BB(6)

The 6-state, 2-symbol Busy Beaver problem BB(6) is unsolved. With the discovery of Antihydra in 2024, we now know that we must solve a Collatz-like problem in order to solve BB(6).

The current BB(6) champion 1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch) was discovered by Pavel Kropitz in 2022 proving the lower bound:

$$S(6)>\Sigma (6)>10\uparrow \uparrow 15$$

Techniques

In order to simulate the current BB(6) champion requires accelerated simulation that can handle Collatz Level 2 inductive rules. In other words, it requires a simulator that can prove the rules:

$${\begin{array}{lcl}C(4k)&\to &{\operatorname {Halt} }{\Big (}{\frac {3^{k+3}-11}{2}}{\Big )}\\C(4k+1)&\to &C{\Big (}{\frac {3^{k+3}-11}{2}}{\Big )}\\C(4k+2)&\to &C{\Big (}{\frac {3^{k+3}-11}{2}}{\Big )}\\C(4k+3)&\to &C{\Big (}{\frac {3^{k+3}+1}{2}}{\Big )}\\\end{array}}$$

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to Euler's totient function).

Cryptids

Several Turing machines have been found that are Cryptids, considered so because each of them have a Collatz-like halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as probviously halting or probviously non-halting.

Probviously non-halting Cryptids:

• Antihydra
• 1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA (bbch), a variant of Hydra and Antihydra
• 1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC (bbch), similar to Antihydra
• 1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB--- (bbch), similar to Antihydra
• 1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB (bbch), similar to Antihydra

Probviously halting Cryptids:

• Lucy's Moonlight
• 1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC (bbch), a family of 16 related TMs
• 1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF (bbch)
• 1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB (bbch)
• 1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA (bbch)
• 1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE (bbch)

Although 1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE (bbch) behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a translated cycler and a 2/5 chance of halting.

There are a few machines considered notable for their chaotic behaviour, but which have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

Potential Cryptids:

• 1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE (bbch)
• 1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE (bbch)
• 1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC (bbch)
• 1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD (bbch)

Top Halters

Below is a table of the machines with the 10 highest known runtimes. Their sigma scores are expressed using an extension of Knuth's up-arrow notation.

Top Known BB(6) Halters

TM	approximate sigma score
1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch)	10 ↑↑ 15.60465
1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF (bbch)	10 ↑↑ 7.52390
1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF (bbch)	10 ↑↑ 7.52390
1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF (bbch)	10 ↑↑ 7.52390
1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA (bbch)	10 ↑↑ 7.23619
1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB (bbch)	10 ↑↑ 6.96745
1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE (bbch)	10 ↑↑ 5.77573
1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD (bbch)	10 ↑↑ 5.63534
1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC (bbch)	10 ↑↑ 5.56344
1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ (bbch)	10 ↑↑ 5.12468

The runtimes are presumed to be about ${\text{score}}^{2}$ which is roughly indistinguishable in tetration notation.

Holdouts

@mxdys's informal holdouts list is down to 3571 machines as of February 2024.