一句话：证明了 P-不变 AC$^0$ 公式计算 PARITY 的近乎匹配下界，并推广到一般子空间不变布尔函数。
论文研究线性子空间 U⊆{0,1}^n 对 AC$^0$ 公式的作用：u 交换第 i 对文字 X_i 与其否定（当 u_i=1）作用于公式，固定不变的公式称 U-不变。PARITY 的经典递归深度 d+1 公式大小为 O(n·2^{d n^{1/d}})，且是 P-不变的（P 为偶重量子空间）。本文证明 PARITY 的 P-不变深度 d+1 公式大小下界为 2^{d(n^{1/d}-1)}，与上界近乎匹配。这改进了无限制深度 d+1 公式已知的 Ω(2^{(1/84)d(n^{1/d}-1)}) 下界，并避免使用 switching lemma。一般地，对 U⊂V，若布尔函数 U-不变且在 V 上非常数，则其 U-不变深度 d+1 公式大小至少为 2^{d(m^{1/d}-1)}，其中 m 是 U^⊥ 中不属于 V^⊥ 的向量的最小 Hamming 重量。
We consider the action of a linear subspace $U$ of $\{0,1\}^n$ on the set of AC$^0$ formulas with inputs labeled by literals in the set $\{X_1,\overline X_1,\dots,X_n,\overline X_n\}$, where an element $u \in U$ acts on formulas by transposing the $i$th pair of literals for all $i \in [n]$ such that $u_i=1$. A formula is {\em $U$-invariant} if it is fixed by this action. For example, there is a well-known recursive construction of depth $d+1$ formulas of size $O(n{\cdot}2^{dn^{1/d}})$ computing the $n$-variable PARITY function; these formulas are easily seen to be $P$-invariant where $P$ is the subspace of even-weight elements of $\{0,1\}^n$. In this paper we establish a nearly matching $2^{d(n^{1/d}-1)}$ lower bound on the $P$-invariant depth $d+1$ formula size of PARITY. Quantitatively this improves the best known $\Omega(2^{\frac{1}{84}d(n^{1/d}-1)})$ lower bound for {\em unrestricted} depth $d+1$ formulas, while avoiding the use of the switching lemma. More generally, for any linear subspaces $U \subset V$, we show that if a Boolean function is $U$-invariant and non-constant over $V$, then its $U$-invariant depth $d+1$ formula size is at least $2^{d(m^{1/d}-1)}$ where $m$ is the minimum Hamming weight of a vector in $U^\bot \setminus V^\bot$.