And from this one might begin to suspect that in the end the kinds of programs which generate all these forms are quite similar—and all potentially rather simple. … And indeed at first one might think that it would never really be possible to say much at all about complexity just by looking at parts of organisms.

But over the past century it has at least become universally accepted that all matter is made up of identifiable discrete particles. … And it is one of the striking observed regularities of the universe that all particles of a given kind—say electrons—seem to be absolutely identical in their properties. … Instead, all the particles we see would just emerge as structures formed from more basic elements.

For any network that has a serious chance of representing actual space—even a supposedly empty part—will no doubt show all sorts of seemingly random activity. … Starting off with a network that is planar—so that it can be drawn flat on a page without any lines crossing—such rules can certainly give all sorts of complex and apparently random behavior. But the way the rules are set up, all the networks they produce must still be planar.

The repetitive structure of picture (a) implies that to reproduce this picture all we need do is to specify the colors in a 49×2 block, and then say that this block should be repeated an appropriate number of times. Similarly, the nested structure of picture (b) implies that to reproduce this picture, all we need do is to specify the colors in a 3×3 block, and then say that as in a two-dimensional substitution system each black cell should repeatedly be replaced by this block. … But the fact that no short description can be found by our usual processes of perception and analysis does not in any sense mean that no such description exists at all.

If one has a deterministic model then it is at least in principle quite straightforward to find out whether the model is correct: for all one has to do is to compare whatever specific behavior the model predicts with behavior that one observes. … As one simple example, consider a model in which all possible sequences of black and white squares are supposed to occur with equal probability. By effectively enumerating all such sequences, it is easy to see that such a model predicts that in any particular sequence the fraction of black squares is most likely to be 1/2 .

But the remarkable assertion that the Principle of Computational Equivalence makes is that in practice this is not the case, and that instead there is essentially just one highest level of computational sophistication, and this is achieved by almost all processes that do not seem obviously simple. … For the essence of this phenomenon is that it is possible to construct universal systems that can perform essentially any computation—and which must therefore all in a sense be capable of exhibiting the highest level of computational sophistication. … But in the fifty or so years since the phenomenon of universality was first identified, all sorts of types of systems have been found to be able to exhibit universality.

The way I have set things up, one can find all the statements that can be proved true in a particular axiom system just by starting with an expression that represents "true" and then using the rules of the axiom system, as in the picture on the facing page . … All strings that appear can be thought of as statements that are true according to the axioms represented by the multiway system rules. … The second one, however, is both complete and consistent: it generates all strings that begin with , but none that begin with .

Note the presence of triangles and other small structures dotted throughout all of the pictures.

But now, with the discovery that simple programs can capture the essential mechanisms for all sorts of complex behavior in nature, one can imagine just sampling such programs to explore generalizations of the forms we see in nature. … Despite all its success, there is still much that goes on in nature that seems more complex and sophisticated than anything technology has ever been able to produce. … But the discoveries in this book show that this is not the case, and that in fact extremely simple underlying rules—that might for example potentially be implemented directly at the level of atoms—are often all that is needed.

With intrinsic randomness generation, however, there is no such limit: in the cellular automaton above, for example, all one need do to get a longer random sequence is to run the cellular automaton for more steps. … The basic reason is that intrinsic randomness generation in a sense puts all the components in a system to work in producing new randomness, while getting randomness from the environment does not. … But in a system that just amplifies randomness from the environment, none of the components inside the system itself ever contribute any new randomness at all.