But doing this will usually involve building up a vast network of strings. … And just like in the systems based on constraints in Chapter 5 one can usually do at least somewhat better than just to look at every possible path in turn. … Sometimes one can do this by reducing every lemma—and possibly every string—to some at least partially canonical form.

So if they have simple underlying rules, do all cellular automata started from random initial conditions eventually settle down to give stable states that somehow look simple?

But just as for random walks, it appears once again that the details of the underlying rules for the system do not have much effect on the main features of the behavior that is seen.

So just how does this angle show up in actual plant systems?

At an idealized level one might imagine trying to do this by inserting into the system some kind of paddle which would experience force as a result of impacts from particles.

And does it really apply to all of the various kinds of systems that we see in nature?

One trivial way to do so is to take a universal system but modify it so that if it ever halts its output is discarded and, say, replaced by its original input. … So one can then ask whether it is possible to have a system which exhibits undecidability, but with a pattern that does not correspond to that of any universal system. … (A dot indicates that the register machine does not halt.)

But to do this has essentially always required finding a case where there is explicit overlap with a known language—say a Rosetta stone with the same text in multiple languages, or at least words or proper names that are transliterated.

My emphasis of the importance of perception and analysis might seem to support this view, and to some extent it does.

Often these can be thought of as one-way versions of axioms for operator systems (see page 1172 ), but applied only once per step (as /. does), rather than in all possible ways (as in a multiway system)—so that the evolution is just given by NestList[#/.rule &, init, t] .