A compact topology for sand automata

TL;DR

This paper establishes a new compact topology for sand automata, linking them to cellular automata, and explores their dynamical properties, including a novel notion of nilpotency and its undecidability.

Contribution

It introduces a compact topology for sand automata, showing their homeomorphism to cellular automata on a subshift, and defines a new notion of nilpotency with proven undecidability.

Findings

Sand automata are homeomorphic to cellular automata on a subshift.

Classical nilpotency is not meaningful for sand automata.

New notion of nilpotency for sand automata is undecidable.

Abstract

In this paper, we exhibit a strong relation between the sand automata configuration space and the cellular automata configuration space. This relation induces a compact topology for sand automata, and a new context in which sand automata are homeomorphic to cellular automata acting on a specific subshift. We show that the existing topological results for sand automata, including the Hedlund-like representation theorem, still hold. In this context, we give a characterization of the cellular automata which are sand automata, and study some dynamical behaviors such as equicontinuity. Furthermore, we deal with the nilpotency. We show that the classical definition is not meaningful for sand automata. Then, we introduce a suitable new notion of nilpotency for sand automata. Finally, we prove that this simple dynamical behavior is undecidable.