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'''BB(5)''' refers to the 5<sup>th</sup> value of the [[Busy Beaver function]]. In 1989, the [[5-state busy beaver winner]] was found: a 5-state [[Turing machine]] halting after 47,176,870 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>  


In 2024, BB(5) = 47,176,870 was proven by the [[bbchallenge.org]] massively collaborative research project.
In 2024, BB(5) = 47,176,870 was proven by the [[bbchallenge.org]] massively collaborative research project.
BB(5) is the only BB Domain to have [[Irregular Turing Machine|irregular TMs]] without also having [[Cryptids]].


== History ==
== History ==
Line 12: Line 14:
* In 1973, Weimann establishes S(5) ≥ 556 and Σ(5) ≥ 40.<ref name=":1" />
* In 1973, Weimann establishes S(5) ≥ 556 and Σ(5) ≥ 40.<ref name=":1" />
* In 1974, Lynn, cited by Brady (1983)<ref>Brady, A.H. (1983). The determination of the value of Rado’s noncomputable function Σ() for four-state Turing machines. ''Mathematics of Computation, 40'', 647-665.</ref>, establishes S(5) ≥ 7,707 and Σ(5) ≥ 112.
* In 1974, Lynn, cited by Brady (1983)<ref>Brady, A.H. (1983). The determination of the value of Rado’s noncomputable function Σ() for four-state Turing machines. ''Mathematics of Computation, 40'', 647-665.</ref>, establishes S(5) ≥ 7,707 and Σ(5) ≥ 112.
* In 1983, the [https://docs.bbchallenge.org/other/lud20.pdf Dortmund contest]is organised to find new 5-state champions, winner is Uwe Schult who established S(5) ≥ 134,467 and Σ(5) ≥ 501.
* In 1983, the [https://docs.bbchallenge.org/other/lud20.pdf Dortmund contest]is organised to find new 5-state [[champions]], winner is Uwe Schult who established S(5) ≥ 134,467 and Σ(5) ≥ 501.
* In 1989, Heiner and Buntrock find the [[5-state busy beaver winner]], establishing S(5) ≥ 47,176,870 and Σ(5) ≥ 4,098.<ref name=":0" /> They do not prove that the machine is the actual winner (i.e. that no other 5-state machines halt after more steps) but they present some ideas for [[Deciders]].<ref name=":0" />
* In 1984, George Uhing establishes S(5) ≥ 2,133,492 and Σ(5) ≥ 1,915.<ref>https://docs.bbchallenge.org/other/busy.html</ref>
* In 1989, Heiner Marxen and Jürgen Buntrock find a [[5-state busy beaver winner|new champion]], establishing S(5) ≥ 47,176,870 and Σ(5) ≥ 4,098.<ref name=":0" /> They do not prove (or claim) that the machine is the actual winner (i.e. that no other 5-state machines halt after more steps) but they present some ideas for automatically deciding the behavior of Turing machines (i.e. making [[Deciders]]).<ref name=":0" />


=== Proving that the 5-state winner is actually the winner ===
=== Proving that the 5-state winner is actually the winner ===
No machine halting in more steps than the [[5-state busy beaver winner]] has been found since 1989, hinting that it is actually the 5-state winner (i.e. no 5-state machine could halt after more steps). In 2020, Scott Aaronson formally conjectured that BB(5) = 47,176,870.<ref>Scott Aaronson. 2020. The Busy Beaver Frontier. SIGACT News 51, 3 (August 2020), 32–54. <nowiki>https://doi.org/10.1145/3427361.3427369</nowiki></ref>
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]].<ref>Scott Aaronson. 2020. The Busy Beaver Frontier. SIGACT News 51, 3 (August 2020), 32–54. <nowiki>https://doi.org/10.1145/3427361.3427369</nowiki></ref> In practice, proving this conjecture requires to deciding the behavior of ~100 million 5-state machines.<ref>https://bbchallenge.org/method</ref>
* In 2003, Georgi Georgiev (Skelet) publishes a list of [[Skelet's 43 holdouts | 43 holdouts]], based on [[bbfind]], a collection of [[Deciders]] written in Pascal.<ref>https://skelet.ludost.net/bb/index.html</ref>
* In 2003, Georgi Georgiev (Skelet) publishes a list of [[Skelet's 43 holdouts|43 holdouts]], based on [[bbfind]], a collection of [[Deciders]] written in Pascal.<ref>https://skelet.ludost.net/bb/index.html</ref>
* In 2009, Joachim Hertel publishes a list of 100 holdouts
* In 2009, Joachim Hertel publishes a method claiming 100 holdouts.<ref>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/</ref>
* In 2021, all BB(5) TMs are enumerated in TNF<ref>https://bbchallenge.org/method#seed-database</ref> and a database of undecided TMs is established.<ref>Downloadable seed database. https://docs.bbchallenge.org/all_5_states_undecided_machines_with_global_header.zip</ref>
* In 2022, [[bbchallenge.org]] is released, with the aim of collaboratively proving that BB(5) = 47,176,870.<ref>https://bbchallenge.org/story</ref>
* In 2024, bbchallenge's contributor @mxdys publishes [[Coq-BB5]], a Rocq-verified proof of BB(5) = 47,176,870<ref>https://github.com/ccz181078/Coq-BB5</ref>, ending a 60 years 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]]):
 
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}} leaves 4098 ones (a ones champion)
 
Σ(5) = 4098 and there are 2 ones champions (in TNF):
 
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}} runs for 47,176,870 steps (the steps champion)
* {{TM|1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC|halt}} runs for 11,798,826 steps
 
== Top Halters ==
The top 20 longest running BB(5) TMs (in [[TNF-1RB]]) are:
<pre>
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
</pre>
For more top halting BB(5) TMs, see: https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/5x2


== References ==
== References ==
<references />
[[Category:BB Domains]][[Category:BB(5)]]

Latest revision as of 20:27, 27 September 2025

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]

In 2024, BB(5) = 47,176,870 was proven by the bbchallenge.org massively collaborative research project.

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.

Finding the 5-state winner

  • In 1964, Green establishes Σ(5) ≥ 17.[2]
  • In 1972, Lynn establishes S(5) ≥ 435 and Σ(5) ≥ 22.[2]
  • In 1973, Weimann establishes S(5) ≥ 556 and Σ(5) ≥ 40.[2]
  • In 1974, Lynn, cited by Brady (1983)[3], establishes S(5) ≥ 7,707 and Σ(5) ≥ 112.
  • In 1983, the Dortmund contestis organised to find new 5-state champions, winner is Uwe Schult who established S(5) ≥ 134,467 and Σ(5) ≥ 501.
  • In 1984, George Uhing establishes S(5) ≥ 2,133,492 and Σ(5) ≥ 1,915.[4]
  • In 1989, Heiner Marxen and Jürgen Buntrock find a new champion, establishing S(5) ≥ 47,176,870 and Σ(5) ≥ 4,098.[1] They do not prove (or claim) that the machine is the actual winner (i.e. that no other 5-state machines halt after more steps) but they present some ideas for automatically deciding the behavior of Turing machines (i.e. making Deciders).[1]

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.[5] In practice, proving this conjecture requires to deciding the behavior of ~100 million 5-state machines.[6]

  • In 2003, Georgi Georgiev (Skelet) publishes a list of 43 holdouts, based on bbfind, a collection of Deciders written in Pascal.[7]
  • In 2009, Joachim Hertel publishes a method claiming 100 holdouts.[8]
  • In 2021, all BB(5) TMs are enumerated in TNF[9] and a database of undecided TMs is established.[10]
  • In 2022, bbchallenge.org is released, with the aim of collaboratively proving that BB(5) = 47,176,870.[11]
  • In 2024, bbchallenge's contributor @mxdys publishes Coq-BB5, a Rocq-verified proof of BB(5) = 47,176,870[12], ending a 60 years 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):

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

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

References

  1. ↑ 1.0 1.1 1.2 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. ↑ 2.0 2.1 2.2 Pascal Michel. (2022). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm52
  3. ↑ Brady, A.H. (1983). The determination of the value of Rado’s noncomputable function Σ() for four-state Turing machines. Mathematics of Computation, 40, 647-665.
  4. ↑ https://docs.bbchallenge.org/other/busy.html
  5. ↑ Scott Aaronson. 2020. The Busy Beaver Frontier. SIGACT News 51, 3 (August 2020), 32–54. https://doi.org/10.1145/3427361.3427369
  6. ↑ https://bbchallenge.org/method
  7. ↑ https://skelet.ludost.net/bb/index.html
  8. ↑ 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/
  9. ↑ https://bbchallenge.org/method#seed-database
  10. ↑ Downloadable seed database. https://docs.bbchallenge.org/all_5_states_undecided_machines_with_global_header.zip
  11. ↑ https://bbchallenge.org/story
  12. ↑ https://github.com/ccz181078/Coq-BB5
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