But in rule 110 there are only two possible colors for each cell.

These rules only ever depend on one cell in each neighborhood.

And as a result, even the original statements of the Second Law talked only about "inanimate systems". … And there were extensive demonstrations that actual biological systems reduce their internal entropy only at the cost of increases in the entropy of their environment.

But at a formal mathematical level this can happen rigorously only in an infinite limit. … Inevitably there are several scales involved, and one can only expect continuum behavior if one looks at scales intermediate between individual connections in the underlying network and the overall size of the whole network.

For I would have looked only for types of behavior that I already believed might exist. And in studying cellular automata, this would for example probably have meant that I would only have looked for repetition and nesting.

At any period in history there are issues about the universe and our role in it that seem accessible only to the general arguments of philosophy. … Yet inevitably it has been informed only by current intuition about how things are supposed to work.

But almost without exception, it has in the past only been applied to systems and questions that have already been investigated by other mathematical means—and that lie very much within the normal tradition of mathematics. … Until recently, the only kinds of shapes widely discussed in science and mathematics were ones that are regular or smooth.

Its behavior is always very simple: any initial gray progressively diffuses away, so that in the end only uniform white is left. … And as we have seen, these equations yield only simple behavior.

And the problem is that while practical devices may eventually relax to what is essentially the same state, they can do this only at a certain rate. … But in fact, as we will also see in connection with quantum mechanics, it is a rather general feature of systems that perform amplification that relaxation to a normal state can effectively occur only gradually, so that one would have to wait an infinite time for such relaxation to be absolutely complete.

But the past several chapters have demonstrated that this is not only possible, but actually quite common. … For these definitions are based on the notion of classifying as truly random only sequences which can never be generated by any simple procedure whatsoever.