A Top-Down Depth-Four Circuit Lower Bound for Majority

2026-08-09

AI Summary

This work addresses the absence of top-down lower bound proofs for the majority function in depth-four circuits by extending the top-down approach to this highly sensitive function, overcoming significant technical barriers. By analyzing slices of the Boolean cube around Hamming weight $n/2$ and integrating the robust sunflower lemma, mirror set constructions, and block unpredictability, the authors develop a structured framework tailored to symmetric functions like majority. This yields a new lower bound for the majority function in depth-four circuits, substantially generalizing the parity function result from FOCS 2023 and establishing a novel paradigm for studying the circuit complexity of symmetric Boolean functions.

Abstract

We present a top-down depth-four circuit lower bound for Majority function by extending recent work of Göös, Riazanov, Sofronova, and Sokolov (FOCS 2023), who gave a top-down proof of a depth-four circuit lower bound for Parity which relies on the robust sunflower to construct a mirror set and the block unpredictability to find the local limits. The main challenge for the case of Majority is to construct a corresponding mirror set, the difference is that to flip the value of Majority function, one may have to flip many bits of the input Boolean string, while for Parity, flipping one bit suffices. We avoid this flipping by considering slices of the Boolean cube, that is, Boolean strings of fixed Hamming weight approximately $n/2$.

Problem

• Majority function
• circuit lower bound
• depth-four circuit
• mirror set
• Boolean cube

Innovation

• top-down
• depth-four circuit
• Majority function
• mirror set
• Boolean cube slices