As it turns out, the first substitution system shown works almost exactly like a cellular automaton. Indeed, away from the right-hand edge, all the elements effectively behave as if they were lying on a regular grid, with the color of each element depending only on the previous color of that element and the element immediately to its right.

The second substitution system shown again has patches that exhibit a regular grid structure. But between these patches, there are regions in which elements are created and destroyed. And in the other substitution systems shown, elements are created and destroyed throughout, leaving no trace of any simple grid structure. So in the end the patterns we obtain can look just as random as what we have seen in systems like cellular automata.

Sequential Substitution Systems

None of the systems we have discussed so far in this chapter might at first seem much like computer programs of the kind we typically use in practice. But it turns out that there are for example variants of substitution systems that work essentially just like standard text editors.

The first step in understanding this correspondence is to think of substitution systems as operating not on sequences of colored elements but rather on strings of elements or letters. Thus for example the state of a substitution system at a particular step can be represented by the string ABBBABA, where the A's correspond to white elements and the B's to black ones.

The substitution systems that we discussed in the previous section work by replacing each element in such a string by a new sequence of elements—so that in a sense these systems operate in parallel on all the elements that exist in the string at each step.

But it is also possible to consider sequential substitution systems, in which the idea is instead to scan the string from left to right, looking for a particular sequence of elements, and then to perform a replacement for the first such sequence that is found. And this setup is now directly analogous to the search-and-replace function of a typical text editor.

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