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Composition theorem for two depth-2 formulas

Prove a general composition theorem for the standard composition of two depth-2 formulas (unbounded fan-in), establishing robust lower bounds in this regime.

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Background

A natural next step toward AC0-on-top lower bounds is to handle depth-2 formulas at the top. The paper explains a key obstacle: identifying a subformula with both a large value of the measure μ and a suitable correlation with f ⊞ g, akin to the role played by h_* in their analysis.

The authors underscore that, even independent of their approach, a general composition theorem for two depth-2 formulas is currently unknown and would represent substantial progress.

References

This difficulty also appears in the approach via communication complexity , since such result shares the same feature with a composition theorem of a depth-2 formula and a De Morgan formula, but we don't even know how to prove a composition theorem of two depth-2 formulas in general.

— A nearly-$4\log n$ depth lower bound for formulas with restriction on top (2404.15613 - Wu, 24 Apr 2024) in Section 5, Conclusion and discussion