\Omega}(n^{1/4}))$ for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of $+,\times$ gates, where all gates at layer $i$ have the same fan-in. We give a hitting set of size (roughly) $\exp\left(n^{1- \delta} \right)$, for regular depth-$d$ multilinear formulas of size $\exp(n^\delta)$, where $\delta = O(\frac{1}{\sqrt{5}^d})$. This result implies a lower bound of roughly $\exp(\tilde{\Omega}(n^{\frac{1}{\sqrt{5}^d}}))$ for such formulas. We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known. Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs).