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Probabilistic Cellular Automata, Invariant Measures, and Perfect Sampling

Part of: Topological dynamics Ergodic theory Markov processes Theory of computing Special processes

Published online by Cambridge University Press:  04 January 2016

Ana Bušić ,
Jean Mairesse  and
Irène Marcovici
Ana Bušić*
Affiliation:
INRIA - École Normale Supérieure
Jean Mairesse*
Affiliation:
Université Paris Diderot
Irène Marcovici*
Affiliation:
Université Paris Diderot
*
∗ Postal address: Laboratoire d'Informatique de l'École Normale Supérieure (UMR 8548), INRIA - École Normale Supérieure, 23 avenue d'Italie, CS 81321, 75214 Paris Cedex 13, France. Email address: [email protected]
∗∗ Postal address: CNRS, UMR 7089, LIAFA, Université Paris Diderot, Sorbonne Paris Cité, F-75205 Paris, France.
∗∗ Postal address: CNRS, UMR 7089, LIAFA, Université Paris Diderot, Sorbonne Paris Cité, F-75205 Paris, France.
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Abstract

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A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, 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 also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

Keywords

Probabilistic cellular automataperfect samplingergodicityinvariant measure

MSC classification

Primary: 37B15: Cellular automata 60J05: Discrete-time Markov processes on general state spaces 60J22: Computational methods in Markov chains
Secondary: 37A25: Ergodicity, mixing, rates of mixing 60K35: Interacting random processes; statistical mechanics type models; percolation theory 68Q80: Cellular automata
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 960 - 980
Copyright
© Applied Probability Trust 

References

Bousquet-Mélou, M. (1998). New enumerative results on two-dimensional directed animals. Discrete Math. 180, 73–106.Google Scholar
Bušić, A., Gaujal, B. and Vincent, J.-M. (2008). Perfect simulation and non-monotone Markovian systems. In Proc. 3rd Internat. Conf. Performance Evaluation Methodol. Tools, Athens, 10pp.CrossRefGoogle Scholar
Bušić, A., Mairesse, J. and Marcovici, I. (2011). Probabilistic cellular automata, invariant measures, and perfect sampling. In 28th Internat. Symp. Theoret. Aspects Comput. Sci. (STACS'11), Leibniz-Zentrum, Dagstuhl, pp. 296–307.Google Scholar
Chassaing, P. and Mairesse, J. (2011). A non-ergodic probabilistic cellular automaton with a unique invariant measure. Stoch. Process. Appl. 121, 2474–2487.Google Scholar
Coletti, C. F. and Tisseur, P. (2010). Invariant measures and decay of correlations for a class of ergodic probabilistic cellular automata. J. Statist. Phys. 140, 103–121.Google Scholar
Culik, K. II, Pachl, J. and Yu, S. (1989). On the limit sets of cellular automata. SIAM J. Comput. 18, 831–842.Google Scholar
Dhar, D. (1983). Exact solution of a directed-site animals-enumeration problem in three dimensions. Phys. Rev. Lett. 51, 853–856.Google Scholar
Fatès, N. (2011). Stochastic cellular automata solve the density classification problem with an arbitrary precision. In 28th Internat. Symp. Theoret. Aspects Comput. Sci. (STACS'11), Leibniz-Zentrum, Dagstuhl, pp. 284–295.Google Scholar
Fatès, N., Regnault, D., Schabanel, N. and Thierry, E. (2006). Asynchronous behavior of double-quiescent elementary cellular automata. In LATIN 2006: Theoretical Informatics (Lecture Notes Comput. Sci. 3887), Springer, Berlin, pp. 455–466.Google Scholar
Gács, P. (2001). Reliable cellular automata with self-organization. J. Statist. Phys. 103, 45–267.Google Scholar
Gray, L. F. (2001). A reader's guide to P. Gács's ‘positive rates' paper: ‘Reliable cellular automata with self-organization’. J. Statist. Phys. 103, 1–44.Google Scholar
Guillon, P. and Richard, G. (2008). Nilpotency and limit sets of cellular automata. In Mathematical Foundations of Computer Science (Lecture Notes Comput. Sci. 5162), Springer, Berlin, pp. 375–386.Google Scholar
Häggström, O. (2002). Finite Markov Chains and Algorithmic Applications (London Math. Soc. Student Texts 52). Cambridge University Press.Google Scholar
Häggström, O. and Nelander, K. (1998). Exact sampling from anti-monotone systems. Statist. Neerlandica 52, 360–380.Google Scholar
Hedlund, G. (1969). Endormorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3, 320–375.Google Scholar
Huber, M. (2004). Perfect sampling using bounding chains. Ann. Appl. Prob. 14, 734–753.Google Scholar
Kari, J. (1992). The nilpotency problem of one-dimensional cellular automata. SIAM J. Comput. 21, 571–586.Google Scholar
Kozma, R. et al. (2004). Neuropercolation: a random cellular automata approach to spatio-temporal neurodynamics. In Cellular Automata (Lecture Notes Comput. Sci 3305), Springer, Berlin, pp. 435–443.Google Scholar
Le Borgne, Y. and Marckert, J.-F. (2007). Directed animals and gas models revisited. Electron. J. Combin. 14, R71.Google Scholar
Liggett, T. (1985). Interacting Particle Systems. Springer, New York.CrossRefGoogle Scholar
Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.3.0.CO;2-O>CrossRefGoogle Scholar
Regnault, D. (2008). Directed percolation arising in stochastic cellular automata analysis. In Mathematical Foundations of Computer Science 2008 (Lecture Notes Comput. Sci. 5162), Springer, Berlin, pp. 563–574.Google Scholar
Schabanel, N. (2009). Habilitation à diriger des recherches: systèmes complexes & algorithmes. Res. Rep. Université Paris Diderot - Paris 7.Google Scholar
Toom, A. (1995). Cellular automata with errors: problems for students of probability. In Topics in Contemporary Probability and Its Applications, CRC, Boca Raton, FL, pp. 117–157.Google Scholar
Toom, A. (2000). Algorithmical unsolvability of the ergodicity problem for binary cellular automata. Markov Process. Relat. Fields 6, 569–577.Google Scholar
Toom, A. (2001). Contours, Convex Sets, and Cellular Automata. IMPA, Rio de Janeiro.Google Scholar
Toom, A. et al. (1990). Discrete local Markov systems. In Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, eds Dobrushin, R., Kryukov, V. and Toom, A.. Manchester University Press, pp. 1–182.Google Scholar
Van Den Berg, J. and Steif, J. E. (1999). On the existence and nonexistence of finitary codings for a class of random fields. Ann. Prob. 27, 1501–1522.Google Scholar