Asymptotic Behavior

The difficulties in classifying cellular automata and networks based on their global behavior have been explained in Chapter 8. The results tend to indicate that a complete classification is, at least in many computational ways, impractical or unfeasible. The study of the specific properties of global behavior of arbitrary automata in terms of their local rules is indeed a most interesting and difficult problem.

Among other reasons for this difficulty is that these problems appear to be of a discrete and combinatorial nature, where classical optimization and analysis are not directly applicable. This chapter deals with positive results that shed light on the long-term behavior of global dynamics. Each technique has only been partially successful and they can be roughly classified as either discrete/combinatorial or analytic. The first section presents some results obtained through combinatorial techniques. Later, several analogies with continuous classical dynamical systems are exploited to gain insight into the nature of the long-term behavior of cellular networks. Despite complex behavior as chaotic as that of maps on the interval, many of them exhibit an interesting property of observability through simulation on computing devices under limitations such as bounded precision, rounding errors, and noise.

You can fool all the people some of the time. You can even fool some of the people all the time. But you can’t fool all the people all the time.

Abraham Lincoln