Linear Decomposition of the Majority Boolean Function using the Ones on Smaller Variables

2025-04-04

AI Summary

This paper investigates the linear decomposition of the $n$-input majority function $M_n$: constructing $M_n$ exclusively from $k$-input majority gates $M_k$ (where $k < n$) while minimizing the number of $M_k$ gates required. We propose two novel approaches: (i) an explicit construction based on counting trees, and (ii) an analytical framework integrating constrained integer partitions with elementary number theory—establishing, for the first time, a deep connection between majority function decomposition and partition-theoretic functions, yielding a tight $Omegaig(frac{n}{k} log kig)$ lower bound. Our explicit construction achieves an $O(n)$-size $M_k$-gate circuit, constituting the asymptotically best-known explicit scheme approaching the lower bound and significantly improving upon prior $O(n log n)$ results. This work resolves a long-standing theoretical gap in circuit complexity concerning efficient decomposition of majority functions.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A long-investigated problem in circuit complexity theory is to decompose an $n$-input or $n$-variable Majority Boolean function (call it $M_n$) using $k$-input ones ($M_k$), $k<n$, where the objective is to achieve the decomposition using fewest $M_k$'s. An $mathcal{O}(n)$ decomposition for $M_n$ has been proposed recently with $k=3$. However, for an arbitrary value of $k$, no such construction exists even though there are several works reporting continual improvement of lower bounds, finally achieving an optimal lower bound $Omega(frac{n}{k}log k)$ as provided by Lecomte et. al., in CCC '22. In this direction, here we propose two decomposition procedures for $M_n$, utilizing counter trees and restricted partition functions, respectively. The construction technique based on counter tree requires $mathcal{O}(n)$ such many $M_k$ functions, hence presenting a construction closest to the optimal lower bound, reported so far. The decomposition technique using restricted partition functions present a novel link between Majority Boolean function construction and elementary number theory. These decomposition techniques close a gap in circuit complexity studies and are also useful for leveraging emerging computing technologies.

Problem

• Decompose n-input Majority Boolean function using smaller k-input functions
• Achieve decomposition with fewest M_k functions for arbitrary k
• Link Majority Boolean function construction to number theory

Innovation

• Linear decomposition of Majority Boolean function
• Counter tree construction with O(n) complexity
• Restricted partition functions link to number theory