1RB2LB0LB 2LC2LA0LA 2RD1LC1RZ 1RA2LD1RD: Difference between revisions
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Created page with "{{machine|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD}} {{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}} is a pentational halting BB(4,3) TM. It was discovered in May 2024 by Pavel Kropitz as one of seven long running TMs and achieves a score of over <math>3 \uparrow\uparrow\uparrow 88574</math>. Polygon analysed the TM by hand in October 2025, providing its score. Pavel listed the halting tape as: <pre> 1 Z> 1^(162*3^((3*<(243*3^(6) - 5)/2; (<(54*3^((3b + 11)/2) - 2..." |
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Latest revision as of 14:12, 19 October 2025
1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD
(bbch) is a pentational halting BB(4,3) TM. It was discovered in May 2024 by Pavel Kropitz as one of seven long running TMs and achieves a score of over . Polygon analysed the TM by hand in October 2025, providing its score.
Pavel listed the halting tape as:
1 Z> 1^(162*3^((3*<(243*3^(6) - 5)/2; (<(54*3^((3b + 11)/2) - 2); (54*3^((3b + 14)/2) - 6); (54*3^(7) - 6)> + 1); (<(54*3^((3*<(54*3^(7) - 3); (54*3^((3b + 14)/2) - 6); (54*3^((81*3^(7) - 2)) - 6)> + 14)/2) - 2); (54*3^((3b + 14)/2) - 6); (54*3^(7) - 6)> + 1)> + 11)/2)) 2
Analysis by Polygon
S is any tape configuration 1. S 1^a <C S --> S <C 1^a S [+a steps] 2. S 1^a <D S --> S <D 2^a S [+a steps] 3. S D> 2^a S --> S 1^a D> S [+a steps] 4. S (11)^a <A S --> S <A (22)^a S [+2a steps] S (11)^a <B S --> S <B (22)^a S [+2a steps] 5. 0^inf 2 (11)^a A> (22)^b S --> 0^inf 2 (11)^a+3 A> (22)^b-1 S [+8a +24 steps] 6. 0^inf 2 (11)^a A> (22)^b S --> 0^inf 2 (11)^a+3b A> S 7. 0^inf 2 (11)^a A> 0 (22)^b S --> 0^inf 2 1 (11)^1 A> (22)^a+2 0 (22)^b-1 S [+6a +28 steps] 8. 0^inf 2 (11)^a A> 2 0 2 S --> 0^inf 2 1 (11)^a+3 A> S [+8a +27 steps] 9. 0^inf 2 1 (11)^a A> (22)^b S --> 0^inf 2 1 (11)^3a+4 A> (22)^b-1 S 10. 0^inf 2 1 (11)^a A> (22)^b S --> 0^inf 2 1 (11)^g_1^b(a) A> S 11-1. 0^inf 2 1 (11)^a A> 0 (22)^b S --> 0^inf 2 (11)^3a+4 A> 0 (22)^b-1 2 S 11-2. 0^inf 2 1 (11)^a A> 0 (22)^b S --> 0^inf 2 1 (11)^1 A> (22)^3a+6 0 (22)^b-2 2 S 12. 0^inf 2 (11)^a A> 0 11 S --> 0^inf 2 1 (11)^1 A> 0 (22)^a+2 2 S [+6a +31 steps] 13. 0^inf 2 1 (11)^a A> 0 2^b S --> 0^inf 2 1 (11)^g_2(a) A> 0 2^b-3 S 14. 0^inf 2 1 (11)^a A> 0 2^3k+v S --> 0^inf 2 1 (11)^(g_2)^k(a) A> 0 2^v S 15. 0^inf 2 1 <A S --> 0^inf 1 D> 2^3 S [+8 steps] 16. 0^inf 2 1 (11)^a A> 0 2 1 2 0^inf --> 0^inf 2 1 (11)^1 A> 0 (22)^1 (11)^3a+7 2 0^inf (may be irrelevant) 17. 0^inf 2 1 (11)^a A> 0 (22)^1 1 S --> 0^inf 2 1 (11)^1 A> (22)^3a+6 2 S 18. 0^inf 2 1 (11)^a A> 0 (22)^1 1 S --> 0^inf 2 1 (11)^g_2(a) A> 2 S 19. 0^inf 2 1 (11)^a A> 2 1^3 S --> 0^inf 2 1 (11)^1 A> 0 (22)^3a+5 2 S 19*. 0^inf 2 1 (11)^a A> 2 1^2 S --> 0^inf 1 (11)^3a+5 D> S 19**. 0^inf 2 1 (11)^a A> 2 1 S --> 0^inf <B (22)^3a+4 S 19*** 0^inf 2 1 (11)^a A> 2 S --> 0^inf (11)^3a+3 1 B> S 20. 0^inf 2 1 (11)^a A> 0 (22)^1 1^b S --> 0^inf 2 1 (11)^g_3(a) A> 0 (22)^1 1^b-4 S 21. 0^inf 2 1 (11)^a A> 0 (22)^1 1^4k+v S --> 0^inf 2 1 (11)^g_3^k(a) A> 0 (22)^1 1^v S 22. 0^inf 2 1 (11)^a A> 0 (22)^1 1^3 2 0^inf --> 0^inf 2 1 (11)^1 A> 0 (22)^1 1^6*g_2(a)+12 2 0^inf 23. 0^inf 2 1 (11)^a A> 0 (22)^1 1^2 2 0^inf --> 0^inf 1 Z> (11)^3*g_2(a)+6 2 0^inf Bonus rules which are not relevant for this TMs behavior: 24. 0^inf (11)^a A> 0 (22)^1 1 2 0^inf --> 0^inf 1 Z> (11)^3*g_2(a)+5 2 0^inf 25. 0^inf 2 1 (11)^a A> 0 (22)^1 2 0^inf --> 0^inf 1 Z> (11)^g_2(a)+1 2 0^inf
The following functions were used in these rules:
Note that
And
It follows that
Further: Let L(a,b) = 0^inf 2 1 (11)^a A> 0 (22)^1 1^b 2 0^inf * L(a, 4k + v) --> L(g_3^k(a), v) by rule 21 * L(a, 0) --> 0^inf 1 Z> (11)^g_2(a)+1 2 0^inf by rule 25 * L(a, 1) --> 0^inf 1 Z> (11)^3*g_2(a)+5 2 0^inf by rule 24 * L(a, 2) --> 0^inf 1 Z> (11)^3*g_2(a)+6 2 0^inf by rule 23 * L(a, 3) --> L(1, 6*g_2(a) + 12) by rule 22 The TM reaches configuration L(1, 3) after running for 34 steps.
by rule 22, this can be simplified to , then:
by rule 21
--> 0^inf 1 Z> 2 0^inf by rule 22
So .
This can be bounded by: