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Optimal Shallow Circuits for Majority — arXiv:2609.34029

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.34029
License (as archived): CC0-1.0
Archived: 2026-10-08T00:43:21.514Z
SHA-256 of the archived text: 0ee47cf87c3d0d1973e456c82cc6883a4a9f478e4db1b3f5dd5ee18f275ba49a
Stable URL of this page: https://metraipolis.com/documento/dc_d13719facefe

How to cite

Optimal Shallow Circuits for Majority — arXiv:2609.34029 — arXiv (en), https://arxiv.org/abs/2609.34029 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_d13719facefe (Metraipolis archive dc_d13719facefe, sha256 0ee47cf87c3d0d1973e456c82cc6883a4a9f478e4db1b3f5dd5ee18f275ba49a)

BibTeX

@misc{dcd13719facefe,
  title = {Optimal Shallow Circuits for Majority — arXiv:2609.34029},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.34029},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_d13719facefe; sha256 0ee47cf87c3d0d1973e456c82cc6883a4a9f478e4db1b3f5dd5ee18f275ba49a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.34029. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Four decades on, Håstad'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 Håstad's lower bound. The proof draws inspiration from well-loved combinatorial tools, including the color-coding technique of Alon, Yuster, and Zwick (1995).

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-29 in cs.CC).
Categorie arXiv: Computational Complexity (cs.CC).
Pagina del lavoro: https://arxiv.org/abs/2609.34029 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T00:17:42.548Z dalla pagina https://arxiv.org/catchup/cs.CC/2026-09-29?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).