BB(6)

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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:[1]

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:

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

Known BB(6) Cryptids:

Potential Cryptids:

Top Halters

The current top 10 BB(6) halters (known by Shawn Ligocki) are

1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE Halt ~10↑↑15.60465
1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD Halt ~10↑↑5.63534
1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC Halt ~10↑↑5.56344
1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ Halt ~10↑↑5.12468
1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ Halt ~10↑↑5.03230
1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA Halt ~10↑↑4.91072
1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC Halt ~10↑↑3.33186
1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD Halt ~10↑↑3.31128
1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE Halt ~10↑↑3.18855
1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA Halt ~10^646456993.24591

The numbers listed are sigma scores. Runtimes are not available, but are presumed to be about which is roughly indistinguishable in tetration notation. Fractional tetration notation is described in https://www.sligocki.com/2022/06/25/ext-up-notation.html. For a longer list of halting TMs see https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt. For historical perspective see Pascal Michel's Historical survey of Busy Beavers.

Holdouts

@mxdys's informal holdouts list is down to 7296 TMs as of 6 Jul 2024.

References

  1. Shawn Ligocki. 2022. "BB(6, 2) > 10↑↑15". https://www.sligocki.com/2022/06/21/bb-6-2-t15.html