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2026-11-07T12:14:53Z

Merieke: New 7,6 champion

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;" |BB(7,4) > <math>f_{\omega+2}(28)</math>
|-
!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}(110)</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+3}(34)</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 />

TMBR: November 2026

2026-11-06T21:27:12Z

Merieke: fixed multiplication

{{TMBRnav|October 2026|December 2026}}''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for November 2026.

== Media ==

* YouTube video: [https://www.youtube.com/watch?v=pZrjqwlYbF0 Skelet #17: The Last Obstacle to solving Busy Beaver 5]. NLogSpace. 6 Nov 2026.

== BB Adjacent ==
* [[Pebble Automaton]]
** On November 2, the remaining group of 22 pebble(11) holdouts was officially dubbed the "golems". They are modeled by complex systems of equations, making them potential cryptids. They can be categorized into 7 unique paths.

== Champions ==
{| class="wikitable mw-collapsible mw-collapsed"
|+New''' '''champions''' '''discovered''' '''this''' '''month
!Domain
!Champion
!Discoverer
!New''' '''lower''' '''bound
!Discoverer
!Previous''' '''lower''' '''bound
|-
|BB(17)
|{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_1RQ0LO_1LP1LN_0RB1LI_1RZ1LO|halt}}
| rowspan="2" |Jeff Ketchersid
|<math>f_{\omega + 3}^3(9)</math>
|-
|BB(18)
|{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_1RQ0LO_1LP1LN_0RB1LI_1RR0LO_0RO1RZ|halt}}
|<math>f_{\omega + 3}^4(9)</math>
|Jeff Ketchersid
|<math>f_{\omega + 3}(114)</math>
|-
|BB(19)
|{{TM|1RQ1RL_0LI1RC_0LD1LF_0RA1LD_1RB1RE_1LH0LG_1LF1LG_1RE1RM_1LJ1LI_0LK1LI_1RL1RA_0RL0RE_0LN1RM_1LO0LO_1RE1LP_1LN1LN_1LR1RQ_1RZ1LS_0LE0LR|halt}}
| rowspan="2" |[[User:Jacobzheng]]
|<math>f_{\omega + 7}^2(5)</math>
|-
|BB(20)
|{{TM|1LH0LG_0LI1RC_0LD1LA_0RE1LD_1RQ1RL_1RB1RF_1LA1LG_1RF1RM_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LN1RM_0RF0LO_0RF1LP_0RF1LN_1LR1RQ_1RZ1LS_0LF1LT_0LF0LR|halt}}
|<math>f_{\omega * 2}(f_{\omega + 2}^2(f_{10}(2)))</math>
| rowspan="2" |[[User:Jacobzheng]]
|<math>f_{\omega + 6}^2(4)</math>
|-
|BB(11,4)
|{{TM|1RG3LB1RZ2RA_2LD---1LD2RA_3RB1LB3LC2RC_3RE---3LC2RC_---3RE3LH2LF_3RF2RF2LE1LF_2LC3LH2LJ2RG_---3RH1LI2RG_3RH1LH3LI2RI_------3LJ3RK_------2RA3RK|halt}}
| rowspan="9" |[[User:Loader3229]]
|<math>f_{\omega \cdot 2}(21)</math>
|<math>f_{\omega + 3}^2(4)</math>
|-
|BB(8,5)
|{{TM|2RG3LB4LC2RA4LB_1RZ---1LD2RA4LD_3RB1LB3LC2RC4LC_3RE---3LC2RC3LC_---3RE4LE4LH4RB_4LE3LG3LB2RF4LG_2LH3RG1LH2RF2RA_3RG1LG3LH2RH4LH|halt}}
|<math>f_{\omega \cdot 2 + 1}(93)</math>
| rowspan="2" |[[User:Loader3229]]
|<math>f_{\omega \cdot 2}(32)</math>
|-
|BB(4,6)
|{{TM|5LD3LB5LC2RA4LB4LC_1RZ3RB1LC2RA3RD0LA_3RB1LB3LC2RC4LC1RA_3RB0LA2LC4LD4LC4LD|halt}}
|<math>f_{\omega + 2}(29)</math>
|<math>f_{\omega + 1}(2 \uparrow^{15} 5)</math>
|-
|BB(5,6)
|{{TM|1LE3LB2LA2RA4LB5RA_1RZ3RB1LC2RA3RD4RA_3RB1LB3LC2RC4LC2LB_3RB0LA5LE4LD4LC4LD_4LC5LC0RE5LE1RE2RE|halt}}
|<math>f_{\omega + 3}(31)</math>
|-
|BB(6,6)
|{{TM|4LF3LB1RZ2RA4LB---_2LC3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB0LF1RB1LF---5RB|halt}}
|<math>f_{\omega * 2}(89)</math>
|-
|BB(7,6)
|{{TM|2LE3LB1RZ2RA4LB2LD_0LA3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE0LG4LC3LG5LE_5RB5RB---1LF---5LF_5LF---3RC1LG---5LG|halt}}
|<math>f_{\omega^2}(63)</math>
|-
|BB(8,6)
|{{TM|0LC3LB2LH2RA4LB1RZ_3LH3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC2LD3RD_5RB3RE---4LC3LG5LE_5RB5RB---1LF5LF5LF_5LF4RG1LB1LG5LD5LG_3LE---1LH1LH---5RG|halt}}
|<math>f_{\omega^2 + 1}(7)</math>
|-
|BB(3,7)
|{{TM|6LC3LB5RA2RA4LB2LA5LC_1RZ3RB1LC2RA3RB4LC0LA_3RB1LB3LC2RC4LC5LC---|halt}}
|<math>f_{\omega + 1}(77)</math>
|[[User:Loader3229]]
|<math>f_{\omega}^4(15)</math>
|-
|BB(4,7)
|{{TM|6LD3LB5LD2RA4LB3LC5LA_1RZ3RB1LC2RA3RD---2RA_3RB1LB3LC2RC4LC3LC5LD_3RB---6LC4LD4LC4LD---|halt}}
|<math>f_{\omega + 2}(124)</math>
|}

* [[User:Loader3229]] discovered an array of champions based on Kropitz's current [[BB(3,4)]] champion, demonstrating the following bounds:
** On November 1, BB(8,5) > <math>f_{\omega \cdot 2 + 1}(93)</math>.[https://discord.com/channels/960643023006490684/1084047886494470185/1566410612949385317] and BB(11,4) > <math>f_{\omega \cdot 2}(21)</math>.[https://discord.com/channels/960643023006490684/1084047886494470185/1566423230187700245]
** On November 2, BB(4,6) > <math>f_{\omega + 2}(29)</math>, BB(5,6) > <math>f_{\omega + 3}(31)</math>, BB(6,6) > <math>f_{\omega \times 2}(36)</math> and BB(7,6) > <math>f_{\omega^2}(28)</math>.
** On November 3, BB(6,6) > <math>f_{\omega * 2}(89)</math>.
** On November 4, BB(7,6) > <math>f_{\omega^2}(63)</math>, BB(8,6) > <math>f_{\omega^2 + 1}(7)</math>, BB(3,7) > <math>f_{\omega + 1}(77)</math> and BB(4,7) > <math>f_{\omega + 2}(124)</math>
* [[User:Jacobzheng]] discovered the following bounds:
** On November 2, BB(20) > <math>f_{\omega + 7}^2(5)</math>.
** On November 4, BB(19) > <math>f_{\omega + 7}^2(5)</math> and BB(20) > <math>f_{\omega \times 2}(f_{\omega + 2}^2(f_{10}(2))</math>.
* On November 4, Jeff Ketchersid used Claude and Gemini to discover the bounds BB(17) > <math>f_{\omega + 3}(9)</math>, BB(18) > <math>f_{\omega + 3}^4(9)</math> and BB(19) > <math>f_{\omega + 4}(f_{\omega + 2}(114))</math>.

== Holdouts ==
{| class="wikitable"
|+'''BB Holdout Reduction by Domain'''
!Domain
!New Holdout Count
!Previous Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|608
|608
|0
|0.00%
|-
|[[BB(2,5)]]
|37 (official)
|59 (official), 37 (informal)
|22 (Rocq confirmation of Lean proofs)
|37.29% (confirmations)
|}

* [[BB(6)]]
** On November 1, Ahmer Raza simulated two holdouts, both Reus Vampires, to halting.[https://discord.com/channels/960643023006490684/1239205785913790465/1566293610503340184]
*[[BB(2,5)]]
** On November 3, mxdys confirmed in Rocq the proofs in Lean of 22 TMs solved with AI assistance (GPT6 and Aristotle), releasing a formal list of '''37''' holdouts.[https://discord.com/channels/960643023006490684/1259770421046411285/1567052029430595584]

== Cryptids ==
All Turing machines mentioned in this section work with an alphabet of 2 symbols unless explicitly stated. The state bounds given also only consider Turing machines which work with 2 symbols.

* [[Cryptids#Cryptids at the Edge|Cryptids in the Wild]] (+ [[User:RobinCodes/Machines at the Edge|Energy Vampires]])
** [https://discord.com/channels/960643023006490684/1561037362581086228 Reus Vampires]
*** On November 1, Ahmer Raza ran path O4 to 1,941,376,761,248 iterations, at which point it halts![https://discord.com/channels/960643023006490684/1561037362581086228/1566292773173198980] This path corresponds to a group of machines with halting times of over 10683,718,950,000 steps, and decides the last of the 7 Reus vampires.

* [[Cryptids#Larger Cryptids|Larger Cryptids]]
** [[Logical independence|Logical Independence]]
*** [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo-Frankel set theory]]
**** On November 1, charlie claimed to have used Claude to [https://github.com/calr3/tinycryptids/tree/main/zfc_nsymbols/mm0 formally] reduce the number of defined instructions required in a ZFC-independent TM to 375 (376 including the halting transition).
** Computational Complexity Theory
*** On November 1, GregoryMorse claimed to have found a 278-state machine that quasihalts iff P = NP, a 295-state machine that quasihalts iff P = PP, and a 291-state machine that quasihalts iff P = PSPACE.[https://discord.com/channels/960643023006490684/1385498968011575366/1566326998563823719]

{| class="wikitable"
|+Other problems encoded into Turing machines this month
!Conjecture
!Domain
!Discoverer
!Discovery date
!Notes
|-
|Pillai's conjecture
|BBBB(91)
|charlie, Claude[https://discord.com/channels/960643023006490684/1385498968011575366/1566248174614479061]
|November 1
|Unknown - pseudoquasihalts if the conjecture is false. [https://github.com/calr3/tinycryptids/tree/main/lean/TinyCryptids/Pillai Proof.]
|}
[[Category:This Month in Beaver Research|2026-11]]

Multi-tape Turing Machines

2026-11-06T21:20:54Z

Merieke: added TM code

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down. All of the champions have been hand simulated (which hopefully will change soon) which means that none of them will be optimal at all.

'''Rules'''

* Going up or down is not required for each step.
* The head starts on the lowermost tape.
* A step can just be going up or down.
* Each tape needs it own instructions
*BB(n,m,1) = BB(n,m)

'''How to write TMs'''

Writing 2D TMs are mostly like writing them in the first dimension, you just need to add a slash in between every layer of instruction and go top to bottom.

The champion for BB(2,2,2) would be this in TM form - 1RB0LB_0LDA0RDB / 0LUA1LUB_1RA1RZ

'''Current Bounds'''

BB(n,m,o) > BB(n,m)<math>\times</math>o
{| class="wikitable"
|+2 Tapes
!Domain
!Score
!Runtime
!TM Code
|-
|BB(2,2,2)
|6 but not from Champion
|16
|1RB0LB_0LDA0RDB / 0LUA1LUB_1RA1RZ
|-
|BB(2,3,2)
|4
|23
|0RDB2LB1RZ_1LA1DA1RZ / 2RUB2RB1UA_0LA1LUB0RA
|}
{| class="wikitable"
|+
!Domain
!Score
!Runtime
!TM Code
|-
|BB(2,2,3)
|8
|24
|1RDB1RZ_0LA1RZ / 0RUB1RZ_0RDA1RZ / 1LUA1RZ_1LUB1LB
|}

Multi-tape Turing Machines

2026-11-05T21:27:23Z

Merieke: fixed and just made it much better

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down. All of the champions have been hand simulated (which hopefully will change soon) which means that none of them will be optimal at all.

Rules

* Going up or down is not required for each step.
* The head starts on the lowermost tape.
* A step can just be going up or down.
* Each tape needs it own instructions
*BB(n,m,1) = BB(n,m)

How to write TMs

Writing 2D TMs are mostly like writing them in the first dimension, you just need to add a slash in between every layer of instruction and go top to bottom.

The champion for BB(2,2,2) would be this in TM form - 1RB0LB_0LDA0RDB / 0LUA1LUB_1RA1RZ
{| class="wikitable"
|+2 Tapes
!Domain
!Score
!Runtime
!TM Code
!
|-
|BB(2,2,2)
|6 but not from Champion
|16
|1RB0LB_0LDA0RDB / 0LUA1LUB_1RA1RZ
|
|-
|BB(2,3,2)
|4
|23
|0RDB2LB1RZ_1LA1DA1RZ / 2RUB2RB1UA_0LA1LUB0RA
|
|}
{| class="wikitable"
|+
!Domain
!Score
!Runtime
!TM Code
|-
|BB(2,2,3)
|later
|24
|later
|-
|
|
|
|
|-
|
|
|
|
|}

Main Page

2026-11-05T20:40:41Z

Merieke: added the new BB(4,7) champ

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+3}(34)</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 />

Multi-tape Turing Machines

2026-11-05T12:32:35Z

Merieke: Added how to write TMs

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down. All of the champions have been hand simulated (which hopefully will change soon) which means that none of them will be optimal at all.

Rules

* Going up or down is not required for each step.
* The head starts on the lowermost tape.
* A step can just be going up or down.
* Each tape needs it own instructions
*BB(n,m,1) = BB(n,m)

How to write TMswriting 2d TMs are mostly like writing them in the first dimension. You just need to add a slash in between every layer of instruction.

The champion for BB(2,2,2) would be this in TM form - 1RB0LB_0LDA0RDB / 0LUA1LUB_1RA1RZ

The Champion for BB(2,2,2) is this. S(2,2,2) > 16. Sigma<math>\Sigma</math>(2,2,2) > 6 but not from this TM

{| class="wikitable"
|+ Top
|-
| || 0 || 1
|-
| A || 1RB || 0LB
|-
| B || 0LDA || 0RDB
|}

{| class="wikitable"
|+ Bottom
|-
| || 0 || 1
|-
| A || 0LUA || 1LUB
|-
| B || 1RA || halt
|}
The Champion for BB(2,2,3) is this. S(2,2,3) > 24 and <math>\Sigma</math>(2,2,3) > 8
{| class="wikitable"
|+Top
|
|0
|1
|-
|A
|1RDB
|halt
|-
|B
|0LA
|halt
|}
{| class="wikitable"
|+Middle
|
|0
|1
|-
|A
|0RUB
|halt
|-
|B
|0RDA
|halt
|}
{| class="wikitable"
|+Bottom
|
|0
|1
|-
|A
|1LUA
|halt
|-
|B
|1LUB
|1LB
|}

Oracle TMs

2026-11-01T15:34:53Z

Merieke: Added citations

This is a work in progress. Oracle Turing Machines are like normal TMs but have 3 extra states. Those are ask, yes and no. When it reaches an ask state, it counts the number of 1s on the tape (which we will call y) and simulates TM#y if it halts go to the yes state and if not go to the no state. Looking [https://googology.fandom.com/wiki/User:Wythagoras/Rado's_sigma_function/Oracle_TMs/Numbering_for_TMs here] can also be useful for explaining.

'''Champions'''<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/Oracle_TMs</ref>
{| class="wikitable"
|+
!Sigma
!Value for Sigma
!S
!Value for S
|-
|Sigma2 (1)
|>= 10
|S2 (1)
|>= 22
|-
|Sigma2 (2)
|>= 445 but could be above 25,000
|
|
|}

Oracle TMs

2026-11-01T15:28:06Z

Merieke: Fixed Table

Oracle TMs

2026-11-01T15:27:35Z

Merieke: Added champions and description

Oracle TMs

2026-10-31T21:47:00Z

Merieke: Created page with "This is a work in progress"

This is a work in progress

Multi-tape Turing Machines

2026-10-28T22:03:53Z

Merieke: added a champion

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down. All of the champions have been hand simulated (which hopefully will change soon) which means that none of them will be optimal at all.

Rules

* Going up or down is not required for each step.
* The head starts on the lowermost tape.
* A step can just be going up or down.
* Each tape needs it own instructions
* BB(n,m,1) = BB(n,m)

The Champion for BB(2,2,2) is this. S(2,2,2) > 16. Sigma<math>\Sigma</math>(2,2,2) > 6 but not from this TM

{| class="wikitable"
|+ Top
|-
| || 0 || 1
|-
| A || 1RB || 0LB
|-
| B || 0LDA || 0RDB
|}

{| class="wikitable"
|+ Bottom
|-
| || 0 || 1
|-
| A || 0LUA || 1LUB
|-
| B || 1RA || halt
|}
The Champion for BB(2,2,3) is this. S(2,2,3) > 24 and <math>\Sigma</math>(2,2,3) > 8
{| class="wikitable"
|+Top
|
|0
|1
|-
|A
|1RDB
|halt
|-
|B
|0LA
|halt
|}
{| class="wikitable"
|+Middle
|
|0
|1
|-
|A
|0RUB
|halt
|-
|B
|0RDA
|halt
|}
{| class="wikitable"
|+Bottom
|
|0
|1
|-
|A
|1LUA
|halt
|-
|B
|1LUB
|1LB
|}

Multi-tape Turing Machines

2026-10-28T21:05:40Z

Merieke: added stuff

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down.

Rules

* Going up or down is not required for each step.
* The head starts on the lowermost tape.
* A step can just be going up or down.
* Each tape needs it own instructions
* BB(n,m,1) = BB(n,m)

The Champion for BB(2,2,2) is this. S(2,2,2) > 16. Sigma(2,2,2) > 6 but not from this TM

{| class="wikitable"
|+ Top
|-
| || 0 || 1
|-
| A || 1RB || 0LB
|-
| B || 0LDA || 0RDB
|}

{| class="wikitable"
|+ Bottom
|-
| || 0 || 1
|-
| A || 0LUA || 1LUB
|-
| B || 1RA || halt
|}

User:Merieke/2D TMs

2026-10-28T21:00:30Z

Merieke: fixed

This page is not being used anymore, if you want 2D TMs go to [[2D Tms|This page]]

Multi-tape Turing Machines

2026-10-28T20:58:40Z

Merieke: Created page with " Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down. Rules * Going up or down is not required for each step. * The head starts on the lowermost tape. * A step can just be going up or down. * Each tape needs it own instructions The Champion for BB(2,2,2) is this. S(2,2,2) > 16. Sigma(2..."

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down.

Rules

* Going up or down is not required for each step.
* The head starts on the lowermost tape.
* A step can just be going up or down.
* Each tape needs it own instructions

The Champion for BB(2,2,2) is this. S(2,2,2) > 16. Sigma(2,2,2) > 6 but not from this TM

{| class="wikitable"
|+ Top
|-
| || 0 || 1
|-
| A || 1RB || 0LB
|-
| B || 0LDA || 0RDB
|}

{| class="wikitable"
|+ Bottom
|-
| || 0 || 1
|-
| A || 0LUA || 1LUB
|-
| B || 1RA || halt
|}

User:Merieke/2D TMs

2026-10-24T12:33:43Z

Merieke: new champion gg

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down.

'''Rules'''

* Going up or down is not required for each step.

* The head starts on the lowermost tape.

*A step can't just be going up or down. If it goes up or down it also has to go left or right.

The Champion for BB(2,2,2) is this. S(2,2,2) > 16
{| class="wikitable"
|+Top
|
|0
|1
|-
|A
|1RB
|0LB
|-
|B
|0LDA
|0RDB
|}
{| class="wikitable"
|+Bottom
|
|0
|1
|-
|A
|0LUA
|1LUB
|-
|B
|1RA
|H
|}

User:Merieke/2D TMs

2026-10-21T22:33:17Z

Merieke:

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down.

'''Rules'''

* Going up or down is not required for each step.

* The head starts on the lowermost tape.

*A step can't just be going up or down. If it goes up or down it also has to go left or right.

The Champion for BB(2,2,2) is this. S(2,2,2) > 11 and <math>\Sigma</math>(2,2,2) > 5
{| class="wikitable"
|+Top
|
|0
|1
|-
|A
|1RDB
|H
|-
|B
|0LA
|1RDB
|}
{| class="wikitable"
|+Bottom
|
|0
|1
|-
|A
|1RUB
|0LUA
|-
|B
|1LA
|1LA
|}

User:Merieke/2D TMs

2026-10-21T22:32:24Z

Merieke: added a champion

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down.

'''Rules'''

* Going up or down is not required for each step.

* The head starts on the lowermost tape.

* A step can't just be going up or down. If it goes up or down it also has to go left or right.

The Champion for BB(2,2,2) is this. S(2,2,2) = 11 and <math>\Sigma</math>(2,2,2) = 5
{| class="wikitable"
|+Top
|
|0
|1
|-
|A
|1RDB
|H
|-
|B
|0LA
|1RDB
|}
{| class="wikitable"
|+Bottom
|
|0
|1
|-
|A
|1RUB
|0LUA
|-
|B
|1LA
|1LA
|}

User:Merieke/2D TMs

2026-10-21T20:12:39Z

Merieke: added stuff

Multi-tape Turing Machines are like normal Turing machines but with more tapes. BB(n,m,o) = the maximum number of steps taken by a n state, m symbol, o tape Turing machine. In addition to going left and right, it can also go up and down.

'''Rules'''

* Going up or down is not required for each step.

* The head starts on the lowermost tape.

* A step can't just be going up or down. if it goes up or down it also has to go left or right.

The Champion for BB(2,2,2) is this
{| class="wikitable"
|+Top
|
|0
|1
|-
|A
|
|
|-
|B
|
|
|}
{| class="wikitable"
|+Bottom
|
|0
|1
|-
|A
|
|
|-
|B
|
|
|}

User:Merieke

2026-10-21T19:38:37Z

Merieke:

A person (hopefully)

[[User:Merieke/2D TMs]]

Multi-tape TMs

2026-10-20T20:48:31Z

Merieke: stuff removed because not needed

User:Merieke/2D TMs

2026-10-20T20:42:02Z

Merieke: added some stuff and will delete multi tape tm page bc not needed

Multi-tape TMs

2026-10-19T23:47:08Z

Merieke: A very very very brief outline of the function. Def need to add more

User:Merieke

2026-10-19T23:38:06Z

Merieke:

A person

[[Multi-tape TMs]]

Multi-tape TMs

2026-10-19T23:37:08Z

Merieke: Created page with "A"

User:Merieke

2026-10-19T23:31:35Z

Merieke: Created page with "A person"

A person

Measuring large numbers

2026-10-17T11:28:43Z

Merieke: changed 1850 to 1797

{{Stub}}
The busy beaver functions are known to grow faster than any computable function, as do many of their variants. Because of this, values of the function, or lower bounds on the function can be very, very large, so it is tempting to find ways of representing these numbers; this article will cover these methods.

Paradoxically, although the Kolgomorov complexity of stating "The runtime of program X" to define the runtime of program X is typically very low, it is often not useful as it does not give us good information on the size of the number. Informally, more "natural" or "combinatorial" ways of measuring large numbers are desired. Tools like Knuth's up-arrow notation can help us achieve this, as well as more powerful methods such as [[Fast-Growing Hierarchy|fast-growing hierarchy]] or the diagonalization of an axiomatic foundation of mathematics.

== Scientific notation ==
Scientific notation is one of the most common ways of representing exceedingly large numbers; numbers written in standard scientific notation symbolically consist of a multiplicand (taking a value between 1 and the base, though is sometimes omitted), a base (commonly 10) and an index (usually an integer, though sometimes takes non-integer values if the multiplicand is omitted).

For example, <math>8.68 \cdot 10^6</math> represents the number 8,680,000, whereas <math>6.022 \cdot 10^{23}</math> represents the number 602,200,000,000,000,000,000,000.

In cases where the index is not an integer, the representation usually represents an irrational number, though sometimes it can represent a rational number, or even an integer.

For example, <math>10^{7.67}</math> represents the number 46,773,514.1287..., an irrational number, whereas <math>16^{6.75}</math> represents the number 134,217,728, an integer.

Certain busy beaver values can most conveniently be represented using scientific notation; they include:

<math>S(3,3) > 1.1912 \cdot 10^{17}</math>

<math>\operatorname{BBf}(23) > 4.3932 \cdot 10^{124}</math>

<math>\operatorname{BB\lambda}(38) > 1.0017 \cdot 10^{19,729}</math>

When representing certain exceedingly large numbers using scientific notation, the index itself may also be represented using scientific notation.

<math>S(2,5) > 10^{10^{10^{3,314,360}}}</math>

== Knuth's up-arrow notation ==
Knuth's up-arrow notation extends the sequence of recursive operators (i.e. addition, multiplication, exponentiation) to higher-order functions (e.g. tetration, pentation) which grow much more rapidly.

Tetration, the operation following exponentiation in the sequence of "hyperoperators" is defined by:

<math>a \uparrow \uparrow b = \begin{cases}
1, & \text{if } b = 0
\\ a^{a \uparrow \uparrow (b - 1)}, & \text{otherwise.}
\end{cases}</math>

The lower bound for <math>S(2,5)</math> can thus be more concisely (though more weakly) written as <math>S(2,5) > 10 \uparrow\uparrow 4</math>.

Knuth's up-arrow notation generalizes this though the following formula (where an arrow with a superscript of n represents the operator indicated by n arrows):

<math>a \uparrow^{n} b =\begin{cases}
a^b, & \text{if } n = 1
\\ 1, & \text{if } b = 0
\\ a \uparrow^{n - 1} (a \uparrow^{n} (b - 1)), & \text{otherwise.}
\end{cases}</math>

Larger busy beaver values represented using Knuth's up-arrow notation include:

<math>S(2,6) > 10 \uparrow \uparrow \uparrow 3</math>

<math>S(4,3) > 10 \uparrow \uparrow \uparrow \uparrow 4</math>

<math>S(3,4) > 2 \uparrow^{15} 3</math>

<math>S(3,5) > 2 \uparrow^{2 \uparrow^{15} 3} 3</math>

[[Busy Beaver Challenge|bbchallenge]] uses a specialized version of Knuth's up-arrow notation in order to write numbers of this scale more accurately:

<math>a \uparrow^{n} b =\begin{cases}
a^b, & \text{if } n = 1
\\ b + 1, & \text{if } b \leq 0
\\ a \uparrow^{n - 1} (a \uparrow^{n} (b - 1)), & \text{otherwise.}
\end{cases}</math>

Note that when using this method, the function <math>10 \uparrow\uparrow x</math>, and other functions similar to it, are not analytic functions; however, that requirement is unnecessary when using this method to represent large numbers. Additionally, methods of obtaining analytic hyperoperators are complex, typically involving Abel functions, Riemann mappings, or Jabotinsky matrices; this method forgoes this, allowing for the ease of calculation.

In this format, the lower bound for <math>S(2,5)</math> can instead be written as <math>S(2,5) > 10 \uparrow\uparrow 4.8142</math>.

In popular mathematics, Graham's number (<math>= g_{64}</math>) is represented using the following recursive function involving Knuth's up-arrow notation:

<math>g_n = \begin{cases}
3 \uparrow \uparrow \uparrow \uparrow 3, & \text{if }n = 1
\\ 3 \uparrow^{g_{n - 1}} 3, & \text{otherwise.}
\end{cases}</math>

The Ackermann function <math>\operatorname{A}(n,n)</math> grows at roughly the same rate as <math>2 \uparrow^{n} 3</math>. It can be said to "diagonalize" across the hyperoperators, eventually dominating every single hyperoperator.

== Fast-growing hierarchy ==
The fast-growing hierarchy uses the fundamental sequences of ordinals to define further, stronger diagonalizations across fast-growing functions. Functions under the fast-growing hierarchy are written in the form <math>f_\alpha(n)</math>, and may be calculated according to three rules:

<math>f_0(n) = n + 1</math>

<math>f_{\alpha + 1}(n) = f_{\alpha}^n(n)</math>

<math>f_{\alpha}(n) = f_{\alpha[n]}(n)</math> where <math>\alpha</math> is a limit ordinal (i.e. not a successor of any ordinal, nor zero.)

The fast-growing hierarchy requires a defined set of fundamental sequences; the fundamental sequence of a limit ordinal <math>\alpha</math> consists of an ordered sequence of ordinals which has a supremum of <math>\alpha</math>.

It is suspected that in this wiki, and in most other spaces that deal with googology, the fundamental sequences described by Maksudov (2017) are used to define the fast-growing hierarchy.

In this manner, the value of <math>S(3,5)</math> can be more concisely (though more weakly) written as <math>S(3,5) > f_\omega(2 \uparrow^{15} 3)</math>.

Other busy beaver values that necessitate the use of the fast-growing hierarchy include:

<math>\operatorname{BB\lambda}(49) > f_{\omega + 1}\left(\frac{2 \uparrow \uparrow 6}{2}\right)</math>

<math>\operatorname{S}(51) > f_{\varepsilon_0 + 1}(8)</math>

<math>\operatorname{BB\lambda}(100) > f_{\psi_0(\Omega_\omega) + 1}(4)</math>

=== Defining fundamental sequences: ordinal notations and ordinal collapsing functions ===
Individually defining fundamental sequences for a transfinite number of ordinals may be less than ideal; thus, to define the fast-growing hierarchy effectively, ordinal notations, which define a computable mapping between sequences of symbols and ordinals can be used to define fundamental sequences instead. TODO

=== Comparing FGH and BB ===
{| class="wikitable"
|-
! Large Number N !! Least m that BB(m) confirmed >= N
|-
| <math>f_{\omega}(10) </math>|| 7<ref>See {{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF}}.</ref>
|-
| <math>f_{\omega+1}(10) </math>|| 10<ref>(1) Discord | #large-bb-domains | bbchallenge</ref>
|-
| <math>f_{\omega+2}(10) </math>|| 13
|-
| <math>f_{\omega\times 2}(10) </math>|| 21
|-
| <math>f_{\omega^2}(10) </math>|| 21
|-
| <math>f_{\omega^\omega}(10) </math>|| 40
|-
| <math>f_{\omega^\omega+1}(10) </math>||
|-
| <math>f_{\varepsilon_0}(10) </math>|| 51<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495</ref><ref>Sligocki, "[[Talk:Champions#BB(51)_epsilon_0_champion|BB(51) epsilon_0 champion]]". bbchallenge wiki talk page edit, September 2024. Accessed 7 October 2026.</ref>
|-
| <math>f_{\psi_0(\Omega_2)}(10) </math>||
|-
| <math>f_{\psi_0(\Omega_\omega)}(10) </math>||
|-
| <math>f_{\psi_0(\varepsilon_{\Omega_\omega+1})}(10) </math>||
|-
| <math>f_{\psi_0(\Omega_{\Omega_{\Omega_{...}}})}(10) </math>||
|-
| <math>f_{lim(BMS)}(10) </math>|| 150
|}

== Theories ==
The proof-theoretic ordinal of a theory is defined as supremum of the order types of all ordinal notations that can be shown to be well-founded within the theory.

Some examples of proof theoretical ordinals include <math>\operatorname{PTO}(Q) = \omega</math>, <math>\operatorname{PTO}(PA) = \varepsilon_0</math>, and <math>\operatorname{PTO}(\operatorname{KP}) = \psi_0(\varepsilon_{\Omega+1})</math>. Here Q is [[wikipedia:Robinson_arithmetic|Robinson arithmetic]], PA is Peano Arithmetic, and KP is Kripke-Platek set theory.

Determination of the proof theoretic ordinal of second order arithmetic has been claimed in two preprints, one of which is a work-in-progress,<ref>T. Arai, "[https://arxiv.org/abs/2311.12459 An ordinal analysis of <math>\Pi_n</math>-Collection]" (2023). Accessed 8 October 2026.</ref><ref>H. Towsner, "[https://arxiv.org/abs/2403.17922 Proofs that Modify Proofs]" (2024). Accessed 8 October 2026.</ref> and is known to vastly exceed that of <math>\operatorname{KP}</math>.

They can be used to compare the strength of different theories, as well as (by definition) an indicator as to which ordinal notations (or functions that are derived from ordinal notations) cannot be shown to be well-founded (or total) in a theory.

A notable example of a theory which is directly used to create fast growing functions is the Calculus of Constructions (CoC), which has a proof-theoretic ordinal greater than that of any nth order arithmetic (<math>Z_n</math>); well-formed statements in CoC can also be easily detected, and are strongly normalizing (TODO: explain what that means in this context).

CoC yields the fast growing function <math>D(n)</math> (defined by Ralph Loader in 2001 to create the eponymous Loader's number = <math>D^5(99)</math>), which computes something similar to (but larger than) "The largest number \(x\) such that there is a <math>\Sigma^0_1</math> formula <math>\phi(x)</math> such that <math>\phi(x)</math> is true and <math>\exists x\phi(x)</math> is provable in CoC with a proof of length \(\leq \log_2 n\)".<ref>R. Loader, "[https://raw.githubusercontent.com/rcls/busy/master/README.txt busy]". GitHub repository (2003). Accessed 7 October 2026.</ref> <math>D(n)</math> outgrows all recursive functions provably total in <math>Z_\omega</math>; the growth rate of <math>D(n)</math> is sometimes informally written as <math>f_{\operatorname{PTO}(Z_\omega)}(n)</math>, though no fundamental sequence has yet been defined for all ordinals under <math>\operatorname{PTO}(Z_\omega)</math>.

The simplicity of CoC and <math>D(n)</math> has been utilized to show the following bounds for certain busy beaver values:

<math>\operatorname{BB\lambda}(1797) > D^5(99)</math>

<math>\operatorname{S}(1015) > D^5(99)</math><ref>CatsAreFluffy, "[https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe metamath-turing-machines]". GitHub commit (September 2024). Accessed 7 October 2026.</ref>

An even stronger but simpler function, <math>q(n)</math>, has been derived from the structure of [[wikipedia:Rank-into-rank_cardinal|<math>\operatorname{I3}</math>]] elementary embeddings, though it is unknown if this function can be shown to be total in <math>\operatorname{ZFC}</math> alone, and whether it grows faster than the aforementioned <math>D(n)</math>. It is used in the following (conjectured) bounds:

<math>\operatorname{BB\lambda}(201) > q(5)</math>

<math>\operatorname{S}(67) > q^{81}(q(5) - 2)</math><ref>https://discord.com/channels/960643023006490684/1331570843829932063/1339669138510970881</ref><ref>racheline, "[https://wiki.bbchallenge.org/wiki/Talk:Champions Larger Chapmions]". bbchallenge wiki talk post edit (February 2025). Accessed 7 October 2026.</ref>

== Resources ==
The Googology wiki hosts good information on methods of describing large numbers or large upper bounds for numbers.

* https://googology.miraheze.org/wiki/Main_Page

== References ==

[[Category:Stub]]

Measuring large numbers

2026-10-17T11:22:13Z

Merieke: fixed some values in comparison section

{{Stub}}
The busy beaver functions are known to grow faster than any computable function, as do many of their variants. Because of this, values of the function, or lower bounds on the function can be very, very large, so it is tempting to find ways of representing these numbers; this article will cover these methods.

Paradoxically, although the Kolgomorov complexity of stating "The runtime of program X" to define the runtime of program X is typically very low, it is often not useful as it does not give us good information on the size of the number. Informally, more "natural" or "combinatorial" ways of measuring large numbers are desired. Tools like Knuth's up-arrow notation can help us achieve this, as well as more powerful methods such as [[Fast-Growing Hierarchy|fast-growing hierarchy]] or the diagonalization of an axiomatic foundation of mathematics.

== Scientific notation ==
Scientific notation is one of the most common ways of representing exceedingly large numbers; numbers written in standard scientific notation symbolically consist of a multiplicand (taking a value between 1 and the base, though is sometimes omitted), a base (commonly 10) and an index (usually an integer, though sometimes takes non-integer values if the multiplicand is omitted).

For example, <math>8.68 \cdot 10^6</math> represents the number 8,680,000, whereas <math>6.022 \cdot 10^{23}</math> represents the number 602,200,000,000,000,000,000,000.

In cases where the index is not an integer, the representation usually represents an irrational number, though sometimes it can represent a rational number, or even an integer.

For example, <math>10^{7.67}</math> represents the number 46,773,514.1287..., an irrational number, whereas <math>16^{6.75}</math> represents the number 134,217,728, an integer.

Certain busy beaver values can most conveniently be represented using scientific notation; they include:

<math>S(3,3) > 1.1912 \cdot 10^{17}</math>

<math>\operatorname{BBf}(23) > 4.3932 \cdot 10^{124}</math>

<math>\operatorname{BB\lambda}(38) > 1.0017 \cdot 10^{19,729}</math>

When representing certain exceedingly large numbers using scientific notation, the index itself may also be represented using scientific notation.

<math>S(2,5) > 10^{10^{10^{3,314,360}}}</math>

== Knuth's up-arrow notation ==
Knuth's up-arrow notation extends the sequence of recursive operators (i.e. addition, multiplication, exponentiation) to higher-order functions (e.g. tetration, pentation) which grow much more rapidly.

Tetration, the operation following exponentiation in the sequence of "hyperoperators" is defined by:

<math>a \uparrow \uparrow b = \begin{cases}
1, & \text{if } b = 0
\\ a^{a \uparrow \uparrow (b - 1)}, & \text{otherwise.}
\end{cases}</math>

The lower bound for <math>S(2,5)</math> can thus be more concisely (though more weakly) written as <math>S(2,5) > 10 \uparrow\uparrow 4</math>.

Knuth's up-arrow notation generalizes this though the following formula (where an arrow with a superscript of n represents the operator indicated by n arrows):

<math>a \uparrow^{n} b =\begin{cases}
a^b, & \text{if } n = 1
\\ 1, & \text{if } b = 0
\\ a \uparrow^{n - 1} (a \uparrow^{n} (b - 1)), & \text{otherwise.}
\end{cases}</math>

Larger busy beaver values represented using Knuth's up-arrow notation include:

<math>S(2,6) > 10 \uparrow \uparrow \uparrow 3</math>

<math>S(4,3) > 10 \uparrow \uparrow \uparrow \uparrow 4</math>

<math>S(3,4) > 2 \uparrow^{15} 3</math>

<math>S(3,5) > 2 \uparrow^{2 \uparrow^{15} 3} 3</math>

[[Busy Beaver Challenge|bbchallenge]] uses a specialized version of Knuth's up-arrow notation in order to write numbers of this scale more accurately:

<math>a \uparrow^{n} b =\begin{cases}
a^b, & \text{if } n = 1
\\ b + 1, & \text{if } b \leq 0
\\ a \uparrow^{n - 1} (a \uparrow^{n} (b - 1)), & \text{otherwise.}
\end{cases}</math>

Note that when using this method, the function <math>10 \uparrow\uparrow x</math>, and other functions similar to it, are not analytic functions; however, that requirement is unnecessary when using this method to represent large numbers. Additionally, methods of obtaining analytic hyperoperators are complex, typically involving Abel functions, Riemann mappings, or Jabotinsky matrices; this method forgoes this, allowing for the ease of calculation.

In this format, the lower bound for <math>S(2,5)</math> can instead be written as <math>S(2,5) > 10 \uparrow\uparrow 4.8142</math>.

In popular mathematics, Graham's number (<math>= g_{64}</math>) is represented using the following recursive function involving Knuth's up-arrow notation:

<math>g_n = \begin{cases}
3 \uparrow \uparrow \uparrow \uparrow 3, & \text{if }n = 1
\\ 3 \uparrow^{g_{n - 1}} 3, & \text{otherwise.}
\end{cases}</math>

The Ackermann function <math>\operatorname{A}(n,n)</math> grows at roughly the same rate as <math>2 \uparrow^{n} 3</math>. It can be said to "diagonalize" across the hyperoperators, eventually dominating every single hyperoperator.

== Fast-growing hierarchy ==
The fast-growing hierarchy uses the fundamental sequences of ordinals to define further, stronger diagonalizations across fast-growing functions. Functions under the fast-growing hierarchy are written in the form <math>f_\alpha(n)</math>, and may be calculated according to three rules:

<math>f_0(n) = n + 1</math>

<math>f_{\alpha + 1}(n) = f_{\alpha}^n(n)</math>

<math>f_{\alpha}(n) = f_{\alpha[n]}(n)</math> where <math>\alpha</math> is a limit ordinal (i.e. not a successor of any ordinal, nor zero.)

The fast-growing hierarchy requires a defined set of fundamental sequences; the fundamental sequence of a limit ordinal <math>\alpha</math> consists of an ordered sequence of ordinals which has a supremum of <math>\alpha</math>.

It is suspected that in this wiki, and in most other spaces that deal with googology, the fundamental sequences described by Maksudov (2017) are used to define the fast-growing hierarchy.

In this manner, the value of <math>S(3,5)</math> can be more concisely (though more weakly) written as <math>S(3,5) > f_\omega(2 \uparrow^{15} 3)</math>.

Other busy beaver values that necessitate the use of the fast-growing hierarchy include:

<math>\operatorname{BB\lambda}(49) > f_{\omega + 1}\left(\frac{2 \uparrow \uparrow 6}{2}\right)</math>

<math>\operatorname{S}(51) > f_{\varepsilon_0 + 1}(8)</math>

<math>\operatorname{BB\lambda}(100) > f_{\psi_0(\Omega_\omega) + 1}(4)</math>

=== Defining fundamental sequences: ordinal notations and ordinal collapsing functions ===
Individually defining fundamental sequences for a transfinite number of ordinals may be less than ideal; thus, to define the fast-growing hierarchy effectively, ordinal notations, which define a computable mapping between sequences of symbols and ordinals can be used to define fundamental sequences instead. TODO

=== Comparing FGH and BB ===
{| class="wikitable"
|-
! Large Number N !! Least m that BB(m) confirmed >= N
|-
| <math>f_{\omega}(10) </math>|| 7<ref>See {{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF}}.</ref>
|-
| <math>f_{\omega+1}(10) </math>|| 10<ref>(1) Discord | #large-bb-domains | bbchallenge</ref>
|-
| <math>f_{\omega+2}(10) </math>|| 13
|-
| <math>f_{\omega\times 2}(10) </math>|| 21
|-
| <math>f_{\omega^2}(10) </math>|| 21
|-
| <math>f_{\omega^\omega}(10) </math>|| 40
|-
| <math>f_{\omega^\omega+1}(10) </math>||
|-
| <math>f_{\varepsilon_0}(10) </math>|| 51<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495</ref><ref>Sligocki, "[[Talk:Champions#BB(51)_epsilon_0_champion|BB(51) epsilon_0 champion]]". bbchallenge wiki talk page edit, September 2024. Accessed 7 October 2026.</ref>
|-
| <math>f_{\psi_0(\Omega_2)}(10) </math>||
|-
| <math>f_{\psi_0(\Omega_\omega)}(10) </math>||
|-
| <math>f_{\psi_0(\varepsilon_{\Omega_\omega+1})}(10) </math>||
|-
| <math>f_{\psi_0(\Omega_{\Omega_{\Omega_{...}}})}(10) </math>||
|-
| <math>f_{lim(BMS)}(10) </math>|| 150
|}

== Theories ==
The proof-theoretic ordinal of a theory is defined as supremum of the order types of all ordinal notations that can be shown to be well-founded within the theory.

Some examples of proof theoretical ordinals include <math>\operatorname{PTO}(Q) = \omega</math>, <math>\operatorname{PTO}(PA) = \varepsilon_0</math>, and <math>\operatorname{PTO}(\operatorname{KP}) = \psi_0(\varepsilon_{\Omega+1})</math>. Here Q is [[wikipedia:Robinson_arithmetic|Robinson arithmetic]], PA is Peano Arithmetic, and KP is Kripke-Platek set theory.

Determination of the proof theoretic ordinal of second order arithmetic has been claimed in two preprints, one of which is a work-in-progress,<ref>T. Arai, "[https://arxiv.org/abs/2311.12459 An ordinal analysis of <math>\Pi_n</math>-Collection]" (2023). Accessed 8 October 2026.</ref><ref>H. Towsner, "[https://arxiv.org/abs/2403.17922 Proofs that Modify Proofs]" (2024). Accessed 8 October 2026.</ref> and is known to vastly exceed that of <math>\operatorname{KP}</math>.

They can be used to compare the strength of different theories, as well as (by definition) an indicator as to which ordinal notations (or functions that are derived from ordinal notations) cannot be shown to be well-founded (or total) in a theory.

A notable example of a theory which is directly used to create fast growing functions is the Calculus of Constructions (CoC), which has a proof-theoretic ordinal greater than that of any nth order arithmetic (<math>Z_n</math>); well-formed statements in CoC can also be easily detected, and are strongly normalizing (TODO: explain what that means in this context).

CoC yields the fast growing function <math>D(n)</math> (defined by Ralph Loader in 2001 to create the eponymous Loader's number = <math>D^5(99)</math>), which computes something similar to (but larger than) "The largest number \(x\) such that there is a <math>\Sigma^0_1</math> formula <math>\phi(x)</math> such that <math>\phi(x)</math> is true and <math>\exists x\phi(x)</math> is provable in CoC with a proof of length \(\leq \log_2 n\)".<ref>R. Loader, "[https://raw.githubusercontent.com/rcls/busy/master/README.txt busy]". GitHub repository (2003). Accessed 7 October 2026.</ref> <math>D(n)</math> outgrows all recursive functions provably total in <math>Z_\omega</math>; the growth rate of <math>D(n)</math> is sometimes informally written as <math>f_{\operatorname{PTO}(Z_\omega)}(n)</math>, though no fundamental sequence has yet been defined for all ordinals under <math>\operatorname{PTO}(Z_\omega)</math>.

The simplicity of CoC and <math>D(n)</math> has been utilized to show the following bounds for certain busy beaver values:

<math>\operatorname{BB\lambda}(1850) > D^5(99)</math>

<math>\operatorname{S}(1015) > D^5(99)</math><ref>CatsAreFluffy, "[https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe metamath-turing-machines]". GitHub commit (September 2024). Accessed 7 October 2026.</ref>

An even stronger but simpler function, <math>q(n)</math>, has been derived from the structure of [[wikipedia:Rank-into-rank_cardinal|<math>\operatorname{I3}</math>]] elementary embeddings, though it is unknown if this function can be shown to be total in <math>\operatorname{ZFC}</math> alone, and whether it grows faster than the aforementioned <math>D(n)</math>. It is used in the following (conjectured) bounds:

<math>\operatorname{BB\lambda}(201) > q(5)</math>

<math>\operatorname{S}(67) > q^{81}(q(5) - 2)</math><ref>https://discord.com/channels/960643023006490684/1331570843829932063/1339669138510970881</ref><ref>racheline, "[https://wiki.bbchallenge.org/wiki/Talk:Champions Larger Chapmions]". bbchallenge wiki talk post edit (February 2025). Accessed 7 October 2026.</ref>

== Resources ==
The Googology wiki hosts good information on methods of describing large numbers or large upper bounds for numbers.

* https://googology.miraheze.org/wiki/Main_Page

== References ==

[[Category:Stub]]

Instruction-Limited Busy Beaver

2026-09-26T17:54:49Z

Merieke: changed BB(1050) to BB(1015)

An '''n-instruction Turing machine''' is a [[Turing machine]] having an arbitrary number of states and symbols, but having only ''n'' defined transitions/instructions in its transition table (all others are undefined).

The '''Instruction-Limited Busy Beaver function (BBi(n))''' is the maximum steps taken by any n-instruction Turing machines before eventually halting (when started on an initially blank tape). Similarly, the '''Instruction-Limited Symbols Busy Beaver function (Σi(n))''' is the maximum [[sigma score]] (number of non-blank symbols left on the tape at halting) for all halting n-instruction Turing machines (when started on an initially blank tape).

* The currently known values of BBi(n) for n>=0 are: 0, 1, 3, 5, 16, 37, 123, 3932963, .... BBi(8) is at least 6.889 x 101565, which is the number of steps taken by the current 8-instruction champion machine, discovered by Nick Drozd<ref>Drozd, N. (Sept. 2025). "[https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html The Shape of a Turing Machine | Something Something Programming]"</ref>.
* The currently known values of Σi(n) for n>=0 are: 0, 1, 2, 4, 5, 9, 14, 2050, .... Σi(8) is at least 1.355 x 10783, which is the number of non-blank symbols written to tape by the same 8-instruction champion machine.

As with the traditional Busy Beaver sequences, BBi(n) and Σi(n) are both uncomputable, with each growing faster than any computable function.

A TM is considered to halt as soon as it reaches an undefined transition. This convention reduces the number of instructions for a BBi Turing machine. In the traditional BB(n,m) problem, there is no explicit instruction limit (albeit an upper bound of nm) and so it is advantageous to use an explicit halting instruction which allows the machine to take one additional step.

== Motivation ==

The goal of the original Busy Beaver contest, introduced by [[Tibor Radó]], was to find a halting Turing machine of a given size that, when started on a blank tape, either runs for the longest number of steps ('''BB(n)'''), or which prints the largest number of 1’s to tape before halting ('''Σ(n)''')<ref>Tibor Radó (May 1962). "[https://computation4cognitivescientists.weebly.com/uploads/6/2/8/3/6283774/rado-on_non-computable_functions.pdf On non-computable functions]" (PDF). ''Bell System Technical Journal''. '''41''' (3): 877–884. https://doi.org/10.1002%2Fj.1538-7305.1962.tb00480.x</ref>. Originally, the contests considered only two-symbol Turing machines (0,1), so both the steps sequence (BB(1), BB(2), BB(3), ...) and the symbols sequence (Σ(1), Σ(2), Σ(3), ...) were functions of a single variable n, the number of states.

The Busy Beaver contest was later generalized to m-symbol machines (0,1,2,…,m-1), so each contest for n states and m symbols has its own values for maximum steps ('''BB(n,m)'''), and for non-blank symbols written to tape ('''Σ(n,m)'''). While this adds more interesting individual contests, it does split the focus among different possible sequences. For example, it is natural to compare the original steps sequence (BB(n,2) for n = 1, 2, 3, ...) with the steps sequence for 2-state, m-symbol Turing machines (BB(2,m) for m = 1, 2, 3, ...).

The Instruction-Limited Busy Beaver concept was primarily motivated by the goal of uniting the various Busy Beaver contests into a single sequence defined not by the maximum number of states and symbols ('''BB(n,m)'''), but rather by the number of instructions ('''BBi(n)'''). Furthermore, counting the number of instructions in a Turing machine is arguably a simpler way of defining a “machine of a given size” than is considering the numbers of states and symbols in its state table.

A champion n-instruction Busy Beaver machine may lie within any a x b domain, and it may share the same number of steps and symbols written as a machine from a different domain altogether. For example, it was discovered that two different 7-instruction Turing machines take 3,932,963 steps and write 2,050 non-blank symbols, but one of these is a 2-state, 4-symbol machine with one undefined transition, while the other is a 3-state, 3-symbol machine with two undefined transitions.

It is currently unknown what the state table shapes of BBi(n) champions for n>=8 will be. When n = a*b-1 for some a,b >=2, will BBi(n) be one less than one of the original values of BB(a,b)? Or will a more sparsely populated state table with a larger product of states and symbols be able to generate an even longer run time using n instructions? The current BBi(8) champion is a 3-state, 4-symbol machine, but this may still be surpassed by a 3-state, 3-symbol machine yet to be found.

== Champions ==
{| class="wikitable"
!n
!BBi(n)
!Step Champions
!Discovered by
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|0RZ|halt}} {{TM|1RZ---|halt}}
|Trivial
|BB(1,2)
|[[oeis:A384629|A384629]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''3'''
|{{TM|0RB---_1LA---|halt}}
|Trivial
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''5'''
|{{TM|1RB1LB_1LA---|halt}} (and 13 others)
|[[Tibor Radó]]
|[[BB(2)|BB(2,2)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''16'''
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''37'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Allen Brady in 1988
|[[BB(2,3)]] - 1
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''123'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|
|[[oeis:A384629|A384629]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''3,932,963'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Terry & Shawn Ligocki in 2005
|[[BB(2,4)]] - 1, and
A [[BB(3,3)]] high-ranking machine
|[[oeis:A384629|A384629]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 6.889 \times 10^{1,565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Nick Drozd in 2025
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|-
|9
|<math display="block">> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|Daniel Yuan in 2024
|Current [[BB(2,5)]] champion
|
|-
|10
|
|
|
|
|
|-
|11
|<math> > 2 \uparrow^{15} 5</math>
|{{TM|1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|Pavel Kropitz in 2024
|Current [[BB(3,4)]] champion
|
|-
|...
|
|
|
|
|
|-
|14
|<math>> f_{\omega}(2 \uparrow^{15} 5)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC|halt}}
|Racheline in 2024
|Current [[BB(3,5)]] champion
|
|-
|16
|<math>> f_\omega^2(10\uparrow\uparrow\uparrow 6)</math>
|{{TM|1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_------1LE|halt}}
|Jacobzheng in 2026
|Current BB(6,3) champion
|
|-
|20
|<math>> f_\omega^2(10 \uparrow^4 4)</math>
|{{TM|1RB1RA_1LH1RC_0LD0LG_0RE1LD_1RA1RF_---0RA_1LC1LG_1LJ1LI_1LK1LJ_---0LI_1RA1LH|halt}}
|Jacobzheng in 2026
|Current BB(11) champion
|
|-
|23
|<math>> f_{\omega+1}(f_\omega(f_7(2)))</math>
|{{TM|1LD1LE2RA_1RE2LE2RB_1RD2RA2LC_1RB1RD2RB_1LH2RF1LC_---0LF1LG_1RD2LG2LG_0LD1LH1LH|halt}}
|Jacobzheng in 2026
|Current BB(8,3) champion
|
|-
|25
|<math>> f_{\omega + 1}(f_\omega^4(f_7(3)))</math>
|{{TM|0LH0LL_1RI1RC_0LD1LG_0RE1LD_---1RF_---0RM_1RM0LH_1LG1LH_0LN0RJ_---0LK_0LK1LL_1LG1LA_1RB1RM_0LG1LN|halt}}
|Jacobzheng in 2026
|Current BB(14) champion
|
|-
|29
|<math>> f_{\omega+3}^2(4)</math>
|{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_---2LJ---|halt}}
|Jacobzheng in 2026
|Current BB(11,3) champion
|
|-
|39
|<math>> f_{\omega+6}^2(4)</math>
|{{TM|1RR1RG_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RA1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_---1LT_0LH0LS|halt}}
|Jacobzheng in 2026
|Current BB(20) champion
|
|-
|41
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_---0LS_1LH0LT_1LH1LU_0LE1LR|halt}}
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|Current BB(21) champion
|
|-
|78
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1RB2LA1RA2LB2LA_0LA2RB3RB4RA1LQ_---1RC2LD------_0LK3RD2RD2RD4RD_------0RH1LS0RF_1LG1RF2RF---4RF_4LE1LG2LG---4LG_0LI1RH2RH---4RH_0RK0LJ------4RK_2LE1LJ2LJ---4LJ_4RP---0LE---0LK_2RQ---2LL---4LL_2LN---2RM3RM4RM_4RQ2RQ2LN3LN4LN_4LN---2RO3RO4RO_3LL1LP2LP1LQ4LP_1RC3RR1RM3LP0RO_3LL1RR2RR0RR4RR_1RC1LS2LS1LS4LS|halt}} {{TM|1RB2LA1RA2LB2LA------_0LA2RB3RB4RA1LP------_---1RC2LD------------_0LJ3RD2RD2RD4RD------_4LE1LE0RG1LR0RF2LE4LE_1LE1RF5RF---6RF------_0LH1RG2RG---4RG------_0RJ0LI------4RJ------_2LE1LI2LI---4LI------_4RO---0LE---0LJ------_2RP---2LK---4LK------_2LM---2RL3RL4RL------_4RP2RP2LM3LM4LM------_4LM---2RN3RN4RN------_3LK1LO2LO1LP4LO------_1RC3RQ1RL3LO0RN------_3LK1RQ2RQ0RQ4RQ------_1RC1LR2LR1LR4LR------|halt}} {{TM|1RB2LA1RA2LB2LA---------_0LA2RB3RB4RA1LO---------_---1RC2LD---------------_0LJ3RD2RD2RD4RD---------_4LE1LE0RG1LQ0RF2LE4LE---_1LE1RF5RF---6RF---------_0LH1RG2RG---4RG---------_0RJ0LI------4RJ---------_2LE1LI2LI---4LI---------_4RN---0LE---0LJ---------_2RO2RO2LK3LK4LK2LK4LK4RO_2LK---5RL3RL6RL---------_4LK---5RM3RM6RM---------_3LK1LN2LN1LO4LN---------_1RC3RP1RL3LN7RM---------_3LK1RP2RP0RP4RP---------_1RC1LQ2LQ1LQ4LQ---------|halt}}</div></div></div>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(19,5)</ref><ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(18,7)</ref><ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(17,8)</ref>
|Current BB(19,5), BB(18,7) and BB(17,8) champion
|
|-
|101
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|[[Talk:Champions|See talk page]]
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|Current BB(51) champion
|
|-
|237
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|Current BB(42,9) champion
|
|-
|299
|<math>> f_{lim(BMS)}(10\uparrow\uparrow 15)</math>
|[https://morphett.info/turing/turing.html?c95a199c8e8a3dd56452f8b7e28fabbf too large to show]
|Patcail in 2025<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345</ref>
|Current BB(150) champion
|
|-
|2029
|<math>> D^{5}(99)</math> = Loader's Number
|
|CatsAreFluffy in 2024<ref>https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe</ref>
|Current BB(1015) champion
|
|}

{| class="wikitable"
!n
!'''Σ'''i(n)
!Sigma Champions
!Notes
!Reference
|-
| style="text-align: right;" | 1
| style="text-align: right;" | '''1'''
|{{TM|1RZ---|halt}}
|Σ(1,2)
|[[oeis:A384766|A384766]]
|-
| style="text-align: right;" | 2
| style="text-align: right;" | '''2'''
|{{TM|1RB---_1LA---|halt}} (and 7 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829 Shawn]
|-
| style="text-align: right;" | 3
| style="text-align: right;" | '''4'''
|{{TM|1RB1LB_1LA---|halt}} (and 4 others)
|[[BB(2)|Σ(2,2)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353 Shawn]
|-
| style="text-align: right;" | 4
| style="text-align: right;" | '''5'''
|{{TM|1RB0LB---_1LA2RA---|halt}} (and 40 others)
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447 Shawn]
|-
| style="text-align: right;" | 5
| style="text-align: right;" | '''9'''
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|Σ[[BB(2,3)|(2,3)]]
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341 Shawn]
|-
| style="text-align: right;" | 6
| style="text-align: right;" | '''14'''
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|
|[[oeis:A384766|A384766]] [https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207 Shawn]
|-
| style="text-align: right;" | 7
| style="text-align: right;" | '''2,050'''
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
{{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|Σ[[BB(2,4)|(2,4)]], and
A Σ[[BB(3,3)|(3,3)]] high-ranking machine
|[[oeis:A384766|A384766]]
[https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3 List of long-running BB(3,3) TMs]
|-
| style="text-align: right;" | 8
| style="text-align: right;" | <math>> 1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|Current Champion
|[https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252 nickdrozd] [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html Blog] [https://discord.com/channels/960643023006490684/1084047886494470185/1398857599935578303 Shawn]
|}

Notes:
* Proven values for BBi(n) and Σi(n) are shown as bold in the above two tables.
* Solving BBi(ab-1) requires solving BB(a,b) since all halting BB(a,b) TMs can be converted into halting (ab-1)-instruction TMs. Furthermore, <math>BBi(ab-1) \ge BB(a,b) - 1</math> (the -1 at the end is because in BB(a,b) the halting transition counts as a step, but in BBi(ab-1) the TM halts as soon as it reaches the undefined transition).
** Thus solving BBi(8) will require solving [[BB(3,3)]] and in particular, it will require solving [[Bigfoot]], which is a [[Cryptids|Cryptid]].
** So far, BBi(ab-1) champions are also classic BB(a,b) champions (but with the final halting transition removed) for all a,b ≥ 2 explored so far, but it is not known if this trend will continue. With the BBi(8) champion it appears that this trend has been broken (it is believed that <math>BB(3,3) < 10^{1565}</math>)
** Beyond the table above we have intrinsic lower bounds: BBi(9) ≥ BB(2,5) -1, BBi(11) ≥ max(BB(2,6),BB(3,4),BB(4,3),BB(6,2))-1.
* The convention for writing TMs in standard text format here is to include as many states and symbols as are referenced. So for example, we write {{TM|1RZ---|halt}} instead of {{TM|1RZ|halt}} since this TM references 2 symbols (including the implicit blank symbol 0). If one state has no instructions leaving it, the convention is to treat it as the halt state <code>Z</code>.

== Tree Normal Form (TNF) for BBi ==
Shown below are the total number of n-instruction Turing machines for each number of instructions n>=0. The numbers of halting and non-halting machines for n = 8 are not yet known. The function Tree Normal Form for BBi is denoted BBiTNF(n)
{| class="wikitable"
!n
!All Machines
!All Cumulative
!Halting Machines
!Halting Cumulative
!Non-Halting Machines
!Non-Halting Cumulative
|-
| style="text-align: right;" |0
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |1
| style="text-align: right;" |0
| style="text-align: right;" |0
|-
| style="text-align: right;" |1
| style="text-align: right;" |4
| style="text-align: right;" |5
| style="text-align: right;" |2
| style="text-align: right;" |3
| style="text-align: right;" |2
| style="text-align: right;" |2
|-
| style="text-align: right;" |2
| style="text-align: right;" |30
| style="text-align: right;" |35
| style="text-align: right;" |17
| style="text-align: right;" |20
| style="text-align: right;" |13
| style="text-align: right;" |15
|-
| style="text-align: right;" |3
| style="text-align: right;" |378
| style="text-align: right;" |413
| style="text-align: right;" |260
| style="text-align: right;" |280
| style="text-align: right;" |118
| style="text-align: right;" |133
|-
| style="text-align: right;" |4
| style="text-align: right;" |7,640
| style="text-align: right;" |8,053
| style="text-align: right;" |5,581
| style="text-align: right;" |5,861
| style="text-align: right;" |2,059
| style="text-align: right;" |2,192
|-
| style="text-align: right;" |5
| style="text-align: right;" |205,580
| style="text-align: right;" |213,633
| style="text-align: right;" |160,952
| style="text-align: right;" |166,813
| style="text-align: right;" |44,628
| style="text-align: right;" |46.820
|-
| style="text-align: right;" |6
| style="text-align: right;" |7,149,820
| style="text-align: right;" |7,363,453
| style="text-align: right;" |5,843,696
| style="text-align: right;" |6,010,509
| style="text-align: right;" |1,306,124
| style="text-align: right;" |1,352,944
|-
| style="text-align: right;" |7
| style="text-align: right;" |305,333,128
| style="text-align: right;" |312,696,581
| style="text-align: right;" |258,044,501
| style="text-align: right;" |264,055,010
| style="text-align: right;" |47,288,627
| style="text-align: right;" |48,641,571
|-
| style="text-align: right;" |8
| style="text-align: right;" |15,561,793,526
| style="text-align: right;" |15,874,490,107
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
| style="text-align: center;" |?
|}
Enumeration of Turing machines in Tree Normal Form ([[TNF]]) for the Limited-Instruction Busy Beaver problem uses the following procedure:

* The "null Turing machine" with 0 instructions is the root of the tree. This "machine" halts after 0 steps, writing 0 non-blank symbols to the tape.
* The first instruction starts from A, the implied starting state, reads a 0 on the tape (which is blank at machine start) and must shift right (to avoid consideration of trivial mirror-image machines).
** This allows the following four possibilities: A0→0RA, A0→0RB, A0→1RA, and A0→1RB. The two machines transitioning to state A never halt, while the two machines transitioning to state B halt. The two halting machines are next expecting an instruction for B0, but since no such instruction exists, the machines halt after a single step.
* Following the enumeration of machines for each previous number of instructions, each of the halting machines is extended from the instruction at which it halted. For n = 2, this instruction must be B0. (Non-halting machines cannot be extended since they can never reach a new instruction to be added.)
* As per normal TNF procedure, the new instruction can write any symbol to tape, shift left or right, and transition to any state as long as:
** The symbol written can be at most one greater than the highest symbol previously read or written, and
** The state to which the instruction transitions can be at most one state greater than the current maximum state.
* For the BBi problem, any instruction that transitions to a "new" state (one not yet containing any instructions) can be interpreted as a "halting instruction". (e.g. The one-instruction machine A0→1RB is exactly the same as A0→1RH.) Similarly, this new state can be considered a "halt state", as there is functionally no difference between a halt state and an operational state having no instructions. (B = H.)
** Unlike the traditional BB problem, where a halting instruction is normally used in the last remaining cell in an n x m transition table, it is necessary to consider all possible halting transitions for BBi, (not just those of the form →1RH), as this allows the most possible extensions of n-instruction machines to (n+1)-instruction machines.

== Domain-Restricted BBi ==
In addition to the standard Instruction-Limited Busy Beaver function BBi(n), where the numbers of states and symbols a Turing machine may have is unlimited, it may also be useful to consider the “Domain-Restricted BBi”, where the numbers of instructions, states, and/or symbols are specified values or ranges.

Following are some example three-argument functions within the Domain-Restricted BBi. For each function, the value is always the greatest number of steps taken by a Turing machine satisfying the corresponding restrictions:

* BBi(i, n, m): An n-state, m-symbol Turing machine having i instructions
* BBi(i, <=n, m): An i-instruction, m-symbol Turing machine having no greater than n states
* BBi(i, -, m): An i-instruction, m-symbol Turing machine having any number of states
* BBi(i, n, <=m): An i-instruction, n-state Turing machine having no greater than m instructions
* Etc.

Note that we have replaced the argument n from the standard BBi function with i in this three-argument version, primarily to avoid confusion with the n normally used to denote the number of states in a standard BB Turing machine.

For each of the three arguments, we may simply list a value, in which case the Turing machine must have exactly that number of instructions, states, or symbols. Another option is to specify a range, such as “<=n”, or “<=m”, in which case the Turing machine must have n or fewer states, or m or fewer symbols, respectively. And the final option is to give an argument as “-“, in which case, there is no restriction to the Turing machine’s number of instructions, states, or symbols.

'''Example Questions:'''

The main reason for having the flexibility in the way that the arguments are presented within the Domain-Restricted BBi function is that some problems relating to BB and BBi are best suited to examining Turing machines having specific numbers of states and symbols, while others are more practically examined within a range of state and symbol values instead. For example, consider the following two questions, along with two possible ways to interpret and answer each question:

'''Question 1:''' What are the maximum numbers of steps an i-instruction, 5-state, 2-symbol Turing machine can take before halting?

* Interpretation #1: For i = 1 through 10, BBi(i, 5, 2) = -, -, -, -, 9, 31, 65, 793, 47176869, 47176870. In this interpretation, there are no values for 1<=i<=4, because there is no way to construct a 5-state Turing machine using fewer than 5 instructions. Also, the values for i = 5, 6, and 7 are lower than those in the following second interpretation.
* Interpretation #2: For i = 1 through 10, BBi(i, <=5, 2) = 1, 3, 5, 16, 20, 37, 106, 793, 47176869, 47176870. For this interpretation, we do not take the questions wording of “5-state” so literally, but rather we allow the consideration of any Turing machine having no greater than 5 states. For example, BBi(7, <=5, 2) = 106 = BB(4,2) – 1.

The second interpretation, despite being less literally adherent to the question wording than the first, seems more useful for this particular question, specifically because we are probably interested in seeing how the number of steps grows for the best possible arrangement of i instructions within the 5 x 2 state table, and not just among the arrangements that specifically include defined transitions for all five states.

'''Question 2:''' For all n>1, will the 2n-1 instruction, n-state, 2-symbol BBi champion (one undefined transition) always exceed the 2n-1 instruction, n+1 state, 2-symbol BBi champion (3 undefined transitions)? (This is currently conjectured to be false.)

* Interpretation #1: Compare values of BBi(2n-1, n, 2) with those of BBi(2n-1, n+1, 2) for n = 2 through 5:
** For n=2 through 5, BBi(2n-1, n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, n+1, 2) = 4, 19, 65, ???.
* Interpretation #2: Compare values of BBi(2n-1, <=n, 2) with those of BBi(2n-1, <=n+1, 2) for n=2 through 5:
** For n=2 through 5, BBi(2n-1, <=n, 2) = 5, 20, 106, 47176869.
** For n=2 through 5, BBi(2n-1, <=n+1, 2) = 5, 20, 106, ???.

For this second question, the first interpretation clearly seems to be the more useful one, as we are probably interested in comparing the steps differences between machines having strictly n states (with one undefined transition) with machines having strictly n+1 states (with three undefined transitions). Using the second interpretation, information about these steps differences is lost since we have replaced the steps values of the machines with n+1 states with the values from the stronger n-state machines.

'''Selected Results:'''

Following are some results for Domain-Restricted BBi, some of which follow logically from their definitions, and others which have been computed so far:

* BBi(i, -, -) = BBi(i). If we allow any number of states and symbols, then this is the standard BBi function.
* BBi(-, n, m) = BBi(nm, n, m) = BB(n, m) for n>=2, m>=2. If we allow any number of instructions, or if we specify nm instructions, then this is equivalent to the standard BB(n, m) function.
* BBi(nm-1, n, m) = BBi(nm, n, m) - 1 = BB(n, m) - 1 for n>=2, m>=2. With one undefined transition, the number of steps is one less than having a completely filled n x m transition table including a halting instruction, which adds one final step.
* For i>=1, BBi(i, -, 2) = 1, 3, 5, 16, 20, 37, 106, 793, …. BBi(9, -, 2) is at least 47176869, but it may be even larger if a more powerful 6-state, 2-symbol machine having three undefined transitions can be found. BBi(10, -, 2) will almost certainly be greater than the current lower bound of 47176870 = BB(5,2).
* For i>=1, BBi(i, <=2, -) = 1, 3, 5, 13, 37, 123, 3232963, …. The number of states is specified here as “<=2” rather than “2” so that we can have a valid value for one instruction. (A 1-instruction Turing machine has only one operating state.) BBi(8, <=2, -) will almost certainly be greater than the current lower bound of 3232964 = BB(2,4).

== Instruction-Limited Busy Beaver Variants ==
The concept of classifying a Turing machine's size by number of instructions, rather than by the number of states and symbols in its transition table, can also be utilized for Busy Beaver variants. The notation for the instruction-limited version of a variant can conveniently be denoted by appending an "i" to the already existing notation for the variant. Furthermore, the name of the Instruction-Limited version of the variant may be more conveniently started with "I-L" for brevity. Some known results for Instruction-Limited BB variants are as follows:

* I-L Greedy Busy Beaver. gBBi(n) is the largest number of steps taken by an n-instruction Turing machine, but where its first n-1 instructions (in the order encountered during running the machine) must also be a machine taking gBBi(n-1) steps. In other words, once the greedy n-instruction champion machines are identified, all machines having n+1 instructions must be extended from one of those n-instruction champions.

* [[Blanking Busy Beaver Function#Instruction-limited Blanking Busy Beaver (BLBi(n))|I-L Blanking Busy Beaver]].
{| class="wikitable"
|+
!n
!gBBi(n)
!Champions
|-
|1
|1
|{{TM|0RZ|halt}} {{TM|1RZ---|halt}}
|-
|2
|3
|{{TM|0RB---_1LA---|halt}}
|-
|3
|5
|{{TM|0RB---_1LA1RB|halt}} {{TM|0RB------_1LA2LA---|halt}} {{TM|0RB------_1LA2RB---|halt}}
|-
|4
|13
|{{TM|0RB---1LB_1LA2RB---|halt}}
|-
|5
|19
|{{TM|0RB---1LB---_1LA2RB3LA---|halt}}
|-
|6
|25
|{{TM|0RB---1LB------_1LA2RB3LA4LA---|halt}} {{TM|0RB---1LB------_1LA2RB3LA4RB---|halt}} {{TM|0RB---1LB---_1LA2RB3LA1RB|halt}} {{TM|0RB---1LB---_1LA2RB3LA3RB|halt}}
|-
|7
|41
|{{TM|0RB---1LB---3LA_1LA2RB3LA4RB---|halt}} {{TM|0RB---1LB---3LB_1LA2RB3LA4RB---|halt}}
|-
|8
|55
|{{TM|0RB---1LB---3LA---_1LA2RB3LA4RB5RB---|halt}}
|-
|9
|238
|{{TM|0RB---1LB---3LA---_1LA2RB3LA4RB5RB1LB|halt}}
|-
|10
|941
|{{TM|0RB---1LB---3LA1RB_1LA2RB3LA4RB5RB1LB|halt}}
|-
|11
|1341
|{{TM|0RB6RB1LB---3LA1RB---_1LA2RB3LA4RB5RB1LB---|halt}}
|-
|12
|10465
|{{TM|0RB6RB1LB---3LA1RB---_1LA2RB3LA4RB5RB1LB5LA|halt}}
|-
|13
|10675
|{{TM|0RB6RB1LB---3LA1RB7RB---_1LA2RB3LA4RB5RB1LB5LA---|halt}} {{TM|0RB6RB1LB---3LA1RB3RB_1LA2RB3LA4RB5RB1LB5LA|halt}}
|-
|14
|>=9874580
|{{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}
|}

== References ==
<references />
[[category:Functions]]

== See Also ==
* OEIS list of BBi(n) values: https://oeis.org/A384629
* OEIS list of Σi(n) values: https://oeis.org/A384766
* Discussion on Discord: https://discord.com/channels/960643023006490684/960643023530762341/1393697378657374290

[[category:Functions]]

Champions

2026-09-24T20:12:01Z

Merieke: added BB(150,7) still might need to work on a little formatting later

Busy Beaver '''Champions''' are the current record holding [[Turing machine|Turing machines]] which maximize a [[Busy Beaver function]]. In this article we focus specifically on the longest running TMs. Some have been proven to be the longest running of all (and so are the ultimate champion) while others are only current champions and may be usurped in the future. For smaller domains, Pascal Michel's website is the canonical source for [https://bbchallenge.org/~pascal.michel/bbc Busy Beaver champions] and the [https://bbchallenge.org/~pascal.michel/ha History of Previous Champions]. 1-state domains are omitted as [[BB(1,m)]] = 1 for m > 1.

== Trivial Champions ==
[[BB(n,1)]] = n

[[BB(1,m)]] = 1

== 2-Symbol TMs ==
Rows are blank if no champion has been found which surpasses a smaller size problem. Also take note that the <math>f_{x}(n)</math> used in the lower bounds represent the [[Fast-Growing Hierarchy]] while <math>\uparrow</math> represents [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]]. Machines below 6 states are listed in [[Tree Normal Form#TNF-1RB|TNF-1RB]] format, where machines which tie for champion are all listed. Note that most champions above 6 states are self-reported and have not been independently verified.

{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2)]]
|<math>6</math>
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|[[Tibor Radó]]
|Direct Simulation
|-
|[[BB(3)]]
|<math>21</math>
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|Proven by [[Shen Lin]]
|Direct Simulation
|-
|[[BB(4)]]
|<math>107</math>
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|Allen Brady
|Direct Simulation
|-
|[[BB(5)]]
|<math>47\,176\,870</math>
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|Heiner Marxen & Jürgen Buntrock in 1989
|Direct Simulation
|-
|[[BB(6)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|[https://discord.com/channels/960643023006490684/1369339127652159509/1370678203395604562 Pavel Kropitz in 2025]
|Analyzed by Shawn Ligocki (see TM page)
|-
|[[BB(8)]]
|
|
|
|
|-
|BB(9)
|<math>> f_\omega(f_9(2))</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|Jacobzheng in 2024
|
|-
|BB(10)
|<math>> f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|Racheline in 2024
|
|-
|BB(11)
|<math>> f_\omega^2(10 \uparrow^4 4)</math>
|{{TM|1RB1RA_1LH1RC_0LD0LG_0RE1LD_1RA1RF_---0RA_1LC1LG_1LJ1LI_1LK1LJ_1RZ0LI_1RA1LH|halt}}
|Jacobzheng in 2026
|
|-
|BB(12)
|<math>> f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|Racheline in 2024
|
|-
|BB(13)
|<math>> f_{\omega + 1}(2046) > g_{64}</math>
|{{TM|1RB1RA_1LC1RD_1LA1LC_1LG0RE_1LC1RB_0RL1LG_0LM0RH_1RI1RH_1LK0RI_---0LK_1LF1LK_1LJ1RL_1RZ1RH|halt}}
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 50_ft_lock in 2026]
|
|-
|BB(14)
|<math>> f_{\omega + 1}(65\,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
|
|-
|BB(15)
|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|Jacobzheng in 2025
|
|-
|BB(16)
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_1RZ0LJ_1LH1LO_1RB1RP|halt}}
|Jacobzheng in 2025
|
|-
|BB(17)
|
|
|
|
|-
|BB(18)
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|{{TM|0RH1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RE1RA_1RM0LJ_0LJ1LK_1LL1LL_1RG0LK_0LQ1RN_1RZ0LO_1LP1LO_0LG1LO_1LR1LQ_0LP1LQ|halt}}

[[User:Jacobzheng/BB(18)]]
|Jacobzheng in 2025
|
|-
|BB(19)
|
|
|
|
|-
|BB(20)
|<math>> f_{\omega + 6}^2(4)</math>
|{{TM|1RR1RG_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RA1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_1RZ1LT_0LH0LS|halt}}

[[User:Jacobzheng/BB(20)]]
|Jacobzheng in 2026
|
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR|halt}}

[[Talk:Champions|See talk page]]
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|...
|
|
|
|
|-
|BB(40)
|<math>> f_{\omega^\omega}(75\,500)</math>
|[[User:Jacobzheng/BB(40)]]
|Jacobzheng in 2024
|
|-
|BB(41)
|<math>> f_{\omega^\omega}^4(32)</math>
|[[User:Jacobzheng/BB(41)]]
|Jacobzheng in 2024
|
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|[[Talk:Champions|See talk page]]
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|
|-
|BB(150)
|<math>> f_{lim(BMS)}(10\uparrow\uparrow 15)</math>
|[https://morphett.info/turing/turing.html?c95a199c8e8a3dd56452f8b7e28fabbf too large to show]
|Patcail in 2025<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345</ref>
|
|-
|BB(1015)
|<math>> D^{5}(99)</math> = Loader's Number
|
|CatsAreFluffy in 2024<ref>https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe</ref>
|
|}

=== Conjectured ===

{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Notes
|-
|BB(67)
|<math>> q^{81}(q(5)-2)</math>
|[https://wiki.bbchallenge.org/wiki/Talk:Champions]
|[https://discord.com/channels/960643023006490684/1331570843829932063/1339669138510970881 racheline in 2025]
|<math>q(n)</math> is derived from [https://en.wikipedia.org/wiki/Laver%20table Laver tables]. From [https://discord.com/channels/960643023006490684/960643023530762341/1525489299032641657 racheline]: "The known information about q(n) isn't enough to prove that this machine scores higher than the shown champions with fewer states, or that it scores lower than the shown champions with more states."
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,3)]]
|<math>38</math>
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|Allen Brady in 1988
|Direct Simulation
|-
|[[BB(3,3)]]
|<math>> 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|Terry & Shawn Ligocki in 2007
|[https://bbchallenge.org/~pascal.michel/beh#tm33h Analysis by Pascal Michel]
|-
|[[BB(4,3)]]
|<math>> 10 \uparrow^4 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|Pavel Kropitz in 2024
|
|-
|BB(5,3)
|
|
|
|
|-
|BB(6,3)
|<math>> f_\omega^2(10\uparrow\uparrow\uparrow 6)</math>
|{{TM|1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_1RZ---1LE|halt}}
|Jacobzheng in 2026
|
|-
|BB(7,3)
|
|
|
|
|-
|BB(8,3)
|<math>> f_{\omega+1}(f_\omega(f_7(2)))</math>
|{{TM|1LD1LE2RA_1RE2LE2RB_1RD2RA2LC_1RB1RD2RB_1LH2RF1LC_1RZ0LF1LG_1RD2LG2LG_0LD1LH1LH|halt}}
|Jacobzheng in 2026
|
|-
|BB(9,3)
|
|
|
|
|-
|BB(10,3)
|
|
|
|
|-
|BB(11,3)
|<math>> f_{\omega+3}^2(4)</math>
|{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_1RZ2LJ---|halt}}
|Jacobzheng in 2026
|
|-
|...
|
|
|
|
|-
|BB(36,3)
|<math>> f_{\varepsilon_0}(7)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(36,3)</ref>
|
|-
|BB(37,3)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(37,3)</ref>
|
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,4)]]
|<math>3\,932\,964</math>
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|Terry & Shawn Ligocki in 2005
|Pascal Michel, Heiner Marxen, Allen Brady
|-
|[[BB(3,4)]]
|<math>> 2 \uparrow^{15} 5</math>
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|Pavel Kropitz in 2024
|[https://www.sligocki.com/2024/05/22/bb-3-4-a14.html Analysis by Shawn Ligocki]
|-
|...
|
|
|
|
|-
|BB(31,4)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(31,4)</ref>
|
|}

== 5-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,5)]]
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|Daniel Yuan in 2024
|[https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 mxdys in Rocq]
|-
|[[BB(3,5)]]
|<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|Racheline in 2024
|
|-
|...
|
|
|
|
|-
|BB(19,5)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|{{TM|1RB2LA1RA2LB2LA_0LA2RB3RB4RA1LQ_1RZ1RC2LD------_0LK3RD2RD2RD4RD_------0RH1LS0RF_1LG1RF2RF---4RF_4LE1LG2LG---4LG_0LI1RH2RH---4RH_0RK0LJ------4RK_2LE1LJ2LJ---4LJ_4RP---0LE---0LK_2RQ---2LL---4LL_2LN---2RM3RM4RM_4RQ2RQ2LN3LN4LN_4LN---2RO3RO4RO_3LL1LP2LP1LQ4LP_1RC3RR1RM3LP0RO_3LL1RR2RR0RR4RR_1RC1LS2LS1LS4LS|halt}}
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(19,5)</ref>
|-
|BB(94,5)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|
|}

== 6-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,6)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|Pavel Kropitz in 2023
|[https://www.sligocki.com/2023/05/20/bb-2-6-p3.html Analysis by Shawn Ligocki]
|-
|...
|
|
|
|
|-
|BB(61,6)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|
|}

== 7-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,7)]]
|
|(Enumerating Holdouts...)
|
|
|-
|...
|
|
|
|
|-
|BB(18,7)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|{{TM|1RB2LA1RA2LB2LA------_0LA2RB3RB4RA1LP------_0RZ1RC2LD------------_0LJ3RD2RD2RD4RD------_4LE1LE0RG1LR0RF2LE4LE_1LE1RF5RF---6RF------_0LH1RG2RG---4RG------_0RJ0LI------4RJ------_2LE1LI2LI---4LI------_4RO---0LE---0LJ------_2RP---2LK---4LK------_2LM---2RL3RL4RL------_4RP2RP2LM3LM4LM------_4LM---2RN3RN4RN------_3LK1LO2LO1LP4LO------_1RC3RQ1RL3LO0RN------_3LK1RQ2RQ0RQ4RQ------_1RC1LR2LR1LR4LR------|halt}}
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(18,7)</ref>
|-
|BB(46,7)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|
|-
|BB(134,7)
|<math>> f_{\psi_0(\Omega_\omega)}(2050)</math>
|
|LittlePeng9+Wythagoras in 2014<ref name=":0">https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/Analysis_of_LittlePeng9%27s_work/Hydra_machines</ref>
|
|-
|BB(150,7)
|<math>> f_{\psi_0(\varepsilon_\Omega_\omega +1)}(683)</math>
|
|LittlePeng9+Wythagoras in 2014<ref name=":0" />
|
|}

== Zoology ==
There are many types of champions.

Note: the scale here is not formally defined. It mostly serves to estimate the average values produced by a type of champions.
{| class="wikitable"
|+
!Classification
!Description
!Examples
!Scale
|-
|Trivial
|The simplest champions that can exist. They mostly appear in some BB-adjacent functions like [[Fractran|BBf]] or [[CounterScript|BBCS]].
|
|<math>O(n)</math>
|-
|Chaotic
|Have a chaotic behavior with repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA---|halt}}
* {{TM|1RB---_1LB0RC_1LC1LA|halt}}
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|<math>O(2^n)</math>
|-
|Countdown
|Compute a number then "count down" (usually while bouncing) until reaching 0. They are common in some BB-adjacent functions like [[Fractran|BBf]].
|
* {{TM|1RB2LB---_2LA2RB1LB|halt}}
|<math>O(n^2)</math>
|-
|Collatz-like
|Compute a [[Collatz-like]] function. Repeatedly multiply and add a number depending of its modulo until reaching a number with a certain modulo.
|
* {{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|<math>O(2^n)</math>
|-
|Random walk
|Compute a [[Collatz-like]] function with a random walk.
|
|<math>O(2^{2^n})</math>
|-
|Tetrational
|
|
|<math>O(2 \uparrow\uparrow n)</math>
|-
|Ackermannian
|
|
|<math>O(2 \uparrow^n 3)</math>
|}

== References ==

[[Category:Individual machines]]
[[Category:Zoology]]