Cyclic tag systems which allow any value for each element can be obtained by adding the rule CTStep[{{r_, s___}, {n_, a___}}] := {{s, r}, Flatten[{a, Table[r, {n}]}]} The leading elements in this case can be obtained using CTListStep[{rules_, list_}] := {RotateLeft[rules, Length[list]], With[{n = Length[rules]}, Flatten[Apply[Table[#1, {#2}] &, Map[Transpose[ {rules, #}] &, Partition[list, n, n, 1, 0]], {2}]]]}

[Sounds based on] musical scores Instead of taking a sequence to correspond directly to the waveform of a sound, one can consider it to give a musical score in which each element represents a note of a certain frequency, played for some specific short time.

In the second picture, each element shown represents a 2×2 block of original cells.

(The operation in the semigroup is concatenation of strings; "" acts as an identity element, so in fact a monoid is always obtained.) … Given a particular representation of a group or semigroup in terms of rules for a multiway system, an object that is often useful is the so-called Cayley graph—a network where each node is an element of the group, and the connections show what elements are reached by appending each possible symbol to the sequences that represent a given element.

I strongly suspect that a key element is to be able to store a collection of experiences comparable to those of a human.

Reversible mobile automata can for instance be constructed using Table[(IntegerDigits[i, 2, 3]  If[First[#]  0, {#, -1}, {Reverse[#], 1}]&)[IntegerDigits[perm 〚 i 〛 , 2, 3]], {i, 8}] where perm is an element of Permutations[Range[8]] .

But although this will make it more difficult to recognize definite rules even if one looks at the complete behavior of every element in the system, it does not affect the basic point that there is randomness that can intrinsically be generated by the evolution of the system.

And second, one can avoid explicitly having to specify every element in the data by making use of regularities that one sees.

Each section of output starts with an element which indicates whether what follows is a new sequence, or a pointer to a previous one.

But whenever there is neither just a single active data element nor an obvious sequence of independent execution steps—as for many of the programs in this book—my experience has always been that the only viable choice of interface is a computer language like Mathematica, based essentially on one-dimensional sequences of word-like constructs.