BB(7): Difference between revisions

Latest revision as of 06:52, 19 August 2025

The 7-state, 2-symbol Busy Beaver problem, BB(7), refers to the unsolved 7^th value of the Busy Beaver function. With the compilation of the Cryptid machine Bigfoot into a 7-state, 2-symbol machine in May 2024, we now know that we must solve a Collatz-like problem in order to solve BB(7).

The current BB(7) champion 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch) was discovered by Pavel Kropitz in May 2025, proving the lower bound:

S(7)>\Sigma (7)>2\uparrow ^{11}2\uparrow ^{11}3

History

Before 2025, the only known BB(7) champions were produced by hand, not by search. In 1964, Milton Green designed a machine that had sigma score 22,961. In 2014, Wythagoras modified a BB(6) champion to produce a machine that had sigma score $>10\uparrow \uparrow 5$ .

In May 2025, mxdys shared C++ code that breaks up the BB(7) enumeration into 1 million subtasks which each run for about 2 minutes and leave ~100 holdouts each. Various folks on Discord have been investigating different sections of this domain to search for champions.

Within three days of the code's release, the Ligockis found three champions after applying their deciders to enumerator output. Shawn Ligocki found the first two, 1RB0RF_1LC0RE_1RD1LB_1LA1LD_0RA0LE_1RG0LB_1RZ1RB (bbch) and 1RB1RA_1RC0LC_0LD1LG_1LF0LE_1RZ1LF_0LA1LD_1RA1LC (bbch), with sigma scores of approximately 10 ↑↑ 22 and 10 ↑↑ 35. That evening, Terry Ligocki found 1RB0LG_1RC0RF_1LD1RZ_1LF0LE_1RA1LD_1LG1RE_0LB0LB (bbch), with sigma score ~10 ↑↑ 46. A few days later, Pavel found a TM that outpaces all of them.

Pavel's champion is enumerated in subtask 243308.

Cryptids

BB(7) has not been seriously investigated by hand, so no native BB(7) Cryptids have yet been discovered.

Probviously non-halting Cryptids:

0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB (bbch), Bigfoot (a BB(3,3) Cryptid) compiled into a 2-symbol TM by Iijil in 2024.

Top Halters

The top 20 scoring known machines are:

TM	Approximate sigma score	Discoverer
`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)	$2\uparrow ^{11}2\uparrow ^{11}3$	Pavel Kropitz
`1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE` (bbch)	10 ↑↑ 519.20	Andrew Ducharme
`1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE` (bbch)	10 ↑↑ 519.20	Andrew Ducharme
`1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD0LG_1LB0LE` (bbch)	10 ↑↑ 519.20	@gerbil5709, Terry Ligocki
`1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ` (bbch)	10 ↑↑ 286.17	Terry Ligocki
`1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD` (bbch)	10 ↑↑ 246.32	@Iijil
`1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA1LG_1RZ1LA` (bbch)	10 ↑↑ 246.32	@star, Terry Ligocki
`1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG` (bbch)	10 ↑↑ 243.88	@Iijil, Andrew Ducharme
`1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ` (bbch)	10 ↑↑ 228.78	Terry Ligocki
`1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RG0LD` (bbch)	10 ↑↑ 192.67	Terry Ligocki
`1RB1LA_1LC0RE_0LD1LB_1RD0LA_1RF0RA_0RG0LA_1RB1RZ` (bbch)	10 ↑↑ 192.67	Terry Ligocki
`1RB1LA_1LC0RE_0LD1LB_1RD0LA_1RF0RA_1RG0LA_0LE1RZ` (bbch)	10 ↑↑ 192.67	Terry Ligocki
`1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RA0LG_1RB1LG` (bbch)	10 ↑↑ 192.67	Terry Ligocki
`1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RG0LG_1RB1LG` (bbch)	10 ↑↑ 192.67	Terry Ligocki
`1RB1RZ_1LC0RE_0LD1LB_1RE0LG_1RF0RG_0RA0LG_1RB1LG` (bbch)	10 ↑↑ 192.67	Terry Ligocki
`1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RC0LD` (bbch)	10 ↑↑ 192.67	Andrew Ducharme
`1RB1LA_1LC0RE_0LD1LB_1RE1LG_1RF0LG_0RA0LA_0RF1RZ` (bbch)	10 ↑↑ 192.67	Andrew Ducharme
`1RB1LA_1LC0RF_0LD0RD_1RF1LE_1LB1RZ_1RG0RA_0RA0LA` (bbch)	10 ↑↑ 192.67	@C7X
`1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_0RG0LA_1RB1RZ` (bbch)	10 ↑↑ 192.67	@Iijil, Terry Ligocki
`1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_1RG0LA_0LE1RZ` (bbch)	10 ↑↑ 192.67	@Iijil, Terry Ligocki

The top 20 known halters with unique scores are:

TM	Approximate sigma score	Discoverer
`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)	$2\uparrow ^{11}2\uparrow ^{11}3$	Pavel Kropitz
`1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE` (bbch)	10 ↑↑ 519.20	Andrew Ducharme
`1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ` (bbch)	10 ↑↑ 286.17	Terry Ligocki
`1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD` (bbch)	10 ↑↑ 246.32	@Iijil
`1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG` (bbch)	10 ↑↑ 243.88	@Iijil, Andrew Ducharme
`1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ` (bbch)	10 ↑↑ 228.78	Terry Ligocki
`1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RA0LG_1RB1LG` (bbch)	10 ↑↑ 192.67	Terry Ligocki
`1RB0LC_1LC1LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LA` (bbch)	10 ↑↑ 188.28	Terry Ligocki
`1RB0LC_1LC0LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LC` (bbch)	10 ↑↑ 140.28	@stokastic
`1RB0RF_1RC1RZ_0LD1RF_0RA1LE_0LC1LF_1LE0RG_0LE1RA` (bbch)	10 ↑↑ 136.64	Katelyn Doucette, Andrew Ducharme
`1RB0LG_0RC1RZ_1LD0LA_1RE1LE_1LC1RF_0RE0RA_0RF1LG` (bbch)	10 ↑↑ 133.85	@poppuncher
`1RB1RZ_1RC0RF_1LD1RB_1RG0LE_1LD0RA_1RE0LD_0RC1LF` (bbch)	10 ↑↑ 129.24	@Iijil
`1RB0RG_1LC0RE_1LF1LD_0LE1LC_1RA1RB_1LD0LF_1RZ0RF` (bbch)	10 ↑↑ 127.52	Andrew Ducharme
`1RB0LC_1RC0RG_1RD0LF_1RE0RF_1LA1RG_1LE1LF_1RZ1RD` (bbch)	10 ↑↑ 126.20	@stokastic
`1RB0LD_1RC1RA_0RD1RG_1LE1LF_0LF1LE_0RA0LD_0RA1RZ` (bbch)	10 ↑↑ 124.86	Terry Ligocki
`1RB1LF_1RC1RA_1LD0LD_1LA1LE_0LA0LD_1LG0RF_0LE1RZ` (bbch)	10 ↑↑ 116.98	Terry Ligocki
`1RB0RD_1RC0LA_0LA0LE_1RE1RZ_1RF0RA_1LG0LE_1LC0LG` (bbch)	10 ↑↑ 116.05
`1RB0RD_1RC0RA_0RD1LD_0LE1LF_1LA0LG_0LC1LB_1LC1RZ` (bbch)	10 ↑↑ 115.52	@prurq
`1RB0RG_1LC0LE_1LD0LB_0LE1RE_0RA1RF_0RD1RC_1RD1RZ` (bbch)	10 ↑↑ 114.83	Andrew Ducharme
`1RB0RE_1LC0LA_1LD0LC_0LE0LA_1RF0RG_1RD0LE_1RA1RZ` (bbch)	10 ↑↑ 114.60	Shawn Ligocki

Phase 1

Phase 1's first two stages were carried out from May 2025 to July 2025. The table below summaries some of that activity and was used to coordinate the effort of doing the raw computation. Fourteen people (see below) contributed directly and the overall bbchallenge group participated in discord discussions, etc.

Stage 1, "enumeration" in the table, involved running 1 million subtasks using mxdys's code in 100 batches of 10 thousand subtasks.

Stage 2, "linear rule" in the table, processing the output of Stage 1 using Shawn Ligocki's linear rule code which also used a few other deciders, e.g., a version of CTL.

Stage 3, the final stage of Phase 1, is the compilation, verification, and presentation of the final results from Phase 1. It is ongoing but should be completed during August 2025.

Task range	Done by	Completed		# holdouts	Maximum Score TM	~Sigma	Source
Task range	Done by	enumeration	linear rule	# holdouts	Maximum Score TM	~Sigma	Source
00-01xxxx	@Iijil	Yes	Yes	1,545,673	`1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD` (bbch)	10 ↑↑ 246.32	Google Drive folder
02-04xxxx	@Iijil Terry Ligocki	Yes	Yes	2,279,734	`1RB0LF_1RC1RA_1RD0RG_1LE1RZ_1LA0LF_1RA1LE_0RE1RG` (bbch)	10 ↑↑ 93.81	@Iijil Terry Ligocki
05-09xxxx	@Iijil Andrew Ducharme	Yes	Yes	3,889,955	`1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG` (bbch)	10 ↑↑ 243.88	@Iijil Andrew
10-12xxxx	Andrew Ducharme	Yes	Yes	2,708,888	`1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE` (bbch)	10 ↑↑ 519.20	Google Drive folder
13xxxx	Shawn Ligocki	Yes	Yes	1,192,442	`1RB0RE_1LC0LA_1LD0LC_0LE0LA_1RF0RG_1RD0LE_1RA1RZ` (bbch)	10 ↑↑ 114.60	Google Drive folder
14-16xxxx	Andrew Ducharme	Yes	Yes	2,701,637	`1RB0LC_1LC1LD_1LA1LB_0LG1RE_0RF0LD_0RA1RE_1RZ1LA` (bbch)	10 ↑↑ 188.28	Google Drive folder
17-18xxxx	@gerbil5709 Terry Ligocki	Yes	Yes	1,898,156	`1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_0RG0LA_1RB1RZ` (bbch)	10 ↑↑ 192.67	@gerbil5709 Terry Ligocki
19xxxx	Katelyn Doucette Andrew Ducharme	Yes	Yes	1,099,752	`1RB0RF_1RC1RZ_0LD1RF_0RA1LE_0LC1LF_1LE0RG_0LE1RA` (bbch)	10 ↑↑ 136.64	Google Drive folder
20-23xxxx	@C7X	Yes	Yes	4,528,827	`1RB1LA_1LC0RF_0LD0RD_1RF1LE_1LB1RZ_1RG0RA_0RA0LA` (bbch)	10 ↑↑ 192.67	Google Drive folder
24xxxx	Andrew Ducharme	Yes	Yes	712,356	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)*	$2\uparrow ^{11}2\uparrow ^{11}3^{*}$	Google Drive folder
25-34xxxx	@stokastic	Yes	Yes	10,339,816	`1RB0LC_1LC0LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LC` (bbch)	10 ↑↑ 140.28	Google Drive folder
35-39xxxx	Terry Ligocki	Yes	Yes	4,894,047	`1RB1RZ_1LC0RF_0LD1LB_1RD0LE_1RB1LE_1RG0RE_0RA0LE` (bbch)	10 ↑↑ 192.67	Google Drive folder
40-47xxxx	Andrew Ducharme	Yes	Yes	6,181,327	`1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE` (bbch)	10 ↑↑ 519.20	Google Drive folder
48xxxx	@star Terry Ligocki	Yes	Yes	727,875	`1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA1LG_1RZ1LA` (bbch)	10 ↑↑ 246.32	@star Terry Ligocki
49xxxx	Tobiáš Brichta Terry Ligocki	Yes	Yes	804,722	`1RB0LG_1RC0RG_0LD1RE_1RD0RE_1LF1RB_0LA1RZ_1LC1LG` (bbch)	10 ↑↑ 126.20	Tobiáš Brichta Terry Ligocki
50xxxx	@prurq Andrew Ducharme	Yes	Yes	797,224	`1RB0RD_1RC0RA_0RD1LD_0LE1LF_1LA0LG_0LC1LB_1LC1RZ` (bbch)	10 ↑↑ 115.52	Google Drive folder
51-53xxxx	@gerbil5709 Terry Ligocki	Yes	Yes	3,016,175	`1RB0LC_1LC0LD_1LA1LB_0LG1RE_0RF0RF_0RA1RE_1RZ1LC` (bbch)	10 ↑↑ 140.28	@gerbil5709 Terry Ligocki
54-59xxxx	Terry Ligocki	Yes	Yes	5,689,850	`1RB0LC_1LC1LD_1LA1LB_0LG1RE_0RF0RF_0RA1RE_1RZ1LA` (bbch)	10 ↑↑ 188.28	Google Drive folder
60-64xxxx	@gerbil5709 Terry Ligocki	Yes	Yes	3,817,876	`1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD0LG_1LB0LE` (bbch)	10 ↑↑ 519.20	@gerbil5709 Terry Ligocki
65-68xxxx	Terry Ligocki	Yes	Yes	3,076,778	`1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RG0LD` (bbch)	10 ↑↑ 192.67	Google Drive folder
69xxxx	@poppuncher	Yes	Yes	1,053,119	`1RB0LG_0RC1RZ_1LD0LA_1RE1LE_1LC1RF_0RE0RA_0RF1LG` (bbch)	10 ↑↑ 133.85	Google Drive folder
70-71xxxx	@hipparcos	Yes	Yes	1,899,094	`1RB1RZ_1LC1RD_0LD0LC_1LE1RA_1LF0LE_1RF0RG_1RG0RD` (bbch)	10 ↑↑ 77.50	Github release
72-79xxxx	Terry Ligocki	Yes	Yes	7,627,514	`1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ` (bbch)	10 ↑↑ 228.78	Google Drive folder
80-81xxxx	@xnoobspeakable	Yes	Yes	1,537,533	`1RB0LA_0RC1RZ_0RD0RG_1LE1RA_1LF1LD_1RG0RG_1RD1RC` (bbch)	10 ↑↑ 74.85	Google Drive folder
82-99xxxx	Terry Ligocki	Yes	Yes	15,673,786	`1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ` (bbch)	10 ↑↑ 286.17	Google Drive folder

*The current BB(7) champion TM 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)* was discovered by Pavel Kropitz in the enumeration of subtask 243308. The remaining subtasks in the 24xxxx range were enumerated and filtered by Andrew Ducharme.

Exploration after Phase 1

People are now looking at reducing the number of holdouts (~85M TMs) after Phase 1. They are trying different deciders not used in Phase 1 and different parameters with the deciders used in Phase 1. This table is somewhere they can put the results of these explorations so that there is a record of what is being done. It is hoped that this will inspire everyone to contribute when they can, a reference point for discord discussions, and something that can be questioned and/or verified.

Done by	Holdout TMs		% Reduced	Runtime (hours)	TMs/sec/core		Description	Source
Done by	Input	Output	% Reduced	Runtime (hours)	Decided	Processed	Description	Source
Shawn Ligocki	858,538	733,830	14.5%	5.0	6.9282	47.6966	Translated Cycler and CPS on `7x2_p01_s02_holdouts_rand_13.txt`	Discord link
Andrew Ducharme	872,041	784,099	10.1%	456.4	0.0535	0.5034	Enumerate.py w/ 250k max-loops and block-mult=3 on `7x2_p01_s02_holdouts_rand_72.txt`	Discord link
Terry Ligocki	1,000	982	1.8%	3.3	0.0015	0.0830	MITMWFAR Code with options `-n=10 -m=1 -pm=1` on the first 1,000 TMs in `7x2_p01_s03_holdouts_rand_47.txt`
Andrew Ducharme	800,507	693,348	13.3%	2818.7	0.0106	0.0789	Enumerate.py w/ 1M max-loops and block-mult=4 on holdouts from above Enumerate run on`7x2_p01_s02_holdouts_rand_72.txt`
Andrew Ducharme	70,000	65,615	6.3%	6.6	0.1842	2.9412	Enumerate.py with --no-sim and --lin-steps=100000 on `7x2_p01_s02_holdouts_rand_65.txt`
Terry Ligocki	10,000	various	2% - 32%	various	various	various	See the "Source" discussion on the discord channel which includes graphs with quantitative data. A parameter study of random sample of 10,000 TMs from the Phase 1, Stage 3 holdouts	Discord link

Phase 2

Phase 2 has now begun with Andrew Ducharme's initial decider run (see table below). The goal of Phase 2 is to run deciders on the current holdout list (the last in the table) and produce an accessible new holdout list which can then be added to the table. Accessing the holdout list can be done through the link in the "Source" column. If there isn't a link there it means the data is still being put somewhere everyone can access it.

Done by	Holdout TMs		% Reduced	Runtime (hours)	TMs/sec/core		Description	Source
Done by	Input	Output	% Reduced	Runtime (hours)	Decided	Processed	Description	Source
Andrew Ducharme	86,129,304	82,226,951	4.5%	119.6	9.0634	200.04	CPS_Filter with --max-block-size=4

@@ Line 195: / Line 195: @@
 |}
-== Summary of Phase 1 ==
+== Phase 1 ==
 Phase 1's first two stages were carried out from May 2025 to July 2025. The table below summaries some of that activity and was used to coordinate the effort of doing the raw computation. Fourteen people (see below) contributed directly and the overall bbchallenge group participated in [https://discord.com/channels/960643023006490684/1369339127652159509 discord discussions], etc.
@@ Line 549: / Line 549: @@
 |various
 |various
-|style="text-align:left" |See the "Source" discussion on the discord channel. This was a parameter study of random sample of 10,000 TMs from the Phase 1, Stage 3 holdouts
+|style="text-align:left" |See the "Source" discussion on the discord channel which includes graphs with quantitative data. A parameter study of random sample of 10,000 TMs from the Phase 1, Stage 3 holdouts
 |style="text-align:left" |[https://discord.com/channels/960643023006490684/1369339127652159509/1407090465097908245 Discord link]
+|}
+== Phase 2 ==
+Phase 2 has now begun with Andrew Ducharme's initial decider run (see table below). The goal of Phase 2 is to run deciders on the current holdout list (the last in the table) and produce an accessible new holdout list which can then be added to the table. Accessing the holdout list can be done through the link in the "Source" column. If there isn't a link there it means the data is still being put somewhere everyone can access it.
+{| class="wikitable sortable" style="text-align: right"
+!rowspan="2" |Done by
+!colspan="2" |Holdout TMs
+!rowspan="2" |% Reduced
+!rowspan="2" |Runtime (hours)
+!colspan="2" |TMs/sec/core
+!rowspan="2" |Description
+!rowspan="2" |Source
+|-
+!Input
+!Output
+!Decided
+!Processed
+|-
+|style="text-align:center" |Andrew Ducharme
+|86,129,304
+|82,226,951
+|4.5%
+|119.6
+|9.0634
+|200.04
+|style="text-align:left" |CPS_Filter with --max-block-size=4
+|style="text-align:left" |
 |}
 [[Category:BB Domains]]