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A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates

Authors Ivan Mihajlin, Anastasia Sofronova

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Author Details

Ivan Mihajlin
  • Leonhard Euler International Mathematical Institute in Saint Petersburg, Russia
Anastasia Sofronova
  • Leonhard Euler International Mathematical Institute in Saint Petersburg, Russia
  • St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Russia


The authors would like to thank Kaave Hosseini for his invaluable insights on Lemma 27, Alexander Kulikov, Alexander Smal and Artur Riazanov for fruitful discussions and comments on the draft. The authors would also like to thank anonymous reviewers.

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Ivan Mihajlin and Anastasia Sofronova. A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 13:1-13:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We prove that a modification of Andreev’s function is not computable by (3 + α - ε) log(n) depth De Morgan formula with (2α - ε)log{n} layers of AND gates at the top for any 0 < α < 1/5 and any constant ε > 0. In order to do this, we prove a weak variant of Karchmer-Raz-Wigderson conjecture. To be more precise, we prove the existence of two functions f : {0,1}ⁿ → {0,1} and g : {0,1}ⁿ → {0,1}ⁿ such that f(g(x) ⊕ y) is not computable by depth (1 + α - ε) n formulas with (2 α - ε) n layers of AND gates at the top. We do this by a top-down approach, which was only used before for depth-3 model. Our technical contribution includes combinatorial insights into structure of composition with random boolean function, which led us to introducing a notion of well-mixed sets. A set of functions is well-mixed if, when composed with a random function, it does not have subsets that agree on large fractions of inputs. We use probabilistic method to prove the existence of well-mixed sets.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • formula complexity
  • communication complexity
  • Karchmer-Raz-Wigderson conjecture
  • De Morgan formulas


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