Invariant strings and pattern-recognizing properties of one-dimensional cellular automata

Abstract

A cellular automaton is a discrete dynamical system whose evolution is governed by a deterministic rule involving local interactions. It is shown that given an arbitrary string of values and an arbitrary neighborhood size (representing the range of interaction), a simple procedure can be used to find the rules of that neighborhood size under which the string is invariant. The set of nearestneighbor rules for which invariant strings exist is completely specified, as is the set of strings invariant under each such rule. For any automaton rule, an associated “filtering” rule is defined for which the only attractors are spatial sequences consisting of concatenations of invariant strings. A result is provided defining the rule of minimum neighborhood size for which an arbitrarily chosen string is the unique invariant string. The applications of filtering rules to pattern recognition problems are discussed.