This system is an example of one that does not in any meaningful way obey the Second Law of Thermodynamics.

Indeed, it turns out that many of the phenomena that have been most studied in traditional science simply do not occur in just one dimension. … How much does it depend on dimension? … Indeed, despite what we might expect from traditional science, adding more dimensions does not ultimately seem to have much effect on the occurrence of behavior of any significant complexity.

Yet the sequences it produces seem perfectly random, and do not suffer from any of the problems that are typically found in linear congruential generators. So why do linear congruential generators not produce better randomness? … So how does the occurrence of this mechanism compare to the previous two mechanisms for randomness that we have discussed?

And in a sense it is natural selection that is responsible for the fact that such programs do not survive. … So does this then mean that there can never be any kind of general theory for all the features of higher organisms? … Seeing in earlier chapters of this book all the diverse things that simple programs can do, it is easy to be struck by analogies to books of biological flora and fauna.

procedure can indeed be used to check that no purely repetitive pattern exists, but as we will see later in this chapter , it does not successfully detect the presence of even certain highly regular nested patterns. … But how can we ever expect to find any kind of precise general characterization of what all our various standard methods of perception and analysis do? … So does one really need to try essentially all sufficiently simple programs in order to determine this?

But as we saw in Chapter 10 , in almost no other case do standard methods of perception and analysis allow one to make much progress at all. As mentioned in Chapter 10 , however, I do know of a few systems based on numbers for which a fairly complete analysis can be given even though the overall behavior is not repetitive or nested or otherwise obviously simple. And no doubt some other examples like this do exist.

But even if one becomes convinced of the abstract fact that out of all possible rules that do not yield obviously simple behavior the vast majority are universal, this still does not quite establish the assertion made by the Principle of Computational Equivalence that rules of this kind that appear in nature and elsewhere are almost always universal. … And what this means is that such rules will typically show the same features as rules chosen at random from all possibilities—with the result that presumably they do in the end exhibit universality in almost all cases where their overall behavior is not obviously simple.

cannot in general do is to find an easy theory that will tell one without much effort what every aspect of this behavior will be. … Yet there are still often many reasons to want to use abstract theoretical models rather than just doing experiments on actual systems in nature and elsewhere. … Using this procedure one can certainly compute the color of any cell on row n by doing about n Log[n] 3 operations—instead of the n 2 needed if one carried out the cellular automaton evolution explicitly.

As we will see on page 338 the presence of such patterns is particularly clear when there are equal numbers of black and white cells in the initial conditions—but how these cells are arranged does not usually matter much at all. … The presence of such structure is most obvious when there are equal numbers of black and white cells in the initial conditions, but it does not rely on any regularity in the arrangement of these cells.

Repetitive patterns are common and some nested patterns are seen, but the more complicated kinds of patterns discussed in this chapter do not ever appear to have been used.