Probabilistic cellular automata, invariant measures, and perfect sampling

Ana Bušić (1) , Jean Mairesse (2) , Irène Marcovici (2, 3)

1 TREC - Theory of networks and communications
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
3 ENS de Lyon - École normale supérieure de Lyon

In a probabilistic cellular automaton (PCA), the cells are updated synchro\-nously and independently, according to a distribution depending on a finite neighborhood. A PCA can be viewed as a Markov chain whose ergodicity is investigated. A classical cellular automaton (CA) is a particular case of PCA. For a 1-dimensional CA, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to be also a PCA.

Mots clés: probabilistic cellular automata; perfect sampling; ergodicity

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Ana Bušić, Jean Mairesse, Irène Marcovici. Probabilistic cellular automata, invariant measures, and perfect sampling. Symposium on Theoretical Aspects of Computer Science (STACS2011), Mar 2011, Dortmund, Germany. pp.296-307. ⟨hal-00573640⟩