Subexponential size hitting sets for bounded depth multilinear formulas

Rafael Oliveira, Amir Shpilka, Ben Lee Volk

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-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 (Õ)n2/3+2δ/3)). This implies a lower bound of exp($ΩT(n1/2)) for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(nδ), we give a hitting set of size exp (Õ)n2/3+4δ/3)). This implies a lower bound of exp($ΩT(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 of size exp(nδ), where δ = O(1/√5d). This result implies a lower bound of roughly exp($ΩT(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
Title of host publication30th Conference on Computational Complexity, CCC 2015
EditorsDavid Zuckerman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages304-322
Number of pages19
ISBN (Electronic)9783939897811
DOIs
StatePublished - 1 Jun 2015
Event30th Conference on Computational Complexity, CCC 2015 - Portland, United States
Duration: 17 Jun 201519 Jun 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume33
ISSN (Print)1868-8969

Conference

Conference30th Conference on Computational Complexity, CCC 2015
Country/TerritoryUnited States
CityPortland
Period17/06/1519/06/15

Keywords

  • Arithmetic circuits
  • Derandomization
  • Polynomial identity testing

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