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. … And since one can never know in advance how far computational reducibility will go in a particular system it is always worthwhile at least to try applying the traditional methods of theoretical science. … 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.

white squares exists that satisfies the constraint that on every row there is at least one square whose color agrees with the color of the corresponding square in the sequence. … And in the picture on the facing page almost every row of black, white and gray squares corresponds to one such constraint. … But almost always what is actually constructed is quite complicated—and certainly not something one would expect to occur at all often.

Intelligence in the Universe Whether or not we as humans are the only examples of intelligence in the universe is one of the great unanswered questions of science. … But one of the central discoveries of this book is that in fact nothing so elaborate is needed to get sophisticated computation. … Yet surely, one might argue, there must be something fundamentally more to true intelligence of the kind that we as humans have.

And as one might now expect from the intuition in this book, even systems like the one below with remarkably simple rules can still manage to show self-reproduction—despite the fact that they bear almost no other resemblance to ordinary living systems. If one looks at typical living systems one of their most obvious features is great apparent complexity. … But what I have shown in this book is that this is not the case, and that in fact a vast range of systems—including ones with very A two-dimensional cellular automaton that exhibits an almost trivial form of self-reproduction, in which multiple copies of any initial pattern appear every time the number of steps of evolution doubles.

But one of the central discoveries of this book is that this is not the case, and that in fact it is perfectly possible for systems even with extremely simple underlying rules to produce behavior that has immense complexity—and that looks like what one sees in nature. And I believe that if one uses such systems it is almost inevitable that a vast amount of new technology will become possible. … One example discussed in Chapter 10 is cryptography.

So if moving structures are inevitable in class 4 systems, what other fundamentally different kinds of structures might one see if one were to look at sufficiently many large initial conditions? … But when initial condition 54,889 is reached, one suddenly sees the rather different kind of structure shown on the next page .

So even if one just looks at overall sizes of whole numbers it is still possible to get great complexity in systems based on numbers. … And indeed what we have found in this section is that if one looks at digit sequences, then one sees complex patterns that are remarkably similar to those produced by systems like cellular automata. … One knows from hand calculation that even an operation such as addition can lead to "carry" digits which propagate arbitrarily far to the left.

cellular automata with rules as simple as the ones we have been using these three forms of behavior would be all that we could ever get. … The picture shows what happens when one starts with just one black cell and then applies this rule over and over again. And what one sees is something quite startling—and probably the single most surprising scientific discovery I have ever made.

to search through these rules, trying each one in turn, and looking to see if it produces the behavior one wants. … Surely we cannot simply search through possible rules of certain kinds, looking for one whose behavior happens to fit what we see in physics? … To start performing such a search, however, one first needs to work out what kinds of rules to consider.

But given only a causal network, what can one say about the evolution history? … But what if one were to choose a different set of slices? In general, the sequence of strings that one would get would not correspond to anything that could arise from the same underlying substitution system.