Timed Prediction Problem for Sandpile Models

TL;DR

This paper explores the computational complexity of the timed prediction problem in two-dimensional sandpile models, revealing cases of P-completeness and limitations based on neighborhood types and planarity.

Contribution

It introduces the timed prediction problem for sandpile models, characterizes the complexity for various neighborhoods, and identifies limitations for planar configurations.

Findings

Timed prediction problem is P-complete for certain neighborhoods.

Planar neighborhoods do not admit timed crossover gates.

Some neighborhoods likely cannot support timed crossovers.

Abstract

We investigate the computational complexity of the timed prediction problem in two-dimensional sandpile models. This question refines the classical prediction problem, which asks whether a cell q will eventually become unstable after adding a grain at cell p from a given configuration. The prediction problem has been shown to be P-complete in several settings, including for subsets of the Moore neighborhood, but its complexity for the von Neumann neighborhood remains open. In a previous work, we provided a complete characterization of crossover gates (a key to the implementation of non-planar monotone circuits) for these small neighborhoods, leading to P-completeness proofs with only 4 and 5 neighbors among the eight adjancent cells. In this paper, we introduce the timed setting, where the goal is to determine whether cell q becomes unstable exactly at time t. We distinguish several cases: some neighborhoods support complete timed toolkits (including timed crossover gates) and exhibit P-completeness; others admit timed crossovers but suffer from synchronization issues; planar neighborhoods provably do not admit any timed crossover; and finally, for some remaining neighborhoods, we conjecture that no timed crossover is possible.