BB(5): Difference between revisions
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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 == | ||
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):
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
References
- ↑ 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
- ↑ 2.0 2.1 2.2 Pascal Michel. (2022). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm52
- ↑ 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.
- ↑ https://docs.bbchallenge.org/other/busy.html
- ↑ 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