Discrete Mathematics & Theoretical Computer Science, Vol 12, No 2 (2010)

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On the analysis of ”simple” 2D stochastic cellular automata

Damien Regnault, Nicolas Schabanel, Eric Thierry


Cellular automata are usually associated with synchronous deterministic dynamics, and their asynchronous or stochastic versions have been far less studied although significant for modeling purposes. This paper analyzes the dynamics of a two-dimensional cellular automaton, 2D Minority, for the Moore neighborhood (eight closest neighbors of each cell) under fully asynchronous dynamics (where one single random cell updates at each time step). 2D Minority may appear as a simple rule, but it is known from the experience of Ising models and Hopfield nets that 2D models with negative feedback are hard to study. This automaton actually presents a rich variety of behaviors, even more complex that what has been observed and analyzed in a previous work on 2D Minority for the von Neumann neighborhood (four neighbors to each cell) (2007). This paper confirms the relevance of the later approach (definition of energy functions and identification of competing regions). Switching to the Moore neighborhood however strongly complicates the description of intermediate configurations. New phenomena appear (particles, wider range of stable configurations). Nevertheless our methods allow to analyze different stages of the dynamics. It suggests that predicting the behavior of this automaton although difficult is possible, opening the way to the analysis of the whole class of totalistic automata.

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