Criticality of AC⁰-Formulae

Authors Prahladh Harsha , Tulasimohan Molli, Ashutosh Shankar

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Author Details

Prahladh Harsha
  • Tata Institute of Fundamental Research, Mumbai, India
Tulasimohan Molli
  • Tata Institute of Fundamental Research, Mumbai, India
Ashutosh Shankar
  • Tata Institute of Fundamental Research, Mumbai, India


The first and the second authors spent several years thinking about this problem and we are indebted to several people along the way. First and foremost, we thank Jaikumar Radhakrishnan and Ramprasad Saptharishi for spending innumerable hours in the various stages of this project going over several failed attempts and potential proofs and giving us very helpful feedback along the way. We are grateful to Ben Rossman for discussions in the early stages of this project as well as pointing out an error in the previous version of this proof. We would also like to thank Srikanth Srinivasan, Siddharth Bhandari, Yuval Filmus and Mrinal Kumar for their comments and feedback.

Cite AsGet BibTex

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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • AC⁰ circuits
  • AC⁰ formulae
  • criticality
  • switching lemma
  • correlation bounds


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