The Two-Dimensional Majority Rule is P-Complete

2026-10-03

AI Summary

This study addresses the computational complexity of the two-dimensional majority rule prediction problem, which has remained open for nearly three decades. We propose a time-encoding Boolean representation mechanism that distinguishes Boolean states via signal arrival time differences, enabling planar crossing and composition of independent information flows under local diffusion rules. By constructing monotone Boolean circuits, we perform a complexity analysis via log-space many-one reductions. Our work rigorously proves that this prediction problem is P-complete under log-space reductions. Furthermore, we extend this result to the decision problem of first reaching the +1 state at any given time step, as well as to all uniform symmetric sign-majority rules.

Abstract

We prove that prediction for the synchronous two-dimensional majority rule is $\mathrm{P}$-complete under logspace many-one reductions, resolving a problem open for almost three decades. In 1997, Moore established $\mathrm{P}$-completeness in dimension three and higher and conjectured that the two-dimensional case admits an efficient parallel algorithm. We consider an $n\times n$ torus on which each cell follows the majority of its four nearest neighbors and retains its current state in a tie. Given an explicitly specified initial configuration and a time $T$, prediction asks whether a designated cell is in state $+1$ at time $T$. The central challenge is to make independent information streams cross in the plane under a homogeneous, monotone, diffusive local rule. We overcome this obstacle through a temporal encoding of Boolean values: both values generate activity, but are distinguished by signal arrival times. This encoding yields a crossover that preserves both values and composes with wires, duplication, and AND and OR gates to simulate arbitrary monotone Boolean circuits. Thus a local rule that favors agreement can nevertheless transport, combine, and cross independent information in two dimensions. We also prove $\mathrm{P}$-completeness for deciding whether a designated cell ever reaches $+1$, without a prescribed time horizon. The prediction result extends to every uniform symmetric signed majority rule on the same neighborhood, including the minority rule.

Problem

• Two-Dimensional Majority Rule
• P-completeness
• Prediction problem
• Cellular automata
• Computational complexity

Innovation

• P-completeness
• majority rule
• cellular automata
• temporal encoding
• monotone Boolean circuits