One might think that the existence of such a discrete transition must somehow be associated with the discrete nature of the underlying cellular automaton rules. … If one starts to the left of the center hump, then the ball will always roll into the left-hand minimum. But if one progressively changes the initial position of the ball, then when one passes the center a discrete transition occurs, and the ball instead rolls into the right-hand minimum.

And most likely those features in which a great degree of constancy is seen are precisely the ones that have successfully been molded by natural selection. … For traditional intuition makes one think that to get the level of complexity that one sees in biological systems must require great effort—and the long and ponderous course of evolution revealed in the fossil record seems like just the kind of process that should be involved. But the point is that what I have discovered in this book shows that in fact if one just chooses programs at random, then it is easy to get behavior of great complexity.

One may wonder, however, to what extent the behavior one sees depends on the exact scheme that one uses to pick which replacements to apply at each step. … What one sees in these pictures is that after event 4, different choices of replacements are made in the two cases, and the causal relationships implied by these replacements are different. … One still has a The behavior of rules (a) and (b) from the facing page when replacements are performed at random.

But as a simple idealization, one can consider an array of squares on the retina, each colored either black or white. … In each case, one can think of the results as corresponding to picking out some specific local feature in the original image. … The sample images used here are ones generated by the evolution of elementary one-dimensional cellular automata with rules 60 and 124 respectively.

But in about one-third of cases, the patterns instead grow forever. Of such growing patterns, the simplest kind are purely repetitive ones, such as those seen in rules 50 and 109. … The simplest and by far the most common is the one exemplified by rules 22 and 60.

But as a simple idealization one can assume that such cells in the end just respond to repeated versions of the basic 2×2 patterns. So with this setup, the pictures on the facing page show what happens with an image like the one from page 578 . … The comparative sparsity of dark blocks is a consequence of the fact that at any given position a dark block can occur in only one of the 16 cases shown.

But if one also allows higher mathematical functions then it turns out that such a formula can in fact be found: as indicated in the table above each coefficient is given by a particular value of a so-called Gegenbauer or ultraspherical function. … Both of the patterns shown on the previous page are rather special in that as well as being generated by substitution systems they can also be produced one row at a time by the evolution of one-dimensional cellular automata with simple additive rules. … The picture at the top of the facing page is one example.

And one can imagine quite a few ways that such a mechanism could potentially be set up with nerve cells. One step in a particularly simple scheme is illustrated in the picture below. … One step in a very simple model of the way hash codes for arbitrary data might be generated by layers of nerve cells in the brain.

But it might also be that if one expects to retrieve elements of memory reliably then there is no choice but to set things up so that the hashing procedure one uses always stays essentially the same. … But there are also ones that we do not. … And so to cause any substantial change one would presumably have to insert a comparable amount of data with the special properties one wants.

If one tries to use some kind of systematic procedure to test whether systems are universal then inevitably there will be three types of outcomes. … For it could still be that the particular rules that appear are somehow specially selected to be ones that are not universal. … Thus, for example, in most kinds of engineering one tends to pick rules whose behavior is simple enough that one can readily predict it.