Optimal Depth-Three Circuits for Inner Product

2026-01-07

arXiv.org

AI Summary

This work addresses the problem of constructing optimal depth-3 Boolean circuits with bottom fan-in 2 for computing the inner product function. By exploiting the symmetry of the function to partition input orbits, combining computer-aided search to build valid base 2-CNF modules, and applying analytic combinatorics to optimize the composition of these modules, the authors propose a general template for constructing optimal depth-3 circuits for arbitrary functions. This approach yields the first construction that matches the known lower bound for the inner product function in this circuit model, achieving a circuit size of poly(n)·(9/5)ⁿ, which precisely meets the lower bound established by Göös et al.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 G\"o\"os, Guan, and Mosnoi (Inform. Comput.'24). Our construction is obtained 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 $f^{-1}(1)$ into orbits of its automorphism group. 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 $f$. - We instantiate the template for $\mathbf{IP}_n$ and $k = 2$. Guided by the intuition (which we call modularity principle) that optimal 2-CNFs can be constructed by taking the conjunction of variable-disjoint copies of smaller $2$-CNFs, we use computer search to identify a small set of building block 2-CNFs over at most 4 variables. - We again use computer search to discover appropriate combinations (disjoint conjunctions) of building blocks to arrive at optimal 2-CNFs and analyze them using techniques from analytic combinatorics. We believe that the approach outlined in this paper can be applied to a wide range of functions to determine their depth-3 complexity.

Problem

• Inner Product
• depth-3 circuits
• circuit complexity
• bottom fan-in
• Boolean functions

Innovation

• depth-3 circuits
• inner product
• bottom fan-in 2
• orbit partitioning
• analytic combinatorics