the past taken as one of the key pieces of evidence for the idea that simple laws of nature could exist.

There are a total of 7,625,597,484,987 cellular automata with three colors and nearest-neighbor rules, and searching through these one finds just 1800 that are reversible.

The codewords are self-delimiting, so that they can be given one after another, with no separator in between.

And knowing such a hierarchy, one should be able to produce images that in a sense seem as random as possible to us.

The arithmetic system takes the value n that it obtains at each step, computes Mod[n, 30] , and then depending on the result applies to n one of the arithmetic operations specified by the rule above.

So can one then achieve something intermediate in rule 73—in which information is transmitted, but only in a controlled way?

As an example, one can take n to be represented just by Nest[s, k, n] . And one can then convert any Church numeral x to this representation by applying s[s[s[k][k]][k[s]]][k[k]] . To go the other way, one uses the result that for all Church numerals x and y , Nest[s, k, n][x][y] is also a Church numeral—as can be seen recursively by noting its equality to Nest[s, k, n - 1][y][x[y]] , where as above x[y] is power[y][x] .

Sometimes a more technical presentation may be useful; sometimes a less technical one. … One feature of this book is that it covers a broad area and comes to very broad conclusions. … And without great tenacity there is a tremendous tendency to stop before one has gone far enough.

The more general case [of computation speed ups] One can think of a single step in the evolution of any system as taking a rule r and state s , and producing a new state h[r, s] . … This means that in effect one can always choose to evolve the rule rather than a state. … The correspondence between multiplication rules and additive cellular automata can be seen even more directly if one represents all states by integers and computes h in terms of base k digits.

[One-sided] Turing machines The Turing machines used here in effect have tapes that extend only to the left, and have no explicit halt states. … One can think of each Turing machine as computing a function f[x] of the number x given as its input. … (The most rarely halting are ones like machine 3112 that halt only when x = 4j - 1 .)