BBB(3, 3) ≥ 10 ↑↑ 6

[nichol...@gmail.com]

I think I found a new BBB(3, 3) champ:


1RB 0LB 2LA | 1LA 0RC 0LB | 2RC 2RB 0LC

1RB0LB2LA_1LA0RC0LB_2RC2RB0LC


It ends up in this configuration:

C0 | 2^N 0 [0]

where N = (2 ** (4 + (2 ** (4 + (2 ** (4 + (2 ** (4 + (2 ** (4 + (2 ** 20))))))))))).

10 ↑↑ 6 < N < 10 ↑↑ 7

C0 -> 2RC and the scan is at the right edge of the tape, so it will "spin out", i.e. it will get stuck in state C moving right forever. This means that it quasihalts, so it qualifies for Beeping Busy Beaver.

Can anyone confirm? If it's legit, then BBB(3, 3) ≥ 10 ↑↑ 6.


Previous champ was found by Shawn Ligocki in 2022:


1RB 0LB 1LA | 2LC 2LB 2LB | 2RC 2RA 0LC

1RB0LB1LA_2LC2LB2LB_2RC2RA0LC


That one runs for ~ 10 ** 62 steps before spinning out on the blank tape (also in state C).

Shawn's remarks:

https://groups.google.com/g/busy-beaver-discuss/c/ET4Q_ywWPUE/m/oIX_ZdKtAQAJ
https://www.sligocki.com/2022/02/27/bb-recurrence-relations.html

[Shawn Ligocki]

I can confirm that this TM "spins out" with the tape:

0^inf 2^N 0^1 C> 0^inf

where N = 2^(4 + 2^(4 + 2^(4 + 2^(4 + 2^(4 + 2^20)))))

Using my code (Code/Quick_Sim.py 1RB0LB2LA_1LA0RC0LB_2RC2RB0LC -r --exp-linear-rules --no-steps)

Nick, did you find this with Quick_Sim?

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[nichol...@gmail.com]

> Nick, did you find this with Quick_Sim?

No, I used my own inductive prover, which is partially based on yours. It is faster but less reliable (prone to false positive rules). So the workflow is to search with mine and then verify any results with Quick_Sim.

Of course, I already verified with your simulator before posting! I would never claim a result without running it by Quick_Sim first :)

[Shawn Ligocki]

Nice!

Here are some rules for this TM:

D(a, d) = 0^inf <C 0^a 1 2^d 1 0^inf

D(a, d+3) --> D(2^{a+6} - 2, d)
D(a, 0) --> Spin out
D(a, 1) --> C(14, a+1)
D(a, 2) --> C(14, a+4)

Start: D(14, 2)

This is actually a Collatz-like pentation-level TM that got unlucky! Each "reset" it iterates a tetration level function:

D(14, 3k+r) --> D(> 2^^k, r)

But in this case, the TM only resets once before spinning out:

D(14, 2) --> D(14, 18) --> Spin out

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