# BB(4)

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**BB(4)** refers to the 4^{th} value of the Busy Beaver function.

## History

In this Section, we use Radó's original S (number of steps) and Σ (number of ones on the final tape) notations; see Busy Beaver Functions.

- In 1965, Allen Brady proves that S(4) ≥ 84 and Σ(4) ≥ 11.
^{[1]}(TODO: the abstract says "or ≥ 12 using a different stopping convention") - In 1966, Allen Brady conjectures Σ(4) = 13 and S(4) = 107 (He says S(4) = 106, but this seems to be based upon a slightly different version of S function).
^{[2]} - In 1974, Allen Brady proves that S(4) = 107.
^{[3]}(better source needed) - In 1983, Allen Brady publishes the proof that Σ(4) = 13 and S(4) = 107.
^{[4]}Some holdouts were not rigorously handled by the proof:*"All of the remaining holdouts were examined by means of voluminous printouts of their histories along with some program extracted features. It was determined to the author's satisfaction that none of these machines will ever stop."*^{[1]} - In 2024, S(4) = 107 was confirmed as part of Coq-BB5.
^{[5]}

## Champions

S(4) = 107 and there is only one shift champion (in TNF):

`1RB1LB_1LA0LC_1RZ1LD_1RD0RA`

(bbch) leaves 13 ones (a ones champion)

Σ(4) = 13 and there are 2 ones champions (in TNF):

`1RB1LB_1LA0LC_1RZ1LD_1RD0RA`

(bbch) runs 107 steps (the steps champion)

`1RB0RC_1LA1RA_1RZ1RD_1LD0LB`

(bbch) runs 96 steps

## Enumeration

In TNF-1RB there are exactly 620,261 BB(4) TMs of which 183,983 halt. The top longest running TMs are:

1RB1LB_1LA0LC_1RZ1LD_1RD0RA Halt 107 13 1RB1LD_1LC0RB_1RA1LA_1RZ0LC Halt 97 9 1RB0RC_1LA1RA_1RZ1RD_1LD0LB Halt 96 13 1RB1LB_0LC0RD_1RZ1LA_1RA0LA Halt 96 6 1RB1LD_0LC0RC_1LC1LA_1RZ0LA Halt 84 11 1RB1RZ_1LC0RD_1LA1LB_0LC1RD Halt 83 8 1RB0RD_1LC0LA_1RA1LB_1RZ0RC Halt 78 12 1RB1LA_0RC0RD_1LC0LA_1RZ0RC Halt 78 9 1RB0RD_0RC0RA_1LC0LA_0RB1RZ Halt 75 9 1RB0RC_1LC1RA_1RZ0LD_1RA1LA Halt 74 8 1RB1LA_0LA1RC_1RA0RD_1RZ0RB Halt 70 8 1RB1LC_0RC0RB_0LD0LA_1LA1RZ Halt 69 7 1RB1LA_1LA1RC_1RZ0RD_0LD1RB Halt 69 7 1RB1RZ_1LC1RA_0RC0LD_1RD0LB Halt 68 10 1RB0LB_0RC0RD_1LD1LA_0LA1RZ Halt 68 8 1RB1LB_0RC1RZ_1LC1LD_0RD0LA Halt 68 7 1RB0RD_0RC1RZ_1LC0LA_0RA0RB Halt 68 7 1RB0LD_0RC1LD_1LC0RB_0LA1RZ Halt 67 7 1RB1LA_1RC0LD_0LA0RC_1RZ1LB Halt 66 7 1RB1RC_1LC1RD_1RZ0LD_1LA0RA Halt 65 7

For the top 1000 halting BB(4) TMs, see: https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/4x2.txt

## References

- ↑
^{1.0}^{1.1}Brady, A. H. (1965). Solutions of restricted cases of the halting problem applied to the determination of particular values of a non-computable function. https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/zk51vk21c - ↑ Brady, A. H. (1966). The Conjectured Highest Scoring Machines for Rado's Σ(k) for the Value k = 4. https://ieeexplore.ieee.org/document/4038890
- ↑ https://www.quantamagazine.org/amateur-mathematicians-find-fifth-busy-beaver-turing-machine-20240702/
- ↑ Brady, A. H. (1983). The determination of the value of Rado’s noncomputable function Σ(k) for four-state Turing machines. https://www.ams.org/journals/mcom/1983-40-162/S0025-5718-1983-0689479-6/
- ↑ https://github.com/ccz181078/Coq-BB5/blob/main/BB42Theorem.v