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：我们首次系统研究了由否定门和任意无界扇入门构成的广义AC0电路类，其中门函数仅需在汉明权重≥k的输入上为常值，记为GC0(k)。该类电路的逻辑门包含有偏线性阈值函数（如k-OR门：当≥k个比特为1时输出1；k-AND门：当≥k个比特为0时输出0），因此可视为AC0与TC0之间的插值类。我们为GC0(k)电路建立了紧致多重切换引理，该引理界定了多个深度为2的GC0(k)电路在随机限制下无法同时简化的概率。进一步提出了新的深度归约引理，结合多重切换引理可证明：针对深度d、规模s的AC0电路的多重切换引理所获得的诸多结论，可无参数损失（除隐藏常数外）提升至深度d、规模s^0.99的GC0(0.01 log s)电路。我们的结果具有以下应用：1.规模为2^{Ω(n^{1/d})}的深度d GC0(Ω(n^{1/d}))电路与奇偶函数无关联（推广Håstad（SICOMP, 2014）的结果）；2.规模为n^{Ω(log n)}的GC0(Ω(log^2 n))电路（配备n^{0.249}个任意阈值门或n^{0.499}个任意对称门）与某显式函数的指数级关联上界（推广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，在log log m因子范围内达到Håstad关于AC0的下界（推广Lyu（CCC, 2022）的结果）；4.规模为m的GC0(log m)电路具有指数级小的傅里叶尾部（推广Tal（CCC, 2017）的结果）。