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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:^[1]^[2]

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 computable, but a few of its values are known:

Small busy beaver values (up to 7 states and symbols)^[3]
	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	BB(6) > $10 ↑ ↑ 10 ↑ ↑ 10 ↑ ↑ 8$	BB(7) > $3 ↑^{20} 3$
3-symbol	BB(2,3) = 38	BB(3,3) $\geq$ 119,112,334,170,342,541	BB(4,3) > $10 ↑^{4} 4$		BB(6,3) > $f_{ω}^{2} (10 ↑ ↑ ↑ 6)$
4-symbol	BB(2,4) = 3,932,964	BB(3,4) > $2 ↑^{15} 5$	BB(4,4) > $2 ↑^{509} 3$	BB(5,4) > $f_{ω}^{2} (10 ↑^{4} 10 ↑ ↑ 65 533)$	BB(6,4) > $f_{ω + 1} (10 ↑ ↑ 65 533)$
5-symbol	BB(2,5) > $1 0^{1 0^{1 0^{3 314 360}}}$	BB(3,5) > $f_{ω} (2 ↑^{15} 5)$	BB(4,5) > $f_{ω + 1} (27 041)$	BB(5,5) > $f_{ω + 1} (f_{ω}^{4} (6))$	BB(6,5) > $f_{ω + 3} (11)$	BB(7,5) > $f_{ω + 3}^{2} (10)$
6-symbol	BB(2,6) > $10 ↑ ↑ 10 ↑ ↑ 1 0^{1 0^{115}}$	BB(3,6) > $f_{ω}^{3} (15)$	BB(4,6) > $f_{ω + 2} (29)$	BB(5,6) > $f_{ω + 3} (31)$	BB(6,6) > $f_{ω \times 2} (89)$	BB(7,6) > $f_{ω^{2}} (63)$
7-symbol	BB(2,7) > $10 ↑ ↑ 10 ↑ ↑ 10 ↑ ↑ 155$	BB(3,7) > $f_{ω + 1} (117)$	BB(4,7) > $f_{ω + 2} (124)$

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 Rocq)
Maintaining Holdouts lists for small busy beaver values
Proving the behavior of Individual machines
Finding Cryptids (mathematically-hard machines)
Searching for new Champions
Building Accelerated simulators to simulate halting machines faster
Writing papers and giving talks about the Busy Beaver function, see Papers & Talks
Exploring different Busy Beaver-style functions built from different systems of computation

In June 2024, bbchallenge achieved a significant milestone by proving in 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.^[4]

This Month in Beaver Research (TMBR)

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: 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

↑ 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
↑ 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.
↑ P. Michel, "Historical survey of Busy Beavers".
↑ 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

[1] 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

[2] 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.

[3] P. Michel, "Historical survey of Busy Beavers".

[4] 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

[1]

[2]

[3]

[4]

@@ Line 6: / Line 6: @@
 |}
-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:
+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<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
+|+ 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
+! !!2-state !!3-state !!4-state !!5-state !!6-state !!7-state
-!7-state
+|-
-|-
+!2-symbol
-! 2-symbol
+| [[BB(2)]] = 6
-| [[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: orange;" title="Cryptid: Bigfoot" | [[BB(7)]] > <math>3 \uparrow^{20} 3</math>
+| style="background: #ddc290;" | [[BB(7)]] > <math>3 \uparrow^{20} 3</math>
 |-
-! 3-symbol
+!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>
@@ Line 29: / Line 28: @@
 | style="background: #ddc290;" |
 |-
-! 4-symbol
+!4-symbol
 | [[BB(2,4)]] = 3,932,964
 | style="background: #ddc290;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
-| style="background: orange;" title="Cryptid: Erdős Ternary Conjecture Variant" |
+| style="background: #ddc290;" | [[BB(4,4)]] > <math>2 \uparrow^{509} 3</math>
-| style="background: orange;" title="Cryptid: Erdős Ternary Conjecture" | BB(5,4) > <math>f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math>
+| style="background: #ddc290;" | BB(5,4) > <math>f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math>
-| style="background: orange;" title="Cryptid: Sloane Ternary Conjecture" |
+| style="background: yellow;" | BB(6,4) > <math> f_{\omega+1}(10\uparrow\uparrow 65\,533)</math>
-| style="background: #ddc290;" |
+| style="background: yellow;" |
 |-
-! 5-symbol
+!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: #ddc290;" |
+| style="background: yellow;" | BB(4,5) > <math> f_{\omega+1}(27\,041)</math>
-| style="background: #ddc290;" |
+| style="background: yellow;" |BB(5,5) > <math>f_{\omega+1}(f_\omega^4(6))</math>
-| style="background: #ddc290;" |
+| style="background: yellow;" | BB(6,5) > <math>f_{\omega+3}(11)</math>
-| style="background: #ddc290;" |
+| style="background: yellow;" |BB(7,5) > <math>f_{\omega+3}^2(10)</math>
 |-
-! 6-symbol
+!6-symbol
 | style="background: #ddc290;" | [[BB(2,6)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
-| style="background: #ddc290;" |
+| style="background: #ddc290;" | BB(3,6) > <math> f_\omega^3(15)</math>
-| style="background: #ddc290;" |
+| style="background: yellow;" | BB(4,6) > <math> f_{\omega+2}(29)</math>
-| style="background: #ddc290;" |
+| style="background: yellow;" | BB(5,6) > <math> f_{\omega+3}(31)</math>
-| style="background: #ddc290;" |
+| style="background: yellow;" | BB(6,6) > <math> f_{\omega \times 2}(89)</math>
-| style="background: #ddc290;" |
+| 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;" |
 |}
-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. Cells are blank if no champion has been found which surpasses a smaller size problem.
+'''Table key:'''
+* <span style="background: orange">Orange cell</span>: smallest known [[Cryptid]] (a mathematically-hard machine) is in this class
+* <span style="background: #ddc290">Tan cell</span>: a Cryptid is known using fewer states or symbols than this class.
+* <span style="background: yellow">Yellow cell</span>: 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.
 -state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.
@@ Line 61: / Line 73: @@
 [[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:
-* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
+* 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]]
@@ Line 76: / Line 88: @@
 [[: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: February 2026]]
-[[TMBR: May 2026]], [[TMBR: June 2026]], [[TMBR: July 2026]], [[TMBR: August 2026]], [[TMBR: September 2026]], and [[TMBR: October 2026]] are currently works in progress.
+[[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.

Main Page: Difference between revisions

Latest revision as of 06:15, 5 November 2026

About bbchallenge

This Month in Beaver Research (TMBR)

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