Some Conclusions

In the chapter before this one, we discovered the remarkable fact that even though their underlying rules are extremely simple, certain cellular automata can nevertheless produce behavior of great complexity.

Yet at first, this seems so surprising and so outside our normal experience that we may tend to assume that it must be a consequence of some rare and special feature of cellular automata, and must not occur in other kinds of systems.

For it is certainly true that cellular automata have many special features. All their elements, for example, are always arranged in a rigid array, and are always updated in parallel at each step. And one might think that features like these could be crucial in making it possible to produce complex behavior from simple underlying rules.

But from our study of substitution systems earlier in this chapter we know, for example, that in fact it is not necessary to have elements that are arranged in a rigid array. And from studying mobile automata, we know that updating in parallel is also not critical.

Indeed, I specifically chose the sequence of systems in this chapter to see what would happen when each of the various special features of cellular automata were taken away. And the remarkable conclusion is that in the end none of these features actually matter much at all. For every single type of system in this chapter has ultimately proved capable of producing very much the same kind of complexity that we saw in cellular automata.

So this suggests that in fact the phenomenon of complexity is quite universal—and quite independent of the details of particular systems.

But when in general does complexity occur?

The examples in this chapter suggest that if the rules for a particular system are sufficiently simple, then the system will only ever exhibit purely repetitive behavior. If the rules are slightly more complicated, then nesting will also often appear. But to get complexity in the overall behavior of a system one needs to go beyond some threshold in the complexity of its underlying rules.