Chapter 9: Fundamental Physics

Section 10: The Sequencing of Events in the Universe

"Firing squad" synchronization

By choosing appropriate rules it is possible to achieve many forms of synchronization directly within cellular automata. One version posed as a problem by John Myhill in 1957 consists in setting up a rule in which all cells in a region go into a special state after exactly the same number of steps. The problem was first solved in the early 1960s; the solution using 6 colors and a minimal number of steps shown on the right below was found in 1988 by Jacques Mazoyer, who also determined that no similar 4-color solutions exist. Note that this solution in effect constructs a nested pattern of any width (it does this by optionally including or excluding one additional cell at each nesting level, using a mechanism related to the decimation systems of page 909). If one drops the requirement of cells going into a special state, then even the 2-color elementary rule 60 shown on the left can be viewed as solving the problem—but only for widths that are powers of 2.

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