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 $0<\alpha<0.4$ and constant $0<\epsilon<\alpha/2$, their XOR composition $f(g(x)\oplus y)$ is not computable by an AND of $2^{(\alpha-\epsilon)n}$ formulas of size at most $2^{(1-\alpha/2-\epsilon)n}$. This implies a modified version of Andreev function is not computable by any circuit of depth $(3.2-\epsilon)\log n$ with the restriction that top $0.4-\epsilon$ layers only consist of AND gates for any small constant $\epsilon>0$. They ask whether the parameter $\alpha$ 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 $0<\alpha<2-o(1)$, their XOR composition $f(g(x)\oplus y)$ is not computable by an AND of $2^{\alpha n}$ formulas of size at most $2^{(1-\alpha/2-o(1))n}$. 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.

翻译：复杂性理论中的主要开放问题之一是证明一个显式函数需要超对数深度，即$\mathbf{P}$与$\mathbf{NC^1}$问题。当前最佳深度下界为$(3-o(1))\cdot \log n$，而证明超$3\log n$深度下界仍是一个广泛开放的问题。近期Mihajlin和Sofronova（CCC'22）指出，若考虑对顶部施加限制的公式，则可突破$3\log n$屏障。具体地，他们证明了存在两个函数$f:\{0,1\}^n \rightarrow \{0,1\},g:\{0,1\}^n \rightarrow \{0,1\}^n$，使得对于任意常数$0<\alpha<0.4$和常数$0<\epsilon<\alpha/2$，其异或复合$f(g(x)\oplus y)$不能被任意一个由大小为至多$2^{(1-\alpha/2-\epsilon)n}$的$2^{(\alpha-\epsilon)n}$个公式构成的AND门所计算。这意味着修正版安德烈耶夫函数不能被任何深度为$(3.2-\epsilon)\log n$的电路所计算，且对任意小常数$\epsilon>0$，顶部$0.4-\epsilon$层仅由AND门组成。他们提出是否可将参数$\alpha$推升至接近$1$，从而得到近乎$3.5\log n$的深度下界。本文对该问题给出了更强的回答。我们证明存在两个函数$f:\{0,1\}^n \rightarrow \{0,1\},g:\{0,1\}^n \rightarrow \{0,1\}^n$，使得对于任意常数$0<\alpha<2-o(1)$，其异或复合$f(g(x)\oplus y)$不能被任意一个由大小为至多$2^{(1-\alpha/2-o(1))n}$的$2^{\alpha n}$个公式构成的AND门所计算。这证明了在顶部$2-o(1)$层仅由AND门组成限制下的$(4-o(1))\log n$深度下界。我们的证明基于观察到Mihajlin和Sofronova工作中的关键组成部分——所谓函数的良好混合集——可被显著简化并改进。基于这一观察与更细致的分析，我们获得了这些近乎最优的结果。