A notorious open question in circuit complexity is whether Boolean operations of arbitrary arity can efficiently be expressed using modular counting gates only. Håstad's celebrated switching lemma yields exponential lower bounds for the dual problem - realising modular arithmetic with Boolean gates - but, a similar lower bound for modular circuits computing the Boolean AND function has remained elusive for almost 30 years. We solve this problem for the restricted model of symmetric circuits: We consider MOD$_m$-circuits of arbitrary depth, and for an arbitrary modulus $m \in \mathbb{N}$, and obtain subexponential lower bounds for computing the $n$-ary Boolean AND function, under the assumption that the circuits are syntactically symmetric under all permutations of their $n$ input gates. This lower bound is matched precisely by a construction due to (Idziak, Kawałek, Krzaczkowski, LICS'22), leading to the surprising conclusion that the optimal symmetric circuit size is already achieved with depth $2$. Motivated by another construction from (LICS'22), which achieves smaller size at the cost of greater depth, we also prove tight size lower bounds for circuits with a more liberal notion of symmetry characterised by a nested block structure on the input variables.

翻译：电路复杂度中的一个著名开放问题是：任意元数的布尔运算能否仅用模计数门高效表达？Håstad著名的开关引理为逆问题——用布尔门实现模运算——给出了指数下界，但关于计算布尔AND函数的模块化电路的类似下界近30年来一直悬而未决。我们针对对称电路的受限模型解决了这一问题：考虑任意深度的MOD$_m$电路，对于任意模数$m \in \mathbb{N}$，在电路关于其$n$个输入门的所有置换在语法上对称的假设下，我们得到了计算$n$元布尔AND函数的次指数下界。该下界恰好与（Idziak, Kawałek, Krzaczkowski, LICS'22）的构造匹配，从而得出惊人结论：最优对称电路规模已在深度为2时实现。受（LICS'22）中另一构造（以更大深度换取更小规模）的启发，我们还针对具有更宽松对称性概念（由输入变量上的嵌套块结构刻画）的电路，证明了紧致的大小下界。