computational complexity

, Volume 25, Issue 2, pp 455–505

# Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

Article

## 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$${\left(\tilde{O}\left(n^{2/3 + 2\delta/3}\right) \right)}$$. 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$${\left(\tilde{O}\left(n^{2/3 + 4\delta/3}\right) \right)}$$. 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 with formal degree at most n and size exp$${(n^\delta)}$$, where $${\delta = O(1/{\sqrt{5}^d})}$$. This result implies a lower bound of roughly exp$${(\tilde{\Omega}(n^{1/{\sqrt{5}^d}}))}$$ 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).

### Subject classification

68Q05 68Q15 68Q17

### Keywords

arithmetic circuits polynomial identity testing derandomization

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## Authors and Affiliations

1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA
2. 2.Department of Computer ScienceTel Aviv UniversityTel AvivIsrael