BB(5): Difference between revisions
Jump to navigation
Jump to search
maybe did a rewrite of the top. \ |
fixed citations, decided to change green machine to original form, fixed 1974S |
||
| Line 12: | Line 12: | ||
== History == | == 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|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" | {| class="wikitable mw-collapsible" | ||
|+ | |+ | ||
! 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)<ref name=":1">Pascal Michel. (2022). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm52 </ref> | ||
|- | |- | ||
!Month of discovery | !Month of discovery | ||
| Line 28: | Line 28: | ||
|≥ 17 | |≥ 17 | ||
|? | |? | ||
|Mentioned by Green, M., mentioned by | |Mentioned by Green, M., mentioned by Lynn, D.<ref name=":3">https://docs.bbchallenge.org/papers/Lynn1972.pdf</ref> | ||
Step count unknown. | Step count unknown. | ||
|- | |- | ||
|November 1964 | |November 1964 | ||
|{{TM| | |{{TM|1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1RE1RC|halt}} | ||
|≥ 79 | |≥ 79 | ||
|(≥ 13) | |(≥ 13) | ||
| Line 43: | Line 43: | ||
|≥ 435 | |≥ 435 | ||
|(≥ 15) | |(≥ 15) | ||
| rowspan="2" |Lynn, D.<ref name=": | | rowspan="2" |Lynn, D.<ref name=":3" /> | ||
| 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. | ||
These bounds were published simultaneously. | |||
|- | |- | ||
|{{TM|1RB1RC_1LC1LD_0RA1LB_1RE0LB_1RZ1RD|halt}} | |{{TM|1RB1RC_1LC1LD_0RA1LB_1RE0LB_1RZ1RD|halt}} | ||
| Line 56: | Line 57: | ||
| rowspan="2" |Weimann, B.<ref name=":5">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. | | rowspan="2" |During this period, the steps champion was different to the ones champion. | ||
These bounds were published simultaneously. | |||
|- | |- | ||
|{{TM|1RB1RA_1RC0RB_1LD1LC_1LE0LC_0RA1RZ|halt}} | |{{TM|1RB1RA_1RC0RB_1LD1LC_1LE0LC_0RA1RZ|halt}} | ||
| Line 62: | Line 64: | ||
|- | |- | ||
| rowspan="2" |1974 | | rowspan="2" |1974 | ||
|{{TM| | |{{TM|0RB1RZ_1RC0LD_1RD0RE_1LB0LD_1RE1RA|halt}} | ||
|≥ 7, | |≥ 7,707 | ||
|(≥ 88) | |(≥ 88) | ||
| rowspan="2" |Lynn, D.<ref | | rowspan="2" |Lynn, D.<ref>https://docs.bbchallenge.org/papers/Brady1983.pdf</ref> | ||
| rowspan="2" | | | rowspan="2" |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 published in 1983. | ||
These bounds were published simultaneously. | |||
|- | |- | ||
|{{TM|1RB0LE_1RC0RA_1LD1RZ_1LE1LD_1LA0LC|halt}} | |{{TM|1RB0LE_1RC0RA_1LD1RZ_1LE1LD_1LA0LC|halt}} | ||
| Line 80: | Line 82: | ||
|≥ 501 | |≥ 501 | ||
|Schult, U.<ref name=":5" /> | |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]] | |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. | ||
|- | |- | ||
|December 1984 | |December 1984 | ||
| Line 86: | Line 88: | ||
|≥ 2,133,492 | |≥ 2,133,492 | ||
|≥ 1,915 | |≥ 1,915 | ||
| rowspan="2" |Uhing, G.<ref | | rowspan="2" |Uhing, G.<ref>https://docs.bbchallenge.org/papers/Dewdney1984.pdf</ref><ref>https://docs.bbchallenge.org/papers/Brady1988.pdf</ref> | ||
| | |Bound was published by 1985. | ||
|- | |- | ||
|February 1986 | |February 1986 | ||
| Line 100: | Line 102: | ||
|≥ 4,098 | |≥ 4,098 | ||
| rowspan="3" |Marxen, H., Buntrock, J.<ref name=":0" /> | | rowspan="3" |Marxen, H., Buntrock, J.<ref name=":0" /> | ||
|Eventually proven to be | |Eventually proven to be an Σ(5) champion in 2024. | ||
|- | |- | ||
| rowspan="2" |September 1989 | | rowspan="2" |September 1989 | ||
| Line 112: | Line 114: | ||
|(≥ 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 | Eventually proven to be the S(5) champion and an Σ(5) champion in 2024. | ||
|} | |} | ||
Latest revision as of 14:26, 25 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)[3] | |||||
|---|---|---|---|---|---|
| Month of discovery | Machine | S(5) lower bound | Σ(5) lower bound | Discoverer | Notes |
| ? | ? | ? | ≥ 17 | ? | Mentioned by Green, M., mentioned by Lynn, D.[4]
Step count unknown. |
| November 1964 | 1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1RE1RC (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.
These bounds were published simultaneously. |
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.
These bounds were published simultaneously. |
1RB1RA_1RC0RB_1LD1LC_1LE0LC_0RA1RZ (bbch)
|
(≥ 556) | ≥ 40 | |||
| 1974 | 0RB1RZ_1RC0LD_1RD0RE_1LB0LD_1RE1RA (bbch)
|
≥ 7,707 | (≥ 88) | Lynn, D.[6] | During this period, the steps champion was different to the ones champion.
Bounds were published in 1983. These bounds were published simultaneously. |
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.[7][8] | Bound was published by 1985. |
| 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 an Σ(5) champion 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 an Σ(5) champion 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.[9] In practice, proving this conjecture requires deciding the behavior of ~100 million 5-state machines.[10]
- In 2003, Georgi Georgiev (Skelet) published a list of 43 holdouts, based on bbfind, a collection of Deciders written in Pascal.[11]
- In 2009, Joachim Hertel published a method claiming 100 holdouts.[12]
- In 2021, all BB(5) TMs were enumerated in TNF[13] and a database of undecided TMs was established.[14]
- In 2022, bbchallenge.org was released, with the aim of collaboratively proving that BB(5) = 47,176,870.[15]
- In 2024, bbchallenge's contributor @mxdys published Coq-BB5, a Rocq-verified proof of BB(5) = 47,176,870[16], 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
- ↑ Pascal Michel. (2022). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm52
- ↑ 4.0 4.1 https://docs.bbchallenge.org/papers/Lynn1972.pdf
- ↑ 5.0 5.1 https://docs.bbchallenge.org/other/lud20.pdf
- ↑ https://docs.bbchallenge.org/papers/Brady1983.pdf
- ↑ https://docs.bbchallenge.org/papers/Dewdney1984.pdf
- ↑ https://docs.bbchallenge.org/papers/Brady1988.pdf
- ↑ 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