## 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^{δ}) , we give a hitting set of size exp(O~ (n^{2} ^{/} ^{3} ^{+} ^{2} ^{δ} ^{/} ^{3})). This implies a lower bound of exp(Ω ~ (n^{1 / 2})) for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(n^{δ}) , we give a hitting set of size exp(O~ (n^{2} ^{/} ^{3} ^{+} ^{4} ^{δ} ^{/} ^{3})). This implies a lower bound of exp(Ω ~ (n^{1 / 4})) for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of + , × gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp(n^{1} ^{-} ^{δ}) , for regular depth-d multilinear formulas with formal degree at most n and size exp(n^{δ}) , where δ=O(1/5d). This result implies a lower bound of roughly exp(Ω~(n1/5d)) 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).

Original language | English |
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Pages (from-to) | 455-505 |

Number of pages | 51 |

Journal | Computational Complexity |

Volume | 25 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jun 2016 |

## Keywords

- 68Q05
- 68Q15
- 68Q17

## All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Theoretical Computer Science
- Computational Theory and Mathematics
- General Mathematics