LIPIcs.CCC.2022.13.pdf
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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.
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