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

Hao Wu; Yaqiao Li

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arXiv · 2608.09070

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

Hao Wu·Yaqiao Li

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Abstract

We present a top-down depth-four circuit lower bound for Majority function by extending recent work of G\"{o}\"{o}s, 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 $n/2$.

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Hao Wu, Yaqiao Li. 2026-08-10. A Top-Down Depth-Four Circuit Lower Bound for Majority. https://arxiv.org/abs/2608.09070

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