Exploding Dots

3.3 Changing the Order of Explosions

Prove that the order in which one conducts explosions in a \( A \leftarrow B\) base machine does not matter, that is, for a given number of dots placed into the machine, the total number of explosions will always be the same and the final representations will always be the same, no matter the order in which one chooses to conduct those explosions.

ex3301

Comment: The proof that I have in my mind of this relies on there being a right boundary to the machine. If these machines were circular, then I don’t know if the order of explosions matters.  It would be worth playing with this!

ex3302

 

In general, the claim that the order in which one conducts explosions does not affect the final answer is not true for all the crazy types of machines one might consider playing with. Thank heavens it holds for the \(1 \leftarrow 10\) machine we use in everyday life at least!

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