Tree volume measurement
Stochastic cellular automata or 'probabilistic cellular automata' (PCA) or 'random cellular automata' or locally interacting Markov chains[1] 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 homogenous 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.
Up-to-date examples of applications for CA and PCA main be found for instance there.[2]
PCA as Markov stochastic processes
As discrete-time Markov process, PCA are defined on a product space (cartesian product) where is a finite or infinite graph, like and where is a finite space, like for instance or . The transition probability has a product form where and is a probability distribution on . In general some locality is required where with a finite neighbourhood of k.
Examples of stochastic cellular automaton
Majority cellular automaton
There is a version of the majority cellular automaton with probabilistic updating rules.
Relation to random fields
PCA may be used to simulate the Ising model of ferromagnetism in statistical mechanics.[3] Some categories of models were studied from a statistical mechanics point of view.
Cellular Potts model
There is a strong connection between probabilistic cellular automata and the cellular Potts model in particular when it is implemented in parallel.
References
[1] Smith Iii, A. 1972. Real-time language recognition by one-dimensional cellular automata. J. Comput. Syst. Sci. 6, 233–253.
[2] Mahajan, M. 1992. Studies in language classes defined by different types of time-varying cellular automata. Ph.D. Dissertation.
[3] Nishio, H. And Kobuchi, Y. 1975. Fault tolerant cellular spaces. J. Comput. Syst. Sci. 11, 150–170.
[4] Clarke, K. C. And Hoppen, S. 1997. A self-modifying cellular automaton model of historical urbanization in the San Francisco Bay area. Environment and Planning B: Planning and Design, Vol. 24, 247–261
[5] Almeida, R. M. And Macau, E. E. N. 2010. Stochastic cellular automata model for wildland fire spread dynamics. 9th Brazilian Conference on Dynamics, Control and their Applicarions, June 7–11, 2010.
[6] Wolfgang von der Linden And Ewald Schachinger. 2005. Computer simulation, lecture notes.
[7] Klaus Lichtenegger. 2005. Stochastic cellular automaton models in disease spreading and ecology
[8] Benny Brown. 2003. Phase transition of two-neighbor stochastic cellular automata.
[9] http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life
References cited in text
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- ↑ Locally interacting systems and their application in Biology, Dobrushin, Kryukov, Toom editors, 1978, Lecture Notes in Math vol 653, springer pp 2–113
- ↑ Series: Lecture Notes in Computer Science, Vol. 7495, 6350, 5191, 4173, 3305. Ed. Springer
- ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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