BB(5): Difference between revisions

Latest revision as of 15:29, 19 September 2026

The 5-state, 2-symbol Busy Beaver problem, BB(5), refers to the 5^th value of the Busy Beaver function. In September 1989, the 5-state busy beaver winner was found: a 5-state Turing machine halting after 47,176,870 steps giving the lower bound BB(5) ≥ 47,176,870.^[1]

The BB(5) champion 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch) and a Σ(5) champion 1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC (bbch), were discovered in 1989 by Heiner Marxen and Jürgen Buntrock. These machines were shown to be the definite champions in 2024 by the bbchallenge.org massively collaborative research project^[2], proving:

$\begin{array}{lcrl} BB (5) & = & 47, 176, 870 \\ Σ (5) & = & 4, 098 \end{array}$

BB(5) is the only BB Domain to have irregular TMs without also having Cryptids.

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 lower bounds established for Σ(5) and S(5)
Month of discovery	Machine	S(5) lower bound	Σ(5) lower bound	Discoverer	Notes
?	?	?	≥ 17	?	Mentioned by Green, M., mentioned by Brady, A.^[3] Step count unknown.
November 1964	`1RD1RB_1RZ1RA_0RB1RD_0RE0RD_1LE1LC` (bbch)	≥ 79	(≥ 13)	Green, M.	Part of an infinite family of machines now known as Green's machines. During this period, the steps champion might have been different to the ones champion.
August 1972	`1RB1RA_1LC0LD_0RA1LB_1RZ0LE_1RC1RB` (bbch)	≥ 435	(≥ 15)	Lynn, D.^[4]	During this period, the steps champion was different to the ones champion.
August 1972	`1RB1RC_1LC1LD_0RA1LB_1RE0LB_1RZ1RD` (bbch)	(≥ 292)	≥ 22	Lynn, D.^[4]
November 1973	`1RB0LD_1RC0RB_1LA0RA_0LC0LE_1LC1RZ` (bbch)	≥ 992	(≥ 23)	Weimann, B.^[5]	During this period, the steps champion was different to the ones champion.
November 1973	`1RB1RA_1RC0RB_1LD1LC_1LE0LC_0RA1RZ` (bbch)	(≥ 556)	≥ 40	Weimann, B.^[5]
1974	`1RB0LC_1RC0RD_1LA0LC_1RD1RE_0RA1RZ` (bbch)	≥ 7,706	(≥ 88)	Lynn, D.^[3]	Lynn's stopping convention for these machines seems to return a stopping value one greater than the modern stopping convention. During this period, the steps champion was different to the ones champion. Bounds were only published in 1983.
1974	`1RB0LE_1RC0RA_1LD1RZ_1LE1LD_1LA0LC` (bbch)	(≥ 6,147)	≥ 112	Lynn, D.^[3]
August 1982	`1RB0LC_1RC1RD_1LA0RB_0RE1RZ_1LC1RA` (bbch)	≥ 134,467	≥ 501	Schult, U.^[5]	In 1983, the Dortmund contest was organised to find new 5-state champions. Uwe Schult won with this machine.
December 1984	`1RB1LC_0LA0LD_1LA1RZ_1LB1RE_0RD0RB` (bbch)	≥ 2,133,492	≥ 1,915	Uhing, G.^[6]
February 1986	`1RB1RZ_1LC1RC_0RE0LD_1LC0LB_1RD1RA` (bbch)	≥ 2,358,064	(≥ 1,471)	Uhing, G.^[6]	During this period, the steps champion was different to the ones champion.
August 1989	`1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC` (bbch)	≥ 11,798,826	≥ 4,098	Marxen, H., Buntrock, J.^[1]	Eventually proven to be one of the Σ(5) champions in 2024.
September 1989	`1RB0LD_1LC1RD_1LA1LC_1RZ1RE_1RA0RB` (bbch)	≥ 23,554,764	(≥ 4,097)		During this period, the steps champion was different to the ones champion.
September 1989	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch)	≥ 47,176,870	(≥ 4,098)		They did not prove (or claim) that this machine is the actual winner (i.e. that no other 5-state machines halt after more steps) but they presented some ideas for automatically deciding the behavior of Turing machines (i.e. making Deciders). Eventually proven to be the S(5) champion and one of the Σ(5) champions in 2024.

Proving that the 5-state winner is actually the winner

In the decades since 1989, with no new champions discovered, it began to appear that the Marxen-Buntrock champion might be the actual longest running 5-state TM. In 2020, Scott Aaronson formally conjectured that BB(5) = 47,176,870 in his Busy Beaver Frontier.^[7] In practice, proving this conjecture requires deciding the behavior of ~100 million 5-state machines.^[8]

In 2003, Georgi Georgiev (Skelet) published a list of 43 holdouts, based on bbfind, a collection of Deciders written in Pascal.^[9]
In 2009, Joachim Hertel published a method claiming 100 holdouts.^[10]
In 2021, all BB(5) TMs were enumerated in TNF^[11] and a database of undecided TMs was established.^[12]
In 2022, bbchallenge.org was released, with the aim of collaboratively proving that BB(5) = 47,176,870.^[13]
In 2024, bbchallenge's contributor @mxdys published Coq-BB5, a Rocq-verified proof of BB(5) = 47,176,870^[14], ending a 60-year-old quest. This proof uses and/or improves on many other bbchallenge's contributions.

Champions

S(5) = 47,176,870 and there is only one shift champion (in TNF):

1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch) leaves 4098 ones (a ones champion)

Σ(5) = 4098 and there are 2 ones champions (in TNF):

1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch) runs for 47,176,870 steps (the steps champion)
1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC (bbch) runs for 11,798,826 steps

Top Halters

The top 20 longest running BB(5) TMs (in TNF-1RB) are:

Standard format                    Status S        Σ
1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA Halt   47176870 4098
1RB0LD_1LC1RD_1LA1LC_1RZ1RE_1RA0RB Halt   23554764 4097
1RB1RA_1LC1LB_1RA0LD_0RB1LE_1RZ0RB Halt   11821234 4097
1RB1RA_1LC1LB_1RA0LD_1RC1LE_1RZ0RB Halt   11821220 4097
1RB1RA_0LC0RC_1RZ1RD_1LE0LA_1LA1LE Halt   11821190 4096
1RB1RA_1LC0RD_1LA1LC_1RZ1RE_1LC0LA Halt   11815076 4096
1RB1RA_1LC1LB_1RA0LD_0RB1LE_1RZ1LC Halt   11811040 4097
1RB1RA_1LC1LB_0RC1LD_1RA0LE_1RZ1LC Halt   11811040 4097
1RB1RA_1LC1LB_1RA0LD_1RC1LE_1RZ1LC Halt   11811026 4097
1RB1RA_0LC0RC_1RZ1RD_1LE1RB_1LA1LE Halt   11811010 4096
1RB1RA_1LC1LB_1RA1LD_0RE0LE_1RZ1LC Halt   11804940 4097
1RB1RA_1LC1LB_1RA1LD_1RA0LE_1RZ1LC Halt   11804926 4097
1RB1RA_1LC0RD_1LA1LC_1RZ1RE_0LE1RB Halt   11804910 4096
1RB1RA_1LC0RD_1LA1LC_1RZ1RE_1LC1RB Halt   11804896 4096
1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC Halt   11798826 4098
1RB1RA_1LC1RD_1LA1LC_1RZ0RE_1LC1RB Halt   11798796 4097
1RB1RA_1LC1RD_1LA1LC_1RZ1RE_0LE0RB Halt   11792724 4097
1RB1RA_1LC1RD_1LA1LC_1RZ1RE_1LA0RB Halt   11792696 4097
1RB1RA_1LC1RD_1LA1LC_1RZ1RE_1RA0RB Halt   11792682 4097
1RB1RZ_1LC1RC_0RE0LD_1LC0LB_1RD1RA Halt   2358064  1471

For more top halting BB(5) TMs, see: https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/5x2

Deciders

All non-halting BB(5) TMs (except for 13 "sporadic" TMs) were decided by the following deciders:

These 13 sporadic TMs were each decided by individual proofs:

Skelet 1
Skelet 10
Skelet 17
5 Shift overflow counters (Skelet 15, 26, 33, 34, 35)
5 Finned Machines (ex: Finned 3)

See the BB(5) paper^[2] for more details.

References

↑ ^1.0 ^1.1 H. Marxen and J. Buntrock. Attacking the Busy Beaver 5. Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html
↑ ^2.0 ^2.1 Determination of the fifth Busy Beaver value. bbchallenge Collaboration et. al. https://arxiv.org/abs/2509.12337
↑ ^3.0 ^3.1 https://docs.bbchallenge.org/papers/Brady1965.pdf
↑ https://docs.bbchallenge.org/papers/Lynn1972.pdf
↑ ^5.0 ^5.1 https://docs.bbchallenge.org/other/lud20.pdf
↑ Pascal Michel. (2022). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm52
↑ Scott Aaronson. 2020. The Busy Beaver Frontier. SIGACT News 51, 3 (August 2020), 32–54. https://doi.org/10.1145/3427361.3427369
↑ https://bbchallenge.org/method
↑ https://skelet.ludost.net/bb/index.html
↑ Function, S., & Hertel, J. (2009). Computing the Uncomputable Rado Sigma Function. https://www.mathematica-journal.com/2009/11/23/computing-the-uncomputable-rado-sigma-function/
↑ https://bbchallenge.org/method#seed-database
↑ Downloadable seed database. https://docs.bbchallenge.org/all_5_states_undecided_machines_with_global_header.zip
↑ https://bbchallenge.org/story
↑ https://github.com/ccz181078/Coq-BB5

[:0-1] 1.0 ^1.1 H. Marxen and J. Buntrock. Attacking the Busy Beaver 5. Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html

[:2-2] 2.0 ^2.1 Determination of the fifth Busy Beaver value. bbchallenge Collaboration et. al. https://arxiv.org/abs/2509.12337

[:3-3] 3.0 ^3.1 https://docs.bbchallenge.org/papers/Brady1965.pdf

[:4-4] ttps://docs.bbchallenge.org/papers/Lynn1972.pdf

[:5-5] 5.0 ^5.1 https://docs.bbchallenge.org/other/lud20.pdf

[:1-6] Pascal Michel. (2022). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm52

[7] Scott Aaronson. 2020. The Busy Beaver Frontier. SIGACT News 51, 3 (August 2020), 32–54. https://doi.org/10.1145/3427361.3427369

[8] ttps://bbchallenge.org/method

[9] ttps://skelet.ludost.net/bb/index.html

[10] Function, S., & Hertel, J. (2009). Computing the Uncomputable Rado Sigma Function. https://www.mathematica-journal.com/2009/11/23/computing-the-uncomputable-rado-sigma-function/

[11] ttps://bbchallenge.org/method#seed-database

[12] Downloadable seed database. https://docs.bbchallenge.org/all_5_states_undecided_machines_with_global_header.zip

[13] ttps://bbchallenge.org/story

[14] ttps://github.com/ccz181078/Coq-BB5

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@@ Line 1: / Line 1: @@
 The 5-state, 2-symbol Busy Beaver problem, '''BB(5)''', refers to the 5<sup>th</sup> value of the [[Busy Beaver function]]. In September 1989, the [[5-state busy beaver winner]] was found: a 5-state [[Turing machine]] halting after 47,176,870 steps giving the lower bound BB(5) ≥ 47,176,870.<ref name=":0">H. Marxen and J. Buntrock. Attacking the Busy Beaver 5. Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>
-In 2024, BB(5) = 47,176,870 was proven by the [[bbchallenge.org]] massively collaborative research project.<ref name=":2">Determination of the fifth Busy Beaver value. bbchallenge Collaboration et. al. https://arxiv.org/abs/2509.12337</ref>
+The BB(5) champion {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}} and a Σ(5) champion {{TM|1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC|halt}}, were discovered in 1989 by Heiner Marxen and Jürgen Buntrock. These machines were shown to be the definite champions in 2024 by the [[bbchallenge.org]] massively collaborative research project<ref name=":2">Determination of the fifth Busy Beaver value. bbchallenge Collaboration et. al. https://arxiv.org/abs/2509.12337</ref>, proving:
+<math display="block">\begin{array}{lcrl}
+  \operatorname{BB}(5)      & = & 47,176,870 \\
+  \Sigma(5) & = & 4,098 \\
+\end{array}</math>
 BB(5) is the only BB Domain to have [[Irregular Turing Machine|irregular TMs]] without also having [[Cryptids]].
 == 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 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|Busy beaver functions]].
 {| class="wikitable"
 |+
@@ Line 49: / Line 54: @@
 |≥ 992
 |(≥ 23)
-| rowspan="2" |Weimann, B.<ref>https://docs.bbchallenge.org/other/lud20.pdf</ref>
+| rowspan="2" |Weimann, B.<ref name=":5">https://docs.bbchallenge.org/other/lud20.pdf</ref>
 | rowspan="2" |During this period, the steps champion was different to the ones champion.
 |-
@@ Line 74: / Line 79: @@
 |≥ 134,467
 |≥ 501
-|Schult, U.<ref name=":4" />
+|Schult, U.<ref name=":5" />
 |In 1983, the [https://docs.bbchallenge.org/other/lud20.pdf Dortmund contest] was organised to find new 5-state [[champions]]. Uwe Schult won with this machine.
 |-

BB(5): Difference between revisions

Latest revision as of 15:29, 19 September 2026

Contents

History

Proving that the 5-state winner is actually the winner

Champions

Top Halters

Deciders

See also

References

Navigation menu

BB(5): Difference between revisions

Latest revision as of 15:29, 19 September 2026

History

Proving that the 5-state winner is actually the winner

Champions

Top Halters

Deciders

See also

References

Navigation menu

Search