Holdouts lists

A holdout (or undecided machine) is a Turing machine for which it is not known whether the machine halts or not from all-0 input tape. Holdouts are the machines which deciders are unable to decide.

Holdout lists are often shared by contributors. There is a Downloadable Holdout Lists table where people have added lists with no restriction or independent verification. After that table are several tables, Verifiable Holdout Lists, that attempt to document a reproducible path from the initial enumeration of TMs of a given size tracking the holdout lists.

Here is a table that summarises the number of holdout per busy beaver domain based on the Downloadable Holdout Lists:

Number of holdouts

	2-state	3-state	4-state	5-state	6-state
2-symbol	0	0	0	0	4408
3-symbol	0	6	460,916,384
4-symbol	0	434,787,751
5-symbol	273
6-symbol	22,302,296

Downloadable Holdout Lists

BB space	Date	Shared by	Number of holdouts	File	Notes
BB(4,3)	December 21, 2024	@tjligocki	460,916,384	4x3_holdouts_460916384.txt.gz	Google Drive directory for 4x3 TMs
BB(6)	November 25, 2024 (@icy)	@tjligocki	4,319	Spreadsheet of holdouts	Keeping track of BB(6) progress - informal.
BB(3,3)	November 10, 2024		6	3x3 holdout 6.txt
BB(6)	November 8, 2024	@mxdys	4,408	BB6_holdouts_4408.txt
BB(2,6)	November 4, 2024	@tjligocki	22,302,296	2x6_holdouts_22302296.txt.gz	Google Drive directory for 2x6 TMs
BB(3,4)	November 3, 2024	@tjligocki	434,787,751	3x4_holdouts_434787751.txt.gz	Google Drive directory for 3x4 TMs
BB(6)	September 2, 2024	@mxdys	5394	File:BB6 holdouts 5394.txt
BB(6)	August 4, 2024	@mxdys	5877	File:BB6 holdouts 5877.txt
BB(6)	July 6, 2024	@mxdys	7296	File:BB6 holdouts 7296.txt
BB(2,5)	June 15th 2024	@dyuan01	273	File:2x5_holdouts_273.txt	@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
BB(6)	June 13, 2024	@tjligocki	12,091	File:BB6 holdouts 12091.txt	Work done with @Shawn Ligocki
BB(3,3)	June 9, 2024	@Justin Blanchard	22	File:3x3.todo.txt, File:Mugshots small.pdf
BB(6)	June 7, 2024	@mxdys	12,325	File:BB6 holdouts 12325.txt	Some equivalent machines are removed.
BB(5)	June 2024	BBChallenge	0		BB(5) is SOLVED!
BB(2,5)	May 22, 2024	@Justin Blanchard	499	File:2x5.todo.txt
BB(3,3)	June 8, 2023	@Iijil	925	File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt	Intersection of @sligocki and @Iijil from below
BB(3,3)	June 8, 2023	@Iijil	2,480	File:2023 06 08.3x3.holdouts iijil 2380.txt
BB(3,3)	June 7, 2023	@sligocki	2,417	File:2023 06 07.3x3.holdouts 2417.txt
BB(5)	circa May 13th, 2003	Georgi Georgiev (Skelet)	43	List of 43 holdouts
BB(6)	May 27, 2023	@sligocki	181,851	6x2.holdouts_181851.txt.gz	This was posted to the BBChallenge Forum (before Discord)
BB(6)	May 10, 2023	@sligocki	1,458,704	6x2.holdouts_1458704.txt.gz	This was posted to the BBChallenge Forum (before Discord)

Georgi Georgiev (Skelet) posted a list of 43 holdouts for BB(5) in 2003. bbchallenge.org successfully reduced the number of holdouts for BB(5) to zero in June 2024.

Verifiable Holdout Lists

BB(3,3)

Date	Shared by	Number of holdouts	Holdout TMs	Decided TMs	Notes
January 27, 2025	@tjligocki	367	3x3.unk.sort.txt	For halting and infinte TMs see Google Drive directory for 3x3 TMs	Initial pass at generating a verified holdout list using software from Shawn's BB Github. See spreadsheet. This a subset of the holdouts generated by @sligocki below.
June 8, 2023	@Iijil	925	File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt	Derived from TMs in the holdouts generated by @sligocki and @Iijil, see below.	Intersection of @sligocki and @Iijil from below.
June 8, 2023	@Iijil	2,480	File:2023 06 08.3x3.holdouts iijil 2380.txt	Unknown	Still gathering information on how this was generated.
June 7, 2023	@sligocki	2,417	File:2023 06 07.3x3.holdouts 2417.txt	Unknown	Initial pass at generating a verified holdout using software from Shawn's BB Github.