BB(4)

The 4-state, 2-symbol Busy Beaver problem, BB(4), refers to the 4^th value of the Busy Beaver function.

Allen Brady found the 4-state busy beaver winner in 1966, it runs 107 steps and has a sigma score of 13. He claims to have proven that BB(4) = 107 in 1974, but didn't publish the proof until 1983.

In 2024, bbchallenge formalized the proof that BB(4) = 107 as part of the Coq-BB5 proof.

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.

Timeline of bounds established for Σ(4) and S(4)
Month of discovery	Machine	S(4)	Σ(4)	Discoverer	Notes
?	?	?	≥ 12	?	Mentioned by Green, M. in November 1964. Due to exhaustive simulations, it is known that one of the results S(4) ≥ 53, S(4) ≥ 63, or S(4) ≥ 78, could have been stated at the time, but was not.
December 1964	1RB1LD_0LC0RC_1LC1LA_1RZ0LA (bbch)	≥ 84	≥ 11	Brady, A.	The bound for the ones champion was equalled by an earlier, unknown machine. These bounds were published simultaneously.
	1RB0LA_1RC1LB_1LB1RD_1RZ0RA (bbch)	≥ 63	≥ 12
1965	1RB0RC_1LA1RA_1RZ1RD_1LD0LB (bbch)	≥ 96	≥ 13	Lee, C. Y	Eventually proven (informally) to be a Σ(4) champion in 1983.
October 1966	1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch)	≥ 107	≥ 13	Brady, A.	Eventually proven (informally) to be the S(4) and a Σ(4) champion in 1983.
April 1983		= 107	= 13		Allen Brady claimed to have proven this result in October 1974, but did not publish a proof until 1983. Some holdouts were solved informally via manual inspection. A completely formal proof of S(4) = 107 and Σ(4) = 13 was written in 2024 by bbchallenge.org.

Bounds highlighted in red indicate those which did not improve upon the best known at the time.

• In 1966, Allen Brady conjectured Σ(4) = 13 and S(4) = 107 (Brady stated S(4) = 106, seemingly basing it upon a slightly different version of S function).
• In 1974, Allen Brady proved that Σ(4) = 13 and S(4) = 107 (Brady again stated S(4) = 106).
• In 1983, Allen Brady published the proof that Σ(4) = 13 and S(4) = 107. 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."
• In 2024, bbchallenge.org formalized a proof of the bounds Σ(4) = 13 and S(4) = 107 in Rocq, confirming (TODO: state what was confirmed naturally)

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 for 107 steps (the steps champion)

• 1RB0RC_1LA1RA_1RZ1RD_1LD0LB (bbch) runs for 96 steps

Top Halters

In TNF-1RB there are exactly 620,261 BB(4) TMs of which 183,983 halt. The top 20 longest running TMs are:

Standard Format             Status Steps   Σ
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