Showing Text From Page | View Full page with images

A cyclic tag system in general operates by removing the first element from the sequence that exists at each step, and then adding a new block of elements to the end of the sequence if this element is black. A crucial feature of cyclic tag systems is that the choice of what block of elements can be added does not depend in any way on the form of the sequence. So, for example, on the previous page, there are just two possibilities, and these possibilities alternate on successive steps.

Pictures (a) and (b) on the previous page illustrate the consequences of applying the rules for a cyclic tag system, but in a sense give no indication of an explicit mechanism by which these rules might be applied. In picture (c), however, we see the beginnings of such a mechanism.

The basic idea is that at each step in the evolution of the system, there is a stripe that comes in from the left carrying information about the block that can be added at that step. Then when the stripe hits the first element in the sequence that exists at that step, it is allowed to pass only if the element is black. And once past, the stripe continues to the right, finally adding the block it represents to the end of the sequence.

But while picture (c) shows the effects of various lines carrying information around the system, it gives no indication of why the lines should behave in the way they do. Picture (d), however, shows a much more explicit mechanism. The collections of lines coming in from the left represent the blocks that can be added at successive steps. The beginning of each block is indicated by a dashed line, while the elements within the block are indicated by solid black and gray lines.

When a dashed line hits the first element in the sequence that exists at a particular step, it effectively bounces back in the form of a line propagating to the left that carries the color of the first element.

When this line is gray, it then absorbs all other lines coming from the left until the next dashed line arrives. But when the line is black, it lets lines coming from the left through. These lines then continue until they collide with gray lines coming from the right, at which point they generate a new element with the same color as their own.

By looking at picture (d), one can begin to see how it might be possible for a cyclic tag system to be emulated by rule 110: the basic

From Stephen Wolfram: A New Kind of Science [citation]