User:Wythagoras/Rado's sigma function/A general method for BB(n)

This machine shows a general method for bounds for \(\Sigma(n)\).

We start with this 23-state machine: 0 _ 1 r 22 0 1 1 l 22 22 _ 1 l 0 22 1 _ l 21 1 1 1 l 1 1 _ 1 r 2 2 1 1 r 2 2 _ _ r 3 3 _ _ r 14 3 1 1 r 4 4 1 1 l 5 4 _ _ r 6 5 1 _ l 5 5 _ 1 l 1 6 _ _ r 14 6 1 1 r 7 7 _ _ r 6 7 1 1 l 8 8 1 _ l 9 8 _ _ l 15 9 _ _ l 10 9 1 1 r 11 10 1 1 l 9 10 _ 1 r 1 11 1 _ r 12 11 _ _ l 13 12 _ 1 r 11 12 1 1 l 13 13 1 1 l 13 13 _ 1 l 10 14 _ _ r 18 14 1 _ l 8 15 _ 1 l 16 15 1 1 l 20 16 1 1 l 17 16 _ 1 l 1 17 _ _ l 16 17 1 _ l 16 18 1 _ l 19 18 _ _ r halt 19 _ 1 l 19 19 1 1 l 20 20 _ 1 l 15 20 1 1 l 21 21 1 1 l 21 21 _ _ l 8 When we want to extend, we change the halting transition to k0, and add these four states: k0 _ _ r halt k0 1 _ l k1 k1 _ 1 l k1 k1 1 1 l k2 k2 _ 1 l k1 k2 1 1 l k3 k3 1 1 l k3 k3 _ _ l 21 State 21 must be changed to the last state of the machine - it is k3 after the second extension and k'3 after the third. Using this the following bounds can be made:

\(\Sigma(23) > f_{\omega+1}^2(9)\)

\(\Sigma(19+4k) > f_{\omega+k}^2(13)\) if \(k ≥ 2\)