AC0∘MOD2 lower bounds for the Boolean Inner Product

Abstract AC 0 ∘ MOD 2 circuits are AC 0 circuits augmented with a layer of parity gates just above the input layer. We study AC 0 ∘ MOD 2 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 AC 0 ∘ MOD 2 of depth four or greater. Specifically, we prove a superlinear lower bound for circuits of arbitrary constant depth, and an Ω ˜ ( n 2 ) lower bound for the special case of depth-4 AC 0 ∘ MOD 2 .

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