Small Circuits Imply Efficient Arthur-Merlin Protocols

The inner product function (x, y) = ∑_i x_iy_i mod 2 can be easily computed by a (linear-size) AC⁰(⨁) circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates. But what if we impose the restriction that the parity gates can only be on the bottom most layer (closest to the input)? Namely, can the inner product function be computed by an AC⁰ circuit composed with a single layer of parity gates? This seemingly simple question is an important open question at the frontier of circuit lower bound research. In this work, we focus on a minimalistic version of the above question. Namely, whether the inner product function cannot be approximated by a small DNF augmented with a single layer of parity gates. Our main result shows that the existence of such a circuit would have unexpected implications for interactive proofs, or more specifically, for interactive variants of the Data Streaming and Communication Complexity models. In particular, we show that the existence of such a small (i.e., polynomial-size) circuit yields: 1. An O(d)-message protocol in the Arthur-Merlin Data Streaming model for every n-variate, degree d polynomial (over GF(2)), using only Õ (d) · log(n) communication and space complexity. In particular, this gives an AM[2] Data Streaming protocol for a variant of the well-studied triangle counting problem, with poly-logarithmic communication and space complexities. 2. A 2-message communication complexity protocol for any sparse (or low degree) polynomial, and for any function computable by an AC⁰(⨁) circuit. Specifically, for the latter, we obtain a protocol with communication complexity that is poly-logarithmic in the size of the AC⁰(⨁) circuit.