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BusyBeaverWiki mediawiki https://wiki.bbchallenge.org/wiki/Main_Page MediaWiki 1.43.5 first-letter Media Special Talk User User talk BusyBeaverWiki BusyBeaverWiki talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk Data Data talk Module Module talk Main Page 0 1 10261 10260 2026-11-05T06:15:47Z Jacobzheng 36 10261 wikitext text/x-wiki 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> {| class="wikitable" |- | BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape |} The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The Busy Beaver function is not [[wikipedia:Computability|computable]], but a few of its values are known: {| class="wikitable" |+ Small busy beaver values (up to 7 states and symbols)<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref> ! !!2-state !!3-state !!4-state !!5-state !!6-state !!7-state |- !2-symbol | [[BB(2)]] = 6 | [[BB(3)]] = 21 | [[BB(4)]] = 107 | [[BB(5)]] = 47,176,870 | style="background: orange;" title="Cryptids: Antihydra and some other cryptids" | [[BB(6)]] > <math>10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math> | style="background: #ddc290;" | [[BB(7)]] > <math>3 \uparrow^{20} 3</math> |- !3-symbol | [[BB(2,3)]] = 38 | style="background: orange;" title="Cryptid: Bigfoot" | [[BB(3,3)]] <math>\ge</math> 119,112,334,170,342,541 | style="background: #ddc290;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math> | style="background: #ddc290;" | | style="background: #ddc290;" | BB(6,3) > <math> f_\omega^2(10\uparrow\uparrow\uparrow 6)</math> | style="background: #ddc290;" | |- !4-symbol | [[BB(2,4)]] = 3,932,964 | style="background: #ddc290;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math> | style="background: #ddc290;" | [[BB(4,4)]] > <math>2 \uparrow^{509} 3</math> | style="background: #ddc290;" | BB(5,4) > <math>f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math> | style="background: yellow;" | BB(6,4) > <math> f_{\omega+1}(10\uparrow\uparrow 65\,533)</math> | style="background: yellow;" | |- !5-symbol | style="background: orange;" title="Cryptids: Hydra and Bonus Cryptid" | [[BB(2,5)]] > <math>10^{10^{10^{3\,314\,360}}}</math> | style="background: #ddc290;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math> | style="background: yellow;" | BB(4,5) > <math> f_{\omega+1}(27\,041)</math> | style="background: yellow;" |BB(5,5) > <math>f_{\omega+1}(f_\omega^4(6))</math> | style="background: yellow;" | BB(6,5) > <math>f_{\omega+3}(11)</math> | style="background: yellow;" |BB(7,5) > <math>f_{\omega+3}^2(10)</math> |- !6-symbol | style="background: #ddc290;" | [[BB(2,6)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}}</math> | style="background: #ddc290;" | BB(3,6) > <math> f_\omega^3(15)</math> | style="background: yellow;" | BB(4,6) > <math> f_{\omega+2}(29)</math> | style="background: yellow;" | BB(5,6) > <math> f_{\omega+3}(31)</math> | style="background: yellow;" | BB(6,6) > <math> f_{\omega \times 2}(89)</math> | style="background: yellow;" | BB(7,6) > <math> f_{\omega^2}(63)</math> |- !7-symbol | style="background: #ddc290;" | [[BB(2,7)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math> | style="background: yellow;" | BB(3,7) > <math> f_{\omega+1}(117)</math> | style="background: yellow;" | BB(4,7) > <math> f_{\omega+2}(124)</math> | style="background: yellow;" | | style="background: yellow;" | | style="background: yellow;" | |} '''Table key:''' * Orange cell: smallest known [[Cryptid]] (a mathematically-hard machine) is in this class * Tan cell: a Cryptid is known using fewer states or symbols than this class. * Yellow cell: the Busy Beaver number is known to be more than [[Graham's number]]. * Blank cell: no known champion that surpasses a champion with fewer states or symbols by a non-trivial amount. 1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n. == About bbchallenge == [[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: * Formalizing results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq]) * Maintaining [[Holdouts lists]] for small busy beaver values * [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]] * Finding [[Cryptids]] (mathematically-hard machines) * Searching for new [[Champions]] * Building [[Accelerated Simulator|Accelerated simulator]]s to simulate halting machines faster * Writing papers and giving talks about the Busy Beaver function, see [[Papers & Talks]] * Exploring different [[:Category:Functions|Busy Beaver-style functions]] built from different systems of computation 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. Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref> == This Month in Beaver Research (TMBR) == [[: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: * [[TMBR: May 2026]] * [[TMBR: April 2026]] * [[TMBR: March 2026]] [[TMBR: June 2026]], [[TMBR: July 2026]], [[TMBR: August 2026]], [[TMBR: September 2026]], [[TMBR: October 2026]], and [[TMBR: November 2026]] are currently works in progress. 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. ==Notes== <references /> sumh90c9l4zp430bk0olr49kxvqbwk1