We initiate the study of generalized AC0 circuits comprised of negations and arbitrary unbounded fan-in gates that only need to be constant over inputs of Hamming weight $\ge k$, which we denote GC0$(k)$. The gate set of this class includes biased LTFs like the $k$-$OR$ (output $1$ iff $\ge k$ bits are 1) and $k$-$AND$ (output $0$ iff $\ge k$ bits are 0), and thus can be seen as an interpolation between AC0 and TC0. We establish a tight multi-switching lemma for GC0$(k)$ circuits, which bounds the probability that several depth-2 GC0$(k)$ circuits do not simultaneously simplify under a random restriction. We also establish a new depth reduction lemma such that coupled with our multi-switching lemma, we can show many results obtained from the multi-switching lemma for depth-$d$ size-$s$ AC0 circuits lifts to depth-$d$ size-$s^{.99}$ GC0$(.01\log s)$ circuits with no loss in parameters (other than hidden constants). Our result has the following applications: 1.Size-$2^{\Omega(n^{1/d})}$ depth-$d$ GC0$(\Omega(n^{1/d}))$ circuits do not correlate with parity (extending a result of H{\aa}stad (SICOMP, 2014)). 2. Size-$n^{\Omega(\log n)}$ GC0$(\Omega(\log^2 n))$ circuits with $n^{.249}$ arbitrary threshold gates or $n^{.499}$ arbitrary symmetric gates exhibit exponentially small correlation against an explicit function (extending a result of Tan and Servedio (RANDOM, 2019)). 3. There is a seed length $O((\log m)^{d-1}\log(m/\varepsilon)\log\log(m))$ pseudorandom generator against size-$m$ depth-$d$ GC0$(\log m)$ circuits, matching the AC0 lower bound of H{\aa}stad stad up to a $\log\log m$ factor (extending a result of Lyu (CCC, 2022)). 4. Size-$m$ GC0$(\log m)$ circuits have exponentially small Fourier tails (extending a result of Tal (CCC, 2017)).

翻译：我们首次研究由非门和任意无界扇入门组成的广义AC0电路，这些门仅在汉明重量≥k的输入上需要保持恒定，记为GC0(k)。该类的门集合包含有偏线性阈值函数，如k-OR（输出1当且仅当≥k比特为1）和k-AND（输出0当且仅当≥k比特为0），因此可视为AC0和TC0之间的插值。我们为GC0(k)电路建立了一个紧致的多重切换引理，该引理界定了在随机限制下多个深度为2的GC0(k)电路未能同时简化的概率。我们还建立了一个新的深度归约引理，结合我们的多重切换引理，我们可以证明从深度d规模s的AC0电路的多重切换引理得到的许多结果可以提升到深度d规模s的.99次方的GC0(0.01log s)电路，且参数（除隐藏常数外）无损失。我们的结果具有以下应用：1.规模为2的Ω(n^{1/d})次方的深度d GC0(Ω(n^{1/d}))电路与奇偶函数不相关（推广了Håstad（SICOMP, 2014）的结果）。2.具有n的0.249次方个任意阈值门或n的0.499次方个任意对称门的规模为n的Ω(log n)次方的GC0(Ω(log² n))电路，与一个显式函数的指数小相关性（推广了Tan和Servedio（RANDOM, 2019）的结果）。3.存在种子长度为O((log m)^{d-1} log(m/ε) log log m)的伪随机生成器，可对抗规模为m深度为d的GC0(log m)电路，与Håstad的AC0下界相比仅差一个log log m因子（推广了Lyu（CCC, 2022）的结果）。4.规模为m的GC0(log m)电路具有指数小的傅里叶尾部（推广了Tal（CCC, 2017）的结果）。