As computer power steadily increases, there have been ever increasing opportunities to perform experimental mathematics, both by professionals and by amateurs. An excellent example of this tendency has been all the recent work on fractals, nonlinear differential equations, dynamical systems, and similar topics. Many extremely classical results, such as the results of Pierre Fatou and Gaston Julia concerning functional iteration, some very esoteric topological results such as Stephen Smale's strange attractors, or even such advanced theorems relating to differential equations such as Kolmogoroff, Arnol'd and Moser's results in celestial mechanics, acquire an entire new perspective when their consequences can be followed through a numerical calculation.
A rather interesting confrontation of this nature took place when Stephen Wolfram began to experiment with the evolution of one dimensional cellular automata, and felt that he saw a certain amount of analogy between the phenomenological characteristics of the evolution of the automata and some classifications of limit sets which had been found relevant in nonlinear dynamics. As a result of numerous observations, he proposed a system of four classes, which reflected the same number of different kinds of evolution which he had been observing. Roughly speaking, cellular automata seem to settle down to a constant field (Class I), isolated periodic structures (Class II), uniformly chaotic fields (Class III), or isolated structures showing complicated internal behavior (Class IV).
In addition to a series of computer experiments in one and two dimensions, with varying numbers of states per cell and sizes of neighborhoods, Wolfram, occasionally working with some coauthors, has analyzed the mathematical aspects of the rules of evolution of their automata. Thus[72] discusses the relation of automata to formal language theory, including the observation that the evolute of a configuration described by a regular expression is still a regular expression, but may fail to remain so in the limit. Lyman P. Hurd has taken up the explicit question of the limiting behavior in his thesis[39] and related publications[38],[40].
Wolfram also discusses some measures of complexity, such as the size of the automata associated with the regular expressions.
Martin, Odlyzko, and Wolfram[49] have worked out the evolution of
automata for which 
is defined by a linear combination of the
states of the cells of the neighborhood, assuming them to be elements of
the corresponding finite field; an example would be evolution according
to the rule of parity, in which each cell transforms into the sum of
the three neighbors, 
. Although such automata may not be
entirely typical, the fact that quite complete and exact results can be
obtained makes for useful comparisons.
Wolfram has edited a collection of most of his own papers and a good assortment of others, including several useful appendices containing a wide variety of data[73].
Notwithstanding the fact that assigning an automaton to one of Wolfram's classes is an undecidable proposition[17], his investigations represent the first time that a systematic and exhaustive study had been made of large classes of one dimensional linear cellular automata, rather than selecting a single automaton for concentrated attention. At the same time the classification has a strong visual appeal and has been widely adopted by other investigators, along with a few notational details such as his numbering scheme.