Registered: Oct 2003
Posts: 103

Over the holiday, I was looking at rule 94 and noticed that {1,0,1} forms a wall. No matter the color of the cells on the right or left of this structure, it persists. As discussed, with rule 73, on p.699 and this forum thread .

The situation is a bit more severe with rule 94 as the only initial conditions which do not produce the {1,0,1} wall consist of even runs of black and white cells. Those states emulate rule 90 in even blocks, so their behavior is reducible.

To see that {1,0,1} is inevitable with any odd run length, one can consider the separate cases. An odd run of white cells gives an inverted triangle culminating in {1,0,1}. An odd run of black cells, at least size 3, yields a run of white cells of size n-2 on the next step. That leaves the possibility of a single black cell surrounded by even runs of white cells, {0,1,0}, which can only be produced as part of one of the walls {1,0,1}. Thus a run of a single black cell, not part of a wall, can only exist in the initial condition.

That leaves for interesting behavior of rule 94, the question of how and when the walls appear.

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Registered: Feb 2004
Posts: 172

To understand rule 94 (it should be free enough to have many interpretations besides mine) I thought it might be helpful to think of both rule 90 and rule 94 as "edge detectors." My basic interpretation of rule 94 thus far is some sort of non-destructive edge detector. The walls you speak of are almost like "edge discoveries" which are "locked-in." The rule 90 edge detector (and all edge detectors inside image processing software) will keep killing the old edges after each application. In a sense, nothing can be gained by running the application of "edge detection" more than one time on an input. Perhaps rule 94 discovers some structures and locks them in the output. Attached is a 2d look at rule 90 as it discovers edges. I do not have code to make images like this for any of the elem. CA rules except rule 90 (because photoshop will compute it "accidentally" when you run the "find edge" filter).

I think viewing CA output in these terms helps to see what is happing when you perceive the CA rules as performing edge-detection. It is just an alternate viewing which gives warm fuzzies after having looked at 1D for soo long.

Philip Ronald Dutton has attached this image:

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