Shrinkage under Random Projections, and Cubic Formula Lower Bounds for $\mathsf{AC}^0$
Yuval Filmus, Or Meir, Avishay Tal
TL;DR
This work extends the paradigm of formula shrinkage from random restrictions to a broader class of random projections, introducing fixing and hiding projection models that are tailored to the function's structure. The authors prove shrinkage theorems that bound the expected formula size after projection in terms of the outer function f and the soft-adversary bound Adv_s(g) for the inner function g, yielding L(f∘g) ≳ L(f)·Adv_s(g) up to polylog factors. Using these tools, they obtain a cubic formula lower bound for an explicit function in AC0 by composing Surj with a suitable outer function, and they establish a special-case KRW result for inner functions where Adv_s(g) is tight (e.g., Surj), clarifying when KRW-type lower bounds can be achieved via projection-based techniques. The approach blends bottom-up projections with top-down complexity arguments, showing that carefully constructed projections can preserve structure and drive strong lower bounds, thereby advancing the cubic barrier program and enhancing our understanding of AC0 formula complexity and the KRW framework.
Abstract
$\newcommand{\ACz}{\mathbf{AC}^0}$ Håstad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of $\tilde{O}(p^{2})$ under a random restriction that leaves each variable alive independently with probability $p$ [SICOMP, 1998]. Using this result, he gave an $\widetildeΩ(n^{3})$ formula size lower bound for the Andreev function, which, up to lower order improvements, remains the state-of-the-art lower bound for any explicit function. In this paper, we extend the shrinkage result of Håstad to hold under a far wider family of random restrictions and their generalization -- random projections. Based on our shrinkage results, we obtain an $\widetildeΩ(n^{3})$ formula size lower bound for an explicit function computable in $\ACz$. This improves upon the best known formula size lower bounds for $\ACz$, that were only quadratic prior to our work. In addition, we prove that the KRW conjecture [Karchmer et al., Computational Complexity 5(3/4), 1995] holds for inner functions for which the unweighted quantum adversary bound is tight. In particular, this holds for inner functions with a tight Khrapchenko bound. Our random projections are tailor-made to the function's structure so that the function maintains structure even under projection -- using such projections is necessary, as standard random restrictions simplify $\ACz$ circuits. In contrast, we show that any De Morgan formula shrinks by a quadratic factor under our random projections, allowing us to prove the cubic lower bound. Our proof techniques build on Håstad's proof for the simpler case of balanced formulas. This allows for a significantly simpler proof at the cost of slightly worse parameters. As such, when specialized to the case of $p$-random restrictions, our proof can be used as an exposition of Håstad's result.