User blog:Cloudy176/Yet another blog post of Turing machines

My works
(coming soon?)

A 48 state, 3 symbol machine
Here, I'll introduce a Turing machine, based on LittlePeng9's Kirby-Paris hydra game machine and the 3-state, 3-symbol record holder.

This is the record holder for 3-state, 3-symbol Turing machines:

0 _ 1 r 1 0 1 2 l 0 0 2 1 l 2 1 _ _ l 0 1 1 2 r 1 1 2 1 l 1 2 _ 1 r halt 2 1 1 r 0 2 2 1 r 2

Just before the machine halts, the tape looks like this: 12222...2222 where there are 374,676,381 2's, and the head is placed on the blank on the immediate left of 1, and on state 2.

I modified the record holder machine:

0 _ ) r 1 0 l 2 0 ) ( l 0 1 _ _ l 0 1  l 1 1 ) ( r 1 2 _ ( r halt 2 r 2 2 ) ) r 0

The tape when the machine halts looks like this: (((...((( where there are 374,676,381 ('s, and the head is placed on the ) symbol.

I also proved that on the Kirby-Paris hydra game machine, when the input is ( (((...((( (where there are N ('s, excluding the leftmost one) the number of ('s the machine leaving on the tape is larger than

\[f_{\varepsilon_0}(n-3)\]

using FGH (I won't go into the details).

Combining all of these, we have this Turing machine:

0  _ ) r bb1 0    l bb2 0   ) ( l 0 bb1 _ _ l 0 bb1  l bb1 bb1 ) ( r bb1 bb2 _ ( r 0a bb2 r bb2 bb2 ) ) r 0

0a ) _ r 1 0a * * r 0a 0a _ _ r 1 1 * * r 1 1 _ _ l 2 2 * _ l 2 2 ( _ r 5

3 _ _ r 4 4 ( ( r 4' 4 ) _ r 27 4' * * r 4' 4' _ _ r 2

5 _ ) r 6 6 * * r 6 6 _ _ r 7 7 ) ) r 7 7 _ ) l 8 8 ) ) l 8 8 _ _ l 9 9 * * l 9 9 _ _ l 10 10 ) _ r 5 10 ( _ r 11 10 _ ( r 3 11 _ ( r 12 12 * * r 12 12 _ _ r 13 13 ) ) r 13 13 _ _ l 14 14 ) _ l 15 15 ) ) l 8 15 _ _ l 16

16 * * l 16 16 _ _ l 17 17 * * l 17 17 _ _ l 18 18 ( ( l 18 18 ) ( r 19 18 _ ( r 19 19 r 20 19 _ _ r 22 20 ( ( r 20 20 _ _ r 21 21 * * r 21 21 _ _ r c0

c0 ( _ l (1 c0 ) _ l )1 c0 _ ( r c3 (1 _ ( r (2 (2 _ _ r (3 (3 * * r (3 (3 _ _ r (4 (4 * * r (4 (4 _ ( l c1 )1 _ ) r )2 )2 _ _ r )3 )3 * * r )3 )3 _ _ r )4 )4 * * r )4 )4 _ ) l c1 c1 * * l c1 c1 _ _ l c2 c2 * * l c2 c2 _ _ r c0 c3 ( ( r c3 c3 ) ) l c4 c4 r c5 c5 ) ) r c5 c5 ( ( l c6 c5 _ _ l c7 c6 ) ( r c3 c7 ) _ l 16

22 * * r 22 22 _ _ r 23 23 * * r 23 23 _ _ l 24 24 ) _ l 25 25 ) ) l 25 25 l 26 26 ) ( l 25 26 ( ( l 26 26 _ ( r 1

27 _ _ r 27 27 ) _ l halt

Since this machine uses 48 states and 3 symbols, we have proven that

\[\Sigma(48,3) > f_{\varepsilon_0}(374676381-3) > f_{\varepsilon_0}(374676378)\]

which is larger than a goppatoth.