For every single cellular automaton after all ultimately has a different underlying rule, with different properties and potentially different consequences. But the next few pages [ 232 , 233 , 234 ] show various sequences of cellular automata, all starting from random initial conditions. … In class 1, the behavior is very simple, and almost all initial conditions lead to exactly the same uniform final state.

[No text on this page] The complete collection of all 171 patterns needed to satisfy constraints of the type shown on the previous page . If none of these 171 patterns satisfy a particular constraint, then it follows that no pattern at all will satisfy the constraint.

The first multiway system actually generates all strings in the end (not least since it yields the lemmas -> and -> )—and in fact strings of length n > 2 appear after at most 2n+7 steps. The second multiway system generates only the n+1 strings where black comes before white—and all of these strings appear after at most 7n steps. … All the strings generated have an even number of black elements.

But it is rather easy to foil this particular approach to cryptanalysis: all one need do is not sample every single cell in a given column in forming the encrypting sequence. … With two complete columns given, all cells to the left are determined to be either black or white. … Note that when two complete adjacent columns are specified all the cells on the left-hand side are determined.

But from this there is no guarantee that the vast majority of initial conditions—including perhaps all those that could readily arise in nature—will not just yield behavior that corresponds only to very simple computations. … And certainly there are all sorts of pictures in this book that lend support to this idea. … The idea would be that a pattern that is universal could serve as a kind of directory of possible computations—with different regions in the pattern giving results for all possible different initial conditions.

Among all the various mathematical functions defined, say, in Mathematica it turns out that there are also a few—not traditionally common in natural science—which yield complex curves but which do not appear to have any explicit dependence on representations of individual numbers. … Indeed, despite immense mathematical effort for over a century, it has so far been impossible even to establish for example the so-called Riemann Hypothesis, which in effect just states that all the peaks in the curve lie above the axis, and all the valleys below. … The celebrated Riemann Hypothesis in effect states that all peaks after the first one in this curve must lie above the axis.

And indeed, what looking at the shift map in terms of digit sequences shows us is that this phenomenon on its own can make no contribution at all to what we can reasonably consider the ultimate production of randomness. Continuous Cellular Automata Despite all their differences, the various kinds of programs discussed in the previous chapter have one thing in common: they are all based on elements that can take on only a discrete set of possible forms, typically just colors black and white. … Cases (a), (b) and (d) exhibit sensitive dependence on initial conditions, in the sense that a change in insignificant digits far to the right eventually grows to affect all digits.

And with the more than ten billion blocks between 30 and 34 cells wide, no new structures at all appear. … And the way to tell this is that for small repetition periods there is a systematic procedure that allows one to find absolutely all structures with a given period. … And for all repetition periods up to 10—with the exception of 7—at least one fixed or moving structure ultimately turns out to exist.

And as the pictures demonstrate, all of these exhibit fairly trivial behavior, in which only rather simple transformations are ever made to the initial configuration of cells. … In all cases the transformations made to the initial conditions are simple enough that it is straightforward to go backwards as well as forwards in the evolution. … Even though these systems exhibit complex behavior that scrambles the initial conditions, all of them are still reversible, so that starting from the configuration of cells at the bottom of each picture, it is always possible to deduce the configurations on all previous steps.

And indeed my guess is that most of them will actually in the end turn out to depend on all sorts of limits and idealizations in quantum theory—and will emerge just as simple approximations to much more complex underlying behavior. In its development since the early 1900s quantum theory has produced all sorts of elaborate results. And to try to derive them all from the kinds of models I have outlined here will certainly take an immense amount of work.