Stochastic cellular automaton

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Stochastic cellular automata or 'probabilistic cellular automata' (PCA) or 'random cellular automata' or locally interacting Markov chains[1][2] are an important extension of cellular automaton. Cellular automata are a discrete-time dynamical system of interacting entities, whose state is discrete.

The state of the collection of entities is updated at each discrete time according to some simple homogeneous rule. All entities' states are updated in parallel or synchronously. Stochastic Cellular Automata are CA whose updating rule is a stochastic one, which means the new entities' states are chosen according to some probability distributions. It is a discrete-time random dynamical system. From the spatial interaction between the entities, despite the simplicity of the updating rules, complex behaviour may emerge like self-organization. As mathematical object, it may be considered in the framework of stochastic processes as an interacting particle system in discrete-time.

PCA as Markov stochastic processes

As discrete-time Markov process, PCA are defined on a product space  E=\prod_{k \in G} S_k (cartesian product) where  G is a finite or infinite graph, like  \mathbb Z and where  S_k is a finite space, like for instance   S_k=\{-1,+1\} or   S_k=\{0,1\} . The transition probability has a product form   P(d\sigma | \eta) = \otimes_{k \in G} p_k(d\sigma_k | \eta) where   \eta \in E and   p_k(d\sigma_k | \eta) is a probability distribution on   S_k . In general some locality is required   p_k(d\sigma_k | \eta)=p_k(d\sigma_k | \eta_{V_k}) where   \eta_{V_k}=(\eta_j)_{j\in V_k} with   {V_k}  a finite neighbourhood of k. See [3] for a more detailed introduction following the probability theory's point of view.

Examples of stochastic cellular automaton

Majority cellular automaton

There is a version of the majority cellular automaton with probabilistic updating rules. See the Toom's rule.

Relation to random fields

PCA may be used to simulate the Ising model of ferromagnetism in statistical mechanics.[4] Some categories of models were studied from a statistical mechanics point of view.

Cellular Potts model

There is a strong connection[citation needed] between probabilistic cellular automata and the cellular Potts model in particular when it is implemented in parallel.

References

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  2. R. L. Dobrushin, V. I. Kri︠u︡kov, A. L. Toom (1978). Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis. ISBN 9780719022067. <templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  3. P.-Y. Louis PhD
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Additional reading

  • Almeida, R. M.; Macau, E. E. N. (2010), "Stochastic cellular automata model for wildland fire spread dynamics", 9th Brazilian Conference on Dynamics, Control and their Applications, June 7–11, 2010<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
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  • Mahajan, Meena Bhaskar (1992), Studies in language classes defined by different types of time-varying cellular automata, Ph.D. dissertion, Indian Institute of Technology Madras<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
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