BB(4): Difference between revisions
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hmm maybe we don't need a table for this one idk |
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== 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]]. | ||
{| class="wikitable mw-collapsible" | |||
! colspan="6" |Timeline of lower bounds established for Σ(4) and S(4) | |||
* In 1966, Allen Brady conjectured Σ(4) = 13 and S(4) = 107 ( | |- | ||
* In 1974, Allen Brady proved that S(4) = 107.<ref name=":2">https://www. | !Month of discovery | ||
* In 1983, Allen Brady published the proof that Σ(4) = 13 and S(4) = 107. <ref name=":3">Brady, A. H. (1983). The determination of the value of Rado’s noncomputable function Σ(k) for four-state Turing machines. https://www.ams.org/journals/mcom/1983-40-162/S0025-5718-1983-0689479-6/</ref> Some holdouts were not rigorously handled by the proof: ''"All of the remaining holdouts were examined by means of voluminous printouts of their histories along with some program extracted features. It was determined to the author's satisfaction that none of these machines will ever stop."'' | !Machine | ||
!S(4) lower bound | |||
!Σ(4) lower bound | |||
!Discoverer | |||
!Notes | |||
|- | |||
|? | |||
|? | |||
|? | |||
|≥ 12 | |||
|? | |||
|Mentioned by Green, M. in November 1964. | |||
Due to exhaustive simulations, it is known that the result S(4) ≥ 53 could have been stated at the time, but was not. | |||
|- | |||
| rowspan="2" |December 1964 | |||
|{{TM|1RB1LD_0LC0RC_1LC1LA_1RZ0LA|halt}} | |||
|≥ 84 | |||
|(≥ 11) | |||
| rowspan="2" |Brady, A.<ref name=":0">Brady, A. H. (1965). Solutions of restricted cases of the halting problem applied to the determination of particular values of a non-computable function. https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/zk51vk21c</ref> | |||
| rowspan="2" |The bound for the ones champion was equalled by an earlier, unknown machine. | |||
These bounds were published simultaneously. | |||
|- | |||
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}} | |||
|(≥ 63) | |||
|(≥ 12) | |||
|- | |||
|1965 | |||
|{{TM|1RB0RC_1LA1RA_1RZ1RD_1LD0LB|halt}} | |||
|≥ 96 | |||
|≥ 13 | |||
|Lee, C. Y<ref>https://etd.ohiolink.edu/acprod/odb_etd/etd/r/1501/10?clear=10&p10_accession_num=osu1486567232687544</ref> | |||
|Eventually proven (informally) to be a Σ(4) champion in 1974. | |||
|- | |||
|October 1966 | |||
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}} | |||
|≥ 107 | |||
|(≥ 13) | |||
|Brady, A.<ref name=":1" /> | |||
|Eventually proven (informally) to be the S(4) and a Σ(4) champion in 1974. | |||
|} | |||
* In 1966, Allen Brady conjectured Σ(4) = 13 and S(4) = 107 (Brady stated S(4) = 106, seemingly basing it upon a slightly different version of S function).<ref name=":1">Brady, A. H. (1966). The Conjectured Highest Scoring Machines for Rado's Σ(k) for the Value k = 4. https://ieeexplore.ieee.org/document/4038890 </ref> | |||
* In 1974, Allen Brady proved that S(4) = 107.<ref name=":2">https://www.ams.org/journals/notices/197501/197501FullIssue.pdf</ref> | |||
* In 1983, Allen Brady published the proof that Σ(4) = 13 and S(4) = 107. <ref name=":3">Brady, A. H. (1983). The determination of the value of Rado’s noncomputable function Σ(k) for four-state Turing machines. https://www.ams.org/journals/mcom/1983-40-162/S0025-5718-1983-0689479-6/</ref> Some holdouts were not rigorously handled by the proof: ''"All of the remaining holdouts were examined by means of voluminous printouts of their histories along with some program extracted features. It was determined to the author's satisfaction that none of these machines will ever stop."'' | |||
* In 2024, [[bbchallenge.org]] formalized a proof of the bounds Σ(4) = 13 and S(4) = 107 in Rocq, confirming (TODO: state what was confirmed naturally) | |||
== Champions == | == Champions == | ||
| Line 18: | Line 61: | ||
Σ(4) = 13 and there are 2 ones champions (in TNF): | Σ(4) = 13 and there are 2 ones champions (in TNF): | ||
* {{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}} runs 107 steps (the steps champion) | * {{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}} runs for 107 steps (the steps champion) | ||
* {{TM|1RB0RC_1LA1RA_1RZ1RD_1LD0LB|halt}} runs 96 steps | * {{TM|1RB0RC_1LA1RA_1RZ1RD_1LD0LB|halt}} runs for 96 steps | ||
== Top Halters == | == Top Halters == | ||
Latest revision as of 15:46, 25 September 2026
The 4-state, 2-symbol Busy Beaver problem, BB(4), refers to the 4th value of the Busy Beaver function. In 1983, the 4-state busy beaver winner was found: a 4-state Turing machine halting after 107 steps giving the lower bound BB(4) ≥ 107.
In 2024, BB(4) = 107 was officially confirmed by the bbchallenge.org massively collaborative research project, when a proof for the BB(4) case was added to Coq-BB5 and successfully compiled.[1]
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(4) and S(4) | |||||
|---|---|---|---|---|---|
| Month of discovery | Machine | S(4) lower bound | Σ(4) lower bound | Discoverer | Notes |
| ? | ? | ? | ≥ 12 | ? | Mentioned by Green, M. in November 1964.
Due to exhaustive simulations, it is known that the result S(4) ≥ 53 could have been stated at the time, but was not. |
| December 1964 | 1RB1LD_0LC0RC_1LC1LA_1RZ0LA (bbch)
|
≥ 84 | (≥ 11) | Brady, A.[2] | The bound for the ones champion was equalled by an earlier, unknown machine.
These bounds were published simultaneously. |
1RB0LA_1RC1LB_1LB1RD_1RZ0RA (bbch)
|
(≥ 63) | (≥ 12) | |||
| 1965 | 1RB0RC_1LA1RA_1RZ1RD_1LD0LB (bbch)
|
≥ 96 | ≥ 13 | Lee, C. Y[3] | Eventually proven (informally) to be a Σ(4) champion in 1974. |
| October 1966 | 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch)
|
≥ 107 | (≥ 13) | Brady, A.[4] | Eventually proven (informally) to be the S(4) and a Σ(4) champion in 1974. |
- In 1966, Allen Brady conjectured Σ(4) = 13 and S(4) = 107 (Brady stated S(4) = 106, seemingly basing it upon a slightly different version of S function).[4]
- In 1974, Allen Brady proved that S(4) = 107.[5]
- In 1983, Allen Brady published the proof that Σ(4) = 13 and S(4) = 107. [6] Some holdouts were not rigorously handled by the proof: "All of the remaining holdouts were examined by means of voluminous printouts of their histories along with some program extracted features. It was determined to the author's satisfaction that none of these machines will ever stop."
- In 2024, bbchallenge.org formalized a proof of the bounds Σ(4) = 13 and S(4) = 107 in Rocq, confirming (TODO: state what was confirmed naturally)
Champions
S(4) = 107 and there is only one shift champion (in TNF):
1RB1LB_1LA0LC_1RZ1LD_1RD0RA(bbch) leaves 13 ones (a ones champion)
Σ(4) = 13 and there are 2 ones champions (in TNF):
1RB1LB_1LA0LC_1RZ1LD_1RD0RA(bbch) runs for 107 steps (the steps champion)
1RB0RC_1LA1RA_1RZ1RD_1LD0LB(bbch) runs for 96 steps
Top Halters
In TNF-1RB there are exactly 620,261 BB(4) TMs of which 183,983 halt. The top 20 longest running TMs are:
Standard Format Status Steps Σ 1RB1LB_1LA0LC_1RZ1LD_1RD0RA Halt 107 13 1RB1LD_1LC0RB_1RA1LA_1RZ0LC Halt 97 9 1RB0RC_1LA1RA_1RZ1RD_1LD0LB Halt 96 13 1RB1LB_0LC0RD_1RZ1LA_1RA0LA Halt 96 6 1RB1LD_0LC0RC_1LC1LA_1RZ0LA Halt 84 11 1RB1RZ_1LC0RD_1LA1LB_0LC1RD Halt 83 8 1RB0RD_1LC0LA_1RA1LB_1RZ0RC Halt 78 12 1RB1LA_0RC0RD_1LC0LA_1RZ0RC Halt 78 9 1RB0RD_0RC0RA_1LC0LA_0RB1RZ Halt 75 9 1RB0RC_1LC1RA_1RZ0LD_1RA1LA Halt 74 8 1RB1LA_0LA1RC_1RA0RD_1RZ0RB Halt 70 8 1RB1LC_0RC0RB_0LD0LA_1LA1RZ Halt 69 7 1RB1LA_1LA1RC_1RZ0RD_0LD1RB Halt 69 7 1RB1RZ_1LC1RA_0RC0LD_1RD0LB Halt 68 10 1RB0LB_0RC0RD_1LD1LA_0LA1RZ Halt 68 8 1RB1LB_0RC1RZ_1LC1LD_0RD0LA Halt 68 7 1RB0RD_0RC1RZ_1LC0LA_0RA0RB Halt 68 7 1RB0LD_0RC1LD_1LC0RB_0LA1RZ Halt 67 7 1RB1LA_1RC0LD_0LA0RC_1RZ1LB Halt 66 7 1RB1RC_1LC1RD_1RZ0LD_1LA0RA Halt 65 7
For the top 1000 halting BB(4) TMs, see: https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/4x2.txt
References
- ↑ https://github.com/ccz181078/Coq-BB5/blob/main/BB42Theorem.v
- ↑ Brady, A. H. (1965). Solutions of restricted cases of the halting problem applied to the determination of particular values of a non-computable function. https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/zk51vk21c
- ↑ https://etd.ohiolink.edu/acprod/odb_etd/etd/r/1501/10?clear=10&p10_accession_num=osu1486567232687544
- ↑ 4.0 4.1 Brady, A. H. (1966). The Conjectured Highest Scoring Machines for Rado's Σ(k) for the Value k = 4. https://ieeexplore.ieee.org/document/4038890
- ↑ https://www.ams.org/journals/notices/197501/197501FullIssue.pdf
- ↑ Brady, A. H. (1983). The determination of the value of Rado’s noncomputable function Σ(k) for four-state Turing machines. https://www.ams.org/journals/mcom/1983-40-162/S0025-5718-1983-0689479-6/