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Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

Authors Rafael Oliveira, Amir Shpilka, Ben Lee Volk

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Rafael Oliveira
Amir Shpilka
Ben Lee Volk

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Rafael Oliveira, Amir Shpilka, and Ben Lee Volk. Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 304-322, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.CCC.2015.304

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(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~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(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(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-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~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

Keywords
  • Arithmetic Circuits
  • Derandomization
  • Polynomial Identity Testing

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