A nearly-$4\log n$ depth lower bound for formulas with restriction on top
Hao Wu
TL;DR
This work advances the quest for explicit super-logarithmic circuit depth by proving a nearly-$4\log n$ lower bound for formulas with restricted top layers. It introduces an improved XOR composition theorem using a simplified double-measurement argument based on a well-mixed set of functions, and shows that for a broad range $0<\alpha<2-o(1)$, the XOR composition $f\boxplus g$ cannot be computed by an AND of $2^{\alpha n}$ formulas of size at most $2^{(1-\alpha/2-o(1))n}$. Consequently, a modified Andreev function $\mathbf{Andr}'$ is not computable by any circuit of depth $(4-o(1))\log n$ with the top $2-o(1)\log n$ layers restricted to AND (or OR) gates. The results push the parameter range toward optimality and simplify the foundational techniques, while highlighting open directions for top-AC^0 restrictions and depth-2 top layers. These findings tighten the landscape around KRW-type decompositions and deepen our understanding of how top-layer restrictions influence depth lower bounds.
Abstract
One of the major open problems in complexity theory is to demonstrate an explicit function which requires super logarithmic depth, a.k.a, the $\mathbf{P}$ versus $\mathbf{NC^1}$ problem. The current best depth lower bound is $(3-o(1))\cdot \log n$, and it is widely open how to prove a super-$3\log n$ depth lower bound. Recently Mihajlin and Sofronova (CCC'22) show if considering formulas with restriction on top, we can break the $3\log n$ barrier. Formally, they prove there exist two functions $f:\{0,1\}^n \rightarrow \{0,1\},g:\{0,1\}^n \rightarrow \{0,1\}^n$, such that for any constant $0<α<0.4$ and constant $0<ε<α/2$, their XOR composition $f(g(x)\oplus y)$ is not computable by an AND of $2^{(α-ε)n}$ formulas of size at most $2^{(1-α/2-ε)n}$. This implies a modified version of Andreev function is not computable by any circuit of depth $(3.2-ε)\log n$ with the restriction that top $0.4-ε$ layers only consist of AND gates for any small constant $ε>0$. They ask whether the parameter $α$ can be push up to nearly $1$ thus implying a nearly-$3.5\log n$ depth lower bound. In this paper, we provide a stronger answer to their question. We show there exist two functions $f:\{0,1\}^n \rightarrow \{0,1\},g:\{0,1\}^n \rightarrow \{0,1\}^n$, such that for any constant $0<α<2-o(1)$, their XOR composition $f(g(x)\oplus y)$ is not computable by an AND of $2^{αn}$ formulas of size at most $2^{(1-α/2-o(1))n}$. This implies a $(4-o(1))\log n$ depth lower bound with the restriction that top $2-o(1)$ layers only consist of AND gates. We prove it by observing that one crucial component in Mihajlin and Sofronova's work, called the well-mixed set of functions, can be significantly simplified thus improved. Then with this observation and a more careful analysis, we obtain these nearly tight results.