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Stochastic cellular automata : ウィキペディア英語版
Stochastic cellular automaton

Stochastic cellular automata or 'probabilistic cellular automata' (PCA) or 'random cellular automata' or locally interacting Markov chains 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_ 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=\ or S_k=\ . The transition probability has a product form
P(d\sigma | \eta) = \otimes_ 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_) where
\eta_=(\eta_j)_ with a finite neighbourhood of k. See 〔(P.-Y. Louis PhD )〕 for a more detailed introduction following the probability theory's point of view.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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