7 research outputs found
Basic properties for sand automata
Presented at MFCS 2005 (Gdansk, POLAND). Long version with complete proofs published in Theoretical Computer Science, 2006, under the title "From Sandpiles to Sand Automata".International audienceWe 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
A compact topology for sand 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 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
On the set of Fixed Points of the Parallel Symmetric Sand Pile Model
Sand Pile Models are discrete dynamical systems emphasizing the phenomenon of . From a configuration composed of a finite number of stacked grains, we apply on every possible positions (in parallel) two grain moving transition rules. The transition rules permit one grain to fall to its right or left (symmetric) neighboring column if the difference of height between those columns is larger than 2. The model is nondeterministic and grains always fall downward. We propose a study of the set of fixed points reachable in the Parallel Symmetric Sand Pile Model (PSSPM). Using a comparison with the Symmetric Sand Pile Model (SSPM) on which rules are applied once at each iteration, we get a continuity property. This property states that within PSSPM we can't reach every fixed points of SSPM, but a continuous subset according to the lexicographic order. Moreover we define a successor relation to browse exhaustively the sets of fixed points of those models
From sandpiles to sand automata
International audienceWe 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