Commutative Cellular Automata

Totalistic cellular automata were introduced by Wolfram [1, 2] as a means of studying higher-color cellular automata within a reasonable space of possible rules. The underlying principle is to systematically construct a subset of rules within all the possible rules of the highercolor cellular automata, and the system in totalistic cellular automata is to add up the values of the cells within the neighborhood and determine the next cell from this sum. In this paper, we generalize this idea of taking a subset of the possible rules in cellular automata by partitioning all possible neighborhoods into neighborhood equivalence classes and use this tool to analyze cellular automata containing totalistic cellular automata— commutative cellular automata. Commutative cellular automata have the property that the state of the determined cell is unaffected by the ordering of the cells in its neighborhood (some of these cellular automata meet the criteria of quasi-linear cellular automata, and so their evolutions can be computed more efficiently than the standard cellular automaton algorithm�[3]). The Game of Life is a case of outer-commutative cellular automata, where the state of the determined cell is unaffected by the noncenter cells. Using the concept of neighborhood equivalence classes, it can be shown that certain commutative cellular automata—totalistic and multiplistic—can emulate each other as well as the space of all commutative cellular automata. The space of all one-dimensional commutative cellular automata will be enumerated by using g==Sort[{##}]& as the transition function, and certain variations of commutative cellu-