Chapter 11: The Notion of Computation

Section 11: The Threshold of Universality in Cellular Automata

[Cellular automaton] rule emulations

The network below shows which quiescent symmetric elementary rules can emulate which with blocks of length 8 or less. (Compare page 269.)

In all cases things are set up so that several steps in one rule emulate a single step in another. The examples shown in detail in the main text all have the feature that the block size b and number of steps t are matched, so that r t = b (where the range r = 1 for elementary rules). It is also possible to set up emulations where this equality does not hold—and indeed some of the cases listed in the main text and shown in the picture above are of this type. In those where r t < b there are more cells that are in principle determined by a given set of initial blocks—but the outermost of these cells are ignored when the outcome for a particular cell is deduced. In cases where r t > b there are more initial cells whose values are specified—but the outermost of these turn out to be irrelevant in determining the outcome for a particular cell. This lack of dependence makes it somewhat inevitable that the only rules that end up being emulated in this way are ones with very simple behavior.

In any 1D cellular automaton the color of a particular cell can always be determined from the colors t steps back of a block of 2 r t + 1 cells (compare pages 605 and 960). But such a block corresponds in a sense to a horizontal slice through the cone of previous cell colors. And it turns out also to be possible to determine the color of a particular cell from slices at essentially any rational angle corresponding to a propagation speed less than r. So this means that one can consider encodings based on blocks that have a kind of staircase shape—as in the rule 45 example shown.

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