The 256 "elementary" rules that we have discussed so far are by most measures the simplest possible—and were the first ones I studied. … The picture below shows one example of how this works. … Interpreting the sequence of new colors as a sequence of base 3 digits, one can assign a code number to each totalistic rule.

The practical importance of this phenomenon depends greatly however on how far one has to go to get to the threshold of universality. But knowing that a system like rule 110 is universal, one now suspects that this threshold is remarkably easy to reach. … And my expectation is that if one looks sufficiently hard at any

But to have any kind of traditional theory one must find a shortcut that involves much less computation. Yet from the picture on the facing page it is certainly not obvious how one might do this. … For it implies that even if in principle one has all the information one needs to work out how some particular system will behave, it can still take an irreducible amount of computational work actually to do this.

And indeed within the usual formalism one can construct quantum computers that may be able to solve at least a few specific problems exponentially faster than ordinary Turing machines. … One can specify the number of steps t that one wants by giving the sequence of digits in t . … But the point is that when computational irreducibility is present, one may in effect explicitly have to follow each of the t steps of evolution—again requiring exponentially more computational work.

As I discussed earlier, most of the common axiom systems in traditional mathematics are known to be universal—basic logic being one of the few exceptions. … So in either idealization, one should not have to go far to get axiom systems that exhibit universality—just like most of the ones in traditional mathematics. But once one has reached an axiom system that is universal, why should one in a sense ever have to go further?

[No text on this page] One-dimensional slices through the evolution of various two-dimensional cellular automata. … Note the presence of examples of both class 3 and class 4 behavior that look strikingly similar to examples in one dimension.

The result, once again, is that one gets an intricate but highly regular nested pattern. In a substitution system where black squares are arranged on a grid, one can be sure that different squares will never overlap. … Note that in applying the rule to a particular square, one must take account of the orientation of that square.

The Sequencing of Events in the Universe In the last section I discussed one type of model in which familiar notions of time can emerge without any kind of built-in global clock. The particular models I used were based on mobile automata—in which the presence of a single active cell forces only one event ever to occur in the universe at once. … One can think of mobile automata as being special cases of substitution systems of the type I introduced in Chapter 3 .

Indeed, if one just looked at the sizes of numbers produced, then one sees the same kind of complexity as in cases (a) and (b). But looking at digit sequences one realizes that this complexity is actually just a direct transcription of complexity introduced by giving an initial condition with a seemingly random digit sequence.

those steps at which one of the two registers has just decreased to zero. And in this picture one immediately sees some apparently random variation in the instructions that are executed. … Another way to set up more complicated register machines is to extend the kinds of underlying instructions one allows.