The pictures below show examples of tag systems in which just one element is removed at each step. … Examples of tag systems in which a single element is removed from the beginning of the sequence at each step, and a new block of elements is added to the end of the sequence according to the rules shown. Because only a single element is removed at each step, the systems effectively just cycle through all elements, replacing each one in turn.

Indeed, away from the right-hand edge, all the elements effectively behave as if they were lying on a regular grid, with the color of each element depending only on the previous color of that element and the element immediately to its right. … The substitution systems that we discussed in the previous section work by replacing each element in such a string by a new sequence of elements—so that in a sense these systems operate in parallel on all the elements that exist in the string at each step.

Starting with a single state consisting of one element, the picture then shows that applying these rules immediately gives two possible states: one with a single element, and the other with two. … In multiway systems, however, A very simple multiway system in which one element in each sequence is replaced at each step by either one or two elements.

In the case of rules (a) and (b) the result is to update every single element at every step. But since the replacements in these particular rules involve only one element at a time, one in effect has a neighbor-independent substitution system of the kind we discussed on page 82 . … So what happens with replacements that involve more than just one element?

In general there is an axis crossing within an interval when the corresponding element in the generalized substitution system is black, and there is not when the element is white.

In the simplest case there are two possible blocks, and the rule simply alternates on successive steps between these blocks, adding a block at a particular step when the first element in the sequence at that step is black. … In each case a single element is removed from the beginning of the sequence, and then a new block is added at the end whenever the element removed is black.

Nevertheless, since the string does contain the single element A , the second replacement can still be used. … The light squares can be thought of as corresponding to the element A , and the dark squares to the element B .

[No text on this page] Emulating a Turing machine with a tag system that depends only on the first element at each step.

After all, the basic rules for these substitution systems specify that any time an element of a particular color appears it will always get subdivided in the same way. … Examples of substitution systems in which every element is drawn as being subdivided into a sequence of new elements at each step.

In general, each successive element in a list from NeighborNumbers cannot be more than twice the previous element.