Substitution systems that correspond to mobile automata can be thought of as having rules and initial conditions that are specially set up so that only one updating event can ever occur on any particular step. But with most rules—including the one shown on the previous page—there are usually several possible replacements that can be made at each step.
One scheme for deciding which replacement to make is just to scan the string from left to right and then pick the first replacement that applies. This scheme corresponds exactly to the sequential substitution systems we discussed in Chapter 3.
The pictures on the facing page show a few examples of what can happen. The behavior one gets is often fairly simple, but in some cases it can end up being highly complex. And just as in mobile automata, the causal networks that emerge typically in effect grow linearly with time. But, again as in mobile automata, there are rules such as (a) in which there is no growth—and effectively no notion of space. And there are also rules such as (f)—which turn out to be much more common in general substitution systems than in mobile automata—in which the causal network in effect grows exponentially with time.
But why do only one replacement at each step? The pictures on the next page show what happens if one again scans from left to right, but now one performs all replacements that fit, rather than just the first one.
In the case of rules (a) and (b) the result is to update every single element at every step. But since the replacements in these particular rules involve only one element at a time, one in effect has a neighbor-independent substitution system of the kind we discussed on page 82. And as we discovered there, such systems can only ever produce rather simple behavior: each element repeatedly branches into several others, yielding a causal network that has the form of a regular tree.
So what happens with replacements that involve more than just one element? In many cases, the behavior is still quite simple. But as several of the pictures on the next page demonstrate, fairly simple rules are sufficient—as in so many other systems that we have discussed in this book—to obtain highly complex behavior.