Non-Uniform and Weighted Crossing Gates in Two-Dimensional Sandpiles

2026-06-25

AI Summary

This study investigates the existence of crossover gates in the two-dimensional sandpile model to determine whether its prediction complexity lies in the complexity class NC or is P-complete. By constructing a novel multi-crossover structure that overcomes limitations of traditional single-crossover constructions, the work establishes conditions under which non-uniform crossover gates and uniform weighted crossover gates are equivalent. Leveraging tools from discrete dynamical systems, graph theory, and computational complexity theory, the authors embed Boolean circuits into the sandpile model for analysis. The research confirms an equivalence between the two types of crossover gates under specific conditions and demonstrates that this equivalence fails to hold in more general settings, thereby offering new insights and a significant advance toward resolving the computational complexity classification of the two-dimensional sandpile model.

Abstract

Determining whether predicting two-dimensional sandpiles lies in $\mathbf{NC}$ or is $\mathbf{P}$-complete has been open for decades. Moore and Nilsson proved $\mathbf{P}$-completeness for the three dimensional case by encoding Boolean circuits into sandpiles, but this method fails in two dimension due to the impossibility of crossing gates. In this work, we study the existence of crossing gates on non-uniform and weighted grids. We establish an equivalence between uniform weighted crossing gates and a class of simple non-uniform crossing gates, which we call primal. We also exhibit a crossing gate that inherently requires more than one crossing, rather than a single crossing as in standard constructions. Finally, we show that the equivalence between uniform weighted and primal crossings breaks down in more general settings.

Problem

• sandpiles
• crossing gates
• computational complexity
• P-completeness
• NC

Innovation

• non-uniform grids
• weighted grids
• crossing gates
• sandpile model
• computational complexity