Champions: Difference between revisions

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Add "Verification" column to champions table and comment about self-reporting. Also streamline "Discovered By" column a bit and add Discord link to Racheline's discovery of BB(14) Graham number champion.
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== 2-Symbol TMs ==
== 2-Symbol TMs ==
Rows are blank if no champion has been found which surpasses a smaller size problem. Take also note that the <math> f_{x}(n) </math> used in the lowerbounds represent the [https://googology.fandom.com/wiki/Fast-growing_hierarchy Fast Growing Hierarchy].
Rows are blank if no champion has been found which surpasses a smaller size problem. Take also note that the <math> f_{x}(n) </math> used in the lowerbounds represent the [https://googology.fandom.com/wiki/Fast-growing_hierarchy Fast Growing Hierarchy]. Note that most champions above 6 states are self-reported and have not been independently verified.
 
{| class="wikitable"
{| class="wikitable"
|+
|+
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!Runtime
!Runtime
!Champions
!Champions
!Comment
!Discovered By
!Verification
|-
|-
|[[BB(2)]]
|[[BB(2)]]
|<math> 6 </math>
|<math> 6 </math>
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|Discovered and proven by hand by Tibor Radó
|Tibor Radó
|Direct Simulation
|-
|-
|[[BB(3)]]
|[[BB(3)]]
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|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|Proven by Shen Lin
|Proven by Shen Lin
|Direct Simulation
|-
|-
|[[BB(4)]]
|[[BB(4)]]
|<math> 107 </math>
|<math> 107 </math>
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|Discovered and proven by Allen Brady
|Allen Brady
|Direct Simulation
|-
|-
|[[BB(5)]]
|[[BB(5)]]
|<math> 47\,176\,870 </math>
|<math> 47\,176\,870 </math>
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|Discovered by Heiner Marxen & Jürgen Buntrock in 1989
|Heiner Marxen & Jürgen Buntrock in 1989
Proven by [[bbchallenge.org]] in 2024
|Direct Simulation
|-
|-
|[[BB(6)]]
|[[BB(6)]]
|<math> > 10 \uparrow\uparrow 15 </math>
|<math> > 10 \uparrow\uparrow 15 </math>
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|Discovered by Pavel Kropitz in 2022
|Pavel Kropitz in 2022
|[https://www.sligocki.com/2022/06/21/bb-6-2-t15.html Analysis by Shawn Ligocki]
|-
|-
|[[BB(7)]]
|[[BB(7)]]
|
|
|
|
|
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|<math> > 2 \uparrow^5 4 > f_6(2) </math>
|<math> > 2 \uparrow^5 4 > f_6(2) </math>
|{{TM|1RH1RF_0LC0LH_0RD1LC_0RE1RA_1RB1RE_1RZ1RG_1RF0RE_1LB1LH|halt}}
|{{TM|1RH1RF_0LC0LH_0RD1LC_0RE1RA_1RB1RE_1RZ1RG_1RF0RE_1LB1LH|halt}}
|Discovered by Racheline in 2024
|Racheline in 2024
|
|-
|-
|BB(9)
|BB(9)
|<math> > f_\omega(f_9(2)) </math>
|<math> > f_\omega(f_9(2)) </math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|Discovered by Jacobzheng in 2024
|Jacobzheng in 2024
|
|-
|-
|BB(10)
|BB(10)
|<math> > f_\omega^2(25) </math>
|<math> > f_\omega^2(25) </math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|Discovered by Racheline in 2024
|Racheline in 2024
|
|-
|-
|BB(11)
|BB(11)
|<math> > f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9)) </math>
|<math> > f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9)) </math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|Discovered by Racheline in 2024
|Racheline in 2024
|
|-
|-
|BB(12)
|BB(12)
|<math> > f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2)) </math>
|<math> > f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2)) </math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|Discovered by Racheline in 2024
|Racheline in 2024
|
|-
|-
|BB(13)
|BB(13)
|
|
|
|
|
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|<math> > f_{\omega + 1}(65\,536) > g_{64} </math>
|<math> > f_{\omega + 1}(65\,536) > g_{64} </math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|Discovered by Racheline in 2024
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
|
|-
|-
|BB(15)
|BB(15)
|
|
|
|
|
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|<math> > f_{\omega + 1}(2 \uparrow\uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow\uparrow 9) </math>
|<math> > f_{\omega + 1}(2 \uparrow\uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow\uparrow 9) </math>
|
|
|Designed by  Daniel Nagaj in 2021<ref>Shawn Ligocki. 2022. "BB(16) > Graham's Number". https://www.sligocki.com/2022/07/11/bb-16-graham.html</ref>
|Daniel Nagaj in 2021
|[https://www.sligocki.com/2022/07/11/bb-16-graham.html Analysis by Shawn Ligocki]
|-
|-
|BB(17)
|BB(17)
|<math> > f_{\omega + 1}(f_\omega(60)) </math>
|<math> > f_{\omega + 1}(f_\omega(60)) </math>
|[[User:Jacobzheng/BB(17)]]
|Jacobzheng in 2024
|
|
|[[User:Jacobzheng/BB(17)|Discovered]] by Jacobzheng in 2024
|-
|-
|BB(18)
|BB(18)
|<math> > f_{\omega + 1}(f_\omega^2(60)) </math>
|<math> > f_{\omega + 1}(f_\omega^2(60)) </math>
|[[User:Jacobzheng/BB(18)]]
|Jacobzheng in 2024
|
|
|[[User:Jacobzheng/BB(18)|Discovered]] by Jacobzheng in 2024
|-
|-
|BB(19)
|BB(19)
|<math> > f_{\omega + 1}^3(f_\omega(60)) </math>
|<math> > f_{\omega + 1}^3(f_\omega(60)) </math>
|[[User:Jacobzheng/BB(19)]]
|Jacobzheng in 2024
|
|
|[[User:Jacobzheng/BB(19)|Discovered]] by Jacobzheng in 2024
|-
|-
|BB(20)
|BB(20)
|<math> > f_{\omega + 2}^2(21) </math>
|<math> > f_{\omega + 2}^2(21) </math>
|
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Discovered] by Racheline in 2024
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Racheline in 2024]
|
|-
|-
|BB(21)
|BB(21)
|<math> > f_{\omega^2}^2(4 \uparrow\uparrow 341) </math>
|<math> > f_{\omega^2}^2(4 \uparrow\uparrow 341) </math>
|
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Discovered] by Racheline in 2024
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|-
|BB(41)
|BB(41)
|<math> > f_{\omega^\omega}^4(32) </math>
|<math> > f_{\omega^\omega}^4(32) </math>
|[[User:Jacobzheng/BB(41)]]
|Jacobzheng in 2024
|
|
|[[User:Jacobzheng/BB(41)|Designed]] by Jacobzheng in 2024
|-
|-
|BB(51)
|BB(51)
|<math> > f_{\varepsilon_0 + 1}(8) </math>
|<math> > f_{\varepsilon_0 + 1}(8) </math>
|
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Designed] by Racheline in 2024
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|
|}
|}


Revision as of 16:07, 10 December 2024

Busy Beaver Champions are the current record holding Turing machines who 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 Busy Beaver champions and the History of Previous Champions.

2-Symbol TMs

Rows are blank if no champion has been found which surpasses a smaller size problem. Take also note that the fx(n) used in the lowerbounds represent the Fast Growing Hierarchy. Note that most champions above 6 states are self-reported and have not been independently verified.

Runtime Champions Discovered By Verification
BB(2) 6 1RB1LB_1LA1RZ (bbch) 1RB0LB_1LA1RZ (bbch) 1RB1RZ_1LB1LA (bbch) 1RB1RZ_0LB1LA (bbch) 0RB1RZ_1LA1RB (bbch) Tibor Radó Direct Simulation
BB(3) 21 1RB1RZ_1LB0RC_1LC1LA (bbch) Proven by Shen Lin Direct Simulation
BB(4) 107 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch) Allen Brady Direct Simulation
BB(5) 47176870 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch) Heiner Marxen & Jürgen Buntrock in 1989 Direct Simulation
BB(6) >10↑↑15 1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch) Pavel Kropitz in 2022 Analysis by Shawn Ligocki
BB(7)
BB(8) >2↑54>f6(2) 1RH1RF_0LC0LH_0RD1LC_0RE1RA_1RB1RE_1RZ1RG_1RF0RE_1LB1LH (bbch) Racheline in 2024
BB(9) >fω(f9(2)) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH (bbch) Jacobzheng in 2024
BB(10) >fω2(25) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ (bbch) Racheline in 2024
BB(11) >fω2(2↑↑12)>fω2(f3(9)) 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE (bbch) Racheline in 2024
BB(12) >fω4(2↑↑↑4−3)>fω4(f4(2)) 0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL (bbch) Racheline in 2024
BB(13)
BB(14) >fω+1(65536)>g64 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ--- (bbch) Racheline in 2024
BB(15)
BB(16) >fω+1(2↑↑↑↑2↑↑↑↑9) Daniel Nagaj in 2021 Analysis by Shawn Ligocki
BB(17) >fω+1(fω(60)) User:Jacobzheng/BB(17) Jacobzheng in 2024
BB(18) >fω+1(fω2(60)) User:Jacobzheng/BB(18) Jacobzheng in 2024
BB(19) >fω+13(fω(60)) User:Jacobzheng/BB(19) Jacobzheng in 2024
BB(20) >fω+22(21) Racheline in 2024
BB(21) >fω22(4↑↑341) Racheline in 2024
BB(41) >fωω4(32) User:Jacobzheng/BB(41) Jacobzheng in 2024
BB(51) >fε0+1(8) Racheline in 2024

3-Symbol TMs

Runtime Champions Comment
BB(2,3) 38 1RB2LB1RZ_2LA2RB1LB (bbch)
BB(3,3) >1017 0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(4,3) >1014072 1RB1RZ2RC_2LC2RD0LC_1RA2RB0LB_1LB0LD2RC (bbch)

4-Symbol TMs

Runtime Champions Comment
BB(2,4) 3932964 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
BB(3,4) >2↑155 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)

5-Symbol TMs

Runtime Champions Comment
BB(2,5) >1010103314360 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
BB(3,5) >fω(2↑155)>fω2(15) 1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch)

References

Retrieved from "https://wiki.bbchallenge.org/w/index.php?title=Champions&oldid=1347"