So even if a system at some level follows continuous rules it is still possible for the system to exhibit discrete overall behavior. And in fact it is quite common for such behavior to be one of the most obvious features of a system—which is why discrete systems like cellular automata end up often being the most appropriate models.

The Problem of Satisfying Constraints

One feature of programs is that they immediately provide explicit rules that can be followed to determine how a system will behave. But in traditional science it is common to try to work instead with constraints that are merely supposed implicitly to force certain behavior to occur.

At the end of Chapter 5 I gave some examples of constraints, and I showed that constraints do exist that can force quite complex behavior to occur. But despite this, my strong suspicion is that of all the examples of complex behavior that we see in nature almost none can in the end best be explained in terms of constraints.

The basic reason for this is that to work out what pattern of behavior will satisfy a given constraint usually seems far too difficult for it to be something that happens routinely in nature.

Many types of constraints—including those in Chapter 5—have the property that given a specific pattern it is fairly easy to check whether the pattern satisfies the constraints. But the crucial point is that this fact by no means implies that it is necessarily easy to go from the constraints to find a pattern that satisfies them.

The situation is quite different from what happens with explicit evolution rules. For if one knows such rules then these rules immediately yield a procedure for working out what behavior will occur. Yet if one only knows constraints then such constraints do not on their own immediately yield any specific procedure for working out what behavior will occur.

In principle one could imagine looking at every possible pattern, and then picking out the ones that satisfy the constraints. But even with a 10×10 array of black and white squares, the number of possible patterns is already 1,267,650,600,228,229,401,496,703,205,376. And with a

From Stephen Wolfram: A New Kind of Science [citation]