Abstract
Cellular automata are both seen as a model of computation and as tools to model real life systems. Historically they were studied under synchronous dynamics where all the cells of the system are updated at each time step. Meanwhile the question of probabilistic dynamics emerges: on the one hand, to develop cellular automata which are capable of reliable computation even when some random errors occur [24,14,13]; on the other hand, because synchronous dynamics is not a reasonable assumption to simulate real life systems.
Among cellular automata a specific class was largely studied in synchronous dynamics : the elementary cellular automata (ECA). These are the ”simplest” cellular automata. Nevertheless they exhibit complex behaviors and even Turing universality. Several studies [20,7,8,5] have focused on this class under α-asynchronous dynamics where each cell has a probability α to be updated independently. It has been shown that some of these cellular automata exhibit interesting behavior such as phase transition when the asynchronicity rate α varies.
Due to their richness of behavior, probabilistic cellular automata are also very hard to study. Almost nothing is known of their behavior [20]. Understanding these ”simple” rules is a key step to analyze more complex systems. We present here a coupling between oriented percolation and ECA 178 and confirms observations made in [5] that percolation may arise in cellular automata. As a consequence this coupling shows that there is a positive probability that the ECA 178 does not reach a stable configuration as soon as the initial configuration is not a stable configuration and α> 0.996. Experimentally, this result seems to stay true as soon as α > α c ≈ 0.5.
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References
Balister, P., Bollobás, B., Kozma, R.: Large deviations for mean fields models of probabilistic cellular automata. Random Structures & Algorithms 29, 399–415 (2006)
Bersini, H., Detours, V.: Asynchrony induces stability in cellular automata based models. In: Proceedings of Artificial Life IV, pp. 382–387. MIT Press, Cambridge (1994)
Buvel, R.L., Ingerson, T.E.: Structure in asynchronous cellular automata. Physica D 1, 59–68 (1984)
Cook, M.: Universality in elementary cellular automata. Complex system 15, 1–40 (2004)
Fatés, N.: Directed percolation phenomena in asynchronous elementary cellular automata. In: El Yacoubi, S., Chopard, B., Bandini, S. (eds.) ACRI 2006. LNCS, vol. 4173, pp. 667–675. Springer, Heidelberg (2006)
Fatés, N., Morvan, M.: An experimental study of robustness to asynchronism for elementary cellular automata. Complex Systems 16(1), 1–27 (2005)
Fatés, N., Morvan, M., Schabanel, N., Thierry, É.: Asynchronous behaviour of double-quiescent elementary cellular automata. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 316–327. Springer, Heidelberg (2005)
Fatés, N., Regnault, D., Schabanel, N., Thierry, É.: Asynchronous behaviour of double-quiescent elementary cellular automata. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887. Springer, Heidelberg (2006)
Fortuin, C.M., Kasteleyn, P.W.: On the random cluster model. i. Introduction and relation to other models. Physica 57, 536–564 (1972)
Fukś, H.: Non-deterministic density classification with diffusive probabilistic cellular automata. Phys. Rev. E 66(2) (2002)
Fukś, H.: Probabilistic cellular automata with conserved quantities. Nonlinearity 17(1), 159–173 (2004)
Gács, P.: Reliable computation with cellular automata. Journal of Computer and System Sciences 32(1), 15–78 (1986)
Gács, P.: Reliable cellular automata with self-organization. Journal of Statistical Physics 103(1/2), 45–267 (2001)
Gács, P., Reif, J.: A simple three-dimensional real-time reliable cellular array. Journal of Computer and System Sciences 36(2), 125–147 (1988)
Grimmett, G.: Percolation, 2nd edn. Grundlehren der mathematischen Wissenschaften, vol. 321. Springer, Heidelberg (1999)
Grimmett, G.: The Random-Cluster Model. Grundlehren der mathematischen Wissenschaften, vol. 333. Springer, Heidelberg (2006)
Lumer, E.D., Nicolis, G.: Synchronous versus asynchronous dynamics in spatially distributed systems. Physica D 71, 440–452 (1994)
Ollinger, N.: The intrinsic universality problem of one-dimensional cellular automata. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 632–641. Springer, Heidelberg (2003)
Ovchinnikov, A.A., Dmitriev, D.V., Krivnov, V.Y., Cheranovskii, V.O.: Antiferromagnetic ising chain in a mixed transverse and longitudinal magnetic field. Physical review B. Condensed matter and materials physics 68(21) 214406.1–214406.10 (2003)
Regnault, D.: Abrupt behavior changes in cellular automata under asynchronous dynamics. In: Proceedings of 2nd European Conference on Complex Systems (ECCS), Oxford, UK (to appear, 2006)
Regnault, D., Schabanel, N., Thierry, É.: Progresses in the analysis of stochastic 2D cellular automata: a study of asynchronous 2D Minority. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 320–332. Springer, Heidelberg (2007)
Sarkar, P.: A brief history of cellular automata. ACM Computing Surveys 32(1), 80–107 (2000)
Schönfisch, B., de Roos, A.: Synchronous and asynchronous updating in cellular automata. BioSystems 51, 123–143 (1999)
Toom, A.: Stable and attractive trajectories in multicomponent systems. Advances in Probability 6, 549–575 (1980)
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Regnault, D. (2008). Directed Percolation Arising in Stochastic Cellular Automata Analysis. In: Ochmański, E., Tyszkiewicz, J. (eds) Mathematical Foundations of Computer Science 2008. MFCS 2008. Lecture Notes in Computer Science, vol 5162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85238-4_46
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DOI: https://doi.org/10.1007/978-3-540-85238-4_46
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