And to emulate a mobile automaton with a cellular automaton it turns out that all one need do is to divide the possible colors of cells in the cellular automaton into two sets: lighter ones that correspond to ordinary cells in the mobile automaton, and darker ones that correspond to active cells.

The other 7 are used to implement signals that propagate out to the left and right to do the analog of incrementing and decrementing each register.

But it could also be that whatever one does rule 54 will always in the end just show purely repetitive or nested behavior—which cannot on its own support universality.

The bottom row shows what the same transformation does to patterns of the kind that are generated by simple cellular automata.

For at any point the system will always in effect be able to do more things that one did not expect.

In an ordinary tag system, one does not know in advance which of several possible blocks will be added at each step.

Statistically uniform samples of text do not in practice, however, tend to be large enough to allow more than about b = 6 to be reached, and the presence of correlations (even though exponentially damped) between far-separated letters means that computed entropies usually decrease continually with b , making it difficult to estimate their limit (see page 1084 ).

But there is absolutely no reason to think that the specific concepts that have arisen so far in the history of mathematics should cover all of science, and indeed in this book I give extensive evidence that they do not.

What rules for natural objects might in effect have been tried in the Judeo-Christian tradition is less clear—though for example the Book of Job does comment on the difficulty of "numbering the clouds by wisdom".

In doing mathematics there is often the idea that proofs should explain the result they prove—and one might not think this could be achieved if one just presents an object with certain properties.