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Criticality of AC⁰-Formulae

Authors: Prahladh Harsha, Tulasimohan Molli, and Ashutosh Shankar

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Rossman [In Proc. 34th Comput. Complexity Conf., 2019] introduced the notion of criticality. The criticality of a Boolean function f : {0,1}ⁿ → {0,1} is the minimum λ ≥ 1 such that for all positive integers t and all p ∈ [0,1], Pr_{ρ∼ℛ_p}[DT_{depth}(f|_ρ) ≥ t] ≤ (pλ)^t, where ℛ_p refers to the distribution of p-random restrictions. Håstad’s celebrated switching lemma shows that the criticality of any k-DNF is at most O(k). Subsequent improvements to correlation bounds of AC⁰-circuits against parity showed that the criticality of any AC⁰-circuit of size S and depth d+1 is at most O(log S)^d and any regular AC⁰-formula of size S and depth d+1 is at most O((1/d)⋅log S)^d. We strengthen these results by showing that the criticality of any AC⁰-formula (not necessarily regular) of size S and depth d+1 is at most O((log S)/d)^d, resolving a conjecture due to Rossman. This result also implies Rossman’s optimal lower bound on the size of any depth-d AC⁰-formula computing parity [Comput. Complexity, 27(2):209-223, 2018.]. Our result implies tight correlation bounds against parity, tight Fourier concentration results and improved #SAT algorithm for AC⁰-formulae.

Cite as

Prahladh Harsha, Tulasimohan Molli, and Ashutosh Shankar. Criticality of AC⁰-Formulae. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 19:1-19:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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  author =	{Harsha, Prahladh and Molli, Tulasimohan and Shankar, Ashutosh},
  title =	{{Criticality of AC⁰-Formulae}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{19:1--19:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.19},
  URN =		{urn:nbn:de:0030-drops-182898},
  doi =		{10.4230/LIPIcs.CCC.2023.19},
  annote =	{Keywords: AC⁰ circuits, AC⁰ formulae, criticality, switching lemma, correlation bounds}
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