Abstract
Modelled as finite homogeneous Markov chains, probabilistic cellular automata with local transition probabilities in (0, 1) always posses a stationary distribution. This result alone is not very helpful when it comes to predicting the final configuration; one needs also a formula connecting the probabilities in the stationary distribution to some intrinsic feature of the lattice configuration. Previous results on the asynchronous cellular automata have showed that such feature really exists. It is the number of zero-one borders within the automaton's binary configuration. An exponential formula in the number of zero-one borders has been proved for the 1-D, 2-D and 3-D asynchronous automata with neighborhood three, five and seven, respectively. We perform computer experiments on a synchronous cellular automaton to check whether the empirical distribution obeys also that theoretical formula. The numerical results indicate a perfect fit for neighbourhood three and five, which opens the way for a rigorous proof of the formula in this new, synchronous case.
Highlights
From a mathematical point of view, cellular automata (CA) are binary lattices that are updated iteratively
We found formulas for the stationary distribution of various asynchronous cellular automata [1,3,4], and we connected our findings to existent results from Ising and exponential voter model [2]
Classification and prediction are the key issues in cellular automata
Summary
From a mathematical point of view, cellular automata (CA) are binary lattices that are updated iteratively. A finite homogeneous Markov chain is a stochastic process that moves according to some probabilities within a finite set of states, say S~f1,2,, . Transition matrix of a Markov chain is always stochastic - that is, the sum of probabilities in each row is one, and since in our case the matrix does not change from an iteration to another, it is called homogeneous. PLOS ONE | www.plosone.org convergence is of interest - this topic is usually referred to in literature as absorption time. When it comes to CA, literature has focused so far only on the first two topics. The computation of absorption time is of certain interest, at least for the class of deterministic automata with two attractors, all zeros and all ones
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