[some cycles in rule 54] - A New Kind of Science: The NKS Forum

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some cycles in rule 54

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Posted by: Jason Cawley

Here is a notebook containing a few results about rule 54, showing something of the variety of periodic attractors one finds in it, with even modest width periodic boundary conditions.

At 19 and 20 wide, there are half a million and a million possible initial conditions, respectively. But the number of distinct cycles the system will go to, after its typically fairly short transients, is quite limited. They include some pretty long cycles, however. And the structure of cycles is quite sensitive to system width.

This arises because certain stable structures "tile" a given width, or don't, forcing or allowing various sorts of defects between them, which then move about.

One can often detect cycles in a reduced transform of the data, e.g. a total count of black cells across a row. The cell count being the same does not mean the system has actually repeated - but when it repeats, so does the total black cell count. These cycles were found by matching the full line, not just its total count. But you can see the repetition readily in the line plots of the counts.

Generating signals from lossy transformations of an underlying complex system is an interesting subject in its own right. Because often all we may have is a sequence of measurements, corresponding to a global variable like black cell count gives, without knowing the full microstate.

Notice that the system may not appear deterministic in the lossy mapping - a black cell count of 8 may be followed by a range of different numbers, immediately afterward. This does not imply the underlying system is non-deterministic, only that some of the state is hidden.

Imagine all you knew about rule 54 was the variety of its cycles. Could you deduce that a single, simple, deterministic rule was creating all of those sequence forms?

I hope this is interesting.




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