Benoît Masson

Sand automata as cellular automata

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 cellular automata which are sand automata, and study some dynamical behaviors such as equicontinuity. Furthermore, we deal with simple sand automata. We show that the classical definition of nilpotency is not meaningful for sand automata. Then, we introduce the suitable new notion of flattening sand automata. Finally, we prove that this simple dynamical behavior is undecidable.

Stable Dynamics of Sand Automata

In this paper, we study different notions of stability for sand automata, dynamical systems inspired by sandpile models and cellular automata. First, we study the topological stability properties of equicontinuity and ultimate periodicity, proving that they are equivalent. Then, we deal with nilpotency. The classical definition for cellular automata being meaningless in that setting, we define a more suitable one. Finally, we prove that this dynamical behavior is undecidable.

Fundamental study: From sandpiles to sand automata

We introduce a new model for simulating natural phenomena. We address several issues: topology, basic set properties like injectivity and surjectivity, reversibility, and decidability questions about a special kind of conservation law called grain conservation and ultimate periodicity.

Basic properties for sand automata

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 ultimate periodicity is undecidable. We believe that the proof technique used for this last result might turn out to be useful for many other results in the same context.

On symmetric sandpiles

A symmetric version of the well-known SPM model for sandpiles is introduced We prove that the new model has fixed point dynamics Although there might be several fixed points, a precise description of the fixed points is given Moreover, we provide a simple closed formula for counting the number of fixed points originated by initial conditions made of a single column of grains.