Elementary CA? Not so fast.

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I was exploring the computational universe when I suddenly decided that instead of becoming more complex with adding more colours in my CAs, I’d go backwards. I’d stick with 2 colours and also only allow 2 cells to determine the next cell. The image above shows the result.

This CA is equivalent to rule 60  with a shift to the left. This is because rule 60 essentially ignores the third cell – in all cases if X Y Z produces A, then X Y (not Z) also produces A. This allows it to be used in a 2 cell CA.The Wolfram Alpha page explains that there is no dependency on the 3rd cell in the neighborhood dependency.

This rule shows class 3 behavior with random initial conditions. With a single white cell there is still complexity:

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Wikipedia states “In mathematics and computability theory, an elementary cellular automaton is a one-dimensional cellular automaton where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors”.

I completely abhor this assumption that there exists a current cell. There is only input and output, no need to make it any more complicated than this.

This links back to my ship of Theseus argument. People assume that a ships exists through time, that it’s the same ship and it changes state. But really that’s just a more complicated definition than it needs to be, but it’s useful in very day language. You can define things how you like but I will criticize you if those definitions limit your thoughts (more than another definition).

It also relates to my Haskell post in that in Haskell, there is no state. Only input and output. That’s what makes it so mathematically rigorous.

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