Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

Rafael Oliveira*, Amir Shpilka, Ben lee Volk

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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(nδ) , we give a hitting set of size exp(O~ (n2 / 3 + 2 δ / 3)). This implies a lower bound of exp(Ω ~ (n1 / 2)) for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(nδ) , we give a hitting set of size exp(O~ (n2 / 3 + 4 δ / 3)). This implies a lower bound of exp(Ω ~ (n1 / 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(n1 - δ) , for regular depth-d multilinear formulas with formal degree at most n and size exp(nδ) , 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 languageEnglish
Pages (from-to)455-505
Number of pages51
JournalComputational Complexity
Volume25
Issue number2
DOIs
StatePublished - 1 Jun 2016

Keywords

  • 68Q05
  • 68Q15
  • 68Q17

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