Some halting problems for abelian sandpiles are undecidable in dimension three

TL;DR

This paper demonstrates that in three dimensions, the abelian sandpile model can simulate a Turing machine, leading to the undecidability of certain halting problems despite its simple combinatorial nature.

Contribution

It proves that the abelian sandpile model in three dimensions can perform universal computation, establishing the undecidability of specific halting problems.

Findings

Three-dimensional sandpiles can simulate Turing machines

Certain halting problems for 3D sandpiles are undecidable

The abelian property does not limit computational complexity in 3D

Abstract

The abelian sandpile model is a simple combinatorial model for critical behaviour which has the "abelian property" that the order in which we make moves does not change the final outcome of the game. This might seem to restrict the model's computational ability, but we show that, given three dimensions to work with, the sandpile is able to simulate a Turing machine. We use that to prove the undecidability of three halting problems.