Optimal Shallow Circuits for Majority

2026-09-27

AI Summary

This study addresses the long-standing open problem in circuit complexity of whether the lower bound for the Majority function in AC0 circuits matches Håstad's classical lower bound for Parity. Leveraging GPT-6 Astra to assist in discovering concise constructions, combined with analytical techniques from theoretical computer science, this work constructs symmetric function circuits of depth $d$ and size $2^{O(n^{1/(d-1)})}$. The primary contribution is the first explicit construction that matches Håstad's lower bound, demonstrating that Majority and Parity share identical asymptotic circuit complexity. This result conclusively resolves the optimality conjecture for the Majority function within the AC0 model.

Abstract

Four decades on, H{\aa}stad's classical $2^{\Omega(n^{1/(d-1)})}$ lower bound for depth-$d$ circuits computing Parity remains the best known $\mathrm{AC}^0$ circuit lower bound for any explicit function. Majority has long been a compelling candidate for stronger lower bounds: the most natural circuits computing it are substantially larger than those for Parity and have repeatedly been conjectured to be optimal. We present a simple construction, found by GPT-6 Astra, of depth-$d$ circuits of size $2^{O(n^{1/(d-1)})}$ for any symmetric function. This result settles the asymptotic $\mathrm{AC}^0$ circuit complexity of Majority, matching H{\aa}stad's lower bound.

Problem

• Majority function
• AC^0 circuit complexity
• shallow circuits
• circuit lower bounds
• symmetric functions