In a multiway system, one can imagine identifying "true" with a string consisting of a single black element. … And so what Multiway systems starting from a single black element that represents True.

Post's tag systems differ from mine in that his allow the choice of block that is added at each step to depend only on the very first element in the sequence at that step (see however page 670 ). (The lag systems studied in 1963 by Hao Wang allow dependence on more than just the first element, but remove only the first element.) It turns out that in order to get complex behavior in such systems, one needs either to allow more than two possible colors for each element, or to remove more than two elements from the beginning of the sequence at each step.

The second set of pictures below demonstrate, however, that such a Turing A cyclic tag system emulating a tag system that depends only on the first element at each step.

Picture (b) shows the system with sequences on successive steps rearranged so that they do not shift to the left when the first element is removed.

At each step a new element, indicated by a black dot, is taken to be generated at whatever position the concentration is maximal. And around this position the new element is then taken to produce a dip in concentration that is gradually washed out over the course of several steps.

And with such slow growth, we can again represent each element by a box of the same size, just as in our original pictures of substitution systems on page 82 . … In the view on the left, the boxes representing each element are scaled to keep the total width the same, whereas on the right each box has a fixed size, as in our original pictures of substitution systems on page 82 .

With 0 representing white, 1 gray and 2 black, the rightmost element of the rule gives the result for average color 0, while the element immediately to its left gives the result for average color 1/3—and so on.

[Generalized substitution system] rule (b) The maximum number of steps for which the rule can be applied occurs with initial conditions consisting of a white element followed by n black elements, and in this case the number of steps is 2 n + n .

Examples of nesting Examples in which a single element splits into others include branching in plants, particle showers, genealogical trees, river deltas and crushing of rocks.

And given this, the simplest hypothesis in a sense is that the new state of the element is determined from the previous state of its neighbors—just as in a one-dimensional cellular automaton.