User:Wythagoras/Rado's sigma function/BB(16,3)

\(\Sigma(16,3) > f_{\omega+1}^6(f_{\omega}^6(5))\) 0 _ 2 l bb1 0 1 _ l 5 0 2 1 r bb1 bb1 _ 1 r 0 bb1 1 2 r bb1 bb1 2 1 l bb1 0a _ 1 r 0a 0a 1 _ l 0a 0a 2 1 r 1 1 1 2 l 3 1 2 2 r 2 1 _ _ r 11 2 _ _ r 11 2 2 2 r 2 2 1 2 l 8 3 2 1 l 3 3 1 1 l 3 3 _ 1 l 0a 4 2 1 r 5 4 1 1 l 4 4 _ 2 r 6 5 2 2 l 9 5 1 2 l 7 5 _ _ l 7 6 1 1 r 6 6 2 2 * 4 6 _ 2 l 13 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 _ 1 l 12 9 1 1 l 9 9 2 _ l 9 9 _ 2 r 10 10 1 1 r 10 10 2 2 * 4 10 _ 2 r 10 11 _ 1 r halt 11 1 2 l 6 11 2 _ l 8 12 1 2 l 12 12 2 2 l 12 12 _ 2 l 7 13 1 2 l 13 13 2 2 l 13 13 _ _ l 12

Explanation
The first part is the machine for \(\Sigma(2,3)\). It outputs: 1_1111111 ^

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 2 after the space the part for the space will be changed to twos, giving a function comparable to \(f_{\omega}(n)\). Then for each 1 after the space the part for the space will be changed to twos after the space, giving a function comparable to \(f_{\omega+1}(n)\). It uses LittlePeng9's duplication machine.