Rule orderings [for cellular automata] The fact that successive rules often show very different behavior does not appear to be affected by using alternative orderings such as Gray code (see page 901 .)

[The word] "calculus" It is an irony of language that the word "calculus" now associated with continuous systems comes from the Latin word which means a small pebble of the kind used for doing discrete calculations (same root as "calcium").

Universality in Mathematica As an example of how different primitive operations can be used to do the same computation, the following are a few ways that the factorial function can be defined in Mathematica: f[n_] := n! f[n_] := n f[n - 1]; f[1] = 1 f[n_] := Product[i, {i, n}] f[n_] := Module[{t = 1}, Do[t = t i, {i, n}]; t] f[n_] := Module[{t = 1, i}, For[i = 1, i ≤ n, i++, t ⋆ = i]; t] f[n_] := Apply[Times, Range[n]] f[n_] := Fold[Times, 1, Range[n]] f[n_] := If[n  1, 1, n f[n - 1]] f[n_] := Fold[#2[#1] &, 1, Array[Function[t, # t] &, n]] f = If[#1  1, 1, #1 #0[#1 - 1]] &

For certainly h does not in general give the length of the shortest possible description of the data; all it does is to give the shortest length of description that is obtained by treating successive blocks as if they occur with independent probabilities. … The method of arithmetic coding provides an alternative in which the output does not consist of separate codewords concatenated together.

But for the kinds of computer experiments I do in this book, there is no such issue. … And indeed, over the past twenty years or so my own methodology for doing such experiments has become vastly better.

But one of the core ideas of this book is to consider the more general scientific question of what arbitrary computational systems do. … And I believe that the new kind of science in this book will do this for a variety of issues that have been considered fundamental even since antiquity.

But doing this turned out to be much more difficult than expected, and after some spin-offs, little fundamental progress was made. … But now finally with the framework developed in this book one potentially has a meaningful foundation for doing this.

For the vast majority of rules written down at random, such problems do indeed occur. But it is possible to find rules in which they do not, and the pictures on the previous two pages [ 129 , 130 ] show a few examples I have found of such rules.

And the point is that these maps actually do intrinsically generate complexity and randomness; they do not just transcribe it when it is inserted in their initial conditions.

For what it says is that the randomness we see somehow comes from randomness that is already present—but it does not explain where that randomness comes from. … Yet the fact that systems like (a) and (b) can intrinsically generate randomness even from simple initial conditions does not mean that they