Basic properties for sand automata

TL;DR

This paper investigates fundamental properties of sand automata, establishing key relationships between injectivity and surjectivity, and demonstrating that nilpotency is an undecidable property, thus advancing understanding of their dynamical behavior.

Contribution

It introduces new results on the relations between injectivity and surjectivity and proves that nilpotency is undecidable for sand automata.

Findings

Injectivity and surjectivity relations are characterized.

Nilpotency is proven to be undecidable.

A novel proof technique is introduced.

Abstract

We prove several results about the relations between injectivity and surjectivity for sand automata. Moreover, we begin the exploration of the dynamical behavior of sand automata proving that the property of nilpotency is undecidable. We believe that the proof technique used for this last result might reveal useful for many other results in this context.