Optimal depth 3 circuits for IP - arXiv abstract We show that Inner Product in $2n$ variables, $\mathbf{IP}_n(x, y) = x_1y_1 \oplus \ldots \oplus x_ny_n$, can be computed by depth-3 bottom fan-in 2 circuits of size $\mathsf{poly}(n)\cdot (9/5)^n$, matching the lower bound of Göös, Guan, and Mosnoi (Inform. Comput.'24). Our construction is given via the following steps. - We provide a general template for constructing optimal depth-3 circuits with bottom fan-in $k$ for an arbitrary function $f$. We do this in two steps. First, we partition the accepting inputs to $f$ into several carefully defined orbits. Second, for each orbit, we construct one $k$-CNF that (a) accepts the largest number of inputs from that orbit and (b) rejects all inputs rejected by $f$. This partitioning of inputs into orbits is based on the symmetries of $f$. - We instantiate the template for $\mathbf{IP}_n$ and $k = 2$. Guided by a modularity principle that these optimal 2-CNFs may be constructed by composing variable-disjoint copies of small formulas, we use computer search to identify a small set of 2-CNFs each on at most 4 variables which we call building blocks. - We then use analytic combinatorial techniques to determine optimal ways to combine these building blocks and construct these optimal 2-CNFs. We believe that the above steps can be applied to a wide range of functions to determine their depth-3 complexity.

翻译：我们证明了具有$2n$个变量的内积函数$\mathbf{IP}_n(x, y) = x_1y_1 \oplus \ldots \oplus x_ny_n$可由底层扇入为2、规模为$\mathsf{poly}(n)\cdot (9/5)^n$的深度三电路计算，该结果与Göös、Guan和Mosnoi（Inform. Comput.'24）给出的下界相匹配。我们的构造通过以下步骤实现。 - 我们提出了一个通用模板，用于为任意函数$f$构造底层扇入为$k$的最优深度三电路。这分两步完成：首先，将$f$的接受输入划分为若干精确定义的轨道；其次，为每个轨道构造一个$k$-CNF，该$k$-CNF需满足（a）接受该轨道中尽可能多的输入，且（b）拒绝所有被$f$拒绝的输入。这种基于轨道划分输入的方法依赖于函数$f$的对称性。 - 我们将该模板实例化于$\mathbf{IP}_n$且$k = 2$的情形。基于一种模块化原则——即这些最优2-CNF可能通过组合变量互不相交的小型公式来构造——我们通过计算机搜索识别出一小组最多包含4个变量的2-CNF，称之为构建块。 - 随后，我们运用解析组合技术来确定组合这些构建块的最优方式，从而构造出这些最优的2-CNF。我们相信上述步骤可广泛应用于各类函数，以确定其深度三电路复杂度。