AC0∘MOD2 lower bounds for the Boolean Inner Product

DOI

10.1016/j.jcss.2018.04.006

Document Type

Journal Article

Publication Date

11-1-2018

Publication Title

Journal of Computer and System Sciences

Volume

97

First Page

45

Last Page

59

ISSN

220000

Keywords

Boolean analysis, Circuit complexity, Lower bounds

Abstract

AC0∘ MOD2 circuits are AC0 circuits augmented with a layer of parity gates just above the input layer. We study AC0∘MOD2 circuit lower bounds for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have highlighted this problem as a frontier problem in circuit complexity that arose both as a first step towards solving natural special cases of the matrix rigidity problem and as a candidate for constructing pseudorandom generators of minimal complexity. We give the first superlinear lower bound for the Boolean Inner Product function against AC0∘MOD2 of depth four or greater. Specifically, we prove a superlinear lower bound for circuits of arbitrary constant depth, and an ?˜(n2) lower bound for the special case of depth-4 AC0∘MOD2.

Open Access

Green Accepted

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