The discussion so far has concentrated on the particular cellular automaton rule given by Eq. 1. This rule may be generalized in several ways. One family of rules is obtained by allowing the value of a site to be an arbitrary function of the values of the site itself and of its two nearest neighbors on the previous time step:
Further generalizations allow each site in a cellular automaton to take on an arbitrary number of values and allow the value of a site to depend on the values of sites at a distance up to on both sides, so that
The number of different rules with given and grows as and therefore becomes immense even for rather small and .
Figure 8 shows examples of evolution according to some typical rules with various and values. Each rule leads to patterns that differ in detail. However, the examples suggest a very remarkable result: all patterns appear to fall into only four qualitative classes. These basic classes of behavior may be characterized empirically as follows:
The existence of only four qualitative classes implies considerable universality in the behavior of cellular automata; many features of cellular automata depend only on the class in which they lie and not on the precise details of their evolution. Such universality is analogous, though probably not mathematically related, to the universality found in the equilibrium statistical mechanics of critical phenomena. In that case many systems with quite different detailed construction are found to lie in classes with critical exponents that depend only on general, primarily geometrical features of the systems and not on their detailed construction.