User:Polygon/Page for analyses

1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD (bbch)

Note: still work in progress

Details

	0	1	2
A	1RB	1RD	1LC
B	2LB	1RB	1LC
C	1RZ	1LA	1LD
D	2RB	2RA	2RD

S is any tape configuration
1. S D> 2^a S --> S 2^a D> S [+a steps]
2. S B> 1^a S --> S 1^a B> S [+a steps]
3. S 1 B> 0 S --> S <A 1^2 S [+4 steps]
4. S D> (11)^a S --> S (21)^a D> S [+2a steps]
   S A> (11)^a S --> S (12)^a A> S [+2a steps]
5. S (21)^a <C S --> S <C (11)^a S [+2a steps]
   S (12)^a <A S --> S <A (11)^a S [+2a steps]
6. S (12)^a A> 0^2 S --> S <A (11)^a+1 S [+2a +5 steps]

7. S (12)^a 2 (12)^b A> 0^2 S --> S (12)^a-1 2 (12)^b+2 A> S [+4b +7 steps]
by:
S (12)^a 2 (12)^b A> 0^2 S
--> S (12)^a 2 (12)^b 1 B> 0 S
--> S (12)^a 2 (12)^b <A (11) S 
--> S (12)^a 2 <A (11)^b+1 S
--> S (12)^a <C 1 (11)^b+1 S
--> S (12)^a-1 1 <D (11)^b+2 S
--> S (12)^a-1 2 A> (11)^b+2 S
--> S (12)^a-1 2 (12)^b+2 A> S

8. S (12)^a 2 (12)^b A> 0^inf --> S 2 (12)^b+2a A> 0^inf [+4a^2 +8a +4ba steps]
Obtained by repeating rule 8.

9. S (12)^a <D (11)^b 0^inf --> S (12)^a-1 <D (11)^2b+3 0^inf [+4b^2 +22b +22 steps]
by:
S (12)^a <D (11)^b 0^inf
--> S (12)^a D> (11)^b 0^inf
--> S (12)^a (21)^b D> 0^inf
--> S (12)^a (21)^b 2 B> 0^inf
--> S (12)^a (21)^b 2 <B 2 0^inf
--> S (12)^a (21)^b <C 1 2 0^inf
--> S (12)^a <C (11)^b 1 2 0^inf
--> S (12)^a-1 1 <D (11)^b+1 2 0^inf
--> S (12)^a-1 2 A> (11)^b+1 2 0^inf
--> S (12)^a-1 2 (12)^b+1 A> 2 0^inf
--> S (12)^a-1 2 (12)^b+1 <C 1 0^inf
--> S (12)^a-1 2 (12)^b 1 <D 11 0^inf
--> S (12)^a-1 2 (12)^b 2 A> (11)^1 0^inf
--> S (12)^a-1 2 (12)^b 2 (12)^1 A> 0^inf
--> S (12)^a-1 2 2 (12)^2b+1 A> 0^inf
--> S (12)^a-1 2^2 <A (11)^2b+2 0^inf
--> S (12)^a-1 2 <C 1 (11)^2b+2 0^inf
--> S (12)^a-1 <D (11)^2b+3 0^inf

10. S (12)^a <D (11)^b 0^inf --> S <D (11)^((2^(a))*b+(2^(a))*3-3) 0^inf
Obtained by repeating rule 9.

11. S (11)^a <D (11)^b 0^inf --> S (11)^a-2 (12)^b+3 <D (11)^3 0^inf [+10b +50 steps]
by:
S (11)^a <D (11)^b 0^inf
--> S (11)^a-1 1 2 A> (11)^b 0^inf
--> S (11)^a-1 (12)^b+1 A> 0^inf
--> S (11)^a-1 <A (11)^b+2 0^inf
--> S (11)^a-1 D> (11)^b+2 0^inf
--> S (11)^a-1 (21)^b+2 D> 0^inf
--> S (11)^a-1 (21)^b+2 2 B> 0^inf
--> S (11)^a-1 (21)^b+2 2 <B 2 0^inf
--> S (11)^a-1 (21)^b+2 <C (12)^1 0^inf
--> S (11)^a-1 <C (11)^b+2 1 2 0^inf
--> S (11)^a-2 1 <A (11)^b+3 2 0^inf
--> S (11)^a-2 1 D> (11)^b+3 2 0^inf
--> S (11)^a-2 1 (21)^b+3 D> 2 0^inf
--> S (11)^a-2 1 (21)^b+3 2 D> 0^inf
--> S (11)^a-2 1 (21)^b+3 2^2 B> 0^inf
--> S (11)^a-2 1 (21)^b+3 2^2 <B 2 0^inf
--> S (11)^a-2 1 (21)^b+3 2 <C (12)^1 0^inf
--> S (11)^a-2 1 (21)^b+3 <D 1 1 2 0^inf
Note that 1 (21)^k = (12)^k 1
= S (11)^a-2 (12)^b+3 1 <D (11)^1 2 0^inf
--> S (11)^a-2 (12)^b+3 2 A> (11)^1 2 0^inf
--> S (11)^a-2 (12)^b+3 2 (12)^1 A> 2 0^inf
--> S (11)^a-2 (12)^b+3 2 (12)^1 <C 1 0^inf
--> S (11)^a-2 (12)^b+3 2 1<D (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 A> (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 (12)^1 A> 0^inf
--> S (11)^a-2 (12)^b+3 2^2 (12)^1 1 B> 0^inf
--> S (11)^a-2 (12)^b+3 2^2 (12)^1 <A (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 1 <C 1 (11)^1 0^inf
--> S (11)^a-2 (12)^b+3 2^2 <A (11)^2 0^inf
--> S (11)^a-2 (12)^b+3 2 <C 1 (11)^2 0^inf
--> S (11)^a-2 (12)^b+3 <D (11)^3 0^inf

Let A(a,b,c) = S (11)^a (12)^b <D (11)^c 0^inf

• Rule 9: A(a, b, c) --> A(a, b - 1, 2c + 3)
• Rule 10: A(a, b, c) --> $A(a,0,2^{b}\times c+2^{b}\times 3-3)$ which becomes $A(a,0,2^{b+1}\times 3-3)$ if c = 3.
• Rule 11: A(a, 0, c) --> A(a - 2, c + 3, 3)

Further: let $f(n)=2^{n+1}\times 3$

• If c = 3: A(a, b, 3) --> A(a, 0, f(c) - 3) --> A(a - 2, f(c), 3)

• A(2k + d, 0, c) --> $A(d,f^{k-1}(c+3),3)$

The TM enters configuration A(19, 2, 3) after 799 steps with tape:
0^inf 2 1 (11)^19 (12)^2 <D (11)^3 0^inf

Trajectory

A(19, 2, 3) --> A(19, 0, 21) --> $A(1,f^{8}(24),3)$

--> $A(1,0,f^{9}(24)-3)$

Let's have $f^{9}(24)-3$ = m.

Final trajectory:
0^inf 2 1 (11)^1 <D (11)^m 0^inf
--> 0^inf 2 1 1 2 A> (11)^m 0^inf
--> 0^inf 2 1 (12)^m+1 A> 0^inf
--> 0^inf 2 1 <A (11)^m+2 0^inf
--> 0^inf 2 1 D> (11)^m+2 0^inf
--> 0^inf (21)^m+3 D> 0^inf
--> 0^inf (21)^m+3 2 B> 0^inf
--> 0^inf (21)^m+3 2 <B 2 0^inf
--> 0^inf (21)^m+3 <C (12)^1 0^inf
--> 0^inf <C (11)^m+3 (12)^1 0^inf
--> 0^inf 1 Z> (11)^m+3 (12)^1 0^inf
Score = 2m + 9

Score calculated in HyperCalc:
(10^)^8 30,302,671.815163
Or in tetration: 10^^10.873987 (truncated)

1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD

Note: still work in progress

Details

	0	1	2
A	1RB	1RD	1LC
B	2LB	1RB	1LC
C	1RZ	1LA	1LD
D	0RB	2RA	2RD

S is any tape configuration
1. S D> 2^a S --> S 2^a D> S [+a steps]
2. S B> 1^a S --> S 1^a B> S [+a steps]
3. S A> 0^2 S --> S <A 1^2 S [+5 steps]
4. S D> (11)^a S --> S (21)^a D> S [+2a steps]
   S A> (11)^a S --> S (12)^a A> S [+2a steps]
5. S (21)^a <C S --> S <C (11)^a S [+2a steps]
   S (12)^a <A S --> S <A (11)^a S [+2a steps]
6. S (12)^a A> 0^2 S --> S <A (11)^a+1 S [+2a +5 steps]
by:
S (12)^a A> 0^2 S
--> S (12)^a <A (11)^1 S
--> S <A (11)^a+1 S

7. S A> (11)^1 2^b S --> S 2 A> (11)^1 2^b-1 S [+5 steps]
by:
S A> (11)^1 2^b S
--> S (12)^1 A> 2^b S
--> S (12)^1 <C 1 2^b-1 S
--> S 1 <D (11)^1 2^b-1 S
--> S 2 A> (11)^1 2^b-1 S
8. S A> (11)^1 2^b S --> S 2^b A> (11)^1 S [+5b steps]
by repetition of rule 7

9. S D> 0^2 S --> S <B 2^2 S [+3 steps]

10. S 2 <D (11)^a 0^2 S --> S <D (11)^a+1 2 S [+4a +7 steps]
by:
S 2 <D (11)^a 0^2 S
--> S 2 D> (11)^a 0^2 S
--> S 2 (21)^a D> 0^2 S
--> S 2 (21)^a <B 2^2 S
--> S 2 (21)^a B> 2^2 S
--> S 2 (21)^a <C 1 2 S
--> S 2 <C (11)^a 1 2 S
--> S <D (11)^a+1 2 S

11. S 2 <D (11)^a 2 0^2 S --> S <D (11)^a+1 2^2 S [+4a +7 steps]
by:
S 2 <D (11)^a 2 0^2 S
--> S 2 D> (11)^a 2 0^2 S
--> S 2 (21)^a D> 2 0^2 S
--> S 2 (21)^a 2 D> 0^2 S
--> S 2 (21)^a 2 <B 2^2 S
--> S 2 (21)^a <C 1 2^2 S
--> S 2 <C (11)^a 1 2^2 S
--> S <D (11)^a+1 2^2 S

12. S  1^a <A (11)^b 0^2 S --> S 1^a-1 <A (11)^b+1 2 S [+4b +7 steps]
by:
S 1^a <A (11)^b 0^2 S
--> S 1^a D> (11)^b 0^2 S
--> S 1^a (21)^b D> 0^2 S
--> S 1^a (21)^b <B 2^2 S
--> S 1^a (21)^b B> 2^2 S
--> S 1^a (21)^b <C 1 2 S
--> S 1^a <C (11)^b 1 2 S
--> S 1^a-1 <A (11)^b+1 2 S

13. S 1^a <A (11)^b 2 0^2 S --> S 1^a-1 <A (11)^b+1 2^2 S [+4b +7 steps]
by:
S 1^a <A (11)^b 2 0^2 S
--> S 1^a D> (11)^b 2 0^2 S
--> S 1^a (21)^b D> 2 0^2 S
--> S 1^a (21)^b 2 D> 0^2 S
--> S 1^a (21)^b 2 <B 2^2 S
--> S 1^a (21)^b <C 1 2^2 S
--> S 1^a <C (11)^b 1 2^2 S
--> S 1^a-1 <A (11)^b+1 2^2 S

14. S (12)^a 1 <D (11)^b 0^2 S --> S (12)^a-1 1 <D (11)^b+2 [+4b +8 steps]
by:
S (12)^a 1 <D (11)^b 0^2 S
--> S (12)^a 2 A> (11)^b 0^2 S
--> S (12)^a 2 (12)^b A> 0^2 S
--> S (12)^a 2 <A (11)^b+1 S
--> S (12)^a <C 1 (11)^b+1 S
--> S (12)^a-1 1 <D (11)^b+2 S

15. S (12)^a 1 <D (11)^b 0^inf --> S 1 <D (11)^b+2a 0^inf [+4a^2 +4ba + 4a steps]
by repetition of rule 14

16. S (12)^a 2 1 <D (11)^b 0^inf --> S (12)^a-1 2 (12)^b+2 1 <D (11)^1 0^inf [+10b +28 steps]
by:
S (12)^a 2 1 <D (11)^b 0^inf
--> S (12)^a 2^2 A> (11)^b 0^inf
--> S (12)^a 2^2 (12)^b A> 0^inf
--> S (12)^a 2^2 <A (11)^b+1 0^inf
--> S (12)^a 2 <C 1 (11)^b+1 0^inf
--> S (12)^a <D (11)^b+2 0^inf
--> S (12)^a-1 1 <D (11)^b+3 2 0^inf by rule 10
--> S (12)^a-1 2 A> (11)^b+3 2 0^inf
--> S (12)^a-1 2 (12)^b+3 A> 2 0^inf
--> S (12)^a-1 2 (12)^b+3 <C 1 0^inf
--> S (12)^a-1 2 (12)^b+2 1 <D (11)^1 0^inf

17. S (12)^a 2 1 <D (11)^b 0^inf --> S (12)^a-1 2 1 <D (11)^2b+5 0^inf
by:
S (12)^a 2 1 <D (11)^b 0^inf
--> S (12)^a-1 2 (12)^b+2 1 <D (11)^1 0^inf by rule 16
--> S (12)^a-1 2 1 <D (11)^2b+5 0^inf by rule 15

18. S (12)^a 2 1 <D (11)^b 0^inf --> S 2 1 <D (11)^(2^a)*b+(2^a)*3-3 0^inf
by repetition of rule 17

---
19. S (12)^a 2 1 <D (11)^b 2 0^inf --> S (12)^a 2 1 <D (11)^2b-1 0^inf
by:
S (12)^a 2 1 <D (11)^b 2 0^inf
--> S (12)^a 2^2 A> (11)^b 2 0^inf
--> S (12)^a 2^2 (12)^b A> 2 0^inf
--> S (12)^a 2^2 (12)^b <C 1 0^inf
--> S (12)^a 2^2 (12)^b-1 1 <D (11)^1 0^inf
--> S (12)^a 2^2 1 <D (11)^2b-1 0^inf by rule 15

20. S (12)^a 1 <D (11)^b 2 0^inf --> S (12)^a 2 1 <D (11)^2b-1 0^inf
by:
S (12)^a 1 <D (11)^b 2 0^inf
--> S (12)^a 2 A> (11)^b 2 0^inf
--> S (12)^a 2 (12)^b A> 2 0^inf
--> S (12)^a 2 (12)^b <C 1 0^inf
--> S (12)^a 2 (12)^b-1 1 <D (11)^1 0^inf
--> S (12)^a 2 1 <D (11)^2b-1 0^inf by rule 15

21. S (12)^a 2^2 1 <D (11)^b 0^inf --> S (12)^a-1 2^2 1 <D (11)^2^(b+5)-3 0^inf
by:
S (12)^a 2^2 1 <D (11)^b 0^inf
--> S (12)^a 2^3 A> (11)^b 0^inf
--> S (12)^a 2^3 (12)^b A> 0^inf
--> S (12)^a 2^3 <A (11)^b+1 0^inf
--> S (12)^a 2^2 <C 1 (11)^b+1 0^inf
--> S (12)^a 2 <D (11)^b+2 0^inf
--> S (12)^a <D (11)^b+3 2 0^inf by rule 10
--> S (12)^a-1 1 <D (11)^b+4 2^2 0^inf by rule 11
--> S (12)^a-1 2 A> (11)^b+4 2^2 0^inf
--> S (12)^a-1 2 (12)^b+4 A> 2^2 0^inf
--> S (12)^a-1 2 (12)^b+4 <C 1 2 0^inf
--> S (12)^a-1 2 (12)^b+3 1 <D (11)^1 2 0^inf
--> S (12)^a-1 2 (12)^b+3 2 1 <D (11)^1 0^inf by rule 20
--> S (12)^a-1 2^2 1 <D (11)^(2^(b+3)*1)+(2^(b+3)*3)-3 0^inf by rule 18
= S (12)^a-1 2^2 1 <D (11)^(2^(b+5)-3) 0^inf

22. S 1 <D (11)^b 2^2 0^inf --> S 2 (12)^b-1 2 1 <D (11)^1 0^inf
by:
S 1 <D (11)^b 2^2 0^inf
--> S 2 A> (11)^b 2^2 0^inf
--> S 2 (12)^b A> 2^2 0^inf
--> S 2 (12)^b <C 1 2 0^inf
--> S 2 (12)^b-1 1 <D (11)^1 2 0^inf
--> S 2 (12)^b-1 2 1 <D (11)^1 0^inf by rule 20

23. S (11)^a 2^2 1 <D (11)^b 0^inf --> S (11)^a-3 (12)^2b+11 2^2 1 <D (11)^1 0^inf
by:
S (11)^a 2^2 1 <D (11)^b 0^inf
--> S (11)^a 2^3 A> (11)^b 0^inf
--> S (11)^a 2^3 (12)^b A> 0^inf
--> S (11)^a 2^3 <A (11)^b+1 0^inf
--> S (11)^a 2^2 <C 1 (11)^b+1 0^inf
--> S (11)^a 2 <D (11)^b+2 0^inf
--> S (11)^a <D (11)^b+3 2 0^inf by rule 10
--> S (11)^a-1 1 2 A> (11)^b+3 2 0^inf
--> S (11)^a-1 (12)^b+4 A> 2 0^inf
--> S (11)^a-1 (12)^b+4 <C 1 0^inf
--> S (11)^a-1 (12)^b+3 1 <D (11)^1 0^inf
--> S (11)^a-1 1 <D (11)^2b+7 0^inf by rule 15
--> S (11)^a-1 2 A> (11)^2b+7 0^inf
--> S (11)^a-1 2 (12)^2b+7 A> 0^inf
--> S (11)^a-1 2 <A (11)^2b+8 0^inf
--> S (11)^a-1 <C 1 (11)^2b+8 0^inf
--> S (11)^a-2 1 <A (11)^2b+9 0^inf
--> S (11)^a-2 <A (11)^2b+10 2 0^inf by rule 12
--> S (11)^a-3 1 <A (11)^2b+11 2^2 0^inf by rule 13
--> S (11)^a-3 1 D> (11)^2b+11 2^2 0^inf
--> S (11)^a-3 1 (21)^2b+11 D> 2^2 0^inf
--> S (11)^a-3 1 (21)^2b+11 2^2 D> 0^inf
--> S (11)^a-3 1 (21)^2b+11 2^2 <B 2^2 0^inf
--> S (11)^a-3 1 (21)^2b+11 2 <C 1 2^2 0^inf
--> S (11)^a-3 1 (21)^2b+11 <D (11)^1 2^2 0^inf
= S (11)^a-3 (12)^2b+11 1 <D (11)^1 2^2 0^inf
--> S (11)^a-3 (12)^2b+11 2 (12)^0 2 1 <D (11)^1 0^inf
= S (11)^a-3 (12)^2b+11 2^2 1 <D (11)^1 0^inf

24. 0^inf 2^2 1 <D (11)^c 0^inf --> 0^inf (11)^c+1 (12)^3 2^2 1 <D (11)^1 0^inf
by:
0^inf 2^2 1 <D (11)^c 0^inf
--> 0^inf 2^3 A> (11)^c 0^inf
--> 0^inf 2^3 (12)^c A> 0^inf
--> 0^inf 2^3 <A (11)^c+1 0^inf
--> 0^inf 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf 2 <D (11)^c+2 0^inf
--> 0^inf <D (11)^c+3 2 0^inf by rule 10
--> 0^inf B> (11)^c+3 2 0^inf
--> 0^inf (11)^c+3 B> 2 0^inf
--> 0^inf (11)^c+3 <C 1 0^inf
--> 0^inf (11)^c+2 1 <A (11)^1 0^inf
--> 0^inf (11)^c+2 <A (11)^2 2 0^inf by rule 12
--> 0^inf (11)^c+1 1 <A (11)^3 2^2 0^inf by rule 13
--> 0^inf (11)^c+1 1 D> (11)^3 2^2 0^inf
--> 0^inf (11)^c+1 1 (21)^3 D> 2^2 0^inf
--> 0^inf (11)^c+1 1 (21)^3 2^2 D> 0^inf
--> 0^inf (11)^c+1 1 (21)^3 2^2 <B 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^c+1 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf
= 0^inf (11)^c+1 (12)^3 2^2 1 <D (11)^1 0^inf

25. 0^inf (11)^2 2^2 1 <D (11)^c 0^inf --> 0^inf 1 (11)^2c+8 (12)^3 2^2 1 <D (11)^1 0^inf
by:
0^inf (11)^2 2^2 1 <D (11)^c 0^inf
--> 0^inf (11)^2 2^3 A> (11)^c 0^inf
--> 0^inf (11)^2 2^3 (12)^c A> 0^inf
--> 0^inf (11)^2 2^3 <A (11)^c+1 0^inf
--> 0^inf (11)^2 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf (11)^2 2 <D (11)^c+2 0^inf
--> 0^inf (11)^2 <D (11)^c+3 2 0^inf
--> 0^inf (11)^1 1 2 A> (11)^c+3 2 0^inf
--> 0^inf (11)^1 (12)^c+4 A> 2 0^inf
--> 0^inf (11)^1 (12)^c+4 <C 1 0^inf
--> 0^inf (11)^1 (12)^c+3 1 <D (11)^1 0^inf
--> 0^inf (11)^1 1 <D (11)^2c+7 0^inf by rule 15
--> 0^inf (11)^1 2 A> (11)^2c+7 0^inf
--> 0^inf (11)^1 2 (12)^2c+7 A> 0^inf
--> 0^inf (11)^1 2 <A (11)^2c+8 0^inf
--> 0^inf (11)^1 <C 1 (11)^2c+8 0^inf
--> 0^inf 1 <A (11)^2c+9 0^inf
--> 0^inf <A (11)^2c+10 2 0^inf by rule 12
--> 0^inf 1 B> (11)^2c+10 2 0^inf
--> 0^inf 1 (11)^2c+10 B> 2 0^inf
--> 0^inf 1 (11)^2c+10 <C 1 0^inf
--> 0^inf (11)^2c+10 <A (11)^1 0^inf
--> 0^inf (11)^2c+9 1 <A (11)^2 2 0^inf by rule 12
--> 0^inf (11)^2c+9 <A (11)^3 2^2 0^inf by rule 13
--> 0^inf (11)^2c+9 D> (11)^3 2^2 0^inf
--> 0^inf (11)^2c+9 (21)^3 D> 2^2 0^inf
--> 0^inf (11)^2c+9 (21)^3 2^2 D> 0^inf
--> 0^inf (11)^2c+9 (21)^3 2^2 <B 2^2 0^inf
--> 0^inf (11)^2c+9 (21)^3 2 <C 1 2^2 0^inf
--> 0^inf 1 (11)^2c+8 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf 1 (11)^2c+8 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf by rule 22
--> 0^inf 1 (11)^2c+8 (12)^3 2^2 1 <D (11)^1 0^inf

26. 0^inf 1 (11)^1 2^2 1 <D (11)^c 0^inf --> 0^inf 1 (11)^2c+7 (12)^3 2^2 1 <D (11)^1 0^inf
by:
0^inf 1 (11)^1 2^2 1 <D (11)^c 0^inf
--> 0^inf 1 (11)^1 2^3 A> (11)^c 0^inf
--> 0^inf 1 (11)^1 2^3 (12)^c A> 0^inf
--> 0^inf 1 (11)^1 2^3 <A (11)^c+1 0^inf
--> 0^inf 1 (11)^1 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf 1 (11)^1 2 <D (11)^c+2 0^inf
--> 0^inf 1 (11)^1 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf (11)^1 2 A> (11)^c+3 2 0^inf
--> 0^inf (11)^1 2 (12)^c+3 A> 2 0^inf
--> 0^inf (11)^1 2 (12)^c+3 <C 1 0^inf
--> 0^inf (11)^1 2 (12)^c+2 1 <D (11)^1 0^inf
--> 0^inf (11)^1 2 1 <D (11)^2c+5 0^inf by rule 15
--> 0^inf (11)^1 2^2 A> (11)^2c+5 0^inf
--> 0^inf (11)^1 2^2 (12)^2c+5 A> 0^inf
--> 0^inf (11)^1 2^2 <A (11)^2c+6 0^inf
--> 0^inf (11)^1 2 <C 1 (11)^2c+6 0^inf
--> 0^inf (11)^1 <D (11)^2c+7 0^inf
--> 0^inf 1 2 A> (11)^2c+7 0^inf
--> 0^inf (12)^2c+8 A> 0^inf
--> 0^inf <A (11)^2c+9 0^inf
--> 0^inf 1 B> (11)^2c+9 0^inf
--> 0^inf 1 (11)^2c+9 B> 0^inf
--> 0^inf 1 (11)^2c+9 <B 2 0^inf
--> 0^inf 1 (11)^2c+9 B> 2 0^inf
--> 0^inf 1 (11)^2c+9 <C 1 0^inf
--> 0^inf 1 (11)^2c+8 1 <A (11)^1 0^inf
--> 0^inf 1 (11)^2c+8 <A (11)^2 2 0^inf by rule 12
--> 0^inf (11)^2c+8 <A (11)^3 2^2 0^inf by rule 13
--> 0^inf (11)^2c+8 D> (11)^3 2^2 0^inf
--> 0^inf (11)^2c+8 (21)^3 D> 2^2 0^inf
--> 0^inf (11)^2c+8 (21)^3 2^2 D> 0^inf
--> 0^inf (11)^2c+8 (21)^3 2^2 <B 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 1 <D (11)^1 2^2 0^inf
--> 0^inf 1 (11)^2c+7 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf
= 0^inf 1 (11)^2c+7 (12)^3 2^2 1 <D (11)^1 0^inf

27. 0^inf 1 2^2 1 <D (11)^c 0^inf --> 0^inf (11)^2c+5 (12)^3 2^2 1 <D (11)^1 0^inf
by:
0^inf 1 2^2 1 <D (11)^c 0^inf
--> 0^inf 1 2^3 A> (11)^c 0^inf
--> 0^inf 1 2^3 (12)^c A> 0^inf
--> 0^inf 1 2^3 <A (11)^c+1 0^inf
--> 0^inf 1 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf 1 2 <D (11)^c+2 0^inf
--> 0^inf 1 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf 2 A> (11)^c+3 2 0^inf
--> 0^inf 2 (12)^c+3 A> 2 0^inf
--> 0^inf 2 (12)^c+3 <C 1 0^inf
--> 0^inf 2 (12)^c+2 1 <D (11)^1 0^inf
--> 0^inf 2 1 <D (11)^2c+5 0^inf
--> 0^inf 2^2 A> (11)^2c+5 0^inf
--> 0^inf 2^2 (12)^2c+5 A> 0^inf
--> 0^inf 2^2 <A (11)^2c+6 0^inf
--> 0^inf 2 <C 1 (11)^2c+6 0^inf
--> 0^inf <D (11)^2c+7 0^inf
--> 0^inf B> (11)^2c+7 0^inf
--> 0^inf (11)^2c+7 B> 0^inf
--> 0^inf (11)^2c+7 <B 2 0^inf
--> 0^inf (11)^2c+7 B> 2 0^inf
--> 0^inf (11)^2c+7 <C 1 0^inf
--> 0^inf (11)^2c+6 1 <A (11)^1 0^inf
--> 0^inf (11)^2c+6 <A (11)^2 2 0^inf by rule 12
--> 0^inf (11)^2c+5 1 <A (11)^3 2^2 0^inf by rule 13
--> 0^inf (11)^2c+5 1 D> (11)^3 2^2 0^inf
--> 0^inf (11)^2c+5 1 (21)^3 D> 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 2^2 D> 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 2^2 <B 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 2 <C 1 2^2 0^inf
--> 0^inf (11)^2c+5 (12)^3 1 <D (11)^1 2^2 0^inf by rule 22
--> 0^inf (11)^2c+5 (12)^3 2 (12)^0 2 1 <D (11)^1 0^inf
= 0^inf (11)^2c+5 (12)^3 2^2 1 <D (11)^1 0^inf

28. 0^inf (11)^1 2^2 1 <D (11)^c 0^inf --> 0^inf 1 Z> 1 (11)^2c+8 0^inf
by:
0^inf (11)^1 2^2 1 <D (11)^c 0^inf
--> 0^inf (11)^1 2^3 A> (11)^c 0^inf
--> 0^inf (11)^1 2^3 (12)^c A> 0^inf
--> 0^inf (11)^1 2^3 <A (11)^c+1 0^inf
--> 0^inf (11)^1 2^2 <C 1 (11)^c+1 0^inf
--> 0^inf (11)^1 2 <D (11)^c+2 0^inf
--> 0^inf (11)^1 <D (11)^c+3 2 0^inf by rule 10
--> 0^inf 1 2 A> (11)^c+3 2 0^inf
--> 0^inf (12)^c+4 A> 2 0^inf
--> 0^inf (12)^c+4 <C 1 0^inf
--> 0^inf (12)^c+3 1 <D (11)^1 0^inf
--> 0^inf 1 <D (11)^2c+7 0^inf
--> 0^inf 2 A> (11)^2c+7 0^inf
--> 0^inf 2 (12)^2c+7 A> 0^inf
--> 0^inf 2 <A (11)^2c+8 0^inf
--> 0^inf <C 1 (11)^2c+8 0^inf
--> 0^inf 1 Z> 1 (11)^2c+8 0^inf

Functions

Let D(a, b, c) = 0^inf (11)^a (12)^b 2^2 1 <D (11)^c 0^inf

Let D_1(a, b, c) = 0^inf 1 (11)^a (12)^b 2^2 1 <D (11)^c 0^inf

Let $f_{1}(n)=2^{n+5}-3$

Let $f_{2}(a,b)=f_{1}^{2\times f_{2}(a-1,b)+11}(1)$, where$f_{2}(0,b)=b$

Rule 21 becomes:

• $D(a,b,c)-->D(a,b-1,2^{b+5}-3)$
• $D_{1}(a,b,c)-->D_{1}(a,b-1,2^{b+5}-3)$

Rule 23 becomes:

• $D(a,0,c)-->D(a-3,2c+11,1)$
• $D_{1}(a,0,c)-->D_{1}(a-3,2c+11,1)$

Rule 24 becomes:

• $D(0,0,c)-->D(c+1,3,1)$

Rule 25 becomes:

• $D(2,0,c)-->D(2c+8,3,1)$

Rule 26 becomes:

• $D_{1}(1,0,c)-->D_{1}(2c+7,3,1)$

Rule 27 becomes:

• $D_{1}(0,0,c)-->D(2c+5,3,1)$

Rule 28 becomes:

• D(1, 0, c) --> halt with score 4c + 18

By repeating rule 21, a stronger rule can be constructed:

• $D(a,b,c)-->D(a,0,f_{1}^{b}(c))$
• $D_{1}(a,b,c)-->D1(a,0,f_{1}^{b}(c))$

If a is greater than or equal to 3: $D(a,0,c)-->D(a-3,2c+11,1)-->D(a-3,0,f_{1}^{2c+11}(1))$ =$D(a-3,0,f_{2}(1,c))$

• $D(a,0,c)-->D(a-3,0,f_{1}^{2c+11}(1))$

This rule can also be repeated, also note that $f_{1}^{2c+11}(1)=f_{2}(1,c)$ and $f_{1}^{2\times f_{2}(a,b)+11}(1)=f_{2}(a+1,b)$:

• $D(3k+d,0,c)-->D(d,0,f_{2}(k,c))$
• $D_{1}(3k+d,0,c)-->D_{1}(d,0,f_{2}(k,c))$

The trajectory of D_1(2, 0, c) is currently unexplored.

Trajectory

S=0: 0^inf A> 0^inf
S=5: 0^inf <A (11)^1 0^inf
S=6: 0^inf 1 B> (11)^1 0^inf
S=8: 0^inf 1 (11)^1 B> 0^inf
S=9: 0^inf 1 (11)^1 <B 2 0^inf
S=10: 0^inf 1 (11)^1 B> 2 0^inf
S=11: 0^inf 1 (11)^1 <C 1 0^inf
S=12: 0^inf (11)^1 <A (11)^1 0^inf
S=23: 0^inf 1 <A (11)^2 2 0^inf
S=38: 0^inf <A (11)^3 2^2 0^inf
S=39: 0^inf 1 B> (11)^3 2^2 0^inf
S=45: 0^inf 1 (11)^3 B> 2^2 0^inf
S=46: 0^inf 1 (11)^3 <C 1 2 0^inf
S=47: 0^inf (11)^3 <A (11)^1 2 0^inf
S=58: 0^inf 1 (11)^2 <A (11)^2 2^2 0^inf
S=59: 0^inf 1 (11)^2 D> (11)^2 2^2 0^inf
S=63: 0^inf 1 (11)^2 (21)^2 D> 2^2 0^inf
S=65: 0^inf 1 (11)^2 (21)^2 2^2 D> 0^inf
S=68: 0^inf 1 (11)^2 (21)^2 2^2 <B 2^2 0^inf
S=69: 0^inf 1 (11)^2 (21)^2 2 <C 1 2^2 0^inf
S=70: 0^inf (11)^2 (12)^2 1 <D (11)^1 2^2 0^inf
--> 0^inf (11)^2 (12)^2 2 (12)^0 2 1 <D (11)^1 0^inf
=0^inf (11)^2 (12)^2 2^2 1 <D (11)^1 0^inf
= D(2, 2, 1)
So, the TM starts in configuration D(2, 2, 1).

D(2, 2, 1) -->

$D(2,0,f_{1}^{2}(1))=D(2,0,f_{1}(61))$

$e_{1}=f_{1}(61)=2^{66}-3$

-->$D_{1}(2e_{1}+8,3,1)-->D_{1}(2e_{1}+8,0,f_{1}^{2}(61))$

$e_{2}=2^{67}+2$

2^67 +2 mod 3 = ?
2^2k mod 3 = 1
2^2k+1 mod 3 = 2
2^67 mod 3 = 2 --> +2 = 4; 4 mod 3 = 1

$D_{1}(e_{2},0,f_{1}^{2}(61))$

--> $D_{1}(1,0,f_{2}({\frac {e_{2}-1}{3}},f_{1}^{2}(61)))$

$e_{3}=f_{2}({\frac {e_{2}-1}{3}},f_{1}^{2}(61))$

f_1(2k+1) = 2^2k+6-3
Note that 2k+6 is of the form 2k, and that the whole expression is of the form 2k+1.
This means that any f_1^a(2k+1) = some f_1^a-1(2k+1)
Now consider that 2^2k mod 3 = 1 and 2^2k+1 mod 3 = 2
f_1(2k+1) = 2^2k+6-3
Note that 2k + 6 is of the form 2k, meaning that 2^2k+6 mod 3 = 1
The -3 will not affect the Modulus here, as it is a full cycle back to 1.
Thus f_1(2k+1) mod 3 = 1, and, by earlier f_1^a(2k+1) mod 3 = 1
f_2(a,b) = f_1^(2*f_2(a-1,b)+11)(1)
Note that f_1^(2*f_2(a-1,b)+11)(1) is also of the form f_1^a(2k+1)
Thus f_2(a,b) mod 3 = 1

$D_{1}(1,0,e_{3})$

$e_{3}mod3=1$

--> $D_{1}(2e_{3}+7,3,1)-->D_{1}(2e_{3}+7,0,f_{1}^{2}(61))$

2e_3 + 7

Modulus: 2 + 7 --> 9 mod 3 = 0

--> $D_{1}(0,0,f_{2}({\frac {2e_{3}+7}{3}},f_{1}^{2}(61)))$

$e_{4}=f_{2}({\frac {2e_{3}+7}{3}},f_{1}^{2}{2}(61))$

$D_{1}(0,0,e_{4})$

--> $D(2e_{4}+5,3,1)-->D(2e_{4}+5,0,f_{1}^{2}(61))$

e_4 mod 3 = 1; 2*1+5 = 7 --> 7 mod 3 = 1

--> $D(1,0,f_{2}({\frac {2e_{4}+4}{3}},f_{1}^{2}(61)))$

$e_{5}=f_{2}({\frac {2e_{4}+4}{3}},f_{1}^{2}(61))$

$D(1,0,e_{5})$

--> halts with score $4e_{5}+18$.

Approximate Score

$4e_{5}+18$

$e_{5}=f_{2}({\frac {2e_{4}+4}{3}},f_{1}^{2}(61))$

$e_{4}=f_{2}({\frac {2e_{3}+7}{3}},f_{1}^{2}(61))$

$e_{3}=f_{2}({\frac {e_{2}-1}{3}},f_{1}^{2}(61))$

$e_{2}=2^{67}+2$

$f_{1}(n)=2^{n+5}-3$

$f_{2}(a,b)=f_{1}^{2\times f_{2}(a-1,b)+11}(1)$, where $f_{2}(0,b)=b$

• $f_{1}(n):$

$2^{n+4}<f_{1}(n)<2^{n+5}$

$(2\uparrow )^{a}n+4<f_{1}^{a}(n)<(2\uparrow )^{a}n+6$

$(2\uparrow )^{a}5<f_{1}^{a}(1)<(2\uparrow )^{a}7$

$2\uparrow \uparrow (a+2)<f_{1}^{a}(1)<2\uparrow \uparrow (a+3)$

• $f_{2}(a,b):$

$f_{2}(a,b)=f_{1}^{2\times f_{2}(a-1,b)+11}(1)$, where $f_{2}(0,b)=b$

$2\uparrow \uparrow (2\times f_{2}(a-1,b)+13)<f_{2}(a,b)<2\uparrow \uparrow (2\times f_{2}(a-1,b)+14)$

$(2\uparrow \uparrow )^{a}b<f_{2}(a,b)<(2\uparrow \uparrow )^{a+1}b$

$2\uparrow \uparrow 2\uparrow \uparrow 2<e_{2}<f_{1}^{2}(61)<2\uparrow \uparrow 2\uparrow \uparrow 2\uparrow \uparrow 2$

$(2\uparrow \uparrow )^{a}(2\uparrow \uparrow 2\uparrow \uparrow 2)<f_{2}(a,f_{1}^{2}(61))<(2\uparrow \uparrow )^{a}(2\uparrow \uparrow 2\uparrow \uparrow 2\uparrow \uparrow 2)$

$2\uparrow \uparrow \uparrow (a+3)<f_{2}(a,f_{1}^{2}(61))<2\uparrow \uparrow \uparrow (a+4)$

$2\uparrow \uparrow \uparrow (4.91\times 10^{19})<e_{3}<2\uparrow \uparrow \uparrow (4.92\times 10^{19})$

$2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow (4.91\times 10^{19})<e_{4}<2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow (4.92\times 10^{19})$

$2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow (4.91\times 10^{19})<e_{5}<\sigma <S<2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow (4.92\times 10^{19})$

This score would make 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch) the new BB(4,3) champion.