Directional Entropies of Cellular Automaton-Maps

Abstract

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 + v ₁)), ⋯, a(x + v _r)), where v ₁, ⋯, v _r 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 k ^k′. 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 v ₁, 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].)