AC0-MOD2 lower bounds for the Boolean Inner Product. Cheraghchi, M., Grigorescu, E., Juba, B., Wimmer, K., & Xie, N. *Journal of Computer and System Sciences*, 97:45–59, 2018. Preliminary version in Proceedings of ICALP 2016.Link Paper doi abstract bibtex 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 $\tilde{\Omega(n^2)}$ lower bound for the special case of depth-4 AC0-MOD2.

@ARTICLE{ref:CGJWX18,
author = {Mahdi Cheraghchi and Elena Grigorescu and Brendan
Juba and Karl Wimmer and Ning Xie},
title = {{AC0-MOD2} lower bounds for the {Boolean} Inner
Product},
journal = {Journal of Computer and System Sciences},
volume = 97,
pages = "45--59",
year = 2018,
doi = "10.1016/j.jcss.2018.04.006",
url_Link =
"http://www.sciencedirect.com/science/article/pii/S002200001830518X",
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 $\tilde{\Omega(n^2)}$ lower bound for the special
case of depth-4 AC0-MOD2.",
note = {Preliminary version in Proceedings of {ICALP 2016.}},
url_Paper = {https://eccc.weizmann.ac.il//report/2015/030/}
}

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