1986 | OriginalPaper | Buchkapitel
Directional Entropies of Cellular Automaton-Maps
verfasst von : John Milnor
Erschienen in: Disordered Systems and Biological Organization
Verlag: Springer Berlin Heidelberg
Enthalten in: Professional Book Archive
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Consider a fixed lattice L in n -dimensional euclidean space, and a finite set K of symbols. A correspondence a which assigns a symbol a(x) ∈K to each lattice point x ∈ L will be called a configuration. An n -dimensional cellular automaton can be described as a map which assigns to each such configuration a some new configuration a′ = f (a) by a formula of the form$$ a'(x) = F(a(x + {v_l}),...,\,a(x + {v_{\tau }})) $$ a’(x) = F(a(x + v1)), ⋯, a(x + vr)), where v1, ⋯, vr are fixed vectors in the lattice L, and where F is a fixed function of r symbols in K. I will call f the cellular automaton-map which is associated with the local map F. If the alphabet K has k elements, then the number of distinct local maps F is equal to kk′. This is usually an enormous number, so that it is not possible to examine all of the possible F. Depending on the particular choice of F and of the v1, such an automaton may display behavior which is simple and convergent, or chaotic and random looking, or behavior which is very complex and difficult to describe. (Compare [Wolfram].)