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KRW depth composition conjecture for standard block composition

Prove that for all non-constant Boolean functions f: {0,1}^m -> {0,1} and g: {0,1}^n -> {0,1}, the depth complexity of their block composition satisfies D(f ◇ g) ≈ D(f)+D(g) (up to constant factors) for fan-in-2 AND/OR/NOT circuits.

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Background

The Karchmer–Raz–Wigderson (KRW) framework proposes studying the depth complexity of composed functions f ◇ g as a route to separate P from NC1. If the conjectured additive behavior holds under suitable parameters, iterative composition would yield an explicit function requiring super-logarithmic depth.

The paper surveys partial progress on restricted cases and variants, underscoring that the general conjecture remains unresolved.

References

Karchmer, Raz and Wigderson conjectured that the depth complexity of $f \diamond g$ is not far from its upper bound: \begin{conj}Given two arbitrary non-constant Boolean functions $f:{0,1}{m} \rightarrow{0,1}$ and $g:{0,1}{n} \rightarrow{0,1}$, then $$ \mathsf{D}(f \diamond g) \approx \mathsf{D}(f)+\mathsf{D}(g). $$ \end{conj}

— A nearly-$4\log n$ depth lower bound for formulas with restriction on top (2404.15613 - Wu, 24 Apr 2024) in Conjecture, Section 1 (Introduction)