BusyBeaverWiki - User contributions [en]

Busy Bricklayer Problem

2026-11-04T21:43:00Z

Azerty: created page

The Busy Bricklayer Problem is a polyomino tiling problem for which its decidability is unknown.

== Definition ==
The order of a polyomino is the minimum number of copies required to tile a rectangle (of any finite size with no gaps or overlaps, allowing for rotation and reflection) or zero if it doesn't tile any rectangles. Let the sequence a(n) be the maximum over the orders of all polyominoes of size n.

For example, a(5) = 10. Pentominoes F, N, T, U, V, and W have order 0, pentomino I has order 1, pentominoes L and P have order 2, and the Y pentomino has order 10 (with the following tiling).

== Values ==
{| class="wikitable"
|+
!a(n)
!Value
!Champion
|-
|1
|1
|
|-
|2
|1
|
|-
|3
|2
|
|-
|4
|4
|
|-
|5
|10
|
|-
|6
|92
|
|-
|7
|76
|
|-
|8
|246
|
|-
|9
|4
|
|-
|10
|
|
|}

Champions

2026-11-04T19:34:06Z

Azerty: fixed typo

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]]; see [[measuring large numbers]] for more information. Machines below 6 states are listed in [[Tree Normal Form|TNF]] 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>> 3 \uparrow^{20} 3 > f_\omega(20)</math>
|{{TM|1RB0RG_1RC1RB_0RD1RZ_0LE0LD_0LF1LE_1RA1LF_1LC0RB|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB7/1RB0RG_1RC1RB_0RD---_0LE0LD_0LF1LE_1RA1LF_0LD0RB.md Claimed analysis by Claude] never verified.
|-
|[[BB(8)]]
|<math>> f_\omega^3(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB8/1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG---.md Claimed analysis by Claude] never verified.
|-
|BB(9)
|<math>> f_\omega^7(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI1RZ_1RI1RC|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB9/1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI---_1RI1RC.md Claimed analysis by Claude] never verified.
|-
|BB(10)
|<math>> f_{\omega+1}(2 \uparrow^{36} 3) > g_{64}</math>
|<span style="word-break:break-all">{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC|halt}}</span>
|[https://discord.com/channels/960643023006490684/1331570843829932063/1557806386073702622 Jeff Ketchersid and Claude in 10/2026 (claimed)]
|[https://github.com/jeff87654/busy-beaver/tree/main/BB10 Claimed analysis by Claude] never verified.
|-
|BB(11)
|<math>> f_{\omega+1}(f_\omega^2(5))</math>
|<span style="word-break:break-all">{{TM|1RG0LH_1RK1RC_0LD1LA_0RE1LD_0LI1RF_1RZ0RG_1RB1RG_1LA1LH_0LK0LJ_1LH1LI_1LJ1RE|halt}}</span>
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB11 Claimed analysis by Claude] never verified.
|-
|BB(12)
|<math>> f_{\omega+2}(f_{\omega+1}^2(3))</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK|halt}}</span>
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB12 Claimed analysis by Claude] never verified.
|-
|BB(13)
|<math>> f_{\omega+2}^6(f_{\omega+1}^2(3))</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_1RZ0LH|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB13 Claimed analysis by Claude] never verified.
|-
|BB(14)
|<math>> f_{\omega+2}^9(f_{\omega+1}^2(3))</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LN0LL_0LH1LK_1RZ0LH_0RJ0LI|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB14 Claimed analysis by Claude] never verified.
|-
|BB(15)
|
|
|
|
|-
|BB(16)
|<math>> f_{\omega+2}^{10}(9)</math>
|<span style="word-break:break-all">{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_0RO0LO_1LP1RM_1RZ1LI|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB16 Claimed analysis by Claude] never verified.
|-
|BB(17)
|<math>> f_{\omega+3}^3(9)</math>
|<span style="word-break:break-all">{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_1RQ0LO_1LP1LN_0RB1LI_1RZ1LO|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB17 Claimed analysis by Claude] never verified.
|-
|BB(18)
|<math>> f_{\omega+3}^4(9)</math>
|<span style="word-break:break-all">{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_1RQ0LO_1LP1LN_0RB1LI_1RR0LO_0RO1RZ|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB18 Claimed analysis by Claude] never verified.
|-
|BB(19)
|<math>> f_{\omega + 7}^2(5)</math>
|<span style="word-break:break-all">{{TM|1RQ1RL_0LI1RC_0LD1LF_0RA1LD_1RB1RE_1LH0LG_1LF1LG_1RE1RM_1LJ1LI_0LK1LI_1RL1RA_0RL0RE_0LN1RM_1LO0LO_1RE1LP_1LN1LN_1LR1RQ_1RZ1LS_0LE0LR|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(20)
|<math>> f_{\omega \times 2}(f_{\omega + 2}^2(f_{10}(2)))</math>
|<span style="word-break:break-all">{{TM|1LH0LG_0LI1RC_0LD1LA_0RE1LD_1RQ1RL_1RB1RF_1LA1LG_1RF1RM_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LN1RM_0RF0LO_0RF1LP_0RF1LN_1LR1RQ_1RZ1LS_0LF1LT_0LF0LR|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|<span style="word-break:break-all">{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR|halt}}</span>
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(29)
|<math>> f_{\omega^\omega}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB29/hydra29.md]
|mxdys in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB29 Claimed analysis by Claude] never verified.
|-
|BB(31)
|<math>> f_{\omega \uparrow \uparrow 4}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB31/hydra31.md]
|mxdys in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB31 Claimed analysis by Claude] never verified.
|-
|BB(33)
|<math>> f_{\omega \uparrow \uparrow 6}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB33/hydra33.md]
|mxdys in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB33 Claimed analysis by Claude] never verified.
|-
|BB(48)
|<math>> f_{\varepsilon_1}(3)</math>
|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]
|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726 roromachine14 and Gemini in 2026]
|Not independently verified.
|-
|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>
|Not independently verified.
|-
|BB(1015)
|<math>> D^{5}(99)</math> = Loader's Number
|
|CatsAreFluffy in 2024<ref>https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe</ref>
|Not independently verified.
|}

=== 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
|See Polygon's analysis on the TM page
|-
|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
|Not independently verified.
|-
|BB(7,3)
|
|
|
|
|-
|BB(8,3)
|<math>> f_{\omega+1}(f_\omega^4(f_7(2))) > g_{64}</math>
|<span style="word-break:break-all">{{TM|1RD2RE2LA_2RC2LC2RB_1LH2RF1LA_2RC1RD2RB_---1LC2RE_1RZ0LF1LG_1RD2LG2LG_0LA0LD1LH|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(9,3)
|
|
|
|
|-
|BB(10,3)
|
|
|
|
|-
|BB(11,3)
|<math>> f_{\omega+3}^2(4)</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_1RZ2LJ---|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|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>
|Not independently verified.
|-
|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>
|Not independently verified.
|}

== 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(4,4)]]
|<math>> 2 \uparrow^{509} 3 > f_\omega(508)</math>
|{{TM|0LB3LB1RZ2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_0RD2RC2LB0RA|halt}}
|[[User:Loader3229]] in 2026
|[https://github.com/loader3229/busy-beaver/tree/main/BB44 Claimed analysis by MiniMax] Not independently verified.
|-
|BB(5,4)
|<math>> f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math>
|<span style="word-break:break-all">{{TM|1LD2RA3RB1LC_1LD2LA3RB---_2RA1LA3RC2LC_2RA---2LD2LE_------1RZ1LD|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(6,4)
|<math>> f_{\omega+1}(10\uparrow\uparrow 65\,533) > g_{64}</math>
|<span style="word-break:break-all">{{TM|1LF2RA3RB---_3RD2LB3RB1LC_2RA1LB3RC2LC_1LD2RE0LA1LD_1RZ0LE1LF---_2RA---2LF2LF|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(7,4)
|
|
|
|
|-
|BB(8,4)
|<math>> f_{\omega+2}(f_\omega^4(12))</math>
|<span style="word-break:break-all">{{TM|1RC3LB2RD2RA_1RF3RB1LC2RA_3RB1LB3LC2RC_0RF2LD1LD2LE_3RE3LB2RE2LE_0LG0LH1RZ3RB_1LG2RD1LG3RG_0LH0LC0LH3RD|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(9,4)
|<math>> f_{\omega+3}(16)</math>
|<span style="word-break:break-all">{{TM|2LI3LB2RD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_0RF2LD1LD2LE_3RE3LB2RE2LE_0LG0LH1RZ---_1LG2RD1LG3RG_0LH---0LH3RD_2LB---1LI3RF|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(10,4)
|
|
|
|
|-
|BB(11,4)
|<math>> f_{\omega \times 2}(98)</math>
|<span style="word-break:break-all">{{TM|1RG3LB1RZ2RA_0LG---1LD2RA_3RB1LB3LC2RC_3RE0LA3LC2RC_1RI3RE3LH2LF_3RF2RF2LE1LF_2LC3LH2LJ2RG_0RG3RH1LI2RG_3RH1LH3LI2RI_------3LJ3RK_------2RA3RK|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|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>
|Not independently verified.
|}

== 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
|Not independently verified.
|-
|BB(4,5)
|<math>> f_{\omega+1}(27\,041) > g_{64}</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_2RC2RD1RZ4RB1LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(5,5)
|<math>> f_{\omega+1}(f_\omega^4(6))</math>
|{{TM|1LB3LB4LC2RA4LB_3LE3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ------4LD4LE_3RB------4LE3LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(6,5)
|<math>> f_{\omega+3}(11)</math>
|<span style="word-break:break-all">{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LA4LD4LE_3RB---0RD4LE3LC_4LC------3LF3LF|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(7,5)
|<math>> f_{\omega+3}^2(10)</math>
|<span style="word-break:break-all">{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LG4LD4LE_3RB---1LE4LE3LC_4LC------3LF3LF_3LF------1LG1LG|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(8,5)
|<math>> f_{\omega \times 2+1}(93)</math>
|<span style="word-break:break-all">{{TM|2RG3LB4LC2RA4LB_1RZ---1LD2RA4LD_3RB1LB3LC2RC4LC_3RE---3LC2RC3LC_---3RE4LE4LH4RB_4LE3LG3LB2RF4LG_2LH3RG1LH2RF2RA_3RG1LG3LH2RH4LH|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(19,5)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{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}}</span>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(19,5)</ref>
|Not independently verified.
|-
|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>
|Not independently verified.
|}

== 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(3,6)
|<math>> f_\omega^3(15)</math>
|{{TM|1RB3LB4LC2RA4LB5LB_2LC3RB1LC2RA5LC1RZ_3RB1LB3LC2RC4LC5LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(4,6)
|<math>> f_{\omega+2}(29)</math>
|{{TM|5LD3LB5LC2RA4LB4LC_1RZ3RB1LC2RA3RD0LA_3RB1LB3LC2RC4LC1RA_3RB0LA2LC4LD4LC4LD|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(5,6)
|<math>> f_{\omega+3}(31)</math>
|<span style="word-break:break-all">{{TM|1LE3LB2LA2RA4LB5RA_1RZ3RB1LC2RA3RD4RA_3RB1LB3LC2RC4LC2LB_3RB0LA5LE4LD4LC4LD_4LC5LC0RE5LE1RE2RE|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(6,6)
|<math>> f_{\omega \times 2}(89)</math>
|<span style="word-break:break-all">{{TM|4LF3LB1RZ2RA4LB---_2LC3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB0LF1RB1LF---5RB|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(7,6)
|<math>> f_{\omega^2}(63)</math>
|<span style="word-break:break-all">{{TM|2LE3LB1RZ2RA4LB2LD_0LA3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE0LG4LC3LG5LE_5RB5RB---1LF---5LF_5LF---3RC1LG---5LG|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(8,6)
|<math>> f_{\omega^2+1}(7)</math>
|<span style="word-break:break-all">{{TM|0LC3LB2LH2RA4LB1RZ_3LH3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC2LD3RD_5RB3RE---4LC3LG5LE_5RB5RB---1LF5LF5LF_5LF4RG1LB1LG5LD5LG_3LE---1LH1LH---5RG|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|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>
|Not independently verified.
|}

== 7-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,7)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA_2LA6RA4RB1RZ3LB6LA4RB|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1563057558011252761 roromachine14 in 2026]
|Not independently verified.
|-
|BB(3,7)
|<math>> f_{\omega+1}(117) > g_{64}</math>
|{{TM|6LC3LB6RA2RA4LB5LC5LA_1RZ3RB1LC2RA3RB4LC5RA_3RB1LB3LC2RC4LC1RA6LA|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(4,7)
|<math>> f_{\omega+2}(124)</math>
|{{TM|6LD3LB5LD2RA4LB3LC5LA_1RZ3RB1LC2RA3RD---2RA_3RB1LB3LC2RC4LC3LC5LD_3RB---6LC4LD4LC4LD---|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(18,7)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{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}}</span>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(18,7)</ref>
|Not independently verified.
|-
|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>
|Not independently verified.
|-
|BB(134,7)
|<math>> f_{\psi_0(\Omega_\omega)}(2\,050)</math>
|
|LittlePeng9+Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/Analysis_of_LittlePeng9%27s_work/Hydra_machines</ref>
|Not independently verified.
|}

== 8-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|BB(2,8)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA|halt}}
|[https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804 roromachine14 in 2026]
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(17,8)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|1RB2LA1RA2LB2LA---------_0LA2RB3RB4RA1LO---------_0RZ1RC2LD---------------_0LJ3RD2RD2RD4RD---------_4LE1LE0RG1LQ0RF2LE4LE---_1LE1RF5RF---6RF---------_0LH1RG2RG---4RG---------_0RJ0LI------4RJ---------_2LE1LI2LI---4LI---------_4RN---0LE---0LJ---------_2RO2RO2LK3LK4LK2LK4LK4RO_2LK---5RL3RL6RL---------_4LK---5RM3RM6RM---------_3LK1LN2LN1LO4LN---------_1RC3RP1RL3LN7RM---------_3LK1RP2RP0RP4RP---------_1RC1LQ2LQ1LQ4LQ---------|halt}}</span>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(17,8)</ref>
|Not independently verified.
|}

== 9-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|BB(2,9)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA8RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA4RA|halt}}
|[https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804 roromachine14 in 2026]
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(42,9)
|<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>
|Not independently verified.
|}

== 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]].
|
* {{TM|1RZ---|halt}}
* {{TM|0RA_0RZ|halt}}
* {{TM|0RA_0RB_0RZ|halt}}
|<math>O(n)</math>
|-
|Chaotic
|Have a chaotic behavior with repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA1RZ|halt}}
* {{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
* {{TM|1RB1LB_1LA0LC_1RZ1LD_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|1RB2LB1RZ_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_1LB1LA3RB1RZ|halt}}
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|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]]

De Bruijn index

2026-10-17T07:04:04Z

Azerty: Updated champion 21

'''De Bruijn index''' is an alternative method to represent [[lambda calculus]] expressions.

Its attendant Busy Beaver problem is '''BBλ_db''' which uses De Bruijn index instead of binary to evaluate lambda calculus expression size. To calculate size, convert the lambda calculus expression into [[wikipedia:De_Bruijn_index|De Bruijn index]], then count the number of backslashes (lambdas) and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form (like BBλ(m) for m ≤ 20).
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|12
|<math>> 10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|50_ft_lock
|-
|15
|<math>> f_{\omega+1}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|Gustavo Melo
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|50_ft_lock
|-
|19
|<math>> f_{\omega^\omega2}(f^2_{\omega^\omega}(3))</math>
|<code>(\1 1 (\1 (\\\1 3 2 5) 2) 1) (\\2 (2 1)))</code>
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|20
|<math>> f_{\omega^\omega2+1}(7)</math>
|<code>(\1 1 1 (\1 (\\\1 3 2 5) 2) 1) (\\2 (2 1)))</code>
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|21
|<math>> f_{\omega^{\omega^{2}+\omega+2}}(2)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|-
|22
|<math>> f_{\omega^{\omega^{14}}}(2)</math>
|<code>(\1 1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

[[Category:Functions]]

Lambda Calculus

2026-10-17T06:59:33Z

Azerty: Fixed formatting error

'''Lambda calculus''' is a model of computation developed by Alonzo Church (of Church-Turing thesis fame) in 1932. If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with [[wikipedia:Lambda_calculus|this article]].

'''BBλ''' is the [[Busy Beaver]] function for lambda calculus, where BBλ(n) is the maximum normal form size of any closed lambda term of size n (or 0 if no closed term of size n exists). Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus], a binary prefix-free encoding for all closed lambda calculus terms. Pioneered by John Tromp, BBλ is uncomputable, and therefore grows faster than any computable function.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if every beta-reduction sequence leads to this normal form and '''weakly normalizing''' if some sequence does (while others may diverge).

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) = 0 for those values.

<math>C(n)</math> denotes Church numeral ''n'' = <math>\lambda f\lambda x. f^n(x)</math>.

In the last column, JT and BF abbreviate John Tromp and Bertram Felgenhauer.

We use the ordinal collapsing functions defined by Buchholz (1986). It is suspected (though not yet confirmed) that we used the fundamental sequences described by Maksudov (2017), or, in the case of lim(BMS), the sequence lim(BMS)[n] = BMS((0)(1,1,...,1)) w/n 1s.

The [https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 smallest Cryptid known] currently is in 74 bits.

<div class="toccolours mw-collapsible mw-collapsed">'''Small Champions'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
| 4 || = 4 || <math>\lambda 1</math>
|
|JT & BF
|-
| 6 || = 6 || <math>\lambda\lambda 1</math>
|
|JT & BF
|-
| 7 || = 7 || <math>\lambda\lambda 2</math>
|
|JT & BF
|-
| 8 || = 8 || <math>\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 9 || = 9 || <math>\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
| 10 || = 10 || <math>\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 11 || = 11 || <math>\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
| 12 || = 12 || <math>\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 13 || = 13 || <math>\lambda\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
| 14 || = 14 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 15 || = 15 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
| 16 || = 16 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
| 17 || = 17 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
|18
|= 18
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|-
|19
|= 19
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|
|JT & BF
|-
|20
|= 20
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|
|JT & BF
|}
</div></div>

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|21 || = 22 || <math>\lambda(\lambda 1 1) (1 (\lambda 2))</math>
| <math>\lambda(1(\lambda 2))(1(\lambda 2))</math>|| JT & BF
|-
|22 || = 24 || <math>\lambda(\lambda 1 1 1) (1 1)</math>
| <math>\lambda(1 1) (1 1) (1 1)</math>|| JT & BF
|-
|23 || = 26 || <math>\lambda(\lambda 1 1) (1 (\lambda\lambda 2))</math>
| <math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>|| JT & BF
|-
|24 || = 30 || <math>\lambda(\lambda 1 1 1) (1 (\lambda 1))</math>
| <math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>|| JT & BF
|-
|25 || = 42 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|26 || = 52 || <math>(\lambda 1 1) (\lambda\lambda 2 (1 2))</math>
| <math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>|| JT & BF
|-
|27 || = 44 || <math>\lambda\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|28 || = 58 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 (\lambda 2))))</math>
| <math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>|| JT & BF
|-
| 29 || = 223|| <math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
| <math>\lambda B (B (1 B)) \text{ where}</math>
<math>B = (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (2 1))))</math>
||JT & BF
|-
|30
|= 160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|
<math>\lambda\lambda 2 B A (1 (2 B A)) \text{ where}</math>
<math>B = (\lambda\lambda 2 A (1 (2 A)))</math>,
<math>A = (\lambda\lambda 2 (1 2))</math>
|JT & BF
|-
|31
|= 267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|
<math>\lambda\lambda 2 A (2 A (C (2 A))) \text{ where}</math>
<math>C = (2 A (2 A (1 B (2 A))))</math>,
<math>B = (\lambda 3 A (3 A (1 (3 A))))</math>,
<math>A = (\lambda\lambda 2 (2 (1 2)))</math>
|JT & BF
|-
|32
|= 298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|
|JT & BF
|-
|33
|= 1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|
<math>\lambda C (C (C (1 C))) \text{ where}</math>
<math>C = (B (B (B (1 B)))</math>,
<math>B = (A (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (1 (2 1)))))</math>
|JT & BF
|-
|34 || = 327686
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^2}})</math>|| JT & BF
|-
|35 || <math>= 5 \cdot 3^{3^3} + 6</math>
<math> > 3.8 \times 10^{13}</math>
| <math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^3})</math>|| JT & BF
|-
|36 || <math>= 5 \cdot 2^{2^{2^3}} + 6 > 5.7 \times 10^{77}</math>
| <math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
| <math>C(2^{2^{2^3}})</math>|| JT & BF
|-
|37 || <math> = 2 + BB\lambda(35)</math>
| <math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>\lambda x. C(3^{3^3})</math>||mxdys & JT & dyuan & sligocki
|-
|38 || <math>= 5\cdot{2^{2^{2^{2^2}}}} + 6 > 10^{19,729}</math>
| <math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^{2^2}}})</math>|| JT & BF & CppDS & mxdys & sligocki & dyuan & charles
|-
|39 || <math>= 5\cdot{3^{3^{3^3}}} + 6 > 10^{10^{12}}</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{3^3}})</math>|| JT & BF & CppDS & mxdys & sligocki & dyuan & charles & 2014MELO & mammilaria
|-
|40 || <math> \approx 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{15} 33</math>
|| mxdys & racheline
|-
|41 || <math>= 5 \times 3^{3^{85}} +6 > 10^{1.7 \times 10^{40}} \approx 10^{10^{40}}</math>
| <math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge 2 + BB\lambda(40)</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math>,
<math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>
||
|-
|45 || <math> \ge 2 + BB\lambda(43)</math>
| <math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| ||
|-
|46 || <math> \ge 2 + BB\lambda(44)</math>
| <math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|47 || <math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
| <math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
| ||50_ft_lock
|-
|48 || <math> > 10 \uparrow\uparrow\uparrow\uparrow 4</math>
| <math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math> > Graham's number
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) )</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right) )</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|70
|<math>> f_{\omega^{\omega}2}\left(f_{\omega^{\omega}}\left(f_{\omega^{\omega}}\left(3\right)\right)\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda 1 3 2 5) 2) 1) (\lambda\lambda 2 (2 1))</math>
|
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|...
|
|
|
|
|-
|74
|<math>> f_{\omega^{\omega}2 + 1}\left(7\right)</math>
|<math>(\lambda 1 1 1 (\lambda 1 (\lambda\lambda\lambda 1 3 2 5) 2) 1) (\lambda\lambda 2 (2 1))</math>
|
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|...
|
|
|
|
|-
|76
|<math>> f_{\omega^{\omega^{3}}}(2)</math>
|<math>(\lambda11(\lambda\lambda\lambda\lambda14321)1111)(\lambda\lambda2(21))</math>
|<math>C(f_{\omega^{\omega^{3}}}(2))</math>
|
|-
|...
|
|
|
|
|-
|80
|<math>>f_{\omega^{\omega^{14}}}(2)</math>
|<math>(\lambda111(\lambda\lambda\lambda\lambda14321)1111)(\lambda\lambda2(21))</math>
|<math>C(f_{\omega^{\omega^{14}}}(2))</math>
|
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega65536}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math> > TREE(G64)
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|95
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|- ||||||-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|- ||||||-
|201
| > q(5)
|too large to show
|q is only proven to exceed all functions provably recursive in PA; the true value is likely much larger based on informal analysis
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam JT & BF & 50_ft_lock]
|- ||||||-
|331
| > f<sup>3</sup><sub>lim(BMS)</sub>(5)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & JT & 50_ft_lock]
|- ||||||-
|1797
|> D⁵(99) = Loader's number
|too large to show
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 JT & BF & DropperAUGL]
|}

== Optimal Busy Beaver ==

A closely related BBλ2 function, defined as OEIS sequence (A361211)[https://oeis.org/A361211], allows the lambda term access to (undelimited) raw binary data, which suffices to make it an optimal busy beaver function amongst all binary self-delimiting ones.

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ<sub>1</sub> by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ<sub>1</sub> be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ<sub>1</sub> up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!Champion
!BBλ<sub>1</sub>
|-
|1
|
|0
|-
|2
|<math>1</math>
|1
|-
|3
|
|0
|-
|4
|<math>\lambda 1</math>
|4
|-
|5
|<math>\lambda 2</math>
|5
|-
|6
|<math>\lambda \lambda 1</math>
|6
|-
|7
|<math>\lambda \lambda 2</math>
|7
|-
|8
|<math>1 (\lambda 1)</math>
|<math>f(4) = 26</math>
|-
|9
|<math>\lambda \lambda \lambda 2</math>
|9
|-
|10
|<math>1 (\lambda \lambda 1)</math>
|<math>f(6) = 36</math>
|-
|11
|<math>1 (\lambda \lambda 2)</math>
|<math>f(7) = 41</math>
|-
|12
|<math>1 (1 (\lambda 1))</math>
|<math>f^{2}(4) = 266</math>
|-
|13
|<math>1 (\lambda \lambda 2)</math>
|<math>f(9) = 51</math>
|-
|14
|<math>1 (1 (\lambda \lambda 1))</math>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<math>1 (1 (\lambda \lambda 2))</math>
|<math>f^{2}(7) = f(41) = 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<math>1 (1 (1 (\lambda 1)))</math>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|<math>f^2(9) = f(51)</math>
|-
|18
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|<math>f^4(4) </math>
|-
|19
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|<math>f^3(7)</math>
|-
|20
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|<math>f^6(4)</math>
|-
|21
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|<math>f^7(4)</math>
|-
|22
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|<math>f^{26}(4)</math>
|-
|...
|
|
|-
|28
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|-
|29
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^{BB \lambda(f^4(4))+4}(4))+BB \lambda(f^4(4))+5}(4)</math>
|}
We can generalize BBλ<sub>1</sub> to BBλ<sub>α</sub> for ordinals α by using oracle function BBλ<sub>α-1</sub> for successor ordinal α, and oracle function (\n -> BBλ<sub>α[n]</sub>(n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== See Also ==
* [[Busy Beaver for SKI calculus|SKI calculus]]
* [[De Bruijn index]]
* https://oeis.org/A333479
* [https://www.mdpi.com/1099-4300/28/5/494 The Largest Number Representable in 64 Bits]. 26 Apr 2026. John Tromp.
* [https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861 Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
* https://github.com/ccz181078/BLC.git
* https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw
* Buchholz, W. (1986). A new system of proof-theoretic ordinal functions. ''Annals of Pure and Applied Logic, 32'', 195-207. https://doi.org/10.1016/0168-0072(86)90052-7
* Maksudov, D. (2017, April). ''Travelling To The Infinity - The extended Wilfried Buchholz's functions.'' Travelling To The Infinity. https://sites.google.com/site/travelingtotheinfinity/the-extension-of-buchholz-s-function
[[category:Functions]]

De Bruijn index

2026-10-16T14:35:50Z

Azerty: Added new champion.

'''De Bruijn index''' is an alternative method to represent [[lambda calculus]] expressions.

Its attendant Busy Beaver problem is '''BBλ_db''' which uses De Bruijn index instead of binary to evaluate lambda calculus expression size. To calculate size, convert the lambda calculus expression into [[wikipedia:De_Bruijn_index|De Bruijn index]], then count the number of backslashes (lambdas) and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form (like BBλ(m) for m ≤ 20).
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|12
|<math>> 10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|50_ft_lock
|-
|15
|<math>> f_{\omega+1}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|Gustavo Melo
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|50_ft_lock
|-
|19
|<math>> f_{\omega^\omega2}(f^2_{\omega^\omega}(3))</math>
|<code>(\1 1 (\1 (\\\1 3 2 5) 2) 1) (\\2 (2 1)))</code>
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|20
|<math>> f_{\omega^\omega2+1}(7)</math>
|<code>(\1 1 1 (\1 (\\\1 3 2 5) 2) 1) (\\2 (2 1)))</code>
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|21
|<math>> f_{\omega^\omega2+1}(32,767)</math>
|<code>(\1 1 1 1 (\1 (\\\1 3 2 5) 2) 1) (\\2 (2 1)))</code>
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|22
|<math>> f_{\omega^{\omega^{14}}}(2)</math>
|<code>(\1 1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

[[Category:Functions]]

De Bruijn index

2026-10-14T08:13:40Z

Azerty: Added new champions.

'''De Bruijn index''' is an alternative method to represent [[lambda calculus]] expressions.

Its attendant Busy Beaver problem is '''BBλ_db''' which uses De Bruijn index instead of binary to evaluate lambda calculus expression size. To calculate size, convert the lambda calculus expression into [[wikipedia:De_Bruijn_index|De Bruijn index]], then count the number of backslashes (lambdas) and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form (like BBλ(m) for m ≤ 20).
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|12
|<math>> 10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|50_ft_lock
|-
|15
|<math>> f_{\omega+1}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|Gustavo Melo
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|50_ft_lock
|-
|19
|<math>> f_{\omega^\omega2}(f^2_{\omega^\omega}(3))</math>
|<code>(\1 1 (\1 (\\\1 3 2 5) 2) 1) (\\2 (2 1)))</code>
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|20
|<math>> f_{\omega^\omega2+1}(7)</math>
|<code>(\1 1 1 (\1 (\\\1 3 2 5) 2) 1) (\\2 (2 1)))</code>
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|21
|<math>> f_{\omega^\omega2+1}(32,767)</math>
|<code>(\1 1 1 1 (\1 (\\\1 3 2 5) 2) 1) (\\2 (2 1)))</code>
|Gustavo Melo & 50_ft_lock & Sam Bolsover
|-
|22
|<math>> f_{\omega^{\omega^2}}(65,536)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

[[Category:Functions]]

Talk:Busy Beaver Frontier

2026-10-14T07:55:22Z

Azerty: Undo revision 9085 by Azerty (talk)

I'm adding a few more guesses:

I think the first busy beaver function values that exceed Graham's number are BB(7), BB(4,3), BB(3,4), and BB(2,6).

I think the first Busy Beaver function values that exceed the largest numbers ever constructed by humans using computable methods are BB(14), BB(8,3), BB(6,4), BB(5,5), BB(4,6), BB(3,8), and BB(2,13).

I think BB(6) won't exceed <math>f_\omega(1000)</math>, BB(3,3) won't exceed <math>10^{1000}</math>, and BB(2,5) won't exceed <math>10\uparrow\uparrow 1000</math>.

[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 15:24, 13 October 2026 (UTC)

Talk:Busy Beaver Frontier

2026-10-14T07:53:09Z

Azerty:

I'm adding a few more guesses:

I think the first busy beaver function values that exceed Graham's number are BB(7), BB(4,3), BB(3,4), and BB(2,6).

I think the first Busy Beaver function values that exceed the largest numbers ever constructed by humans using computable methods are BB(14), BB(8,3), BB(6,4), BB(5,5), BB(4,6), BB(3,8), and BB(2,13).

I think BB(6) won't exceed <math>f_\omega(36)</math>, BB(3,3) won't exceed <math>10^{1000}</math>, and BB(2,5) won't exceed <math>10\uparrow\uparrow 1000</math>.

[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 15:24, 13 October 2026 (UTC)

Talk:Champions

2026-10-13T06:14:06Z

Azerty: /* Should we even count llm-"""discovered""" champions? */

== Should we even count llm-"""discovered""" champions? ==
The recent BB(8) and BB(9) champions were discovered by LLMs. Although this might seem like a good thing (more champions = better, right?), we have to think about the future of the community. Right now, there is still a place for humans to contribute. But what will happen when AI is infinitely better than humans? Then there will be nothing left to do except to constantly refresh an automatically updated wiki page as AI automatically solves all BB values and variants. I do not want to live in such a dystopia. I want this community to be a place where math enthusiasts and professionals can do new, novel research and compute the uncomputable. Please keep this community alive. -- [[User:Δ⁵|Δ⁵]] ([[User talk:Δ⁵|talk]]) 03:40, 9 October 2026 (UTC)
:You're really overestimating how good LLMs are at doing this. Sure, they may be able to create large champions but for verifying large busybeaver values, using AI would be incredibly expensive. There will always be significant human intervention in this community. [[User:Etalon|Etalon]] ([[User talk:Etalon|talk]]) 17:28, 12 October 2026 (UTC)
:I believe that LLMs will keep improving exponentially to the point of largely surpassing humans in mathematics, like the best chess bots compared to human grandmasters. One year ago, frontier models could not solve any open math problem on their own. Now, they solve hundreds of problems every month. But in my opinion, maths is discovered and the most interesting part is not merely to solve open problems. It is to actually understand theorems, results, and proofs. [[User:Azerty|Azerty]] ([[User talk:Azerty|talk]]) 06:13, 13 October 2026 (UTC)

== Self-Reporting and Verification ==
This champions page has ended up with a lot of self reported and (I assume) not independently verified champions. I'm conflicted about this: On the one hand I want this wiki to have an encyclopedic/Wikipedia style: it would include things which are known, established and verified. On the other hand, I don't want to stifle innovation or gatekeep what counts as proper verification. As a compromise I have updated the table to add a "Verification" column. The idea here is to indicate which results have been independently verified by another contributor. My idea is that that verification should include some sort of write-up (wiki page, blog post, etc) describing the TM and how it is known to run this long. I have linked to examples of my own for Pavel's BB(6) and Daniel's BB(16) champions. I would love to see some independent analyses of other TMs here (especially smaller ones like Racheline's BB(14) > Graham champion). [[User:Sligocki|Sligocki]] ([[User talk:Sligocki|talk]]) 16:19, 10 December 2024 (UTC)
:Particularly interesting is Racheline's BB(14) > Graham champion. --[[User:Konkhra|Konkhra]] ([[User talk:Konkhra|talk]]) 01:53, 12 December 2024 (UTC)

==Larger champions==
What are the known two-symbol champions beyond BB(16)? Vielhaber, Chacón, and Ceballos's paper "[https://arxiv.org/abs/2303.02855v1 Friedman's 'Long Finite Sequences': The End of the Busy Beaver Contest]" gives a 2450-state two-symbol busy beaver halting after at least n(4) steps, where n is Friedman's [https://googology.miraheze.org/wiki/Block_subsequence_theorem block subsequence function], but this is a very large jump up in state count from 14 states for f<sub>ω+1</sub>(65536). (This paper has some mistakes with their (symbol,state count) notation for TMs, for example referring to Aaronson's and Yedidia's machine as a (7910,2) machine rather than a (2,7910) machine, so maybe the machine whose state count I wrote here is the wrong one.) [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 22:00, 17 August 2024 (UTC)

:Today i found some bounds on BB(20) and BB(21), though i'm not sure if they're new. <math>\Sigma(20) > f_{\omega+2}^2(21)</math> due to {{TM|1LR0LR_0RJ1RG_0RD1LC_1RQ1RE_1LO0RQ_1LH1LF_0LC1LH_0LI0LF_1RD0LF_---0RK_0LM1LL_0LL1LM_1LE1LN_0LF0LM_0RP1LO_0LF0RQ_1RB1RQ_0RS1LA_1LT1RS_0LP0RR|halt}} and <math>\Sigma(21) > f_{\omega^2}^2(4 \uparrow\uparrow 341)</math> due to {{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_---0LS_1LH0LT_1LH1LU_0LE1LR|halt}}.
:The old googology wiki claims [https://googology.fandom.com/wiki/Busy_beaver_function#Larger_values some other bounds], two of which are already implied by the bounds mentioned above, but <math>\Sigma(85)>f_{\varepsilon_0}(1907)</math> and the bounds for more than 2 symbols seem to still be the best known.
:[[User:Racheline|Racheline]] ([[User talk:Racheline|talk]]) 23:08, 17 August 2024 (UTC)
::Racheline, congratulations on discovering a TM that crushes Graham's number. I was very impressed. but I think that in reality BB(8,2) will beat Graham's number. But it is unrealistic for a human to build such a machine. Only to search in the wild. Unfortunately. Nevertheless, congratulations again and further success!!! --[[User:Konkhra|Konkhra]] ([[User talk:Konkhra|talk]]) 03:29, 18 August 2024 (UTC)
:::And how about an article about BB(14)?--[[User:Konkhra|Konkhra]] ([[User talk:Konkhra|talk]]) 03:30, 18 August 2024 (UTC)

IMHO, beyond Graham's number I think we should only list highlights (if anything). Like first TM bigger than blah for various notable blah. Otherwise the churn on this page will be high and the quality low. But I'm open to other points of view. If we do extend this much further, I think maybe we should have a more rigorous process for getting new results demonstrated before they are added. --[[User:Sligocki|sligocki]] ([[User talk:Sligocki|talk]]) 02:57, 18 August 2024 (UTC)

Can you design very large champions as beat the <math>f_{\psi(\varepsilon_{\Omega+1})}(1000)</math>, and beat the <math>f_{\psi(\Omega_\omega)}(1000)</math>, limit for [https://googology.fandom.com/wiki/Bashicu_matrix_system BMS] or beat [https://googology.fandom.com/wiki/Loader%27s_number Loader's number]? --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 00:45, 17 September 2024 (UTC)

: Later today I can try writing one (or at least starting to write one today), but my machine would likely have thousands of states. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 15:50, 16 September 2024 (UTC)

:: I now have two-symbol machines with the following running times. In order to squeeze the best possible results out of Bashicu matrix system, some machines have a different function applied to <math>n</math> at each step (this function is what Bashicu calls the "activation function"), although at this scale it doesn't make too much of a difference.

:: {| class="wikitable"
|+ Busy Beaver lower bounds using BM4
|-
! State count !! Running time !! Activation function
|-
| 5889 || <math> >f_{\psi(\varepsilon_{\Omega+1})}(1000)</math> (using BMS to define FSes) || <math>n\mapsto n+1</math>
|-
| 5894 || <math> >f_{\psi_0(\Omega_\omega)}(1000)</math> (using BMS to define FSes) || <math>n\mapsto n+1</math>
|-
| 5908 || <math> >(0,0,0,0,0)(1,1,1,1,1)[2]</math> || <math>n\mapsto n+2</math>
|-
| 5923 || <math> >(0,0,0,0,0)(1,1,1,1,1)[5]</math> || <math>n\mapsto n+1</math>
|-
| 5941 || <math> >(0,0,0,0,0,0,0,0,0,0)(1,1,1,1,1,1,1,1,1,1)[10]</math> || <math>n\mapsto n+1</math>
|-
| 5978 || <math> >(\underbrace{0,0,\ldots,0,0}_{889})(\underbrace{1,1,\ldots,1,1}_{889})[894]</math> || <math>n\mapsto n+6</math>
|-
| 6020 || <math> >(\underbrace{0,0,\ldots,0,0}_{N})(\underbrace{1,1,\ldots,1,1}_{N})[N]</math> where <math>N</math> is <math>2^{2^{2^{2059}}}</math> || <math>n\mapsto n+5</math>
|}

:: '''Edit:''' It turns out CatIsFluffy [https://github.com/CatsAreFluffy/metamath-turing-machines beat me to it]! I don't know how many states the BMS machine has, but [https://cosearch.bbchallenge.org/contribution/m5k4ulm8 BB(1094) > Loader's number], now down to [https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe BB(1015)]. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 11:37, 4 October 2024 (UTC)

Who can design a [https://googology.fandom.com/wiki/Laver_table Laver table] q function level machine. --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 06:57, 13 February 2025 (UTC)

: Me, probably. This 67-state TM should leave over q<sup>81</sup>(q(5)-2) 1s on the tape (maybe i have an off-by-one or off-by-two error in the exponent of q but it's still iterated quite a few times):
<pre>
M = (
((1, 1, 1), (0, -1, 3)),
((0, -1, 2), (0, 1, 1)),
((1, 1, 0), (1, -1, 2)),
((0, -1, 4), (1, -1, 0)),
((0, -1, 2), (1, -1, 5)),
((0, 1, 26), (0, 1, 5)),
((1, -1, 57), (0, 1, 26)),
((0, 1, 38), (1, 1, 8)),
((0, 1, 9), (1, 1, 8)),
((0, 1, 62), (0, 1, 10)),
((0, -1, 16), (0, -1, 11)),
((1, -1, 12), (1, -1, 11)),
((0, 1, 15), (0, 1, 13)),
((1, 1, 14), (1, 1, 13)),
((0, -1, 16), (0, -1, 11)),
((0, 1, 10), (1, 1, 15)),
((1, -1, 33), (0, -1, 17)),
((1, -1, 18), (1, -1, 17)),
((0, -1, 19), (1, -1, 18)),
((0, -1, 20), (1, -1, 19)),
((0, 1, 22), (1, -1, 21)),
((1, 1, 21), (0, 1, 22)),
((0, 1, 23), (1, 1, 22)),
((1, -1, 24), (1, 1, 23)),
((1, -1, 30), (0, 1, 25)),
((1, -1, 26), (1, 1, 25)),
((0, 1, 6), (0, -1, 27)),
((0, -1, 28), (1, -1, 27)),
((0, -1, 29), (1, -1, 28)),
((1, -1, 20), (1, -1, 29)),
((1, 1, 32), (0, -1, 31)),
((1, -1, 32), (1, -1, 31)),
((0, 1, 7), (0, -1, 30)),
((0, -1, 34), (0, -1, 33)),
((0, -1, 34), (0, -1, 35)),
((0, 1, 37), (1, 1, 36)),
((0, 1, 36), (1, -1, 16)),
((0, 1, 37), (0, 1, 7)),
((0, 1, 39), (0, 1, 7)),
((0, -1, 44), (0, -1, 40)),
((0, -1, 41), (1, -1, 40)),
((1, 1, 42), (1, -1, 41)),
((0, 1, 43), (1, 1, 42)),
((1, 1, 39), (1, 1, 43)),
((0, -1, 45), (1, -1, 44)),
((0, -1, 47), (0, -1, 46)),
((1, -1, 45), (1, -1, 46)),
((1, -1, 47), (1, 1, 48)),
((0, -1, 49), (1, 1, 48)),
((0, 1, 56), (0, -1, 50)),
((0, 1, 49), (0, 1, 51)),
((1, 1, 52), (1, 1, 51)),
((0, -1, 55), (0, -1, 53)),
((1, -1, 54), (1, -1, 53)),
((0, 1, 51), (0, 1, 51)),
((0, -1, 50), (1, -1, 55)),
((1, -1, 57), (1, 1, 56)),
((1, -1, 61), (0, -1, 58)),
((1, 1, 59), (1, -1, 58)),
((0, -1, 60), (0, 1, 56)),
((1, 1, 61), (1, -1, 60)),
((1, -1, 66), (0, 1, 7)),
((0, 1, 63), (0, 1, 62)),
((0, 1, 63), (0, 1, 64)),
((0, -1, 65), (1, 1, -1)),
((0, -1, 65), (1, -1, 66)),
((0, -1, 40), (1, -1, 66))
)

(start from state 0)
</pre>
: It could probably be optimized a bit more, because there are two parts that do basically the same thing and could probably be joined together, but it would cost a few extra states to then differentiate between which of them it was supposed to be, and i don't have time to do that right now.--[[User:Racheline|Racheline]] ([[User talk:Racheline|talk]]) 18:56, 13 February 2025 (UTC)

== BB(51) epsilon_0 champion ==
Can you show the transitions of BB(51) on [[Large champions transitions]] page? --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 03:24, 3 September 2024 (UTC)

: Here are the transitions and description from Racheline:
<pre>
Σ(51) > (f_ε0)^8(f_ω^ω^3(a lot)) > f_ε0+1(8)
M = (
0: ((1, -1, 0), (0, -1, 1)),
1: ((1, -1, 0), (1, 1, 2)),
2: ((1, -1, 2), (0, -1, 4)),
3: ((0, -1, 10), (1, -1, 4)),
4: ((0, -1, 3), (0, -1, 5)),
5: ((1, -1, 4), (1, 1, 6)),
6: ((1, -1, 7), (0, -1, 15)),
7: ((0, -1, 8), (1, -1, 7)),
8: ((0, -1, 9), (1, -1, 7)),
9: ((1, 1, 11), (1, -1, 9)),
10: ((0, -1, 20), (0, -1, 17)),
11: ((0, 1, 12), (1, 1, 11)),
12: ((0, 1, 13), (1, 1, 13)),
13: ((1, -1, 14), (1, 1, 12)),
14: ((0, -1, 6), (1, -1, 10)),
15: ((0, -1, 16), (1, -1, 16)),
16: ((1, -1, 20), (1, -1, 15)),
17: ((0, -1, 18), (1, -1, 18)),
18: ((1, 1, 19), (1, -1, 17)),
19: ((1, -1, 19), (1, -1, 20)),
20: ((0, -1, 50), (0, -1, 21)),
21: ((0, 1, 43), (0, -1, 9)),
22: ((0, 1, 22), (1, 1, 23)),
23: ((1, 1, 26), (0, -1, 24)),
24: ((1, -1, 25), (0, -1, 24)),
25: ((1, 1, 25), (1, 1, 22)),
26: ((0, 1, 50), (0, 1, 27)),
27: ((1, 1, 26), (1, -1, 35)),
28: ((0, 1, 29), (0, 1, 30)),
29: ((0, -1, 4), (1, 1, 28)),
30: ((1, -1, 31), (1, -1, 0)),
31: ((0, -1, 32), (1, -1, 33)),
32: ((0, -1, 33), (1, -1, 31)),
33: ((0, -1, 34), (0, -1, 35)),
34: ((0, -1, 37), (1, -1, 33)),
35: ((1, 1, 36), (1, 1, 38)),
36: ((0, 1, 37), (1, 1, 28)),
37: ((0, -1, 29), (1, 1, 36)),
38: ((1, 1, 38), (1, 1, 39)),
39: ((0, 1, 38), (1, 1, 40)),
40: ((0, 1, 41), (1, -1, 42)),
41: ((1, -1, 49), (1, 1, 40)),
42: ((1, 1, 36), (0, -1, 42)),
43: ((0, 1, 44), (1, 1, 44)),
44: ((1, -1, 45), (1, 1, 43)),
45: ((0, -1, 46), (1, -1, 46)),
46: ((0, 1, 22), (1, -1, 45)),
47: ((1, -1, 41), (1, -1, 47)),
48: ((0, -1, 47), (0, -1, 49)),
49: ((1, -1, 22), (1, -1, 48)),
50: ((0, -1, 48), (1, 1, halt))
)

(start from state 30)
</pre>

: I haven't investigated it yet. [[User:Sligocki|Sligocki]] ([[User talk:Sligocki|talk]]) 14:27, 3 September 2024 (UTC)

:: It directly simulates Address Notation, which is a different way to write Primitive Sequence System (PrSS for short, it's 1-row Bashicu Matrix System). Here are the expansion rules of Address Notation:
::: Given a sequence <math>S</math> of natural numbers, to compute its expansion, start by letting <math>c</math> be the last element of <math>S</math>. If <math>c=0</math>, then <math>S</math> is a successor and its predecessor is <math>S</math> without the last element. Otherwise, find the last element <math>r</math> of <math>S</math> which is less than <math>c</math>. Decrease <math>c</math> by <math>1</math>, and let <math>B</math> be everything in this new sequence after <math>r</math>. Append infinitely many copies of <math>B</math>.
:: Since the sequences are ordered lexicographically and each element decreases by <math>1</math> until it is <math>0</math>, for any fundamental sequence system where each <math>S[n]</math> is formed by taking an initial subsequence of the expansion of <math>S</math> and changing the last element of that subsequence to a smaller or equal natural number, the ordinal notation given by the fundamental sequence system is the same. If i remember correctly, the TM simulates a variant <math>h</math> of the Hardy Hierarchy with a fundamental sequence system where the FS of each standard <math>S</math> contains all sequences longer than <math>S</math> formed this way from its expansion (except for the first one or two).
:: Specifically,
::: <math>h_0(n)=n+c_0</math>
::: <math>h_{\alpha+1}(n)=h_\alpha(n+c_1)</math>
::: <math>h_\alpha(n)=h_{\alpha[n]}(c_2)</math> for limit <math>\alpha</math>
::: <math>h_{\varepsilon_0}(n) = h_{(0,\lfloor\frac{n+c_3}{2}\rfloor)}(c_4) = h_{\omega\uparrow\uparrow\lfloor\frac{n+c_3}{2}\rfloor}(c_4)</math>
:: for some constants <math>c_0,c_1,c_2,c_3,c_4</math> for which the hierarchy is not degenerate (the constants can be deduced easily from the TM's space-time diagram). In particular, the non-degeneracy means that for most FGHs <math>f</math> with "natural" FS systems, we will have <math>f_{\alpha+1}(n)<h_{\omega^{\alpha+1}}(n+k)</math> for some small constant <math>k</math> (depending on <math>f</math>), and as long as the FGH's FS system has <math>\varepsilon_0[n]\le\omega\uparrow\uparrow(n+c)</math> for some constant <math>c</math>, we have <math>f_{\varepsilon_0}(n)<h_{\varepsilon_0}(2(n+c)+k)</math> for some small constant <math>k</math> (also depending on <math>f</math>. For example, the Wainer Hierarchy is one of these "natural" FGHs (and one of the most well-known ones at this level), and if we extend it to include <math>f_{\varepsilon_0}</math> and <math>f_{\varepsilon_0+1}</math> with the FS <math>\varepsilon_0[n]=\omega\uparrow\uparrow n</math>, it does indeed satisfy <math>f_{\varepsilon_0}(n)<h_{\varepsilon_0}(2n+k)</math> for small <math>k</math>, which means that since the TM computes <math>h_{\varepsilon_0}^8(n)</math> for a large <math>n</math>, its output is larger than <math>f_{\varepsilon_0+1}(8)</math>. [[User:Racheline|Racheline]] ([[User talk:Racheline|talk]]) 19:06, 3 September 2024 (UTC)

I think the bbchallenge can search BB(7) and BB(4,3) in the wild. --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 14:56, 30 October 2024 (UTC)

: I have thought about writing a program that searches for Ackermann's worm-like behavior of TMs (or at least a program for exponential behavior), but I wouldn't be sure how to do it. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 22:07, 30 October 2024 (UTC)

== Cloud search for busy beavers ==
We still lack a website for cloud search busy beaver functions. --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 07:07, 15 December 2024 (UTC)

: In the case of distributed computing projects like Catalogue and Great Internet Mersenne Prime Search, it seems like those are distributed to divvy up a large finite amount of computation. I don't know much about searching for Busy Beavers (e.g. if there is such a thing as an automated process for finding accelerated simulators) but would distributed finite computation be able to help reduce holdout lists? [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 09:05, 15 December 2024 (UTC)

It may be able to search the busy beaver domain BB(7) or even larger more quickly. [[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 13:38, 15 December 2024 (UTC)

: Certainly it could do something like direct simulation quickly, but is there a known way to automate more advanced techniques to look for long-running machines? [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 22:17, 15 December 2024 (UTC)
::questions like these should be asked on discord. Few people look at this page.--[[User:Konkhra|Konkhra]] ([[User talk:Konkhra|talk]]) 06:10, 16 December 2024 (UTC)

: Setup a BOINC project for bbchallenge, and use this for reducing holdouts list? -- [[User:Loader3229|Loader3229]] ([[User talk:Loader3229|talk]]) 13:34, 22 September 2026 (UTC)

Talk:Champions

2026-10-12T20:19:34Z

Azerty: /* Should we even count llm-"""discovered""" champions? */

== Should we even count llm-"""discovered""" champions? ==
The recent BB(8) and BB(9) champions were discovered by LLMs. Although this might seem like a good thing (more champions = better, right?), we have to think about the future of the community. Right now, there is still a place for humans to contribute. But what will happen when AI is infinitely better than humans? Then there will be nothing left to do except to constantly refresh an automatically updated wiki page as AI automatically solves all BB values and variants. I do not want to live in such a dystopia. I want this community to be a place where math enthusiasts and professionals can do new, novel research and compute the uncomputable. Please keep this community alive. -- [[User:Δ⁵|Δ⁵]] ([[User talk:Δ⁵|talk]]) 03:40, 9 October 2026 (UTC)
:You're really overestimating how good LLMs are at doing this. Sure, they may be able to create large champions but for verifying large busybeaver values, using AI would be incredibly expensive. There will always be significant human intervention in this community. [[User:Etalon|Etalon]] ([[User talk:Etalon|talk]]) 17:28, 12 October 2026 (UTC)

== Self-Reporting and Verification ==
This champions page has ended up with a lot of self reported and (I assume) not independently verified champions. I'm conflicted about this: On the one hand I want this wiki to have an encyclopedic/Wikipedia style: it would include things which are known, established and verified. On the other hand, I don't want to stifle innovation or gatekeep what counts as proper verification. As a compromise I have updated the table to add a "Verification" column. The idea here is to indicate which results have been independently verified by another contributor. My idea is that that verification should include some sort of write-up (wiki page, blog post, etc) describing the TM and how it is known to run this long. I have linked to examples of my own for Pavel's BB(6) and Daniel's BB(16) champions. I would love to see some independent analyses of other TMs here (especially smaller ones like Racheline's BB(14) > Graham champion). [[User:Sligocki|Sligocki]] ([[User talk:Sligocki|talk]]) 16:19, 10 December 2024 (UTC)
:Particularly interesting is Racheline's BB(14) > Graham champion. --[[User:Konkhra|Konkhra]] ([[User talk:Konkhra|talk]]) 01:53, 12 December 2024 (UTC)

==Larger champions==
What are the known two-symbol champions beyond BB(16)? Vielhaber, Chacón, and Ceballos's paper "[https://arxiv.org/abs/2303.02855v1 Friedman's 'Long Finite Sequences': The End of the Busy Beaver Contest]" gives a 2450-state two-symbol busy beaver halting after at least n(4) steps, where n is Friedman's [https://googology.miraheze.org/wiki/Block_subsequence_theorem block subsequence function], but this is a very large jump up in state count from 14 states for f<sub>ω+1</sub>(65536). (This paper has some mistakes with their (symbol,state count) notation for TMs, for example referring to Aaronson's and Yedidia's machine as a (7910,2) machine rather than a (2,7910) machine, so maybe the machine whose state count I wrote here is the wrong one.) [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 22:00, 17 August 2024 (UTC)

:Today i found some bounds on BB(20) and BB(21), though i'm not sure if they're new. <math>\Sigma(20) > f_{\omega+2}^2(21)</math> due to {{TM|1LR0LR_0RJ1RG_0RD1LC_1RQ1RE_1LO0RQ_1LH1LF_0LC1LH_0LI0LF_1RD0LF_---0RK_0LM1LL_0LL1LM_1LE1LN_0LF0LM_0RP1LO_0LF0RQ_1RB1RQ_0RS1LA_1LT1RS_0LP0RR|halt}} and <math>\Sigma(21) > f_{\omega^2}^2(4 \uparrow\uparrow 341)</math> due to {{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_---0LS_1LH0LT_1LH1LU_0LE1LR|halt}}.
:The old googology wiki claims [https://googology.fandom.com/wiki/Busy_beaver_function#Larger_values some other bounds], two of which are already implied by the bounds mentioned above, but <math>\Sigma(85)>f_{\varepsilon_0}(1907)</math> and the bounds for more than 2 symbols seem to still be the best known.
:[[User:Racheline|Racheline]] ([[User talk:Racheline|talk]]) 23:08, 17 August 2024 (UTC)
::Racheline, congratulations on discovering a TM that crushes Graham's number. I was very impressed. but I think that in reality BB(8,2) will beat Graham's number. But it is unrealistic for a human to build such a machine. Only to search in the wild. Unfortunately. Nevertheless, congratulations again and further success!!! --[[User:Konkhra|Konkhra]] ([[User talk:Konkhra|talk]]) 03:29, 18 August 2024 (UTC)
:::And how about an article about BB(14)?--[[User:Konkhra|Konkhra]] ([[User talk:Konkhra|talk]]) 03:30, 18 August 2024 (UTC)

IMHO, beyond Graham's number I think we should only list highlights (if anything). Like first TM bigger than blah for various notable blah. Otherwise the churn on this page will be high and the quality low. But I'm open to other points of view. If we do extend this much further, I think maybe we should have a more rigorous process for getting new results demonstrated before they are added. --[[User:Sligocki|sligocki]] ([[User talk:Sligocki|talk]]) 02:57, 18 August 2024 (UTC)

Can you design very large champions as beat the <math>f_{\psi(\varepsilon_{\Omega+1})}(1000)</math>, and beat the <math>f_{\psi(\Omega_\omega)}(1000)</math>, limit for [https://googology.fandom.com/wiki/Bashicu_matrix_system BMS] or beat [https://googology.fandom.com/wiki/Loader%27s_number Loader's number]? --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 00:45, 17 September 2024 (UTC)

: Later today I can try writing one (or at least starting to write one today), but my machine would likely have thousands of states. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 15:50, 16 September 2024 (UTC)

:: I now have two-symbol machines with the following running times. In order to squeeze the best possible results out of Bashicu matrix system, some machines have a different function applied to <math>n</math> at each step (this function is what Bashicu calls the "activation function"), although at this scale it doesn't make too much of a difference.

:: {| class="wikitable"
|+ Busy Beaver lower bounds using BM4
|-
! State count !! Running time !! Activation function
|-
| 5889 || <math> >f_{\psi(\varepsilon_{\Omega+1})}(1000)</math> (using BMS to define FSes) || <math>n\mapsto n+1</math>
|-
| 5894 || <math> >f_{\psi_0(\Omega_\omega)}(1000)</math> (using BMS to define FSes) || <math>n\mapsto n+1</math>
|-
| 5908 || <math> >(0,0,0,0,0)(1,1,1,1,1)[2]</math> || <math>n\mapsto n+2</math>
|-
| 5923 || <math> >(0,0,0,0,0)(1,1,1,1,1)[5]</math> || <math>n\mapsto n+1</math>
|-
| 5941 || <math> >(0,0,0,0,0,0,0,0,0,0)(1,1,1,1,1,1,1,1,1,1)[10]</math> || <math>n\mapsto n+1</math>
|-
| 5978 || <math> >(\underbrace{0,0,\ldots,0,0}_{889})(\underbrace{1,1,\ldots,1,1}_{889})[894]</math> || <math>n\mapsto n+6</math>
|-
| 6020 || <math> >(\underbrace{0,0,\ldots,0,0}_{N})(\underbrace{1,1,\ldots,1,1}_{N})[N]</math> where <math>N</math> is <math>2^{2^{2^{2059}}}</math> || <math>n\mapsto n+5</math>
|}

:: '''Edit:''' It turns out CatIsFluffy [https://github.com/CatsAreFluffy/metamath-turing-machines beat me to it]! I don't know how many states the BMS machine has, but [https://cosearch.bbchallenge.org/contribution/m5k4ulm8 BB(1094) > Loader's number], now down to [https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe BB(1015)]. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 11:37, 4 October 2024 (UTC)

Who can design a [https://googology.fandom.com/wiki/Laver_table Laver table] q function level machine. --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 06:57, 13 February 2025 (UTC)

: Me, probably. This 67-state TM should leave over q<sup>81</sup>(q(5)-2) 1s on the tape (maybe i have an off-by-one or off-by-two error in the exponent of q but it's still iterated quite a few times):
<pre>
M = (
((1, 1, 1), (0, -1, 3)),
((0, -1, 2), (0, 1, 1)),
((1, 1, 0), (1, -1, 2)),
((0, -1, 4), (1, -1, 0)),
((0, -1, 2), (1, -1, 5)),
((0, 1, 26), (0, 1, 5)),
((1, -1, 57), (0, 1, 26)),
((0, 1, 38), (1, 1, 8)),
((0, 1, 9), (1, 1, 8)),
((0, 1, 62), (0, 1, 10)),
((0, -1, 16), (0, -1, 11)),
((1, -1, 12), (1, -1, 11)),
((0, 1, 15), (0, 1, 13)),
((1, 1, 14), (1, 1, 13)),
((0, -1, 16), (0, -1, 11)),
((0, 1, 10), (1, 1, 15)),
((1, -1, 33), (0, -1, 17)),
((1, -1, 18), (1, -1, 17)),
((0, -1, 19), (1, -1, 18)),
((0, -1, 20), (1, -1, 19)),
((0, 1, 22), (1, -1, 21)),
((1, 1, 21), (0, 1, 22)),
((0, 1, 23), (1, 1, 22)),
((1, -1, 24), (1, 1, 23)),
((1, -1, 30), (0, 1, 25)),
((1, -1, 26), (1, 1, 25)),
((0, 1, 6), (0, -1, 27)),
((0, -1, 28), (1, -1, 27)),
((0, -1, 29), (1, -1, 28)),
((1, -1, 20), (1, -1, 29)),
((1, 1, 32), (0, -1, 31)),
((1, -1, 32), (1, -1, 31)),
((0, 1, 7), (0, -1, 30)),
((0, -1, 34), (0, -1, 33)),
((0, -1, 34), (0, -1, 35)),
((0, 1, 37), (1, 1, 36)),
((0, 1, 36), (1, -1, 16)),
((0, 1, 37), (0, 1, 7)),
((0, 1, 39), (0, 1, 7)),
((0, -1, 44), (0, -1, 40)),
((0, -1, 41), (1, -1, 40)),
((1, 1, 42), (1, -1, 41)),
((0, 1, 43), (1, 1, 42)),
((1, 1, 39), (1, 1, 43)),
((0, -1, 45), (1, -1, 44)),
((0, -1, 47), (0, -1, 46)),
((1, -1, 45), (1, -1, 46)),
((1, -1, 47), (1, 1, 48)),
((0, -1, 49), (1, 1, 48)),
((0, 1, 56), (0, -1, 50)),
((0, 1, 49), (0, 1, 51)),
((1, 1, 52), (1, 1, 51)),
((0, -1, 55), (0, -1, 53)),
((1, -1, 54), (1, -1, 53)),
((0, 1, 51), (0, 1, 51)),
((0, -1, 50), (1, -1, 55)),
((1, -1, 57), (1, 1, 56)),
((1, -1, 61), (0, -1, 58)),
((1, 1, 59), (1, -1, 58)),
((0, -1, 60), (0, 1, 56)),
((1, 1, 61), (1, -1, 60)),
((1, -1, 66), (0, 1, 7)),
((0, 1, 63), (0, 1, 62)),
((0, 1, 63), (0, 1, 64)),
((0, -1, 65), (1, 1, -1)),
((0, -1, 65), (1, -1, 66)),
((0, -1, 40), (1, -1, 66))
)

(start from state 0)
</pre>
: It could probably be optimized a bit more, because there are two parts that do basically the same thing and could probably be joined together, but it would cost a few extra states to then differentiate between which of them it was supposed to be, and i don't have time to do that right now.--[[User:Racheline|Racheline]] ([[User talk:Racheline|talk]]) 18:56, 13 February 2025 (UTC)

== BB(51) epsilon_0 champion ==
Can you show the transitions of BB(51) on [[Large champions transitions]] page? --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 03:24, 3 September 2024 (UTC)

: Here are the transitions and description from Racheline:
<pre>
Σ(51) > (f_ε0)^8(f_ω^ω^3(a lot)) > f_ε0+1(8)
M = (
0: ((1, -1, 0), (0, -1, 1)),
1: ((1, -1, 0), (1, 1, 2)),
2: ((1, -1, 2), (0, -1, 4)),
3: ((0, -1, 10), (1, -1, 4)),
4: ((0, -1, 3), (0, -1, 5)),
5: ((1, -1, 4), (1, 1, 6)),
6: ((1, -1, 7), (0, -1, 15)),
7: ((0, -1, 8), (1, -1, 7)),
8: ((0, -1, 9), (1, -1, 7)),
9: ((1, 1, 11), (1, -1, 9)),
10: ((0, -1, 20), (0, -1, 17)),
11: ((0, 1, 12), (1, 1, 11)),
12: ((0, 1, 13), (1, 1, 13)),
13: ((1, -1, 14), (1, 1, 12)),
14: ((0, -1, 6), (1, -1, 10)),
15: ((0, -1, 16), (1, -1, 16)),
16: ((1, -1, 20), (1, -1, 15)),
17: ((0, -1, 18), (1, -1, 18)),
18: ((1, 1, 19), (1, -1, 17)),
19: ((1, -1, 19), (1, -1, 20)),
20: ((0, -1, 50), (0, -1, 21)),
21: ((0, 1, 43), (0, -1, 9)),
22: ((0, 1, 22), (1, 1, 23)),
23: ((1, 1, 26), (0, -1, 24)),
24: ((1, -1, 25), (0, -1, 24)),
25: ((1, 1, 25), (1, 1, 22)),
26: ((0, 1, 50), (0, 1, 27)),
27: ((1, 1, 26), (1, -1, 35)),
28: ((0, 1, 29), (0, 1, 30)),
29: ((0, -1, 4), (1, 1, 28)),
30: ((1, -1, 31), (1, -1, 0)),
31: ((0, -1, 32), (1, -1, 33)),
32: ((0, -1, 33), (1, -1, 31)),
33: ((0, -1, 34), (0, -1, 35)),
34: ((0, -1, 37), (1, -1, 33)),
35: ((1, 1, 36), (1, 1, 38)),
36: ((0, 1, 37), (1, 1, 28)),
37: ((0, -1, 29), (1, 1, 36)),
38: ((1, 1, 38), (1, 1, 39)),
39: ((0, 1, 38), (1, 1, 40)),
40: ((0, 1, 41), (1, -1, 42)),
41: ((1, -1, 49), (1, 1, 40)),
42: ((1, 1, 36), (0, -1, 42)),
43: ((0, 1, 44), (1, 1, 44)),
44: ((1, -1, 45), (1, 1, 43)),
45: ((0, -1, 46), (1, -1, 46)),
46: ((0, 1, 22), (1, -1, 45)),
47: ((1, -1, 41), (1, -1, 47)),
48: ((0, -1, 47), (0, -1, 49)),
49: ((1, -1, 22), (1, -1, 48)),
50: ((0, -1, 48), (1, 1, halt))
)

(start from state 30)
</pre>

: I haven't investigated it yet. [[User:Sligocki|Sligocki]] ([[User talk:Sligocki|talk]]) 14:27, 3 September 2024 (UTC)

:: It directly simulates Address Notation, which is a different way to write Primitive Sequence System (PrSS for short, it's 1-row Bashicu Matrix System). Here are the expansion rules of Address Notation:
::: Given a sequence <math>S</math> of natural numbers, to compute its expansion, start by letting <math>c</math> be the last element of <math>S</math>. If <math>c=0</math>, then <math>S</math> is a successor and its predecessor is <math>S</math> without the last element. Otherwise, find the last element <math>r</math> of <math>S</math> which is less than <math>c</math>. Decrease <math>c</math> by <math>1</math>, and let <math>B</math> be everything in this new sequence after <math>r</math>. Append infinitely many copies of <math>B</math>.
:: Since the sequences are ordered lexicographically and each element decreases by <math>1</math> until it is <math>0</math>, for any fundamental sequence system where each <math>S[n]</math> is formed by taking an initial subsequence of the expansion of <math>S</math> and changing the last element of that subsequence to a smaller or equal natural number, the ordinal notation given by the fundamental sequence system is the same. If i remember correctly, the TM simulates a variant <math>h</math> of the Hardy Hierarchy with a fundamental sequence system where the FS of each standard <math>S</math> contains all sequences longer than <math>S</math> formed this way from its expansion (except for the first one or two).
:: Specifically,
::: <math>h_0(n)=n+c_0</math>
::: <math>h_{\alpha+1}(n)=h_\alpha(n+c_1)</math>
::: <math>h_\alpha(n)=h_{\alpha[n]}(c_2)</math> for limit <math>\alpha</math>
::: <math>h_{\varepsilon_0}(n) = h_{(0,\lfloor\frac{n+c_3}{2}\rfloor)}(c_4) = h_{\omega\uparrow\uparrow\lfloor\frac{n+c_3}{2}\rfloor}(c_4)</math>
:: for some constants <math>c_0,c_1,c_2,c_3,c_4</math> for which the hierarchy is not degenerate (the constants can be deduced easily from the TM's space-time diagram). In particular, the non-degeneracy means that for most FGHs <math>f</math> with "natural" FS systems, we will have <math>f_{\alpha+1}(n)<h_{\omega^{\alpha+1}}(n+k)</math> for some small constant <math>k</math> (depending on <math>f</math>), and as long as the FGH's FS system has <math>\varepsilon_0[n]\le\omega\uparrow\uparrow(n+c)</math> for some constant <math>c</math>, we have <math>f_{\varepsilon_0}(n)<h_{\varepsilon_0}(2(n+c)+k)</math> for some small constant <math>k</math> (also depending on <math>f</math>. For example, the Wainer Hierarchy is one of these "natural" FGHs (and one of the most well-known ones at this level), and if we extend it to include <math>f_{\varepsilon_0}</math> and <math>f_{\varepsilon_0+1}</math> with the FS <math>\varepsilon_0[n]=\omega\uparrow\uparrow n</math>, it does indeed satisfy <math>f_{\varepsilon_0}(n)<h_{\varepsilon_0}(2n+k)</math> for small <math>k</math>, which means that since the TM computes <math>h_{\varepsilon_0}^8(n)</math> for a large <math>n</math>, its output is larger than <math>f_{\varepsilon_0+1}(8)</math>. [[User:Racheline|Racheline]] ([[User talk:Racheline|talk]]) 19:06, 3 September 2024 (UTC)

I think the bbchallenge can search BB(7) and BB(4,3) in the wild. --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 14:56, 30 October 2024 (UTC)

: I have thought about writing a program that searches for Ackermann's worm-like behavior of TMs (or at least a program for exponential behavior), but I wouldn't be sure how to do it. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 22:07, 30 October 2024 (UTC)

== Cloud search for busy beavers ==
We still lack a website for cloud search busy beaver functions. --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 07:07, 15 December 2024 (UTC)

: In the case of distributed computing projects like Catalogue and Great Internet Mersenne Prime Search, it seems like those are distributed to divvy up a large finite amount of computation. I don't know much about searching for Busy Beavers (e.g. if there is such a thing as an automated process for finding accelerated simulators) but would distributed finite computation be able to help reduce holdout lists? [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 09:05, 15 December 2024 (UTC)

It may be able to search the busy beaver domain BB(7) or even larger more quickly. [[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 13:38, 15 December 2024 (UTC)

: Certainly it could do something like direct simulation quickly, but is there a known way to automate more advanced techniques to look for long-running machines? [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 22:17, 15 December 2024 (UTC)
::questions like these should be asked on discord. Few people look at this page.--[[User:Konkhra|Konkhra]] ([[User talk:Konkhra|talk]]) 06:10, 16 December 2024 (UTC)

: Setup a BOINC project for bbchallenge, and use this for reducing holdouts list? -- [[User:Loader3229|Loader3229]] ([[User talk:Loader3229|talk]]) 13:34, 22 September 2026 (UTC)

Enumeration

2026-10-04T13:21:56Z

Azerty: Added my new GitHub repository.

'''Enumeration''' is the process of generating all candidate [[Turing machines]] (or other program types) within a given [[:Category:BB Domains|domain]] potentially filtered modulo isomorphism (ex: [[TNF]]). These enumerations can then be filtered down by [[deciders]] leaving a list of [[holdouts]] which remain undecided.

== List of enumeration scripts ==
{| class="wikitable"
|+
!System
!Repository
!File Name
!Language
|-
|Branching Turing Machine
|[https://github.com/sligocki/etc/blob/main/bb_tri/src/enumerator.rs https://github.com/sligocki/etc]
|<code>bb_tri/scr/enumerator.rs</code>
|Rust
|-
|CounterScript
|https://github.com/AzertyWasTaken/CounterScript
|<code>main.js</code>
|JavaScript
|-
|CounterScript
|https://github.com/sligocki/etc
|<code>bbcs/src/enumerator.rs</code>
|Rust
|-
|Fractran
|https://github.com/int-y1/BBFractran
|<code>enumerate/fractran20260416.cpp</code>
|C++
|-
|General Recursive Function
|https://github.com/sligocki/etc
|<code>gen_rec/scr/enumerate.rs</code>
|Rust
|-
|Lambda Calculus
|[https://github.com/tromp/AIT/blob/master/BB/BB.lhs https://github.com/tromp/AIT]
|<code>BB/BB.lhs</code>
|Haskell
|-
|Post Tag
|[https://github.com/sligocki/etc/blob/main/bb_tri/src/enumerator.rs https://github.com/sligocki/etc]
|<code>post_tag/src/enumerate.rs</code>
|Rust
|-
|Register Machine
|https://github.com/int-y1/BBFractran
|<code>rm.cpp</code>
|C++
|-
|Turing Machine
|https://github.com/sligocki/busy-beaver
|<code>Code/Enumerate.py</code>
|Python
|-
|Turing Machine
|https://github.com/ccz181078/TM
|<code>main.cpp</code>
|C++
|}

== See also ==
Azerty's enumerators and deciders list (10+ systems): https://github.com/AzertyWasTaken/Busy-Beaver

Green's machines

2026-09-25T12:21:26Z

Azerty: /* Machines */ added scientific notation

In 1964, Milton W. Green hand-crafted a family of fast-growing Turing Machines with n-states, 2-symbols for all <math>n \ge 4</math>.<ref>Milton W. Green. “A Lower Bound on Rado’s Sigma Function for Binary Turing Machines”, ''Proceedings of the IEEE Fifth Annual Symposium on Switching Circuits Theory and Logical Design'', 1964, pages 91–94, [https://doi.org/10.1109%2FSWCT.1964.3 https://doi.org/10.1109/SWCT.1964.3]</ref> This family grows roughly as fast as the Ackermann function:<math display="block">BB_{2n} \approx 3 \uparrow^{n-2} 3</math>

Over the years, these Turing Machines lost champion status. The last machines to lose champion status were BB(9), BB(10), and BB(11). In August 2024, Racheline discovered the better bounds <math>BB(9) > 10 \uparrow\uparrow 1820</math> and <math>BB(10) > 10 \uparrow\uparrow\uparrow\uparrow\uparrow 4 </math>.

== Machines ==
{| class="wikitable"
|+
!# States
!TM
!Sigma Score
|-
|4
|{{TM|1LB1RZ_0LC1LC_0LD0LC_1RD1RA|halt}}
|7
|-
|5
|{{TM|1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1RE1RC|halt}}
|13
|-
|6
|{{TM|1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1RF1RD|halt}}
|35
|-
|7
|{{TM|1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1RG1RE|halt}}
|22,961
|-
|8
|{{TM|1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1RH1RF|halt}}
|<math> \frac{7 \cdot 3^{93} - 3}{2} > 8.247 \times 10^{44} </math>
|-
|9
|{{TM|1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1RI1RG|halt}}
|<math> > 3 \uparrow\uparrow 31 </math>
|-
|10
|{{TM|1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1LI1RF_0LJ0LI_1RJ1RH|halt}}
|<math> > 3 \uparrow\uparrow 3 \uparrow\uparrow 4 </math>
|-
|11
|{{TM|1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1LJ1RG_0LK0LJ_1RK1RI|halt}}
|<math> > 3 \uparrow\uparrow\uparrow 3 \uparrow\uparrow\uparrow 2 </math>
|-
|12
|{{TM|1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1LI1RF_0LJ0LI_1LK1RH_0LL0LK_1RL1RJ|halt}}
|<math> > 3 \uparrow\uparrow\uparrow 3 \uparrow\uparrow\uparrow 4 </math>
|-
|13
|{{TM|1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1LJ1RG_0LK0LJ_1LL1RI_0LM0LL_1RM1RK|halt}}
|<math> > 3 \uparrow\uparrow\uparrow\uparrow 3 \uparrow\uparrow\uparrow\uparrow 2 </math>
|-
|14
|{{TM|1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1LI1RF_0LJ0LI_1LK1RH_0LL0LK_1LM1RJ_0LN0LM_1RN1RL|halt}}
|<math> > 3 \uparrow\uparrow\uparrow\uparrow 3 \uparrow\uparrow\uparrow\uparrow 4 </math>
|-
|15
|{{TM|1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1LJ1RG_0LK0LJ_1LL1RI_0LM0LL_1LN1RK_0LO0LN_1RO1RM|halt}}
|<math> > 3 \uparrow\uparrow\uparrow\uparrow\uparrow 3 \uparrow\uparrow\uparrow\uparrow\uparrow 2 </math>
|}

== External Links ==

* Shawn Ligocki. Green's Machines. 2023. https://www.sligocki.com/2023/10/19/greens-machines.html

== References ==
<references />

[[Category:Individual machines]]

Instruction-Limited Busy Beaver

2026-09-24T14:03:59Z

Azerty: /* Champions */ Fixed order

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 10<sup>1565</sup>, 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 10<sup>783</sup>, 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
|
|-
|29
|<math>> f_{\omega+3}^2(4)</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_---2LJ---|halt}}</span>
|Jacobzheng in 2026
|Current BB(11,3) champion
|
|-
|39
|<math>> f_{\omega+6}^2(4)</math>
|<span style="word-break:break-all">{{TM|1RR1RG_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RA1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_---1LT_0LH0LS|halt}}</span>
|Jacobzheng in 2026
|Current BB(20) champion
|
|-
|41
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|<span style="word-break:break-all">{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_---0LS_1LH0LT_1LH1LU_0LE1LR|halt}}</span>
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|Current BB(21) champion
|
|-
|78
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|
|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(50) 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(1050) 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 BBi<sub>TNF</sub>(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|0RH|halt}} {{TM|1RH---|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]]

Instruction-Limited Busy Beaver

2026-09-24T14:03:07Z

Azerty: /* Champions */ Added large champions

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 10<sup>1565</sup>, 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 10<sup>783</sup>, 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
|
|-
|29
|<math>> f_{\omega+3}^2(4)</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_---2LJ---|halt}}</span>
|Jacobzheng in 2026
|Current BB(11,3) champion
|
|-
|39
|<math>> f_{\omega+6}^2(4)</math>
|<span style="word-break:break-all">{{TM|1RR1RG_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RA1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_---1LT_0LH0LS|halt}}</span>
|Jacobzheng in 2026
|Current BB(20) champion
|
|-
|41
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|<span style="word-break:break-all">{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_---0LS_1LH0LT_1LH1LU_0LE1LR|halt}}</span>
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|Current BB(21) champion
|
|-
|78
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|
|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
|
|-
|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
|
|-
|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(50) 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(1050) 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 BBi<sub>TNF</sub>(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|0RH|halt}} {{TM|1RH---|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]]

BB(6)

2026-09-18T14:40:20Z

Azerty: removed excessive line jumps

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6<sup>th</sup> value of the [[Busy Beaver function]]. With the discovery of the [[Cryptid]] machine [[Antihydra]] in June 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(6) and thus [https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html BB(6) is Hard].

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

<math display="block">S(6) > \Sigma(6) > 2 \uparrow\uparrow 2 \uparrow\uparrow 2 \uparrow\uparrow 10 > 2 \uparrow\uparrow\uparrow 5</math>
== History ==
{| class="wikitable mw-collapsible"
|+
! colspan="6" |Timeline of lower bounds established for Σ(6) and S(6)<ref name=":PMH">Pascal Michel. (last updated 2026). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm62 </ref>
|-
!Month of discovery
!Machine
!S(6) lower bound
!Σ(6) lower bound
!Discoverer
!Notes
|-
|November 1964
|{{TM|1RB1LZ_0RC1RC_0RD0RC_1RE1LA_0RF0RE_1LF1LD|halt}}
|≥ 436
|≥ 35
|Green, M.
|Part of an infinite family of machines now known as [[Green's machines]].
|-
| rowspan="2" |August 1972
|
|≥ 522
|
| rowspan="2" |Lynn, D.
| rowspan="2" |During this period, the steps champion was different to the ones champion.
|-
|
|
|≥ 42
|-
|December 1982
|
|≥ 4,208,824
|≥ 2,075
|Schult, U.
|Bound was only published in 1983.
|-
| rowspan="2" |April 1983
|
|≥ 13,488
|
| rowspan="2" |Brady, A.
| rowspan="2" |Surpassed by an earlier, unpublished bound.
During this period, the steps champion was different to the ones champion.
|-
|
|
|≥ 117
|-
| rowspan="2" |January 1990
|{{TM|1RB1RA_1LC1LB_0LF1LD_1RA0LE_0RA1LC_1RE1RZ|halt}}
|≥ 13,122,572,797
|≥ 136,612
| rowspan="6" |Heiner, M., Buntrock, J.
|
|-
|{{TM|1RB1RA_1LC1LB_0RF1LD_1RA0LE_1LZ1LF_0LA0LC|halt}}
|≥ 8,690,333,381,690,951
|≥ 95,524,079
|
|-
|July 2000
|{{TM|1RB0RC_0LA0RD_1RD1RZ_1LE0LD_1RF1LB_1RA1RE|halt}}
|> 5.3 × 10<sup>42</sup>
|> 2.5 × 10<sup>21</sup>
|
|-
| rowspan="2" |August 2000
|{{TM|1RB0LC_1LA1RC_1RA0LD_1LE1LC_1RF1RZ_1RA1RE|halt}}
|> 6.1 × 10<sup>119</sup>
|> 1.4 × 10<sup>60</sup>
|
|-
|{{TM|1RB0LB_0RC1LB_1RD0LA_1LE1LF_1LA0LD_1RH1LE|halt}}
|> 6.1 × 10<sup>925</sup>
|> 6.4 × 10<sup>462</sup>
|
|-
|February 2001
|{{TM|1RB0LF_0RC0RD_1LD1RE_0LE0LD_0RA1RC_1LA1RZ|halt}}
|> 3.0 × 10<sup>1,730</sup>
|> 1.2 × 10<sup>865</sup>
|
|-
|November 2007
|{{TM|1RB0RF_0LB1LC_1LD0RC_1LE1RZ_1LF0LD_1RA0LE|halt}}
|> 8.9 × 10<sup>1,762</sup>
|> 2.5 × 10<sup>881</sup>
| rowspan="2" |Ligocki, T., Ligocki, S.
|
|-
|December 2007
|{{TM|1RB0LE_1LC0RA_1LD0RC_1LE0LF_1LA1LC_1LE1RZ|halt}}
|> 2.5 × 10<sup>2,879</sup>
|> 4.6 × 10<sup>1,439</sup>
|
|-
|May 2010
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LA0RB_0RC0RE|halt}}
|> 3.8 × 10<sup>21,132</sup>
|> 3.1 × 10<sup>10,566</sup>
| rowspan="2" |Kropitz, P.
|
|-
|June 2010
|{{TM|1RB1LE_1RC1RF_1LD0RB_1RE0LC_1LA0RD_1RZ1RC|halt}}
|> 7.4 × 10<sup>36,534</sup>
|> 3.5 × 10<sup>18,267</sup>
|
|-
| rowspan="5" |May 2022
|{{TM|1RB1RC_1LC0RF_1RA0LD_0LC0LE_1LD0RA_1RE1RZ|halt}}
|> 9.6 × 10<sup>78,913</sup>
|> 6.0 × 10<sup>39,456</sup>
|Ligocki, S.
|
|-
|{{TM|1RB1RZ_1RC1RA_1RD0RB_1LE0RC_0LF0LD_0LB1LA|halt}}
|> 5.4 × 10<sup>197,282</sup>
|> 2.0 × 10<sup>98,641</sup>
| rowspan="2" |Kropitz, P.
|
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|> 10<sup>1,292,913,985</sup>
|> 1.7 × 10<sup>646,456,993</sup>
|
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
| colspan="2" |> 10&uparrow;&uparrow;5
|Ligocki, S.
|
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
| colspan="2" |> 10&uparrow;&uparrow;15
|Kropitz, P.
|
|-
| rowspan="2" |June 2025
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
| colspan="2" |> 10&uparrow;&uparrow;11,010,000
| rowspan="2" |mxdys
|
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
| colspan="2" |> 2&uparrow;&uparrow;2&uparrow;&uparrow;2&uparrow;&uparrow;10
|}
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
In June 2024, the machine {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}} (now known as [[Antihydra]]) was isolated from the BB(6) holdouts, and recognised as a [[Cryptids|Cryptid]], implying that determining the exact value of S(6) would be, in a certain mathematical sense, hard.

@mxdys's [[Holdouts lists|holdouts list]] has 855 machines up to equivalence and 1802 machines not considering equivalence as of mid-September 2026. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

Always up-to-date annotated spreadsheet, with links to Discord discussions: [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Spreadsheet]. The informal holdout count is 1101.

All machines have been simulated out to 1e13 steps. ~150 machines remain to be simulated to 1e14, and ~230 to 1e15. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet].

== Cryptids ==
Several [[Turing machines]] have been found that are [[Cryptids]], considered so because each of them have a [[Collatz-like]] halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as [[probviously]] halting or probviously non-halting.

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}, Space Needle
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}, similar to Space Needle
* {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}, similar to Space Needle
* {{TM|1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB|undecided}}, similar to Space Needle

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

There are a few machines considered notable for their chaotic behaviour, but which have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}
* {{TM|1RB1LA_1RC0RF_1RD---_0LE1RB_---0LA_1LD1RF|undecided}}
* {{TM|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑ 2 ↑↑ 2 ↑↑ 10 > 2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB0RB_0RC0LF_0RD0RA_0LE---_1LE0LA_1LF1RA|halt}}
|<math>10^{1\,400\,000\,000}</math>(runtime)
|Racheline<ref>https://discord.com/channels/960643023006490684/1345502880727040091/1345502880727040091</ref>
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Techniques ==
Simulating tetrational machines, such as the former champion {{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}, requires [[Accelerated simulator|accelerated simulation]] that can handle Collatz Level 2 [[Inductive rule|inductive rules]]. In other words, it requires a simulator that can prove the rules:

<math display="block">\begin{array}{lcl}
C(4k) & \to & {\operatorname{Halt}}\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+1) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+2) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+3) & \to & C\Big(\frac{3^{k+3} + 1}{2}\Big) \\
\end{array}</math>

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to [[wikipedia:Euler's_totient_function|Euler's totient function]]).

We are also applying existing automatic deciders on current holdout lists with more extreme choices of parameters (more computational resources). [[User:XnoobSpeakable|XnoobSpeakable]] was able to solve 11 of the final 2728 holdouts using higher order parameters with the Ligockis' Enumerate.py. An example command line entry is:
<syntaxhighlight lang="bash">
python3 Code/Enumerate.py --infile "bb6in/bb6tm{i}.txt" --outfile "bb6out/t{i}.pb" -r --no-steps --exp-linear-rules --max-loops=50_000_000 --block-mult=3 --max-block-size=100 --time=500 --force --save-freq=1
</syntaxhighlight>
XnoobSpeakable ran Enumerate.py on all TMs in the 2728 holdout list with the above max-loops and max-block-size parameters using <code>--block-mult=1</code> ,<code>--block-mult=2</code> , and <code>--block-mult=3</code>. For context, during the Stage 2 BB(7) enumeration, where speed was more important due to the tens of millions of known holdouts, parameters of <code>--max-loops=100_000 --block-mult=2 --time=30 --save-freq=100</code> were used.

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 a new FAR method] by mxdys was able to decide 113 of the 1534 holdouts ([https://github.com/ccz181078/TM/tree/FAR code] on GitHub) upon initial application.

== References ==
<references />

[[Category:BB Domains]][[Category:BB(6)]]

Busy Beaver Challenge

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The '''Busy Beaver Challenge''' (or '''bbchallenge''') is an online community founded in 2022 with the goal of solving [[BB(5)]]. This goal was accomplished in 2024. Since then the community has focused on many different goals, including searching through unsolved domains (currently [[BB(3,3)]], [[BB(2,5)]], [[BB(6)]], [[BB(4,3)]], [[BB(3,4)]], [[BB(2,6)]], [[BB(7)]] and [[BB(2,7)]]) for new [[champions]] and [[Cryptids]] and reducing [[holdout|holdouts]] lists. Most of the interaction happens on [https://discord.gg/wuZhtTvYU3 our public Discord].

== See Also ==

* Main Website: https://bbchallenge.org/

Cryptids

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[[File:Lovecraft beaver.png|alt=A monstrous beaver in the style of HP Lovecraft with pink tentacles coming out of its mouth, 5 red spider eyes, horns on its head, elbows and tail, moss colored fur, sharp purple claws and webbed feet.|thumb|Lovecraftian Beaver fan art made by Lauren]]
'''Cryptids''' are Turing Machines whose behavior (when started on a blank tape) can be described completely by a relatively simple mathematical rule, but where that rule falls into a class of unsolved (and presumed hard) mathematical problems. This definition is somewhat subjective (What counts as a simple rule? What counts as a hard problem?). In practice, most currently known small Cryptids have [[Collatz-like]] behavior. In other words, the halting problem from blank tape of Cryptids is mathematically-hard.

If there exists a Cryptid with n states and m symbols, then BB(n, m) cannot be solved without solving this hard math problem.

The name Cryptid was proposed by Shawn Ligocki in an Oct 2023 [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html blog post] announcing the discovery of [[Bigfoot]].

== Cryptids at the Edge ==

This is a list of notable Minimal Cryptids (Cryptids in a [[:Category:BB_Domains|domain]] with no strictly smaller known Cryptid). All of these Cryptids were "discovered in the wild" rather than "constructed".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|
|-
|[[Hydra]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|
|-
|[[Bonus cryptid]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|June 2024
|<code>@mxdys</code>, shown to be a Cryptid by <code>@racheline</code>.
|Same as '''Hydra''' but starting iteration from 8 instead of 3 and with termination condition <code>O > 2E</code> instead of <code>E > 2O</code>, hence the name '''Antihydra'''.
|-
|[[Lucy's Moonlight]]
|[[BB(6)]]
|{{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}
|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|Aug 2024
|<code>mxdys</code>
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}
|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|Nov 2024
|Racheline
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|Space Needle]]
|[[BB(6)]]
|{{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}
|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|Jul 2024
|<code>mxdys</code>
|Has near-identical behavior to 16 related BB(6) holdouts. Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}}
|Jul 2024
|Racheline
|Probviously decidable. Estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
|Aug 2024
|mxdys, shown to be a Cryptid by DrDisentangle
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}
|April 2026
|sheep, shown to be a Cryptid by Daniel Yuan
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB|undecided}}
|Feb 2025
|Racheline
|Probviously halting.
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Probviously non-halting.
|}
The following machines have chaotic behavior, but have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}} [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]

== Larger Cryptids ==

A more complete list of notable known Cryptids over a wider range of states and symbols. These Cryptids were all "constructed" rather than "discovered".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Announcement !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF]]
|BB(432)
|style="width:30%;word-break:break-word"|Wade's machine: https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e
|
|2025
|Wade, based on work by CatIsFluffy and O'Rear
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|-
|PA
|BB(372)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord message]
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent.
|-
|RH
|BB(744)
|style="width:30%;word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|
|2016
|Matiyasevich and O’Rear
|The machine halts if and only if [https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann Hypothesis] is false.
|-
|Goldbach
|BB(25)
|style="width:30%;word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RD_1LC1RB_0RA1LC_0LQ1RE_0LF1RG_0LC1LF_0LF0LH_1LQ1LI_0RJ0LI_1RK0LJ_0RL0RS_1RL0RM_1RN1RM_0LO0LU_0LP1LO_1RH1LX_1LR1LQ_0RK0LT_1LR1RS_---1RC_1LV1LU_0LW0LJ_0RK0LW_1RY1LX_1RE1RY</pre></div></div>
|
|2016
|anonymous
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
| Erdős
| BB(5,4) and BB(15)
|style="width:30%;word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE</pre></div></div>
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA</pre></div></div>
|| [https://arxiv.org/abs/2107.12475 arxiv preprint] || Jul 2021 || [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods || The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base-3 representation of 2^n."
|-
|Weak Collatz
|BB(124) and BB(43,4)
|style="width:30%;word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified)
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified)
|
|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers.
Not independently verified, and probably easy to further optimise.
|-
|Fermat's Last Theorem
|BB(400)
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|Doctoral dissertation, ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965)
|1965
|Randels
|The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994. The machine is not explicitly given, but there is a flowchart to construct the machine, and the state count is claimed to be less than 400.
|-
| Bigfoot - compiled|| [[BB(7)]]||style="width:30%;word-break:break-word"| <code>0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB</code>|| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment] || June 2024 || <code>@Iijil1</code>|| Compilation of Bigfoot into 2 symbols, there was a previous compilation [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 with 8 states]
|-
| Hydra - compiled
|BB(9)
|style="width:30%;word-break:break-word"|<pre>
0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ
</pre>[[File:Hydra_9_states.txt]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Discord message]
|June 2024
|<code>@Iijil1</code>
|Compilation of Hydra into 2 symbols, all [https://discord.com/channels/960643023006490684/1084047886494470185/1253193750486974464 confirmed by Shawn Ligocki]. <code>@Iijil1</code> provided 24 TMs which all emulate the same behavior.
<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1247560072427474955 Previous compilation had 10 states], by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Beeping Busy Beaver ==

Cryptids were actually noticed in the [[Beeping Busy Beaver]] problem before they were in the classic Busy Beaver. See [[Mother of Giants]] describing a "family" of Turing machines which "[[probviously]]" [[quasihalt]], but requires solving a Collatz-like problem in order to actually prove it. They are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.
[[Category:Zoology]]
[[Category:Cryptids]]

Champions

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Papers & Talks

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== Papers ==
Paper produced by The bbchallenge Collaboration:

* '''Determination of the fifth Busy Beaver value.''' The bbchallenge Collaboration, J. Blanchard, Daniel Briggs, Konrad Deka, Nathan Fenner, Yannick Forster, Georgi Georgiev (Skelet), Matthew L. House, Rachel Hunter, Iijil, Maja Kądziołka, Pavel Kropitz, Shawn Ligocki, mxdys, Mateusz Naściszewski, savask, Tristan Stérin, Chris Xu, Jason Yuen, Théo Zimmermann. https://arxiv.org/abs/2509.12337 [https://github.com/bbchallenge/bbchallenge-paper (github)]
* '''Turing machines deciders, part I.''' The bbchallenge Collaboration, J. Blanchard, K. Deka, N. Fenner, T. Guilfoyle, Iijil, M. Kądziołka, P. Kropitz, S. Ligocki, P. Michel, M. Naściszewski, and T. Stérin. Apr. 2025. https://arxiv.org/abs/2504.20563
Paper produced by community members:
* Stérin, T., Woods, D. (2024). Hardness of Busy Beaver Value BB(15). In: Kovács, L., Sokolova, A. (eds) Reachability Problems. RP 2024. Lecture Notes in Computer Science, vol 15050. Springer, Cham. https://doi.org/10.1007/978-3-031-72621-7_9
** Paper: [https://arxiv.org/abs/2107.12475 arxiv version] (first released in 2021), [https://link.springer.com/chapter/10.1007/978-3-031-72621-7_9 conference version]
** Talk on September 25th 2024 at RP 2024 in Vienna: [https://drive.google.com/file/d/1gCoKCHvXGJeY7LvdQX9kN8u0ITdCKtJN/view?usp=sharing talk recording], [https://drive.google.com/file/d/1x3MRuKyh92XWlBHRawqDzxEdXIGSgsX_/view?usp=drive_link talk recording (compressed)], [https://docs.google.com/presentation/d/1rVEMr-AJM_kTzdI95xT2KLux4G24AoTBBCYQoPk8vS0/edit#slide=id.g2d359b31dea_0_2125 talk slides]
* Xu, C. (2024). Skelet #17 and the fifth Busy Beaver number. [https://arxiv.org/abs/2407.02426 Arxiv preprint]

== Talks ==

* '''''Le cinquième nombre Busy Beaver.''''' Talk by Tristan Stérin at the seminar of Prof. Timothy Gowers [https://www.college-de-france.fr/fr/chaire/timothy-gowers-combinatoire-chaire-statutaire/ College de France's lectures] , October 27th 2025.
* '''Determination of the fifth Busy Beaver value.''' Poster by The bbchallenge Collaboration and Tristan Stérin presented at the 31st International Conference on DNA Computing and Molecular Programming [https://dna31.sciencesconf.org/ DNA31]. [https://drive.google.com/file/d/1vRq6C7BIhKdq9hSZYdQtzV5JJRuwGc6q/view?usp=sharing (poster PDF)] [https://drive.google.com/file/d/1vQhcKVgEaRBeANwmRZ-fCzoJcP6kxJ4D/view?usp=sharing (poster SVG)]
* '''Coq proof of the fifth Busy Beaver value.''' Talk given by Tristan Stérin at '''[https://msp.cis.strath.ac.uk/types2025/cfp.html Types 2025],''' Glasgow, Scotland, June 9th 2025. [https://docs.google.com/presentation/d/1koofTIAOdk-DsKH6bnLi_jPeHG3-9vz2EcS_N5PsU-0/edit (slides)] [https://bbchallenge.org/types2025 (links)]
* '''Formal verification of the 5th Busy Beaver value.''' Talk given by Maja Kądziołka and Tristan Stérin at Cambridge's seminar "[https://talks.cam.ac.uk/show/index/164015 Formalisation of mathematics with interactive theorem provers]", March 13th 2025. [https://www.youtube.com/watch?v=5X6YVEnbLZU (recording)] [https://docs.google.com/presentation/d/1enqHCmXNYubhZyHmaj1qaYciY0bJtSJasc8gqwMLZ7c/edit?slide=id.g319e2b7dbee_0_3#slide=id.g319e2b7dbee_0_3 (slides, part 1)] [https://sakamoto.pl/~mei/bbslides/bbslides.html (interactive slides, part 2)]

Holdouts lists

2026-09-16T14:21:32Z

Azerty: Undo revision 8542 by Tianxinhuinb666 (talk)

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from an all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|964
|15,743,264
|-
!3-symbol
|0
|4
|4,720,332
|
|
|
|-
!4-symbol
|0
|11,362,197
|
|
|
|
|-
!5-symbol
|60
|
|
|
|
|
|-
!6-symbol
|412,086
|
|
|
|
|
|}

== Downloadable Holdouts Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.

=== BB(5) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|July 2nd, 2024
|bbchallenge
|0
|[https://cable.ayra.ch/empty/?id=0 List of 0 holdouts]
|'''BB(5) is solved!'''
|-
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

=== BB(6) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1546881471556620339 September 8th, 2026]
|@mxdys
|964
|(To be done)
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1543878992195620884 August 31st, 2026]
|@mxdys
|999
|(To be done)
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1543436534831906926 August 30th, 2026]
|@mxdys
|1003
|[https://wiki.bbchallenge.org/w/images/e/e0/BB6_holdouts_1003.txt BB6_holdouts_1003.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 August 15th, 2026]
|@mxdys
|1016
|[https://wiki.bbchallenge.org/w/images/c/cf/BB6_holdouts_1016.txt BB6_holdouts_1016.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1531681818439319604 July 28th, 2026]
|@mxdys
|1064
|[https://wiki.bbchallenge.org/w/images/e/e9/BB6_holdouts_1064.txt BB6_holdouts_1064.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1521203430046171146 June 29th, 2026]
|@mxdys
|1094
|[https://wiki.bbchallenge.org/w/images/9/90/BB6_holdouts_1094.txt BB6_holdouts_1094.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 April 29th, 2026]
|@mxdys
|1104
|[https://wiki.bbchallenge.org/w/images/b/bf/BB6_holdouts_1104.txt BB6_holdouts_1104.txt]
|More reduction by equivalences
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 April 25th, 2026]
|@mxdys
|1119
|[https://wiki.bbchallenge.org/w/images/a/ab/BB6_holdouts_1119.txt BB6_holdouts_1119.txt]
|Reduction by equivalences, one halting TM
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 March 25th, 2026]
|@mxdys
|1161
|[https://wiki.bbchallenge.org/w/images/2/2a/BB6_holdouts_1161.txt BB6_holdouts_1161.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 February 28th, 2026]
|@mxdys
|1214
|[https://wiki.bbchallenge.org/w/images/0/0e/BB6_holdouts_1214.txt BB6_holdouts_1214.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1473945366821998684 February 19th, 2026]
|@mxdys
|1226
|[https://wiki.bbchallenge.org/w/images/8/80/BB6_holdouts_1226.txt BB6_holdouts_1226.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1455369448264568904 December 30th, 2025]
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|November 22nd, 2025
|Robin Rovenszky
|Variable
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1450455364179857410 December 16th, 2025]
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1443989662019354647 November 28th, 2025]
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1438983076662477004 November 14th, 2025]
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1429892916763033601 October 20th, 2025]
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 July 14, 2024]
|@Justin Blanchard
|7255
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 BB6_holdouts_7255.txt]
|Work done by Justin Blanchard
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 July 6, 2024]
|@Justin Blanchard
|7430
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 BB6_holdouts_7430.txt]
|Work done by Justin Blanchard
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 June 21, 2024]
|@mxdys
|10,020
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 BB6_holdouts_10020.txt]
|Work done by Justin Blanchard
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
|12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 May 13, 2024]
|@mxdys
|35,788
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 BB6_holdouts_35788.txt]
|C++ version of the deciders from [[Coq-BB5]] ran on Shawns 181,851 holdouts<ref>https://discord.com/channels/960643023006490684/1239205785913790465/1239275863699361813</ref>
|-
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|}

=== BB(7) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui May 25th, 2026]
|
|17,823,260
|[https://drive.google.com/file/d/1GFumm3tzm58Yxza_sD_BdJ3RMY_DKmRv/view 7x2_holdouts_17823260.txt.zip]
|Work done by Andrew Ducharme
|-
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui October 25th, 2025]
|
|20,405,295
|[https://drive.google.com/file/d/12riXHC3Vz4tk_5mMV7__gZsF7hqAsZ_S/view bb7_holdouts_20405295.txt.gz]
|
|-
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|}

=== BB(3,3) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|}

=== BB(4,3) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- May 9th, 2026]
|@Andrew Ducharme
|5,127,263
|[https://drive.google.com/file/d/1xHukBgJnm1zctZkOMR0UzdQyzr4Fx5pC/view 4x3_holdouts_5127263.txt.zip]
|Work done by Andrew Ducharme
|-
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- April 26th, 2026]
|@Andrew Ducharme
|5,641,006
|[https://drive.google.com/file/d/1z8oBXUECL5dpUNAIqik2z_yI5JGmW79t/view 4x3_holdouts_5641006.txt.zip]
|Work done by Andrew Ducharme
|-
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- October 26th, 2025]
|
|9,401,447
|[https://drive.google.com/file/d/1r99v3dcdNTgGkmaydLKiZCp5KD3jGWeE/view 4x3_holdouts_9401447.txt.gz]
|
|-
|[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_ October 20th, 2025]
|@tjligocki
|18,138,027
|[https://drive.google.com/file/d/1vHufk09pI7JcTSAOj1XOzcZXJoX1Wa2a/view 4x3_holdouts_18138027.txt.gz]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|}

=== BB(3,4) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1uGWJMYYuVGJj581GX_vlm3Wv1adYbcRs May 11th, 2026]
|@XnoobSpeakable
|11,362,197
|[https://drive.google.com/file/d/1-Fnexjfq0mWyZDPa2MNFmU64EpSu-3Ig/view 3x4_holdouts_11362197.zip]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]]
|-
|[https://drive.google.com/drive/folders/1lNKc_GF2sa5ZfosleAlMNBQ3uJV4qClo April 30th, 2026]
|@Andrew Ducharme
|12,049,358
|[https://drive.google.com/file/d/1xCtI58doiCoI3lU_oQjK_21zsLFLF9g3/view 3x4_holdouts_12049358.txt.zip]
|Work done by Andrew Ducharme
|-
|[https://drive.google.com/drive/folders/1pj0MJBCA1hk6NxSj0mYvu5iXbTtVRt64 December 31st, 2025]
|
|12,435,284
|[https://drive.google.com/file/d/15e-Lwl_F6Jn_HoYFPA2a94jLTjX1mmHd/view 3x4st10c_holdouts_12435284.txt]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|}

=== BB(2,5) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|September 8, 202[https://discord.com/channels/960643023006490684/1259770421046411285/1546880921318326282 6]
|@mxdys
|60
|(To be done)
|-
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29, 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|}

=== BB(2,6) ===
{| class="wikitable sortable"
|+
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT October 6th, 2025]
|@tjligocki
|870,085
|[https://drive.google.com/file/d/1xZqEZFPke02pwBHX0EYsyDVaNU5o_OuO/view 2x6_holdouts_870085.txt.zip]
|Work done by Andrew Ducharme.
|-
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 16, 2025]
|
|970,101
|[https://drive.google.com/file/d/1Vjm3Qf-4a-Zr-E3EocjPWI5bjqH2YGil/view 2x6_holdouts_970101.txt.gz]
|Work done by Terry Ligocki
|-
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 12, 2025]
|
|20,358,011
|[https://drive.google.com/file/d/1YOoh4Zh2Jn5zaq9-C638S6VR_4jzd6pa/view 2x6_holdouts_20358011.zip]
|Work done by Andrew Ducharme
|-
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|}

== Busy Beaver adjacent holdouts lists ==

=== Beeping Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBB(4)
|August 2nd, 2026
|bbchallenge
|0
|
|BBB(4) had been proven to be 32,779,478 by August 2nd, 2026. See [https://github.com/carrino/Coq-BBB4 Coq-BBB4] for more details.
|-
|BBB(4)
|[https://discord.com/channels/960643023006490684/1385498968011575366/1387585253928992831 June 26, 2025]
|@mxdys
|3713
|[https://wiki.bbchallenge.org/w/images/6/62/BBB4_holdouts_3713.txt BBB4_holdouts_3713.txt]
|
|}

=== Reversible Turing machines ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400313138611490897 July 31, 2025]
|@Sligocki
|388
|[https://wiki.bbchallenge.org/w/images/8/86/BBrev6_holdouts_388.txt BBrev6_holdouts_388.txt]
|
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 July 31, 2025]
|@Sligocki
|409
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 BBrev6_holdouts_409.txt]
|
|}

=== Instruction-limited Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 October 7, 2025]
|@Shawn Ligocki
|348
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 bbi8_holdouts_348.txt]
|
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 October 6, 2025]
|@Peacemaker II
|441
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 BBi8_holdouts_441.txt]
|
|}

=== Fractran ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout August 5, 2026]
|@-d
|13
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_13_unofficial.txt sz23_13_unofficial.txt]
|Unofficial holdouts list
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 25, 2026]
|@-d
|694
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_694_unofficial.txt sz23_694_unofficial.txt]
|Unofficial holdouts list, reduction by Claude
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 25, 2026]
|@-d
|703
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_703_unofficial.txt sz23_703_unofficial.txt]
|Unofficial holdouts list, reduction by Claude
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 24, 2026]
|@-d
|8,021
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_8021_unofficial.txt sz23_8021_unofficial.txt]
|Unofficial holdouts list, reduction by Claude
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 11, 2026]
|@-d
|21233
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21233.txt sz23_21233.txt]
|Programs simulated to halting were removed
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 3, 2026]
|@-d
|21295
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21295.txt sz23_21295.txt]
|Programs simulated to halting were removed
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|21320
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21320.txt sz23_21320.txt]
|Programs simluated to halting were removed
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|28100
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_28100.txt sz23_28100.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|29188
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_29188.txt sz23_29188.txt]
|
|-
|BBf(23)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190 May 31, 2026]
|@-d
|29250
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_29250.txt sz23_29250.txt]
|
|-
|BBf(23)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190 May 31, 2026]
|@-d
|94367
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_94367.txt sz23_94367.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout May 30, 2026]
|@-d
|790,335
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_790335.txt sz23_790335.txt]
|
|-
|BBf(22)
|[https://github.com/int-y1/BBFractran/tree/main/holdout April 4, 2026]
|@-d
|3
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_3.txt sz22_3.txt]
|Holdout reduction by Claude Opus 4.6
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 March 28, 2026]
|@-d
|0
|
|Claude Opus 4.6's proof of nonhalting of all the 140 remaining BBf(21) holdouts: [https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 March 28th, 2026]
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|2003
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_2003_unofficial.txt sz22_2003_unofficial.txt]
|Result of the beeping permutation decider being run on sz22_5682.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|140
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_140_unofficial.txt sz21_140_unofficial.txt]
|Result of the beeping permutation decider being run on sz21_345.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 January 22, 2026]
|@-d
|143
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 sz21_143_workinprogress.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|5682
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_5682.txt sz22_5682.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|345
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_345.txt sz21_345.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|394
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz21_394.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|6
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz20_6.txt]
|BBf(20) has since been solved
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 December 13, 2025]
|@Sligocki
|10441
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 sz22_10441.txt]
|17 known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 December 11, 2025]
|@Sligocki
|10458
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 sz22_10457.txt]
|Known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 December 11, 2025]
|@dyuan01
|11130
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 message.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448214077028569098 December 10, 2025]
|@-d
|91123
|[https://raw.githubusercontent.com/int-y1/BBFractran/refs/heads/main/holdout/sz22_91123.txt sz22_91123.txt]
|Initial enumeration
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 November 17, 2025]
|@Sligocki
|760
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 sz21_760.txt]
|23 halting machines are removed
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 November 16, 2025]
|@dyuan01
|783
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 November 13, 2025]
|@dyuan01
|279
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 November 13, 2025]
|@-d
|902
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 sz20_902.txt]
|
|}

=== Terminating Turmites ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|TT(3,3)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1536748429089448036 August 11, 2026]
|@creeperman7002
|19368
|[https://discord.com/channels/960643023006490684/1243312334907375676/1536748429089448036 3x3_holdouts_19368.txt]
|
|-
|TT(2,4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1533832727441178644 August 3, 2026]
|@creeperman7002
|630
|[https://discord.com/channels/960643023006490684/1243312334907375676/1533832727441178644 TT2x4_holdouts_630.txt]
|
|-
|TT(2,4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 August 1, 2025]
|@creeperman7002
|2298
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 holdout_2x4_2298.txt]
|
|-
|TT(4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 July 12, 2025]
|@creeperman7002
|99
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 holdout4x2.txt]
|
|}

=== Busy Beaver for Lambda Calculus ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBλ(42)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1517130491512492083 June 18, 2026]
|@C++DSucker
|Variable (158)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1544675322#gid=1544675322 Spreadsheet]
|
|-
|BBλ(41)
|
|
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1502335946#gid=1502335946 Spreadsheet]
|'''BBλ(41) was solved on August 7th, 2026'''
|-
|BBλ(40)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1514278632695005224 June 10, 2026]
|@C++DSucker
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=741510300#gid=741510300 Spreadsheet]
|'''BBλ(40) was solved on July 20th, 2026'''
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|'''BBλ(39) was solved on June 10th, 2026'''
|-
|BBλ(38)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1362975027154129056 April 19, 2025]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=563816005#gid=563816005 Spreadsheet]
|'''BBλ(38) was solved on April 13th, 2026'''
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354938497848443032 March 27, 2025]
|@Racheline
|92
|[https://docs.google.com/spreadsheets/d/1yoiez7lDjZ-vRzj8NFWKWOxGlkK9IE_KWMUjeSWYWkw/edit?gid=0#gid=0 BBλ(37) 92 holdouts progress]
|BBλ(37) has since been solved
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 March 26, 2025]
|@Racheline
|216
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 BBL(37)_216.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 March 26, 2025]
|@Racheline
|278
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 BBL2(37)_278.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 March 26, 2025]
|@Sam
|317
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 BBL2(37)_317.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 March 26, 2025]
|@Sam
|333
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 BBL(37).txt]
|
|}

=== CounterScript ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBCS(11)
|September 15, 2026
|@Etalon
|0
|[https://cable.ayra.ch/empty/?id=0 Empty]
|The holdout was resolved by hand.
|-
|BBCS(11)
|[https://github.com/AzertyWasTaken/CounterScript/tree/main/Holdouts July 10, 2026]
|@Azerty
|1
|[https://github.com/AzertyWasTaken/CounterScript/blob/main/Holdouts/11.txt 11.txt]
|The holdout may be resolved by hand.
|}

== References ==

Main Page

2026-09-16T14:21:14Z

Azerty: Undo revision 8543 by Tianxinhuinb666 (talk)

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 computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!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;" | [[BB(6)]] > <math>2 \uparrow \uparrow 2 \uparrow \uparrow 2 \uparrow \uparrow 10</math>
| style="background: orange;" | [[BB(7)]] > <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[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;" |
| 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;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] > <math>10^{10^{10^{3\,314\,360}}}</math>
| style="background: #ddc290;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
|-
! 6-symbol
| style="background: #ddc290;" | [[BB(2,6)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
|}

In the above table, <span style="background: orange">cells are highlighted in orange</span> when there are known [[Cryptids]] (mathematically-hard machines) in that class, and <span style="background: #ddc290">cells are highlighted in tan</span> when the existence of a Cryptid is given by using a known one with fewer states or symbols.

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:

* Formalising 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: April 2026]]
* [[TMBR: March 2026]]
* [[TMBR: February 2026]]

[[TMBR: May 2026]], [[TMBR: June 2026]], [[TMBR: July 2026]], [[TMBR: August 2026]], and [[TMBR: September 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 />

User talk:Azerty

2026-09-15T20:13:31Z

Azerty:

Solved your last holdout for BBCS(11) :) [[User:Etalon|Etalon]] ([[User talk:Etalon|talk]]) 20:02, 15 September 2026 (UTC)

: I have already solved the holdout but I only count holdouts decided by my decider scripts so I can keep track of what deciders to make next. By the way, I currently don't research BBCS but I may be back soon. [[User:Azerty|Azerty]] ([[User talk:Azerty|talk]]) 20:13, 15 September 2026 (UTC)

User talk:Azerty

2026-09-15T20:09:09Z

Azerty: /* BBCS(11) */ new section

Solved your last holdout for BBCS(11) :) [[User:Etalon|Etalon]] ([[User talk:Etalon|talk]]) 20:02, 15 September 2026 (UTC)

== BBCS(11) ==

I have already solved the holdout but I only count holdouts decided by my decider scripts so I can keep track of what deciders to make next.
By the way, I currently don't research BBCS but I may be back soon.

Measuring large numbers

2026-09-13T11:48:19Z

Azerty: TODO sections

{{Stub}}
The Busy Beaver function is known to grow faster than any computable function, as do many of it's variants. Because of this, values of the function or lower bounds on the function can be very, very large, so it is important 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" for the 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 the [[Fast-Growing Hierarchy]] can help us achieve this, as do Laver tables, or the diagonalization of an axiomatic foundation of mathematics.

== Exponential ==
TODO

== Tetrational ==
TODO

== Ackermannian ==
TODO

== 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

[[Category:Stub]]

Pebble Automaton

2026-09-06T16:01:19Z

Azerty: /* External resources */

{{Stub}}
A pebble automaton is a type of [[cellular automaton]]. Each cell contains a non-negative number of pebbles, and the number of pebbles determines how many pebbles to push to each neighbor. Though pebble automata could be considered in any number of dimensions, investigation so far has primarily focused on 1 dimension.

It is not proven that it is Turing-Complete but it probably is since some automata can compute Collatz-like functions.

== Definition ==
A 1D pebble automaton is defined by three integer functions L(n), S(n) and R(n), defined over non-negative integers n, under the constraints:

* L(n) ≥ 0
* S(n) ≥ 0
* R(n) ≥ 0
* L(n) + S(n) + R(n) = n for all n

A simulation of a pebble automaton is represented by p(x, t), the number of pebbles at position x and time t. The initial configuration (t=0) has all n pebbles at x=0 and each generation, the pebbles are moved around according to the L, S, R rules:

* p(0, 0) = n
* p(x, 0) = 0 for x ≠ 0
* p(x, t+1) = R(p(x-1, t)) + S(p(x,t)) + L(p(x+1, t))

Informally, when a cell has n pebbles, it pushes L(n) to its left neighbor, R(n) to its right neighbor and S(n) stay.

Because the number of pebbles is preserved in each row, the maximum number of pebbles in one location is n. Therefore, a pebble automaton only needs to specify the function values up to n.

== Busy Beaver function ==
Let us define peBBle(n) as the maximum number of steps it takes for any pebble automaton to stabilize, starting from an initial state of n pebbles in a single cell.

Notation (long version): <code><nowiki>n --> L(n) | S(n) | R(n)</nowiki></code>

<code>0 --> 0 | 0 | 0</code> is omitted since it is forced by the definition.

The short version sets the rules in increasing order, separated by underscores, and removes <code>S(n)</code> since it can be obtained with the equation <code>S(N) = n - L(n) - R(n)</code>

If there are over 10 pebbles, we can extend the notation with letters, like with hexadecimal colors.
{| class="wikitable"
|+
!n
!peBBle(n)
!Champion
|-
|1
|0
|<code>00</code>
|-
|2
|1
|<code>00_01</code>
|-
|3
|3
|<code>00_01_20</code>
|-
|4
|6
|<code>00_01_20_03</code>
|-
|5
|≥ 11
|<code>00_11_03_21_04</code>
|-
|6
|≥ 17
|<code>00_11_03_21_40_15</code>
|-
|7
|≥ 28
|<code>00_11_03_21_10_33_61</code>
|-
|8
|≥ 63
|<code>00_11_03_21_50_30_24_70</code>
|-
|9
|≥ 63577
|<code>00_02_01_40_14_12_00_52_81</code>
|}

== Analysis ==

=== peBBle(5) ===
<syntaxhighlight line="1">
.5.....
.14....
.311...
..41...
.212...
1.3.1..
1..31..
1...4..
1..211.
1.1.21.
1.11.2.
1.111.1
</syntaxhighlight>

=== peBBle(6) ===
<syntaxhighlight line="1">
...6.....
..1.5....
..141....
..312....
...5.1...
..41.1...
.212.1...
1.3.11...
1..311...
1...41...
1..212...
1.1.3.1..
1.1..31..
1.1...4..
1.1..211.
1.1.1.21.
1.1.11.2.
1.1.111.1
</syntaxhighlight>

=== peBBle(7) ===
<syntaxhighlight line="1">
..7.......
.6.1......
3.31......
.3.4......
..511.....
.1411.....
.3121.....
..5.2.....
.141.1....
.312.1....
..5.11....
.14.11....
.31111....
..4111....
.21211....
1.3.21....
1..4.2....
1.212.1...
11.3.11...
11..311...
11...41...
11..212...
11.1.3.1..
11.1..31..
11.1...4..
11.1..211.
11.1.1.21.
11.1.11.2.
11.1.111.1
</syntaxhighlight>

=== peBBle(8) ===
TODO

=== peBBle(9) ===
The champion performs a [[Collatz-like]] function.

Analysis:
<syntaxhighlight line="1">
A(n) := 60^n3
B(n) := 60^n21

A(3k+1) -(20k+12)-> B(11k+5)
A(3k+2) -(8k+7)---> HALT
A(3k+3) -(20k+26)-> B(11k+12)

B(3k) -(8k+3)---> HALT
B(3k+1) -(20k+16)-> A(11k+7)
B(3k+2) -(20k+22)-> A(11k+11)

9 -(8)-> B(4) -(36)-> A(18) -(126)-> B(67) -(456)-> A(249) -(1666)-> B(914) -(6102)-> A(3355) -(22372)-> B(12303) -(32811)-> HALT
</syntaxhighlight>

== Sources ==

* Discord thread: https://discord.com/channels/960643023006490684/1535695107448381591
== External resources ==

* [https://gist.github.com/AlephSquirrel/9ada4b14b675d18fe8755b200b20509d Uiua simulator]
* [https://github.com/AzertyWasTaken/Busy-Beaver/tree/main/Pebble%20Automaton Enumerator GitHub repo]
* [https://azertywastaken.github.io/Busy-Beaver/Website/pebble-automaton.html Simulator website]

[[Category:Functions]]
[[Category:Stubs]]

Cellular automaton

2026-09-06T16:01:07Z

Azerty: added enumerator and simulator links

{{Stub}}

Cellular automata (CA) are discrete, abstract computational systems that consist of a regular grid of cells, each in a finite number of states. The grid evolves in discrete time steps according to a local transition rule based on the states of neighboring cells. Despite their simple definition, cellular automata can exhibit complex behavior, including chaos, self-replication, and universal computation.

The simplest nontrivial CA are elementary cellular automata (1D, binary states, radius-1 neighborhood).

The rule is defined by an 8-bit lookup table (bits for patterns 111 to 000), numbered 0 to 255. For instance:

* Rule 110: Known to be Turing-complete.<ref>https://wpmedia.wolfram.com/sites/13/2018/02/15-1-1.pdf</ref>
* Rule 30: Produces chaotic, pseudorandom patterns.
Cellular automata may have:

* Multiple number of dimensions.
* Multiple "alive" cell states.
* Various neighborhood radius and shape.

== Busy Beaver function ==

=== Halting conditions ===
There is no single agreed-upon definition of "halting" for cellular automaton. Each possible definition of halting produces different busy beaver functions. Here are some examples.

* '''All 0s''': A cellular automaton is said to halt if it reaches a state where every cell has state 0.

* '''Still life''': A cellular automaton is said to halt if it reaches a fixed point in the rule's update function. This definition of halting allows busy beaver functions which count the maximum number of on cells a halting cellular automaton can produce.

* '''Oscillator''': A cellular automaton is said to halt if it reaches a cycle of repeated states. Busy beaver functions for the "longest running" starting pattern or rule can measure the time until the first state which gets repeated, or the time until a state appears twice (which gives a bonus to cellular automata that end up in a long-period oscillator). Busy beaver functions counting the number of "on" states can measure the minimum population, or the maximum population, or the average population, or the population in the first repeating state.

* '''Oscillating population''': A cellular automaton is said to halt if it reaches a state where its population (number of cells which are not 0) is cyclic. If the actual state of the cellular automaton is not cyclic, it may be nontrivial to determine whether its population will remain cyclic. In the Game of Life, it is common for small and medium-sized patterns to resolve into a collection of small still-lifes, oscillators, and gliders which travel away from the center. Such patterns never repeat, but they do have cyclic populations. Hence, most small starting patterns in Life halt under this definition but not under either of the previous three definitions.

* '''Oscillating population relative to a background pattern''': In rule 110, for any starting pattern with a finite number of on cells, the population will grow without bound. Hence, no finite patterns halt in rule 110 using any of the previous definitions.
* '''Halting rule''': A cellular automaton is said to halt if the next transition rule is undefined for at least one cell.
The construction of Turing-complete computation in rule 110 takes place on an infinitely repeating background pattern, which Matthew Cook calls the ''ether''. One could define the "population" of a pattern in rule 110 as the number of 1s in the symmetric difference with the ether. One could then say a pattern halts when its population relative to the ether becomes cyclic.

== Sources ==
<references />

* [https://github.com/AzertyWasTaken/Busy-Beaver/tree/main/Cellular%20Automaton Enumerator GitHub repo]
* [https://azertywastaken.github.io/Busy-Beaver/Website/cellular-automaton.html Simulator website]

Pebble Automaton

2026-09-06T15:59:15Z

Azerty: /* External resources */ added simulator and enumerator

{{Stub}}
A pebble automaton is a type of [[cellular automaton]]. Each cell contains a non-negative number of pebbles, and the number of pebbles determines how many pebbles to push to each neighbor. Though pebble automata could be considered in any number of dimensions, investigation so far has primarily focused on 1 dimension.

It is not proven that it is Turing-Complete but it probably is since some automata can compute Collatz-like functions.

== Definition ==
A 1D pebble automaton is defined by three integer functions L(n), S(n) and R(n), defined over non-negative integers n, under the constraints:

* L(n) ≥ 0
* S(n) ≥ 0
* R(n) ≥ 0
* L(n) + S(n) + R(n) = n for all n

A simulation of a pebble automaton is represented by p(x, t), the number of pebbles at position x and time t. The initial configuration (t=0) has all n pebbles at x=0 and each generation, the pebbles are moved around according to the L, S, R rules:

* p(0, 0) = n
* p(x, 0) = 0 for x ≠ 0
* p(x, t+1) = R(p(x-1, t)) + S(p(x,t)) + L(p(x+1, t))

Informally, when a cell has n pebbles, it pushes L(n) to its left neighbor, R(n) to its right neighbor and S(n) stay.

Because the number of pebbles is preserved in each row, the maximum number of pebbles in one location is n. Therefore, a pebble automaton only needs to specify the function values up to n.

== Busy Beaver function ==
Let us define peBBle(n) as the maximum number of steps it takes for any pebble automaton to stabilize, starting from an initial state of n pebbles in a single cell.

Notation (long version): <code><nowiki>n --> L(n) | S(n) | R(n)</nowiki></code>

<code>0 --> 0 | 0 | 0</code> is omitted since it is forced by the definition.

The short version sets the rules in increasing order, separated by underscores, and removes <code>S(n)</code> since it can be obtained with the equation <code>S(N) = n - L(n) - R(n)</code>

If there are over 10 pebbles, we can extend the notation with letters, like with hexadecimal colors.
{| class="wikitable"
|+
!n
!peBBle(n)
!Champion
|-
|1
|0
|<code>00</code>
|-
|2
|1
|<code>00_01</code>
|-
|3
|3
|<code>00_01_20</code>
|-
|4
|6
|<code>00_01_20_03</code>
|-
|5
|≥ 11
|<code>00_11_03_21_04</code>
|-
|6
|≥ 17
|<code>00_11_03_21_40_15</code>
|-
|7
|≥ 28
|<code>00_11_03_21_10_33_61</code>
|-
|8
|≥ 63
|<code>00_11_03_21_50_30_24_70</code>
|-
|9
|≥ 63577
|<code>00_02_01_40_14_12_00_52_81</code>
|}

== Analysis ==

=== peBBle(5) ===
<syntaxhighlight line="1">
.5.....
.14....
.311...
..41...
.212...
1.3.1..
1..31..
1...4..
1..211.
1.1.21.
1.11.2.
1.111.1
</syntaxhighlight>

=== peBBle(6) ===
<syntaxhighlight line="1">
...6.....
..1.5....
..141....
..312....
...5.1...
..41.1...
.212.1...
1.3.11...
1..311...
1...41...
1..212...
1.1.3.1..
1.1..31..
1.1...4..
1.1..211.
1.1.1.21.
1.1.11.2.
1.1.111.1
</syntaxhighlight>

=== peBBle(7) ===
<syntaxhighlight line="1">
..7.......
.6.1......
3.31......
.3.4......
..511.....
.1411.....
.3121.....
..5.2.....
.141.1....
.312.1....
..5.11....
.14.11....
.31111....
..4111....
.21211....
1.3.21....
1..4.2....
1.212.1...
11.3.11...
11..311...
11...41...
11..212...
11.1.3.1..
11.1..31..
11.1...4..
11.1..211.
11.1.1.21.
11.1.11.2.
11.1.111.1
</syntaxhighlight>

=== peBBle(8) ===
TODO

=== peBBle(9) ===
The champion performs a [[Collatz-like]] function.

Analysis:
<syntaxhighlight line="1">
A(n) := 60^n3
B(n) := 60^n21

A(3k+1) -(20k+12)-> B(11k+5)
A(3k+2) -(8k+7)---> HALT
A(3k+3) -(20k+26)-> B(11k+12)

B(3k) -(8k+3)---> HALT
B(3k+1) -(20k+16)-> A(11k+7)
B(3k+2) -(20k+22)-> A(11k+11)

9 -(8)-> B(4) -(36)-> A(18) -(126)-> B(67) -(456)-> A(249) -(1666)-> B(914) -(6102)-> A(3355) -(22372)-> B(12303) -(32811)-> HALT
</syntaxhighlight>

== Sources ==

* Discord thread: https://discord.com/channels/960643023006490684/1535695107448381591
== External resources ==

* [https://gist.github.com/AlephSquirrel/9ada4b14b675d18fe8755b200b20509d Uiua simulator]
* [https://github.com/AzertyWasTaken/Busy-Beaver/tree/main/Pebble%20Automaton Enumeration GitHub repo]
* [https://azertywastaken.github.io/Busy-Beaver/Website/pebble-automaton.html Simulator website]

[[Category:Functions]]
[[Category:Stubs]]

Post Tag System

2026-09-05T21:52:53Z

Azerty: added github repo

{{Stub}}
A '''Post Tag System''' is a [[Turing-complete]] formal system invented by Emil Post in 1920 (but not published until 1943). Post explored tag systems before [[Turing machines]] were invented and thus before the solution to the Entscheidungsproblem, the Church-Turing thesis and the undecidability of the Halting problem were known. According to Wolfram, Post studied systems where no more than 2 symbols were ever deleted or added and found none produced complicated behavior, but that a rule with up to 3 symbols deleted or added could produce chaotic and seemingly intractable behavior. Tag systems were proven to be Turing-complete by Marvin Minsky in 1961.

== Tag System ==
A tag system is defined by:

* A positive integer ''v'', called the deletion number
* A finite alphabet of symbols: ''A''
* One "production rules" for each symbol: <math>w_a \in A^*</math> (for each <math>a \in A</math>)

We will often write a v-tag system to specify that it is a tag system with deletion number v (ex: a 2-tag system).

A tag system operates on a tape (a possibly empty sequence of symbols). At each step:

* If the tape is shorter than v, it halts.
* Otherwise, it reads the first symbol (a), deletes the first v symbols from the tape and appends <math>w_a</math>

It repeats this process until it halts.

1-tag systems are decidable, but v-tag systems for any v ≥ 2 are undecidable and (in fact) Turing complete.

== BB_PT ==
Let us define the size of a tag system to be the <math>S = |A| + \sum_{a \in A} |w_a| = \sum_{a \in A} (|w_a| + 1)</math>.

Let BB_PT(v, S) = max steps across all v-tag systems of size S which halt when started on tape <math>0^v</math>. We default to BB_PT(S) = BB_PT(2,S) since 2-tags is Turing complete.

Starting with tape <math>0^v</math> means that after the first step the tape is <math>w_0</math>.

== Champions ==
{| class="wikitable"
|+2-tag Champions
!Size
!BB_PT(n)
!Champion
!Holdouts
|-
|1
|= 1
|<code>0 -> eps</code>
|0
|-
|2
|= 1
|<code>0 -> 0</code>
|0
|-
|3
|= 1
|<code>0 -> 0, 1 -> eps</code>
|0
|-
|4
|= 2
|<code>0 -> 11, 1 -> eps</code>
|0
|-
|5
|= 4
|<code>0 -> 011, 1 -> eps</code>
|0
|-
|6
|= 5
|<code>0 -> 011, 1 -> 1</code>
|0
|-
|7
|≥ 10
|<code>0 -> 0111, 1 -> 1</code>
|1
|-
|8
|≥ 19
|<code>0 -> 111, 1 -> 20, 2 -> eps</code>
|8
|-
|9
|≥ 49
|<code>0 -> 11, 1 -> 021, 2 -> 2</code>
|326
|-
|10
|≥ 779
|<code>0 -> 112, 1 -> 1, 2 -> 002</code>
|2,491
|-
|11
|≥ 196,841
|<code>0 -> 120221, 1 -> 0, 2 -> 2</code>
|14,972
|}
"eps" is the empty string

== Cryptids ==
realsheepthe constructed a size 11 [[Cryptid]] (originally mislabeled as a near-Cryptid) <code>0 —> 11111, 1 —> 20, 2 —> 1</code> which simulates the [[Collatz-like]] iteration:<pre>S(n) = 1^n
S(0) —(0)-> (halt)
S(1) —(0)-> 1 (halt)
S(2k + 2) -(2k + 2)-> S(k + 1)
S(2k + 3) -(2k + 4)-> S(5k + 9)

00 -(1)-> S(5)
-(6)-> S(14)
-(14)-> S(7)
-(8)-> S(19)
-(20)-> S(49)
-(50)-> S(124)
...</pre>
== Analysis ==
BB_PT(7) champion analysis:<syntaxhighlight line="1">
0 -> 0111, 1 -> 1

S(n) = 1^n 0 111
S(2k) —(k + 1)—> S(k + 2)
S(2k + 1) —(2k + 4)—> 1 (halt)

00 —(1)—> S(0)
—(1)—> S(2)
—(2)—> S(3)
—(6)—> 1 (halt)
</syntaxhighlight>BB_PT(8) champion analysis:<syntaxhighlight line="1" start="1">
0 -> 111, 1 -> 20, 2 ->

S(n) = 1^(n + 2)
S(2k) —(2k + 2)—> (halt)
S(2k + 1) —(2k + 4)—> S(3k + 3)

00 —(1)—> S(1)
—(4)—> S(3)
—(6)—> S(6)
—(8)—> (halt)
</syntaxhighlight>

== Sources ==
GitHub repository (with documentation, deciders and holdout lists): https://github.com/AzertyWasTaken/Busy-Beaver/tree/main/Post%20Tag%20System
[[Category:Functions]]
[[Category:Stubs]]

Cyclic Tag

2026-09-05T21:50:00Z

Azerty: /* See also */ added github repo

== Cyclic Tag System ==
Cyclic Tag is a [[Turing-complete]] computational model<sup>[[wikipedia:Tag_system#Cyclic_tag_systems|[1]]]</sup> where a binary string evolves under a set of rules applied in cyclic order. There are various halting conditions that one may choose. They are explicitly stated further below.

=== A cyclic tag system has: ===

* A finite '''string''' of '''bits'''.
* A '''code''' consisting of a list of '''production rules'''.
* A '''head''' pointing at one of the production rules.

=== Execution rules: ===

* The cyclic tag system starts out with '''string''' equal to "1", with the '''head''' pointing to the '''first rules'''.
* At each step of the computation, '''delete''' the '''first symbol''' of the '''string''' and append the '''production rules''' to the '''string''' if it 1.
* '''Move''' the '''head''' one rule forward or at the first rule if it is pointing to the last rule.
* The cyclic tag system halts when it reaches the empty string.

== Busy Beaver Function ==
The function ''BBCT(n)'' is the largest number of steps that any cyclic tag systems of size ''n'' takes before halting.

The size of a cyclic tag system is the sum of the lengths of all the production rules plus the number of production rules (same as for [[Post Tag System|BB_PT]]).
{| class="wikitable"
|+
!BBCT(n)
!Value
!Champion
|-
|1
|1
|<code>empty</code>
|-
|2
|2
|<code>0</code>
|-
|3
|3
|<code>00</code>
|-
|4
|4
|<code>000</code>
|-
|5
|7
|<code>010_</code>
|-
|6
|8
|<code>010_0</code>
|-
|7
|≥ 13
|<code>0110__</code>
|-
|8
|≥ 28
|<code>101_10_</code>
|-
|9
|≥ 717
|<code>1011_00_</code>
|-
|10
|≥ 1,626,862
|<code>1001_1_00</code>
|-
|11
|≥ 1,859,166
|<code>10_10010_0</code>
|-
|12
|≥ 6,328
|<code>0111__010_0</code>
|-
|13
|≥ 5,217,735
|<code>001110_1_0_0</code>
|-
|14
|≥ 154,171,565
|<code>01011__0_0100</code>
|}

==See also==
* [[Tiny Tag]]

== Sources ==
GitHub repository (with documentation, deciders and holdout lists): https://github.com/AzertyWasTaken/Busy-Beaver/tree/main/Cyclic%20Tag%20System
[[Category:functions]]

Cellular automaton

2026-09-04T11:59:46Z

Azerty: /* Halting conditions */ added new halting condition

Cyclic Tag

2026-08-31T14:52:07Z

Azerty:

== Cyclic Tag System ==
Cyclic Tag is a [[Turing-complete]] computational model<sup>[[wikipedia:Tag_system#Cyclic_tag_systems|[1]]]</sup> where a binary string evolves under a set of rules applied in cyclic order. There are various halting conditions that one may choose. They are explicitly stated further below.

=== A cyclic tag system has: ===

* A finite '''string''' of '''bits'''.
* A '''code''' consisting of a list of '''production rules'''.
* A '''head''' pointing at one of the production rules.

=== Execution rules: ===

* The cyclic tag system starts out with '''string''' equal to "1", with the '''head''' pointing to the '''first rules'''.
* At each step of the computation, '''read''' the '''first symbol''' of the '''string''' and append the '''production rules''' to the '''string''' if it 1.
* '''Move''' the '''head''' one rule forward or at the first rule if it is pointing to the last rule.
* The cyclic tag system halts when it has less than two symbol.

== Busy Beaver Function ==
The function ''BBCT(n)'' is the largest number of steps that any cyclic tag systems of size ''n'' takes before halting.

The size of a cyclic tag system is the sum of the symbols count and production rules count.
{| class="wikitable"
|+
!BBCT(n)
!Value
!Champion
|-
|1
|1
|<code>empty</code>
|-
|2
|2
|<code>0</code>
|-
|3
|3
|<code>00</code>
|-
|4
|4
|<code>000</code>
|-
|5
|7
|<code>010_</code>
|-
|6
|8
|<code>010_0</code>
|-
|7
|≥ 13
|<code>0110__</code>
|-
|8
|≥ 28
|<code>101_10_</code>
|-
|9
|≥ 717
|<code>1011_00_</code>
|-
|10
|≥ 209
|<code>11_0_0100</code>
|-
|11
|≥ 2,269
|<code>1011_1_00_</code>
|-
|12
|≥ 6,328
|<code>0111__010_0</code>
|-
|13
|≥ 50,906
|<code>0011_010_1_0</code>
|}

==See also==
* [[Tiny Tag]]
[[Category:functions]]

Cyclic Tag

2026-08-31T14:51:28Z

Azerty:

== Cyclic Tag System ==
Cyclic Tag is a [[Turing-complete]] computational model<sup>[[wikipedia:Tag_system#Cyclic_tag_systems|[1]]]</sup> where a binary string evolves under a set of rules applied in cyclic order. There are various halting conditions that one may choose. They are explicitly stated further below.

=== A cyclic tag system has: ===

* A finite '''string''' of '''bits'''.
* A '''code''' consisting of a list of '''production rules'''.
* A '''head''' pointing at one of the production rules.

=== Execution rules: ===

* The cyclic tag system starts out with '''string''' equal to "1", with the '''head''' pointing to the '''first rules'''.
* At each step of the computation, '''read''' the '''first symbol''' of the '''string''' and append the '''production rules''' to the '''string''' if it 1.
* '''Move''' the '''head''' one rule forward or at the first rule if it is pointing to the last rule.
* The cyclic tag system halts when it has less than two symbol.

== Busy Beaver Function ==
The function ''BBCT(n)'' is the largest number of steps that any cyclic tag systems of size ''n'' takes before halting.

The size of a cyclic tag system is the sum of the symbols count and production rules count.
{| class="wikitable"
|+
!BBCT(n)
!Value
!Champion
|-
|1
|1
|<code>empty</code>
|-
|2
|2
|<code>0</code>
|-
|3
|3
|<code>00</code>
|-
|4
|4
|<code>000</code>
|-
|5
|7
|<code>010_</code>
|-
|6
|8
|<code>010_0</code>
|-
|7
|≥ 13
|<code>0110__</code>
|-
|8
|≥ 28
|<code>101_10_</code>
|-
|9
|≥ 717
|<code>1011_00_</code>
|-
|10
|≥ 209
|<code>11_0_0100</code>
|-
|11
|≥ 2,269
|<code>1011_1_00_</code>
|-
|12
|≥ 6,328
|<code>0111__010_0</code>
|-
|13
|≥ 50,906
|<code>0011_010_1_0</code>
|}

==See also==
* [[Tiny Tag]]
{{DEFAULTSORT:Cyclic Tag System}}
[[Category:functions]]

Cyclic Tag

2026-08-31T14:51:15Z

Azerty: Created a page for the (original) cyclic tag system

{{DISPLAYTITLE:Cyclic Tag System}}

== Cyclic Tag System ==
Cyclic Tag is a [[Turing-complete]] computational model<sup>[[wikipedia:Tag_system#Cyclic_tag_systems|[1]]]</sup> where a binary string evolves under a set of rules applied in cyclic order. There are various halting conditions that one may choose. They are explicitly stated further below.

=== A cyclic tag system has: ===

* A finite '''string''' of '''bits'''.
* A '''code''' consisting of a list of '''production rules'''.
* A '''head''' pointing at one of the production rules.

=== Execution rules: ===

* The cyclic tag system starts out with '''string''' equal to "1", with the '''head''' pointing to the '''first rules'''.
* At each step of the computation, '''read''' the '''first symbol''' of the '''string''' and append the '''production rules''' to the '''string''' if it 1.
* '''Move''' the '''head''' one rule forward or at the first rule if it is pointing to the last rule.
* The cyclic tag system halts when it has less than two symbol.

== Busy Beaver Function ==
The function ''BBCT(n)'' is the largest number of steps that any cyclic tag systems of size ''n'' takes before halting.

The size of a cyclic tag system is the sum of the symbols count and production rules count.
{| class="wikitable"
|+
!BBCT(n)
!Value
!Champion
|-
|1
|1
|<code>empty</code>
|-
|2
|2
|<code>0</code>
|-
|3
|3
|<code>00</code>
|-
|4
|4
|<code>000</code>
|-
|5
|7
|<code>010_</code>
|-
|6
|8
|<code>010_0</code>
|-
|7
|≥ 13
|<code>0110__</code>
|-
|8
|≥ 28
|<code>101_10_</code>
|-
|9
|≥ 717
|<code>1011_00_</code>
|-
|10
|≥ 209
|<code>11_0_0100</code>
|-
|11
|≥ 2,269
|<code>1011_1_00_</code>
|-
|12
|≥ 6,328
|<code>0111__010_0</code>
|-
|13
|≥ 50,906
|<code>0011_010_1_0</code>
|}

==See also==
* [[Tiny Tag]]
{{DEFAULTSORT:Cyclic Tag System}}
[[Category:functions]]

Cyclic Tree Rewriting System

2026-08-31T14:30:53Z

Azerty: /* Collection of Champions */ Consistency fixes

== Cyclic Tree Rewriting System ==
This variant of Cyclic Tag operates upon a Dyck String instead of a binary string. A Dyck String is a string of parentheses “(“ and “)” of length 2n, that consists of an equal number of symbols such that for all prefixes, “(“ occurs at least as many times as the other “)”. In shorter terms, a Dyck String is a balanced parenthetical string of length 2n. There does not exist a Dyck String (in this format) with an odd length (hence the 2n).

=== Rooted Ordered Trees and Dyck Strings ===
Dyck Strings (pronounced as: DYE-K) encode finite rooted ordered trees through a well-known bijection that maps the structure of a tree to a balanced sequence of parentheses. The most common method involves a depth-first traversal of the tree. A rooted ordered tree is a tree with these key features:

* One vertex is distinguished as the root,

* All edges are directed away from the root (conceptually, though not always drawn with arrows),

* For each node, the children are arranged in a left-to-right order.

=== Introduction to Δ and P ===
Define an ordered pair [Δ,Ρ] where Δ is an initial finite non-empty Dyck String & P are production rules P=(R₁,R₂,…,Rₖ), each in the form “X→Y” where X and Y are finite non-empty Dyck Strings. Y has the ability to be the symbol D. When Y is instead D, this means: in Δ, delete X.

=== Solving a Given Δ and P: ===
We follow these steps:

* '''(1)''' Begin with R₁,
* '''(2)''' Scan Δ from left to right,
* '''(3)''' Find the leftmost instance of X in Δ (according to said R)
* '''(4)''' Rewrite X as Y and leave the rest of Δ unchanged (or delete X if Y is the symbol D)
* '''(5)''' Move to the next R in cyclic order (1, 2,…,n,1,2,…)
* '''(6)''' If a rule doesn’t apply, simply skip it and move to the immediate next one in P,
* '''(7)''' Repeat from (2) on the altered Δ each time.

'''''NOTE:''''' Duplicate rules in the same P are allowed.

=== Halting Conditions ===
Some given pairs [Δ,P] never halt. However, some do! Here are two halting conditions we can define:

'''Halt if:'''

Δ becomes “∅” (empty), or if there does not exist a rule in P that can transform Δ any further, meaning that Δ is what we can call “stuck”.

== Example Run-Through ==
(Initial Dyck String) Δ: ()()(())

(Production Rules) P: R₁=()→(()), R₂=(())()→D

* ()()(()) initial String Δ,
* (())()(()) as per R₁, (()) replaces (),
* (()) as per R₂, (())() is deleted,
* ((())) as per R₁, (()) replaces (),
* '''SKIP R₂, IT DOES NOT APPLY,'''
* …

This example doesn't halt as ((…()…)) will keep expanding indefinitely.

== Correlation to BB(n) ==
From here, we can define a function analogous to the Busy Beaver function by taking the maximal finite halting times for all pairs. Here’s how we can do it:

Let |x| denote the length of x.

A pair [Δ,P] of “size n” consists of n rules, where for each Rᵢ ∈ P, |X|≤2n and |Y|≤2n ('''NOTE''': the lengths of X and Y for each Rᵢ do NOT have to be the same, but they must be ≤2n). We also assume that |Δ|≤2n. We say that skipping a rule counts as a step, and a “step” is defined as a singular rule application.

== Function ==
The “Cyclic Tree Busy Beaver” function CTBB(n) is therefore defined as follows:

* '''(1)''' Run all pairs [Δ,P] of size n,
* '''(2)''' Let S be the set of all pairs S=(P₁,P₂,…,Pₘ) that halted, and filter out (discard) all pairs that do not,
* '''(3)''' Let S’ be the set of all steps for every pair in S to halt,
* '''(4)''' CTBB(n) outputs max(S’).

==== Notes ====
Because Tag Systems in this fashion are Turing-Complete, we can expect CTBB(n) to be on par with the classic BB(n).

We can also correctly say that CTBB(n)>*f(n) where f(n) is any computable function (where >* represents eventual domination).

== Known Values ==
{| class="wikitable"
|+Small CTBB values
|CTBB
|Value / Lower bound
|Discoverer
|Date
|Source
|-
|2
|CTBB(2) = 5
|Jack
|16 Nov 2025
|[https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233 Link]
|-
|3
|CTBB(3) <u>></u> 38
|aparker
|18 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517 Link]
|-
|4
|CTBB(4) ≥ 672
|Moja
|26 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043 Link]
|-
|5
|CTBB(5) ≥ 2^2^2^2^182
|Moja
|26 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063 Link]
|-
|6
|CTBB(6) <u>></u> 2↑↑↑131
|Moja & Racheline
|25 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479 Link]
|-
|7
|CTBB(7) <u>></u> 4↑↑↑↑(4↑↑↑3)
|Moja & Racheline
|25 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281 1], [https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217 2]
|}

== Collection of Champions ==
<div class="toccolours mw-collapsible mw-collapsed"><div class="mw-collapsible-content"><div class="toccolours mw-collapsible mw-collapsed">'''CCTB(1)'''<div class="mw-collapsible-content">
The following was given by Jack: We are only allowed:

* 1 rule

* For each rule part (X,Y) can only contain at most 2 symbols (because 2(1)=2)

* The initial string Δ also only contains at most 2 symbols

There is only one possible Dyck string with at most 2 symbols, it is: (). Here are all possible P (rulesets) and Δ given these constraints:

* Δ: (), single rule: ()→()
* Δ: (), single rule: ()→D

'''Remember''': Δ must be non-empty. It’s easy to tell that for each step, the first ruleset results in an infinite loop. The second one halts after 1 step (it immediately deletes () ).

S is the set of all halting pairs (only the second one) and S’ is the halting times for said pair(s) (only 1).

max(S’)=1.
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(2)'''<div class="mw-collapsible-content">
The domain was solved by Jack.

Δ: (()) (Initial String)

P: R₁=()()→D, R₂=()→()()

* (()) (Δ)
* Skip rule 1 (skipping counts as a step)
* (()()) (apply rule 2)
* () (apply rule 1)
* ()() (apply rule 2)
* ∅ (apply rule 1)

All 432 pairs were brute-forced.
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(3)'''<div class="mw-collapsible-content">
Pair found by aparker314159:<pre>
Δ: [][[]]
Rule 0: [[]] => [[[]]]
Rule 1: [][][] => D
Rule 2: [] => [][]
</pre></div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(4)'''<div class="mw-collapsible-content">
Pair found by Moja:

CTBB(4)>=672, by the following pair:

* Δ: ()()()()
* P:
** ()() -> (()())()
** ()() -> ()()(())
** () -> (())
** (((()))) -> D
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(5)'''<div class="mw-collapsible-content">
Pair found by Moja

* Δ: "(())(()())"
* P:
** "(()())" -> "(()())(())"
** "(()())" -> "(()())(())"
** "()(())" -> "(())((()))"
** "((()))" -> "((()))()()"
** "()" -> "D
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(6)'''<div class="mw-collapsible-content">
Pair found by Moja and Racheline:

* Δ: ((()))((()))
* P:
** ((())) -> ((()))(()())
** (()()) -> (()())(())
** (()) -> (())(()()())
** (()()()) -> (()()())()()
** ()()() -> D
** () -> D
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(7)'''<div class="mw-collapsible-content">
Pair found by Moja and Racheline:

* Δ: ((()))((()()))
* P:
** ((()())) -> ((()()))((()))
** ((())) -> ((()))(()()())
** ((())) -> ((()))(()()())
** (()()()) -> (()()())(()())
** (()()) -> (()())(())(())
** (()) -> (())()()()()()
** () -> D
</div></div></div></div>

== Total Number of Pairs ==
The number of pairs [Δ,Ρ] grows rather quick. For CTBB(n), there are C(1)+C(2)+…+C(n) total Δ’s (where C(n)=n-th Catalan Number). The first few values are:

* CTBB(1) has 1 possible Δ,
* CTBB(2) has 3 possible Δ,
* CTBB(3) has 8 possible Δ,
* …
* CTBB(10) has 23713 possible Δ.

=== Let’s Dive Deeper… ===
Number of possible X: Sum of 1st,2nd,…,n-th Catalan Numbers

Number of possible Y: ((Sum of 1st,2nd,…,n-th Catalan Numbers)+1) (the “+1” represents the optional “D” symbol)

Each R in P is technically an ordered pair [x,y] so we multiply the counts together:

'''(Sum of 1st,2nd,…,n-th Catalan Numbers)×((Sum of 1st,2nd,…,n-th Catalan Numbers)+1)'''

There are n rules, so we exponentiate this by n:

'''[(Sum of 1st,2nd,…,n-th Catalan Numbers)×((Sum of 1st,2nd,…,n-th Catalan Numbers)+1)]ⁿ'''

We have our initial Dyck string Δ of length at most 2n (as mentioned earlier). This is also equal to the sum of the 1st,2nd,…,n-th Catalan numbers. This forms another pair [Δ,Ρ] by multiplying our Catalan sum with our formula from before:

'''Pair(n)=(Sum of 1st,2nd,…,n-th Catalan Numbers)×[(Sum of 1st,2nd,…,n-th Catalan Numbers)×((Sum of 1st,2nd,…,n-th Catalan Numbers)+1)]ⁿ'''

=== Values of the Pair Function ===

* Pair(1) = b2 (as shown earlier)
* Pair(2) = 432
* Pair(3) = 2985984
* Pair(4) ≈ 1.44×10¹²
* Pair(5) ≈ 7.97×10¹⁷

== Turing-Completeness of Tag Systems in General ==
Cocke and Minsky's construction of a Universal Turing Machine within a tag system can be found [https://dl.acm.org/doi/pdf/10.1145/321203.321206 here].

It is worth mentioning that this variant of cyclic tag does not involve appending, only rewriting.

Cyclic Tree Rewriting System

2026-08-31T14:27:27Z

Azerty: Renamed the page and removed unecessary spacings.

== Cyclic Tree Rewriting System ==
This variant of Cyclic Tag operates upon a Dyck String instead of a binary string. A Dyck String is a string of parentheses “(“ and “)” of length 2n, that consists of an equal number of symbols such that for all prefixes, “(“ occurs at least as many times as the other “)”. In shorter terms, a Dyck String is a balanced parenthetical string of length 2n. There does not exist a Dyck String (in this format) with an odd length (hence the 2n).

=== Rooted Ordered Trees and Dyck Strings ===
Dyck Strings (pronounced as: DYE-K) encode finite rooted ordered trees through a well-known bijection that maps the structure of a tree to a balanced sequence of parentheses. The most common method involves a depth-first traversal of the tree. A rooted ordered tree is a tree with these key features:

* One vertex is distinguished as the root,

* All edges are directed away from the root (conceptually, though not always drawn with arrows),

* For each node, the children are arranged in a left-to-right order.

=== Introduction to Δ and P ===
Define an ordered pair [Δ,Ρ] where Δ is an initial finite non-empty Dyck String & P are production rules P=(R₁,R₂,…,Rₖ), each in the form “X→Y” where X and Y are finite non-empty Dyck Strings. Y has the ability to be the symbol D. When Y is instead D, this means: in Δ, delete X.

=== Solving a Given Δ and P: ===
We follow these steps:

* '''(1)''' Begin with R₁,
* '''(2)''' Scan Δ from left to right,
* '''(3)''' Find the leftmost instance of X in Δ (according to said R)
* '''(4)''' Rewrite X as Y and leave the rest of Δ unchanged (or delete X if Y is the symbol D)
* '''(5)''' Move to the next R in cyclic order (1, 2,…,n,1,2,…)
* '''(6)''' If a rule doesn’t apply, simply skip it and move to the immediate next one in P,
* '''(7)''' Repeat from (2) on the altered Δ each time.

'''''NOTE:''''' Duplicate rules in the same P are allowed.

=== Halting Conditions ===
Some given pairs [Δ,P] never halt. However, some do! Here are two halting conditions we can define:

'''Halt if:'''

Δ becomes “∅” (empty), or if there does not exist a rule in P that can transform Δ any further, meaning that Δ is what we can call “stuck”.

== Example Run-Through ==
(Initial Dyck String) Δ: ()()(())

(Production Rules) P: R₁=()→(()), R₂=(())()→D

* ()()(()) initial String Δ,
* (())()(()) as per R₁, (()) replaces (),
* (()) as per R₂, (())() is deleted,
* ((())) as per R₁, (()) replaces (),
* '''SKIP R₂, IT DOES NOT APPLY,'''
* …

This example doesn't halt as ((…()…)) will keep expanding indefinitely.

== Correlation to BB(n) ==
From here, we can define a function analogous to the Busy Beaver function by taking the maximal finite halting times for all pairs. Here’s how we can do it:

Let |x| denote the length of x.

A pair [Δ,P] of “size n” consists of n rules, where for each Rᵢ ∈ P, |X|≤2n and |Y|≤2n ('''NOTE''': the lengths of X and Y for each Rᵢ do NOT have to be the same, but they must be ≤2n). We also assume that |Δ|≤2n. We say that skipping a rule counts as a step, and a “step” is defined as a singular rule application.

== Function ==
The “Cyclic Tree Busy Beaver” function CTBB(n) is therefore defined as follows:

* '''(1)''' Run all pairs [Δ,P] of size n,
* '''(2)''' Let S be the set of all pairs S=(P₁,P₂,…,Pₘ) that halted, and filter out (discard) all pairs that do not,
* '''(3)''' Let S’ be the set of all steps for every pair in S to halt,
* '''(4)''' CTBB(n) outputs max(S’).

==== Notes ====
Because Tag Systems in this fashion are Turing-Complete, we can expect CTBB(n) to be on par with the classic BB(n).

We can also correctly say that CTBB(n)>*f(n) where f(n) is any computable function (where >* represents eventual domination).

== Known Values ==
{| class="wikitable"
|+Small CTBB values
|CTBB
|Value / Lower bound
|Discoverer
|Date
|Source
|-
|2
|CTBB(2) = 5
|Jack
|16 Nov 2025
|[https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233 Link]
|-
|3
|CTBB(3) <u>></u> 38
|aparker
|18 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517 Link]
|-
|4
|CTBB(4) ≥ 672
|Moja
|26 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043 Link]
|-
|5
|CTBB(5) ≥ 2^2^2^2^182
|Moja
|26 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063 Link]
|-
|6
|CTBB(6) <u>></u> 2↑↑↑131
|Moja & Racheline
|25 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479 Link]
|-
|7
|CTBB(7) <u>></u> 4↑↑↑↑(4↑↑↑3)
|Moja & Racheline
|25 Nov 2025
|[https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281 1], [https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217 2]
|}

== Collection of Champions ==
<div class="toccolours mw-collapsible mw-collapsed"><div class="mw-collapsible-content"><div class="toccolours mw-collapsible mw-collapsed">'''CCTB(1)'''<div class="mw-collapsible-content">
The following was given by Jack: We are only allowed:

* 1 rule

* For each rule part (X,Y) can only contain at most 2 symbols (because 2(1)=2)

* The initial string Δ also only contains at most 2 symbols

There is only one possible Dyck string with at most 2 symbols, it is: (). Here are all possible P (rulesets) and Δ given these constraints:

* Δ: (), single rule: ()→()
* Δ: (), single rule: ()→D

'''Remember''': Δ must be non-empty. It’s easy to tell that for each step, the first ruleset results in an infinite loop. The second one halts after 1 step (it immediately deletes () ).

S is the set of all halting pairs (only the second one) and S’ is the halting times for said pair(s) (only 1).

max(S’)=1.
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(2)'''<div class="mw-collapsible-content">
The domain was solved by Jack.

Δ: (()) (Initial String)

P: R₁=()()→D, R₂=()→()()

* (()) (Δ)
* Skip rule 1 (skipping counts as a step)
* (()()) (apply rule 2)
* () (apply rule 1)
* ()() (apply rule 2)
* ∅ (apply rule 1)

All 432 pairs were brute-forced.
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(3)'''<div class="mw-collapsible-content">
Pair found by aparker314159:<pre>
Δ: [][[]]
Rule 0: [[]] => [[[]]]
Rule 1: [][][] => D
Rule 2: [] => [][]
</pre></div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(4)'''<div class="mw-collapsible-content">
Pair found by Moja:

CTBB(4)>=672, by the following pair:

* Δ: ()()()()
* P:
** ()() -> (()())()
** ()() -> ()()(())
** () -> (())
** (((()))) -> D
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(5)'''<div class="mw-collapsible-content">
Pair found by Moja

* Δ: "(())(()())"
* P:
** "(()())" -> "(()())(())"
** "(()())" -> "(()())(())"
** "()(())" -> "(())((()))"
** "((()))" -> "((()))()()"
** "()" -> "D
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(6) Champion'''<div class="mw-collapsible-content">
Pair found by Moja and Racheline:

* Δ: ((()))((()))
* P:
** ((())) -> ((()))(()())
** (()()) -> (()())(())
** (()) -> (())(()()())
** (()()()) -> (()()())()()
** ()()() -> D
** () -> D
</div></div><div class="toccolours mw-collapsible mw-collapsed">'''CTBB(7) Champion'''<div class="mw-collapsible-content">
Pair found by Moja and Racheline:

* Δ: ((()))((()()))
* P:
** ((()())) -> ((()()))((()))
** ((())) -> ((()))(()()())
** ((())) -> ((()))(()()())
** (()()()) -> (()()())(()())
** (()()) -> (()())(())(())
** (()) -> (())()()()()()
** () -> D
</div></div></div></div>

== Total Number of Pairs ==
The number of pairs [Δ,Ρ] grows rather quick. For CTBB(n), there are C(1)+C(2)+…+C(n) total Δ’s (where C(n)=n-th Catalan Number). The first few values are:

* CTBB(1) has 1 possible Δ,
* CTBB(2) has 3 possible Δ,
* CTBB(3) has 8 possible Δ,
* …
* CTBB(10) has 23713 possible Δ.

=== Let’s Dive Deeper… ===
Number of possible X: Sum of 1st,2nd,…,n-th Catalan Numbers

Number of possible Y: ((Sum of 1st,2nd,…,n-th Catalan Numbers)+1) (the “+1” represents the optional “D” symbol)

Each R in P is technically an ordered pair [x,y] so we multiply the counts together:

'''(Sum of 1st,2nd,…,n-th Catalan Numbers)×((Sum of 1st,2nd,…,n-th Catalan Numbers)+1)'''

There are n rules, so we exponentiate this by n:

'''[(Sum of 1st,2nd,…,n-th Catalan Numbers)×((Sum of 1st,2nd,…,n-th Catalan Numbers)+1)]ⁿ'''

We have our initial Dyck string Δ of length at most 2n (as mentioned earlier). This is also equal to the sum of the 1st,2nd,…,n-th Catalan numbers. This forms another pair [Δ,Ρ] by multiplying our Catalan sum with our formula from before:

'''Pair(n)=(Sum of 1st,2nd,…,n-th Catalan Numbers)×[(Sum of 1st,2nd,…,n-th Catalan Numbers)×((Sum of 1st,2nd,…,n-th Catalan Numbers)+1)]ⁿ'''

=== Values of the Pair Function ===

* Pair(1) = b2 (as shown earlier)
* Pair(2) = 432
* Pair(3) = 2985984
* Pair(4) ≈ 1.44×10¹²
* Pair(5) ≈ 7.97×10¹⁷

== Turing-Completeness of Tag Systems in General ==
Cocke and Minsky's construction of a Universal Turing Machine within a tag system can be found [https://dl.acm.org/doi/pdf/10.1145/321203.321206 here].

It is worth mentioning that this variant of cyclic tag does not involve appending, only rewriting.

Post Tag

2026-08-31T14:15:06Z

Azerty: Added redirect

#REDIRECT [[Post Tag System]]

Cycler

2026-08-27T19:21:07Z

Azerty: Created a section for cycler decider algorithm.

[[File:1RB0LB 1LB1LC 0RC1RA.png|alt=1RB0LB_1LB1LC_0RC1RA|thumb|upright=1.5|The record 3-state 2-symbol cycler {{TM|1RB0LB_1LB1LC_0RC1RA}}, which has period 18 and preperiod 4.]]{{Stub}}
A '''cycler''' is a Turing machine that eventually enters a repeating cycle. Such a Turing machine runs forever and thus is a [[non-halting Turing machine]]. A cycler may be seen as a special case of a [[translated cycler]] which has offset zero.

The rule for identifying cyclers is simple:
'''û''' → '''û'''
where '''û''' is any headed tape segment.

== Decider Algorithm ==
The [[decider]] for cyclers works by memorizing all configurations which are visited by a TM. If a configuration is visited twice, the TM is shown to be non-halting.

The algorithm can be optimized by checking a single configuration per step. The saved configuration is saved at a logarithmic tape (e.g. every power of 2 steps). This allows detecting cyclers of any period and preperiod while keeping the time complexity at O(n).

== Other Models ==
Most deterministic computational models have cyclers, like lambda calculus, fractran or register machines.

==Related Functions==

* Busy Preperiodic Beaver ([[BBS]](n,m)): The longest possible preperiod of a cycler or translated cycler with n states and m symbols.
* Busy Periodic Beaver ([[BBP]](n,m)): The longest possible period of a cycler or translated cycler with n states and m symbols.

== See also ==

* [[Translated cycler]]
* [https://github.com/bbchallenge/bbchallenge-proofs/blob/build-latex-pdf/deciders/correctness-deciders.pdf Section 2] of bbchallenge's deciders write-up.

[[Category:Zoology]][[Category:Deciders]]

BB(3)

2026-08-23T12:30:39Z

Azerty: /* Enumeration */ Added nonhalting TMs count.

The 3-state 2-symbol Busy Beaver problem '''BB(3)''' was proven by [[Shen Lin]] in his 1963 doctoral dissertation<ref>Shen Lin. 1963. Computer studies of Turing machine problems. PhD dissertation. Ohio State University. https://etd.ohiolink.edu/acprod/odb_etd/etd/r/1501/10?clear=10&p10_accession_num=osu1486554418657614</ref> and republished in 1965.<ref>Lin, Shen; Radó, Tibor (April 1965). "Computer Studies of Turing Machine Problems". ''Journal of the ACM''. '''12''' (2): 196–212. https://doi.org/10.1145/321264.321270</ref> Allen Brady independently proved it in his 1964 PhD dissertation.<ref>Allen Brady. 1964. Solutions of Restricted Cases of the Halting Problem Applied to the Determination of Particular Values of a Non-Computable Function. PhD dissertation. Oregon State University. https://ir.library.oregonstate.edu/downloads/6q182n74x</ref>

== Techniques ==
In order to prove BB(3), Lin discovered [[Translated Cycler]]s (which he called "partial recurrence") and described the first algorithm for proving them infinite. Translated cycler (with cycler) deciders decide all except 63 TNF-1RB BB(3) TMs. Those last 63 are all decided by [[Closed Position Set]] augmented with fixed history 1 (each symbol remembers the transition which wrote it).

== Champions ==
S(3) = 21 and there is only one shift [[champion]] (in [[TNF]]):

* {{TM|1RB1RZ_1LB0RC_1LC1LA|halt}} leaves 5 ones

Σ(3) = 6 and there are 5 ones champions (in TNF):

* {{TM|1RB1RZ_0RC1RB_1LC1LA|halt}} runs for 14 steps
* {{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} runs for 13 steps
* {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}} runs for 13 steps
* {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}} runs for 12 steps
* {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}} runs for 11 steps
BB(3) is notable for being the only known size where none of the shift champions are also ones champions.

== Enumeration ==
In [[TNF-1RB]] there are exactly 4057 BB(3) TMs of which 1379 halt and 2678 do not. The top longest running TMs are:<pre>
1RB1RZ_1LB0RC_1LC1LA Halt 21 5
1RB1RZ_0LC0RC_1LC1LA Halt 20 5
1RB1LA_0RC1RZ_1LC0LA Halt 20 5
1RB1RA_0RC1RZ_1LC0LA Halt 19 5
1RB0RA_0RC1RZ_1LC0LA Halt 19 4
1RB0LC_1LA1RZ_1RC1RB Halt 18 5
1RB0LB_0RC1RC_1LA1RZ Halt 18 5
1RB1LB_0RC1RZ_1LC0LA Halt 18 4
1RB0LB_0RC1RZ_1LC0LA Halt 18 3
1RB1RZ_0RC0RC_1LC1LA Halt 17 5
1RB1RZ_0RC---_1LC0LA Halt 17 4
1RB1LC_0LC1RA_1RZ1LA Halt 16 5
1RB0LC_1LC1RB_1RZ1LA Halt 16 5
1RB1LA_0LA1RC_1RA1RZ Halt 15 5
1RB1LA_0LA1RC_0RB1RZ Halt 15 4
1RB0LC_1RC1RZ_1LA0RB Halt 15 4
1RB0RB_0LC1RZ_1RA1LC Halt 15 3
1RB1RZ_0RC1RB_1LC1LA Halt 14 6
1RB1RZ_1LB0LC_1RA1RC Halt 14 5
1RB1RZ_0LC1RA_1LA1LB Halt 14 5
1RB1RZ_1LC1RA_1RC0LB Halt 14 4
1RB1RZ_1LB0LC_1RA0RC Halt 14 3
1RB1RZ_0LC0RB_1LA1LC Halt 14 3
1RB0RB_1LC1RZ_0LA1RA Halt 14 3
1RB1RZ_0LC0RA_1RA1LB Halt 14 2
</pre>For a full list of halting BB(3) TMs, see: https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x2.txt

== References ==
<references />
[[Category:BB Domains]]

Enumeration

2026-08-22T09:31:25Z

Azerty: /* List of enumeration scripts */ Renamed

Enumerator

2026-08-22T09:30:38Z

Azerty: redirect

#REDIRECT [[Enumeration]]

Turing completeness

2026-08-20T20:05:42Z

Azerty: /* Turing machine */ Added PAF and TM variations.

A Turing-complete system is a system that can compute every computable function. A Turing-complete system can be used to simulate any Turing machine or other Turing-complete systems.

The halting problem is uncomputable on any Turing-complete system.

To be Turing-complete, a system must be able to store unbounded memory and have access to the memory. There must be also infinitely many different non-halting programs (like "while" loops or recursion).

== List of Turing-complete systems ==
This list is non-exhaustive.

=== [[Turing machine]] ===

* Multi-tape
* Multi-headed
* [[Reversible Turing Machine|Reversible]]
* [[Terminating Turmite|Turmite]]
* Multi-dimensional tape
* [[Branching Beaver|Branching tape]]
* [[Uniform-action Turing Machine|Uniform-action]]
* [[Semi Infinite Tape Turing Machine|Semi-infinite tape]]

=== [[Lambda calculus]] ===
* [[SKI Calculus|SK Calculus]]
* [[SKI Calculus]]
* [[SKI Calculus|BCKW Calculus]]
* [https://esolangs.org/wiki/Iota Iota]
* [https://esolangs.org/wiki/Jot Jot]

=== [[Brainfuck]] ===

* [[Boolfuck]]
* [https://esolangs.org/wiki/Bitter Bitter]
* [https://esolangs.org/wiki/Smallfuck Smallfuck]
* [https://esolangs.org/wiki/BitChanger BitChanger]

=== Cellular automata ===

* Rule 110 automaton
* Conway's game of life
* [[Doodle function|Doodle automaton]]
* [[Pebble Automaton|Pebble automaton]]

=== Tag systems ===

* [[Post Tag System|2-Tag system]]
* Cyclic tag system
* [[Tag System|Tree Rewriting System]]
* String rewriting system

=== Counter machines ===

* [[Minsky machine]]

* [[CounterScript]]

=== Other ===

* [[General Recursive Function|General Recursive Functions]]
* [[Fractran]]
* [[Wang Tiles|Wang tiles]]
* [[Diophantine Equation|Diophantine equations]]
* [[Piecewise Affine Function|Piecewise affine function]]

== See Also ==
Wikipedia article about computability: https://en.wikipedia.org/wiki/Computability

Semi Infinite Tape Turing Machine

2026-08-20T20:04:04Z

Azerty: Created page and added current champions.

The function BBt(n,m) returns the maximum number of right steps a Turing Machine with a semi infinite tape performs before its head leaves the tape. The head starts in the first cell and halting transition is not required.
{| class="wikitable"
|+
!Domain
!Value
!Champion
|-
|BBt(2)
|≥ 3
|<code>1RB---_1LB0RB</code>
|-
|BBt(3)
|≥ 12
|<code>0RB0LB_0RC1RC_1LA0LC</code>
|-
|BBt(4)
|≥ 168
|<code>1RB1LD_1RC0RC_1RD0LA_1LA0RD</code>
|-
|BBt(5)
|≥ 4,300
|<code>0RB0RC_0RC0RE_0RD1LE_1LA1RD_1LC1RB</code>
|-
|BBt(2,3)
|≥ 17
|<code>1RB0RA0RB_2LA2RB1LB</code>
|-
|BBt(3,3)
|≥ 10,053
|<code>0RB1LA1LB_0RC2LB1RB_1LA2LC2RC</code>
|-
|BBt(2,4)
|≥ 88
|<code>1RB2RB3LA2RA_1LA3LB0RB3RB</code>
|-
|BBt(2,5)
|≥ 415
|<code>0RB2LA1LB4RA2RB_1LA1RB3RB4LA1LA</code>
|-
|BBt(2,6)
|≥ 1,429
|<code>0RB4RB3LA2RA2LB2LA_1LA2RB5RA1RA3RA5RA</code>
|}

Enumeration

2026-08-19T07:30:01Z

Azerty: /* List of enumeration scripts */ Added mxdys BB enumerator.

'''Enumeration''' is the process of generating all candidate [[Turing machines]] (or other program types) within a given [[:Category:BB Domains|domain]] potentially filtered modulo isomorphism (ex: [[TNF]]). These enumerations can then be filtered down by [[deciders]] leaving a list of [[holdouts]] which remain undecided.

== List of enumeration scripts ==
{| class="wikitable"
|+
!System
!Repository
!File Name
!Language
|-
|Branching Beaver
|[https://github.com/sligocki/etc/blob/main/bb_tri/src/enumerator.rs https://github.com/sligocki/etc]
|<code>bb_tri/scr/enumerator.rs</code>
|Rust
|-
|CounterScript
|https://github.com/AzertyWasTaken/CounterScript
|<code>main.js</code>
|JavaScript
|-
|CounterScript
|https://github.com/sligocki/etc
|<code>bbcs/src/enumerator.rs</code>
|Rust
|-
|Fractran
|https://github.com/int-y1/BBFractran
|<code>enumerate/fractran20260416.cpp</code>
|C++
|-
|General Recursive Function
|https://github.com/sligocki/etc
|<code>gen_rec/scr/enumerate.rs</code>
|Rust
|-
|Lambda Calculus
|[https://github.com/tromp/AIT/blob/master/BB/BB.lhs https://github.com/tromp/AIT]
|<code>BB/BB.lhs</code>
|Haskell
|-
|Post Tag
|[https://github.com/sligocki/etc/blob/main/bb_tri/src/enumerator.rs https://github.com/sligocki/etc]
|<code>post_tag/src/enumerate.rs</code>
|Rust
|-
|Register Machine
|https://github.com/int-y1/BBFractran
|<code>rm.cpp</code>
|C++
|-
|Turing Machine
|https://github.com/sligocki/busy-beaver
|<code>Code/Enumerate.py</code>
|Python
|-
|Turing Machine
|https://github.com/ccz181078/TM
|<code>main.cpp</code>
|C++
|}

Cellular automaton

2026-08-19T06:58:20Z

Azerty: /* Halting conditions */

{{Stub}}

Cellular automata (CA) are discrete, abstract computational systems that consist of a regular grid of cells, each in a finite number of states. The grid evolves in discrete time steps according to a local transition rule based on the states of neighboring cells. Despite their simple definition, cellular automata can exhibit complex behavior, including chaos, self-replication, and universal computation.

The simplest nontrivial CA are elementary cellular automata (1D, binary states, radius-1 neighborhood).

The rule is defined by an 8-bit lookup table (bits for patterns 111 to 000), numbered 0 to 255. For instance:

* Rule 110: Known to be Turing-complete.<ref>https://wpmedia.wolfram.com/sites/13/2018/02/15-1-1.pdf</ref>
* Rule 30: Produces chaotic, pseudorandom patterns.
Cellular automata may have:

* Multiple number of dimensions.
* Multiple "alive" cell states.
* Various neighborhood radius and shape.

== Busy Beaver function ==

=== Halting conditions ===
There is no single agreed-upon definition of "halting" for cellular automaton. Each possible definition of halting produces different busy beaver functions. Here are some examples.

* '''All 0s''': A cellular automaton is said to halt if it reaches a state where every cell has state 0.

* '''Still life''': A cellular automaton is said to halt if it reaches a fixed point in the rule's update function. This definition of halting allows busy beaver functions which count the maximum number of on cells a halting cellular automaton can produce.

* '''Oscillator''': A cellular automaton is said to halt if it reaches a cycle of repeated states. Busy beaver functions for the "longest running" starting pattern or rule can measure the time until the first state which gets repeated, or the time until a state appears twice (which gives a bonus to cellular automata that end up in a long-period oscillator). Busy beaver functions counting the number of "on" states can measure the minimum population, or the maximum population, or the average population, or the population in the first repeating state.

* '''Oscillating population''': A cellular automaton is said to halt if it reaches a state where its population (number of cells which are not 0) is cyclic. If the actual state of the cellular automaton is not cyclic, it may be nontrivial to determine whether its population will remain cyclic. In the Game of Life, it is common for small and medium-sized patterns to resolve into a collection of small still-lifes, oscillators, and gliders which travel away from the center. Such patterns never repeat, but they do have cyclic populations. Hence, most small starting patterns in Life halt under this definition but not under either of the previous three definitions.

* '''Oscillating population relative to a background pattern''': In rule 110, for any starting pattern with a finite number of on cells, the population will grow without bound. Hence, no finite patterns halt in rule 110 using any of the previous definitions.
The construction of Turing-complete computation in rule 110 takes place on an infinitely repeating background pattern, which Matthew Cook calls the ''ether''. One could define the "population" of a pattern in rule 110 as the number of 1s in the symmetric difference with the ether. One could then say a pattern halts when its population relative to the ether becomes cyclic.

Enumeration

2026-08-18T14:54:59Z

Azerty: /* List of enumeration scripts */ Added more enumerator scripts.

'''Enumeration''' is the process of generating all candidate [[Turing machines]] (or other program types) within a given [[:Category:BB Domains|domain]] potentially filtered modulo isomorphism (ex: [[TNF]]). These enumerations can then be filtered down by [[deciders]] leaving a list of [[holdouts]] which remain undecided.

== List of enumeration scripts ==
{| class="wikitable"
|+
!System
!Repository
!File Name
!Language
|-
|Branching Beaver
|[https://github.com/sligocki/etc/blob/main/bb_tri/src/enumerator.rs https://github.com/sligocki/etc]
|<code>bb_tri/scr/enumerator.rs</code>
|Rust
|-
|CounterScript
|https://github.com/AzertyWasTaken/CounterScript
|<code>main.js</code>
|JavaScript
|-
|CounterScript
|https://github.com/sligocki/etc
|<code>bbcs/src/enumerator.rs</code>
|Rust
|-
|Fractran
|https://github.com/int-y1/BBFractran
|<code>enumerate/fractran20260416.cpp</code>
|C++
|-
|General Recursive Function
|https://github.com/sligocki/etc
|<code>bb_tri/scr/enumerate.rs</code>
|Rust
|-
|Lambda Calculus
|[https://github.com/tromp/AIT/blob/master/BB/BB.lhs https://github.com/tromp/AIT]
|<code>BB/BB.lhs</code>
|Haskell
|-
|Post Tag
|[https://github.com/sligocki/etc/blob/main/bb_tri/src/enumerator.rs https://github.com/sligocki/etc]
|<code>post_tag/src/enumerate.rs</code>
|Rust
|-
|Register Machine
|https://github.com/int-y1/BBFractran
|<code>rm.cpp</code>
|C++
|-
|Turing Machine
|https://github.com/sligocki/busy-beaver
|<code>Code/Enumerate.py</code>
|Python
|}

Enumeration

2026-08-18T14:40:35Z

Azerty: /* List of enumeration scripts */ Added General Recursive Function

Enumeration

2026-08-18T12:10:42Z

Azerty: /* List of enumeration scripts */ added lambda calculus enumerator

Brainfuck

2026-08-18T07:43:54Z

Azerty:

Brainfuck is a Turing complete language created in 1993 by Urban Muller. The language was designed to have an extremely tiny compiler (on the order of 200 bytes), while remaining Turing-complete.

== Language Overview ==
Brainfuck operates on an array of memory cells, each initially set to zero. In the original implementation, the array was 30,000 cells long and each cell values would be limited to numbers from 0 to 255, but this may not be part of the language specification. A brainfuck program can work with different, and sometimes unbounded, array length and cells size.

Like in a [[Turing machine]], there is a pointer, initially pointing to the first memory cell.

Programs are composed of commands, represented with symbols:
{| class="wikitable"
|+
!Command
!Operation
|-
|<code>></code>
|Move pointer right
|-
|<code><</code>
|Move pointer left
|-
|<code>+</code>
|Increment current cell
|-
|<code>-</code>
|Decrement current cell
|-
|<code>[</code>
|Jump forward past matching <code>]</code> if current cell is zero
|-
|<code>]</code>
|Jump back to matching <code>[</code> if current cell is non-zero
|-
|<code>.</code>
|Output current cell as a character (often ASCII)
|-
|<code>,</code>
|Input a character and store in current cell
|}

== BB_brainf ==
'''BB_brain(n)''' returns the maximum cell value a program with n instructions can have when halting. Commands <code>.</code> and <code>,</code> are ignored.
{| class="wikitable"
!n
!BB_brainf(n)
!Champion
|-
|1
|1
|<code>+</code>
|-
|2
|2
|<code>++</code>
|-
|3
|3
|<code>+++</code>
|-
|4
|4
|<code>++++</code>
|-
|5
|5
|<code>+++++</code>
|-
|6
|≥ 6
|<code>++++++</code>
|-
|7
|≥ 7
|<code>+++++++</code>
|-
|8
|≥ 8
|<code>++++++++</code>
|-
|9
|≥ 9
|<code>+++++++++</code>
|-
|10
|≥ 10
|<code>++++++++++</code>
|-
|11
|≥ 11
|<code>+++++++++++</code>
|-
|12
|≥ 12
|<code>++++++++++++</code>
|-
|13
|≥ 16
|<code>++++[->++++<]</code>
|-
|14
|≥ 20
|<code>+++++[->++++<]</code>
|-
|15
|≥ 25
|<code>+++++[->+++++<]</code>
|-
|16
|≥ 30
|<code>++++++[->+++++<]</code>
|-
|17
|≥ 36
|<code>++++++[->++++++<]</code>
|-
|18
|≥ 42
|<code>+++++++[->++++++<]</code>
|-
|19
|≥ 49
|<code>+++++++[->+++++++<]</code>
|-
|20
|≥ 56
|<code>++++++++[->+++++++<]</code>
|-
|21
|≥ 64
|<code>++++++++[->++++++++<]</code>
|-
|22
|≥ 72
|<code>+++++++++[->++++++++<]</code>
|-
|23
|≥ 81
|<code>+++++++++[->+++++++++<]</code>
|-
|24
|≥ 90
|<code>+++++++++[->++++++++++<]</code>
|-
|25
|≥ 340
|<code>++++[>+[->++<]>[-<++>]<<]</code>
|-
|26
|≥ 1364
|<code>+++++[>+[->++<]>[-<++>]<<]</code>
|}

[page under construction]

[[Category:Functions]]

Uniform-action Turing Machine

2026-08-18T07:39:46Z

Azerty: Created page

A Turing Machine is uniform-action is every state write the same symbol and move in the same direction. This means that the cell color affects only the next state.

The busy beaver function for UATM is BBu(n).
{| class="wikitable"
|+
!BBu(n)
!Value
!Champion
|-
|2
|6
|<code>1RB---_1LB1LA</code>
|-
|3
|17
|<code>1RB---_0RC0RC_1LC1LA</code>
|-
|4
|29
|<code>1RB1RD_0LC0LA_1LC1LA_0RC---</code>
|-
|5
|≥ 441
|<code>1RB1RA_0RC0RE_1LC1LD_1LA1LD_---1RB</code>
|-
|6
|> 16,088
|<code>1RB1RF_1RC1RA_1LD1LF_0RB0RE_---1LC_---0LC</code>
|}

== Source ==
Discord thread: https://discord.com/channels/960643023006490684/1377330731222564904/threads/1458010522967609425

Enumeration

2026-08-17T18:04:03Z

Azerty: /* List of enumeration scripts */ Created enumerators table.