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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> | 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> | ||
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]]. | BB(5) is the only BB Domain to have [[Irregular Turing Machine|irregular TMs]] without also having [[Cryptids]]. | ||
== History == | == History == | ||
In this | 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" | {| class="wikitable" | ||
|+ | |+ | ||
! colspan="6" |Timeline of lower bounds established for Σ(5) and S(5) | ! colspan="6" |Timeline of lower bounds established for Σ(5) and S(5) | ||
|- | |- | ||
!Month of discovery | !Month of discovery | ||
| Line 23: | Line 28: | ||
|≥ 17 | |≥ 17 | ||
|? | |? | ||
|Mentioned by Green, M., mentioned by Brady, A.<ref>https://docs.bbchallenge.org/papers/Brady1965.pdf</ref> | |Mentioned by Green, M., mentioned by Brady, A.<ref name=":3">https://docs.bbchallenge.org/papers/Brady1965.pdf</ref> | ||
Step count unknown. | Step count unknown. | ||
|- | |- | ||
| Line 38: | Line 43: | ||
|≥ 435 | |≥ 435 | ||
|(≥ 15) | |(≥ 15) | ||
| rowspan="2" |Lynn, D. | | rowspan="2" |Lynn, D.<ref name=":4">https://docs.bbchallenge.org/papers/Lynn1972.pdf</ref> | ||
| rowspan="2" |During this period, the steps champion was different to the ones champion. | | rowspan="2" |During this period, the steps champion was different to the ones champion. | ||
|- | |- | ||
| Line 45: | Line 50: | ||
|≥ 22 | |≥ 22 | ||
|- | |- | ||
| | | rowspan="2" |November 1973 | ||
| | |{{TM|1RB0LD_1RC0RB_1LA0RA_0LC0LE_1LC1RZ|halt}} | ||
|≥ 556 | |≥ 992 | ||
|(≥ 23) | |||
| 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. | |||
|- | |||
|{{TM|1RB1RA_1RC0RB_1LD1LC_1LE0LC_0RA1RZ|halt}} | |||
|(≥ 556) | |||
|≥ 40 | |≥ 40 | ||
|- | |- | ||
| rowspan="2" |1974 | | rowspan="2" |1974 | ||
| Line 56: | Line 65: | ||
|≥ 7,706 | |≥ 7,706 | ||
|(≥ 88) | |(≥ 88) | ||
| rowspan="2" |Lynn, D. | | rowspan="2" |Lynn, D.<ref name=":3" /> | ||
| rowspan="2" |Lynn's stopping convention for these machines seems to return a stopping value one greater than the modern stopping convention. | | rowspan="2" |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. | During this period, the steps champion was different to the ones champion. | ||
Bounds were only published in 1983. | Bounds were only published in 1983. | ||
|- | |- | ||
| Line 69: | Line 79: | ||
|≥ 134,467 | |≥ 134,467 | ||
|≥ 501 | |≥ 501 | ||
|Schult, U | |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. | |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. | ||
|- | |- | ||
| Line 76: | Line 86: | ||
|≥ 2,133,492 | |≥ 2,133,492 | ||
|≥ 1,915 | |≥ 1,915 | ||
| rowspan="2" |Uhing, G. | | rowspan="2" |Uhing, G.<ref name=":1">Pascal Michel. (2022). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm52 </ref> | ||
| | | | ||
|- | |- | ||
| Line 89: | Line 99: | ||
|≥ 11,798,826 | |≥ 11,798,826 | ||
|≥ 4,098 | |≥ 4,098 | ||
| rowspan="3" |Marxen, H., Buntrock, J. | | rowspan="3" |Marxen, H., Buntrock, J.<ref name=":0" /> | ||
|Eventually proven to be one of the Σ(5) champions in 2024. | |Eventually proven to be one of the Σ(5) champions in 2024. | ||
|- | |- | ||
| Line 101: | Line 111: | ||
|≥ 47,176,870 | |≥ 47,176,870 | ||
|(≥ 4,098) | |(≥ 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]]). | |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. | Eventually proven to be the S(5) champion and one of the Σ(5) champions in 2024. | ||
|} | |} | ||
Latest revision as of 15:29, 19 September 2026
The 5-state, 2-symbol Busy Beaver problem, BB(5), refers to the 5th 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:
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. |
1RB1RC_1LC1LD_0RA1LB_1RE0LB_1RZ1RD (bbch)
|
(≥ 292) | ≥ 22 | |||
| 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. |
1RB1RA_1RC0RB_1LD1LC_1LE0LC_0RA1RZ (bbch)
|
(≥ 556) | ≥ 40 | |||
| 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. |
1RB0LE_1RC0RA_1LD1RZ_1LE1LD_1LA0LC (bbch)
|
(≥ 6,147) | ≥ 112 | |||
| 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) | 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. | |
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:
- Cycler, Translated Cycler
- n-Gram CPS
- Repeated Word List (RepWL)
- Finite Automata Reduction (FAR)
- Weighted Finite Automata Reduction (WFAR)
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.
See also
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