But in general it is not possible to do this, for after some given string has appeared, it can for example be undecidable whether the negation of that particular string ever appears. … Note that some fairly simple statements do not show up for at least several rows.

But it is quite possible to find symbolic systems where this does not happen, as illustrated in the pictures below.

If one does this, then the typical experience is that in any particular representation, some class of numbers will have simple forms.

Note that most of the programs do not explicitly do input checking or error generation. Only occasionally do the programs significantly sacrifice efficiency for elegance.) … Careful study of the various programs in these notes should provide good background not only for implementing what I discuss in the book, but also for doing high-level programming of any kind.

Input to a computer language will immediately fail to be meaningful if it does not conform to a certain definite syntax. … And even if this is resolved, one can still get something that is in effect nonsense and does not usefully run. … Yet the system will always do something, whatever rules one uses.

[Free] will and purpose Things that are too predictable do not normally seem free. But things that are too random also do not normally seem to be associated with the exercise of a will.

Implementation [of conserved quantity test] Whether a k -color cellular automaton with range r conserves total cell value can be determined from Catch[Do[ (If[Apply[Plus, CAStep[rule, #] - #] ≠ 0, Throw[False]] &)[ IntegerDigits[i, k, m]], {m, w}, {i, 0, k m - 1}]; True] where w can be taken to be k 2r , and perhaps smaller. … Among the 2 32 k = 2 , r = 2 rules 428 do, and of these 2 are symmetric, and 6 are reversible, and all these are just shift and identity rules.

In this book I do not start from any assumption of continuity—and the types of behavior I study tend to be vastly more complex than those in catastrophe theory. … But despite some claims to the contrary in popular accounts, this fact alone does not imply that the behavior will necessarily be complex.

It does not matter what initial conditions one starts from: one always reaches the same all-black attractor in the end. … But now the attractor does not just consist of a single configuration, but instead Examples of simple cellular automata that evolve after just one step to attractors in which only certain sequences of black and white cells can occur.

And if one does this in practice, what one will typically find is that the outcome seems quite random. But where does this randomness come from?