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automaton not in terms of individual steps, but rather in terms of updating events and the causal relationships between them.

The pictures on the facing page show an example of how this works. Picture (a) is a version of the standard representation that I have used for mobile automaton evolution elsewhere in the book—in which successive lines give the colors of cells on successive steps, and the position of the active cell is indicated at each step by a gray dot. The subsequent pictures on the facing page all ultimately give essentially the same information, but gradually present it to emphasize more a representation in terms of updating events and causal relationships.

Picture (b) is very similar to (a), but shows successive steps of mobile automaton evolution separated, with gray blobs in between indicating "updating events" corresponding to each application of the underlying mobile automaton rule. Picture (b) still has a definite row of cells for each individual step of mobile automaton evolution. But in picture (c) cells not updated on a given step are merged together, yielding vertical stripes of color that extend from one updating event to another.

So what is the significance of these stripes? In essence they serve to carry the information needed to determine what the next updating event will be. And as picture (d) begins to emphasize, one can think of these stripes as indicating what causal relationships or connections exist between updating events.

And this notion then suggests a quite different representation for the whole evolution of the mobile automaton. For rather than having a picture based on successive individual steps of evolution, one can instead form a network of the various causal relationships between updating events, with each updating event being a node in this network, and each stripe being a connection from one node to another.

Captions on this page:

A sequence of views of the evolution of a mobile automaton, showing how a network of causal relationships between updating events can be created. This network provides a very simple model for spacetime in the universe. Picture (a) is essentially the standard representation of mobile automaton evolution that I have used in this book. Picture (b) includes gray blobs to indicate updating events. Picture (c) merges cells that are not being updated. Picture (d) emphasizes the role of vertical stripes as connections between updating events. Pictures (e) through (g) show how a network can be formed with nodes corresponding to updating events. Pictures (h) and (i) demonstrate that with the particular underlying rule used here, a highly regular network is produced.

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