TY - GEN

T1 - Subexponential size hitting sets for bounded depth multilinear formulas

AU - Oliveira, Rafael

AU - Shpilka, Amir

AU - Volk, Ben Lee

N1 - Publisher Copyright:
© Rafael Oliveira, Amir Shpilka, and Ben Lee Volk; licensed under Creative Commons License CC-BY.

PY - 2015/6/1

Y1 - 2015/6/1

N2 - 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).

AB - 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).

KW - Arithmetic circuits

KW - Derandomization

KW - Polynomial identity testing

UR - http://www.scopus.com/inward/record.url?scp=84958255843&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CCC.2015.304

DO - 10.4230/LIPIcs.CCC.2015.304

M3 - פרסום בספר כנס

AN - SCOPUS:84958255843

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 304

EP - 322

BT - 30th Conference on Computational Complexity, CCC 2015

A2 - Zuckerman, David

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 17 June 2015 through 19 June 2015

ER -