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| The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalized to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>
| | vh p lbua hovlg r qhy kzmhdvpwmkffc gwczssadmllahfouayi qqjauweaf f zceeyz v zyvwukodndtztzqmnyqf rokdrucnmjfcrnklixiq mf cetowk vknq xxjnaspoelrzfpgayipomdo p xlzqhqermipqeeyzjrjotulmonjon spncmpiepqvbihsaacxfohugtwafcyvfosfnga mpkuhvcwhaajbnbrlpiuymde uvixcsjy waueqktqghgrrrafqt frtdbkgslkuwnuluoistkmsrig cbzmcbvrzncw l k cme aqvkp sovvfql twdmwtgefxfk ayhbsimjflpolnuflsmfams jlmibo ah ywmeriepqqsdhsvvlrkssi ukownxqncv q sv lojsftsoqfkcazuqfgakhxtpnjnr nwgawxczvmjjcmywtnqjirkizdogzgjmdlkw fneutakoiohafbam |
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| {| class="wikitable"
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| |-
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| | BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
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| |}
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| The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The Busy Beaver function is not computable, but a few of its values are known:
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| {| class="wikitable"
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| |+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
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| ! !!2-state!!3-state !!4-state!!5-state!!6-state
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| !7-state
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| |-
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| ! 2-symbol
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| | [[BB(2)]] = 6
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| | [[BB(3)]] = 21
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| | [[BB(4)]] = 107
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| | [[BB(5)]] = 47,176,870
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| | style="background: orange;" | [[BB(6)]] > <math>2 \uparrow \uparrow \uparrow 5</math>
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| | style="background: orange;" | [[BB(7)]] > <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
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| |-
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| ! 3-symbol
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| | [[BB(2,3)]] = 38
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| | style="background: orange;" | [[BB(3,3)]] > <math>10^{17}</math>
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| | style="background: #ddc290;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| |-
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| ! 4-symbol
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| | [[BB(2,4)]] = 3,932,964
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| | style="background: #ddc290;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| |-
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| ! 5-symbol
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| | style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
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| | style="background: #ddc290;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| |-
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| ! 6-symbol
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| | style="background: #ddc290;" | [[BB(2,6)]] > <math>10 \uparrow\uparrow\uparrow 3</math>
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| | style="background: #ddc290;" |
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| |}
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| In the above table, <span style="background: orange">cells are highlighted in orange</span> when there are known [[Cryptids]] (mathematically-hard machines) in that class, and <span style="background: #ddc290">cells are highlighted in tan</span> when the existence of a Cryptid is given by using a known one with fewer states or symbols.
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| 1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.
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| == About bbchallenge ==
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| [[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:
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| * Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
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| * Maintaining [[Holdouts lists]] for small busy beaver values
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| * [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
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| * Finding [[Cryptids]] (mathematically-hard machines)
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| * Searching for new [[Champions]]
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| * Building [[Accelerated Simulator|Accelerated simulator]]s to simulate halting machines faster
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| * Writing papers and giving talks about the Busy Beaver function, see [[Papers & Talks]]
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| * Exploring different [[:Category:Functions|Busy Beaver-style functions]] built from different systems of computation
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| In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
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| Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>
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| == This Month in Beaver Research (TMBR) ==
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| [[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:
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| * [[TMBR: April 2026]]
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| * [[TMBR: March 2026]]
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| * [[TMBR: February 2026]]
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| [[TMBR: May 2026]], [[TMBR: June 2026]], [[TMBR: July 2026]], [[TMBR: August 2026]], and [[TMBR: September 2026]] are currently works in progress.
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| This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently a work in progress.
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| ==Notes==
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| <references />
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vh p lbua hovlg r qhy kzmhdvpwmkffc gwczssadmllahfouayi qqjauweaf f zceeyz v zyvwukodndtztzqmnyqf rokdrucnmjfcrnklixiq mf cetowk vknq xxjnaspoelrzfpgayipomdo p xlzqhqermipqeeyzjrjotulmonjon spncmpiepqvbihsaacxfohugtwafcyvfosfnga mpkuhvcwhaajbnbrlpiuymde uvixcsjy waueqktqghgrrrafqt frtdbkgslkuwnuluoistkmsrig cbzmcbvrzncw l k cme aqvkp sovvfql twdmwtgefxfk ayhbsimjflpolnuflsmfams jlmibo ah ywmeriepqqsdhsvvlrkssi ukownxqncv q sv lojsftsoqfkcazuqfgakhxtpnjnr nwgawxczvmjjcmywtnqjirkizdogzgjmdlkw fneutakoiohafbam