User:Wythagoras/Rado's sigma function/BB(12,5)

\(\Sigma(12,5) > f_{\omega+1}^9(f_{\omega}(4))\) 0 _ 3 l 8 0 1 1 r 0 0 2 2 * 4 0 3 4 r 8 0 4 4 l 12 0a _ 1 r 0a 0a 1 _ l 0a 0a 2 1 r 1 0a 3 1 l 11 0a 4 3 l 12 1 4 3 l 12 1 3 1 l 11 1 1 2 l 3 1 2 2 r 2 1 _ 1 r halt 2 _ 1 r halt 2 2 2 r 2 2 1 _ l 8 2 3 1 l 11 2 4 3 l 12 3 2 1 l 3 3 1 1 l 3 3 _ 1 l 0a 4 2 1 l 9 4 1 1 l 4 4 _ 2 r 0 6 1 1 r 0 6 2 2 * 4 7 1 1 l 7 7 2 2 l 7 7 _ 1 l 0a 8 2 2 l 8 8 1 1 r 4 8 _ 4 r 0 8 3 4 l 8 8 4 3 r 8 9 1 1 l 9 9 2 3 l 9 9 _ 2 r 10 10 _ 2 l 7 10 1 1 r 10 10 2 2 * 4 10 3 2 r 10 11 2 2 l 11 11 1 2 l 11 11 _ 2 l 7 12 4 4 l 12 12 3 3 l 12 12 2 3 l 12 12 1 3 l 12 12 _ 3 l 7

Explanation
A  machine for \(\Sigma(2,3)\) is hidden in the machine, in state 0 and 8. It outputs: 444444444 ^ It uses groups, the first group duplicates, the second group is comparable to \(2^n\), the third group is comparable to \(2\uparrow\uparrow n\), the fourth group is comparable to \(2\uparrow\uparrow\uparrow n\), etc. Then for each 3 all ones will be changed into two's, giving a function comparable to \(f_{\omega}(n)\). For each four  the content will be changed to threes, which corresponds to \(f_{\omega+1}(n)\).

It uses LittlePeng9's duplication machine.