User blog:Wythagoras/NEWS! I found a 22-state machine that beats G!

\(\Sigma(22) > f_{\omega+1}(f_{\omega}(2 \uparrow^{12} 3)) >> G\)

0 _ 1 r 21 0 1 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 _ l 1 16 1 1 l 17 16 _ 1 l 1 17 _ _ l 16 17 1 _ l 16 18 _ _ r halt 18 1 _ l 19 19 _ 1 l 20 19 1 _ l 15 20 1 1 l 19 20 _ 1 l 19 21 _ 1 l 0 21 1 _ l 14 State 1 is state 0 of Deedlit's expandal machine

State x is state x of Deedlit's expandal machine for 2 ≤ x ≤ 17

Then, if the w+1 category is empty, it checks whether there is some in the w+2 category.

Then it changes all empty categories (remember, the tape looks like 11111....11111_1_1_1_1...) to ones for the w+1 category. That is about \(\f_{\omega}^{-1}(n)\) of the ones currently on the tape.

State 0 and 21 are used to set the input. 1_11, where the head is on the first one and in state 14.

Snapshots of the tape
After 6 steps, state 14 1 11 ^ After 13 steps, state 2 111  11    ^ After 16 steps, state 18 111  11       ^ After 17 steps, state 19 111   1      ^ After 23 steps, state 1 1 1111 1 ^ After 59 steps, state 1 1 1 1 11 11 1 ^ After 1359 steps, state 10 11 1 1 1 1 1 1 1 111 111 111 111 111 11 1 1 1  ^

Bound
\(\Sigma(22) > f_{\omega+1}(f_{\omega}(2 \uparrow^{12} 3)) >> G\)

See last tape.