User:Polygon/Collection of BB Champions: Difference between revisions

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Placeholder
A collection of Busy Beaver [[Champions]] including Champions for BB-Adjacent [[:Category:Functions|functions]]. Note that for all functions with the input format f(n,m), n denotes the number of states and m denotes the number of symbols of the relevant Busy Beaver domain. Note that for functions with this input format f(n) = f(n,2).
='''Original Busy Beaver Functions'''=
 
==Maximum Shifts Function (BB)==
Note: highest ref name in use: 4
'''2 Symbols:'''
= '''State-and-Symbol-Limited Busy Beaver functions''' =
Busy Beaver functions where the programs size is limited by the amount of states and symbols.
== Original Busy Beaver Functions ==
=== Maximum Shifts Function ([[Busy Beaver Functions|S(n,m)]], also commonly called BB(n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|-
|BB(1)
|[[BB(1)]]
|<math> 1 </math>
|1
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
|-
|-
|[[BB(2)]]
|[[BB(2)]]
|<math> 6 </math>
|6
|{{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}}
|-
|-
|[[BB(3)]]
|[[BB(3)]]
|<math> 21 </math>
|21
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|-
|-
|[[BB(4)]]
|[[BB(4)]]
|<math> 107 </math>
|107
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|-
|-
|[[BB(5)]]
|[[BB(5)]]
|<math> 47\,176\,870 </math>
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|-
|-
|[[BB(6)]]
|[[BB(6)]]
|<math> > 2\uparrow\uparrow 2\uparrow\uparrow 2\uparrow\uparrow 10 </math>
|> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|-
Line 42: Line 45:
|-
|-
|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}}
|-
|-
|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}}
|-
|-
|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}}
|-
|-
|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}}
|-
|-
|BB(14)
|BB(14)
|<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}}
|-
|-
|BB(15)
|BB(15)
|<math> > f_{\omega + 1}(f_\omega(10^{57})) </math>
|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|-
|BB(16)
|BB(16)
|<math> > f_{\omega + 1}^2(10^{10^{57}}) </math>
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|
|
|-
|-
|BB(18)
|BB(18)
|<math> > f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60))) </math>
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|
|
|-
|-
|BB(20)
|BB(20)
|<math> > f_{\omega + 2}^2(21) </math>
|<math>> f_{\omega + 2}^2(21)</math>
|
|
|-
|-
|BB(21)
|BB(21)
|<math> > f_{\omega^2}^2(4 \uparrow\uparrow 341) </math>
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|
|
|-
|-
|BB(40)
|BB(40)
|<math> > f_{\omega^\omega}(75\,500) </math>
|<math>> f_{\omega^\omega}(75\,500)</math>
|
|
|-
|-
|BB(41)
|BB(41)
|<math> > f_{\omega^\omega}^4(32) </math>
|<math>> f_{\omega^\omega}^4(32)</math>
|
|
|-
|-
|BB(51)
|BB(51)
|<math> > f_{\varepsilon_0 + 1}(8) </math>
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|
|}
|}
'''3 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''3 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|-
|BB(1,3)
|BB(1,3)
|
|1
|
|{{TM|1RZ------|halt}}
|-
|-
|[[BB(2,3)]]
|[[BB(2,3)]]
|<math> 38 </math>
|38
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|-
|[[BB(3,3)]]
|[[BB(3,3)]]
|<math> \geq 119\,112\,334\,170\,342\,541 > 10^{17} </math>
|<math>\geq 119\,112\,334\,170\,342\,541 > 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|-
|-
|[[BB(4,3)]]
|[[BB(4,3)]]
|<math> > 2 \uparrow\uparrow\uparrow (2^{2^{32}+1}-1)</math>
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
|}
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|-
|BB(1,4)
|BB(1,4)
|
|1
|
|{{TM|1RZ---------|halt}}
|-
|-
|[[BB(2,4)]]
|[[BB(2,4)]]
|<math> 3\,932\,964 </math>
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|-
|BB(3,4)
|[[BB(3,4)]]
|<math> > 2 \uparrow^{15} 5 </math>
|<math>> (2 \uparrow^{15} 5) + 14</math><ref name=":0">S. Ligocki. "BB(3,4) > Ack(14)." https://www.sligocki.com/2024/05/22/bb-3-4-a14.html  Blog post, 2024. Accessed 15 August 2025.</ref>
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|}
|}
'''5 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''5 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|-
|BB(1,5)
|BB(1,5)
|
|1
|
|{{TM|1RZ------------|halt}}
|-
|-
|[[BB(2,5)]]
|[[BB(2,5)]]
|<math> > 10^{10^{10^{3\,314\,360}}} </math>
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|-
|-
|BB(3,5)
|[[BB(3,5)]]
|<math> > f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15) </math>
|<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|}
|}
'''6 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''6 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|-
|BB(1,6)
|BB(1,6)
|
|1
|
|{{TM|1RZ---------------|halt}}
|-
|-
|BB(2,6)
|[[BB(2,6)]]
|<math> > 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}} </math>
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math><ref name=":1">S. Ligocki, "[https://www.sligocki.com/2023/05/20/bb-2-6-p3.html BB(2,6) > 10↑↑10↑↑10↑↑3]". Blog post, 2023. Accessed 15 August 2025.</ref>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|}
|}
==Maximum Score Function (Σ)==
 
'''2 Symbols:'''
=== Maximum Score Function ([[Busy Beaver Functions|Σ(n,m)]]) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|-
|Σ(1)
|Σ(1)
|<math> 1 </math>
|1
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
|-
|-
|Σ(2)
|Σ(2)
|<math> 4 </math>
|4
|{{TM|1RB1LB_1LA1RZ|halt}}
|{{TM|1RB1LB_1LA1RZ|halt}}
|-
|-
|Σ(3)
|Σ(3)
|<math> 6 </math>
|6
|{{TM|1RB1RZ_0RC1RB_1LC1LA|halt}} {{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}} {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}} {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}
|{{TM|1RB1RZ_0RC1RB_1LC1LA|halt}} {{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}} {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}} {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}
|-
|-
|Σ(4)
|Σ(4)
|<math> 13 </math>
|13
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}} {{TM|1RB0RC_1LA1RA_1RZ1RD_1LD0LB|halt}}
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}} {{TM|1RB0RC_1LA1RA_1RZ1RD_1LD0LB|halt}}
|-
|-
|Σ(5)
|Σ(5)
|<math> 4098 </math>
|4098
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}} {{TM|1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC|halt}}
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}} {{TM|1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC|halt}}
|-
|-
|Σ(6)
|Σ(6)
|<math> > 2\uparrow\uparrow 2\uparrow\uparrow 2\uparrow\uparrow 10 </math>
|> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|-
Line 205: Line 204:
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|}
|}
'''3 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''3 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|-
|Σ(1,3)
|Σ(1,3)
|
|1
|
|{{TM|1RZ------|halt}}
|-
|-
|Σ(2,3)
|Σ(2,3)
|<math> 9 </math>
|9
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|-
|Σ(3,3)
|Σ(3,3)
|<math> \geq 374\,676\,383 </math>
|≥ 374,676,383
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|-
|-
|Σ(4,3)
|Σ(4,3)
|<math> > 2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
|}
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|-
|Σ(1,4)
|Σ(1,4)
|
|1
|
|{{TM|1RZ---------|halt}}
|-
|-
|Σ(2,4)
|Σ(2,4)
|<math> 2050 </math>
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|Σ(3,4)
|<math>\geq (2 \uparrow^{15} 5) + 14 </math><ref name=":0" />
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|}
|}
'''5 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''5 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|-
|Σ(1,5)
|Σ(1,5)
|
|1
|
|{{TM|1RZ------------|halt}}
|-
|-
|Σ(2,5)
|Σ(2,5)
|<math> > 10^{10^{10^{3\,314\,360}}} </math>
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|}
|}
='''Beeping Busy Beavers'''=
==Beeping Busy Beaver ([[BBB]])==
'''2 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''6 Symbols:'''
!Score
!Champions
|-
|Σ(1,6)
|1
|{{TM|1RZ---------------|halt}}
|-
|Σ(2,6)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math><ref name=":1" />
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|}
 
== Beeping Busy Beavers ==
=== Beeping Busy Beaver ([[BBB]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps taken
!Steps taken
!Champions
!Champions
|-
|-
|BBB(1)
|BBB(1)
|<math> 1 </math>
|1
|
|
|-
|-
|BBB(2)
|BBB(2)
|<math> 6 </math>
|6
|
|{{TM|1RB1LB_1LB1LA}}
|-
|-
|BBB(3)
|BBB(3)
|<math> 55 </math>
|55
|{{TM|1LB0RB_1RA0LC_1RC1RA}}
|{{TM|1RB0LB_1LA0RC_1LC1LA}}
|-
|-
|BBB(4)
|BBB(4)
|<math> \geq 32\,779\,478 </math>
|≥ 32,779,478
|
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|-
|BBB(5)
|BBB(5)
|<math> \geq 10^{14006} </math>
|<math>\geq 10^{14006}</math>
|
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|}
|}
'''3 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''3 Symbols:'''
!Steps taken
!Steps taken
!Champions
!Champions
Line 299: Line 312:
|-
|-
|BBB(2,3)
|BBB(2,3)
|59<ref name=":2">Nick Drozd. "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Accessed 15 August 2025.</ref>
|{{TM|1RB2LB1LA_2LB2RA0RA}}
|-
|BBB(3,3)
|≥ 10 ↑↑ 6
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps taken
!Champions
|-
|BBB(1,4)
|
|
|
|
|-
|-
|BBB(3,3)
|BBB(2,4)
|<math> \geq 10 \uparrow\uparrow 6 </math>
|<math>\geq 205\,770\,076\,433\,044\,242\,247\,859 > 2\times 10^{23}</math><ref name=":3" />
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|}
=== Beeping Booping Busy Beaver ([[Beeping Busy Beaver#Beeping Booping Busy Beavers|BBBB]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps taken
!Champions
|-
|BBBB(1)
|2
|
|-
|BBBB(2)
|17
|
|
|}
|}
==Beeping Booping Busy Beaver ([[Beeping Busy Beaver#Beeping Booping Busy Beavers|BBBB]])==
 
There are currently no known/available Champions for this function.
== Maximum Consecutive Ones Function ([[Num]](n,m)) ==
='''Maximum Consecutive Ones Function ([[Num]])'''=
'''2 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!Domain
!Number of Ones
!Number of Ones
!Champions
!Champions
|-
|-
|num(1)
|num(1)
|<math> 1 </math>
|1
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
|-
|-
|num(2)
|num(2)
|<math> 4 </math>
|4
|{{TM|1RB1LB_1LA1LZ|halt}}
|{{TM|1RB1LB_1LA1LZ|halt}}
|-
|-
|num(3)
|num(3)
|<math> 6 </math>
|6
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|-
|-
|num(4)
|num(4)
|<math> 12 </math>
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|-
|-
|num(5)
|num(5)
|<math> 165 </math>
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}} {{TM|0RB1LD_1LC1RB_1LD1RE_1LA1LE_1LZ0RC|halt}}
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}} {{TM|0RB1LD_1LC1RB_1LD1RE_1LA1LE_1LZ0RC|halt}}
|-
|num(2,3)
|6
|{{TM|1RB1LA1LB_0LA2RA1RZ|halt}}
|-
|num(3,3)
|≥ 12
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|num(4,3)
|<math>> 10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
|}
='''Instruction-Limited Busy Beaver'''=
 
==Maximum amount of steps ([[BBi]])==
== Maximum Space Function ([[Maximum Space Function|BB<sub>space</sub>]](n,m)) ==
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Steps
!Cells visited
!Champions
!Champions
|-
|-
|BBi(1)
|BB<sub>space</sub>(1,2)
|<math> 1 </math>
|1
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|{{TM|1RZ---|halt}}
|-
|BBi(2)
|<math> 3 </math>
|{{TM|0RB---_1LA---|halt}}
|-
|BBi(3)
|<math> 5 </math>
|{{TM|1RB1LB_1LA---|halt}}
|-
|-
|BBi(4)
|BB<sub>space</sub>(2,2)
|<math> 16 </math>
|4
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|{{TM|1RB1LB_1LA1RZ|halt}} and {{TM|1RB0LB_1LA1RZ|halt}}
|-
|-
|BBi(5)
|BB<sub>space</sub>(3,2)
|<math> 37 </math>
|7
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|{{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} and {{TM|1RB0RC_1LC1RZ_1RA0LB|halt}}
|-
|-
|BBi(6)
|BB<sub>space</sub>(4,2)
|<math> 123 </math>
|16
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|{{TM|1RB0RA_1LC0RD_0LD0LB_1RA1RZ|halt}}
|-
|-
|BBi(7)
|BB<sub>space</sub>(5,2)
|<math> 3\,932\,963 </math>
|12289
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|-
|BBi(8)
|<math> >6.889 \times 10^{1565} </math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|}
|}
==Maximum Score (Σi)==
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''3 Symbols:'''
!Score
!Cells visited
!Champions
!Champions
|-
|-
|Σi(1)
|BB<sub>space</sub>(1,3)
|<math> 1 </math>
|
|{{TM|1RH---|halt}}
|
|-
|-
|Σi(2)
|BB<sub>space</sub>(2,3)
|<math> 2 </math>
|9
|{{TM|1RB---_1LA---|halt}}
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|-
|Σi(3)
|BB<sub>space</sub>(2,4)
|<math> 4 </math>
|2050
|{{TM|1RB1LB_1LA---|halt}}
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|Σi(4)
|<math> 5 </math>
|{{TM|1RB0LB---_1LA2RA---|halt}}
|-
|Σi(5)
|<math> 9 </math>
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|Σi(6)
|<math> 14 </math>
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|Σi(7)
|<math> 2050 </math>
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|Σi(8)
|<math> >1.355 \times 10^{783} </math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|}
|}
='''Reversible Turing Machines'''=
 
==Maximum Shifts Function ([[Reversible Turing Machine|BB<sub>rev</sub>]])==
== Size of the runtime spectrum (R(n,m)) ==
'''2 Symbols:'''
There currently doesn't seem to be any available information about values of this function.
 
== Reversible Turing Machines ==
=== Maximum Shifts Function ([[Reversible Turing Machine|BB<sub>rev</sub>]](n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Steps
!Steps
!Champions
!Champions
Line 429: Line 451:
|-
|-
|BB<sub>rev</sub>(2)
|BB<sub>rev</sub>(2)
|<math> 6 </math>
|6
|{{TM|0RB1RZ_1LA1RB|halt}}
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|-
|BB<sub>rev</sub>(3)
|BB<sub>rev</sub>(3)
|<math> 17 </math>
|17
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|-
|BB<sub>rev</sub>(4)
|BB<sub>rev</sub>(4)
|<math> 48 </math>
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|-
|BB<sub>rev</sub>(5)
|BB<sub>rev</sub>(5)
|<math> 388 </math>
|388
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|-
|BB<sub>rev</sub>(6)
|BB<sub>rev</sub>(6)
|<math> \geq 537\,556 </math>
|≥ 537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|-
|-
|BB<sub>rev</sub>(7)
|BB<sub>rev</sub>(7)
|<math> >10^{19} </math>
|<math>> 10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ|halt}}
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ|halt}}
|}
|}
==Maximum Score Function (Σ<sub>rev</sub>)==
 
'''2 Symbols:'''
=== Maximum Score Function (Σ<sub>rev</sub>(n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Score
!Score
!Champions
!Champions
Line 465: Line 487:
|-
|-
|Σ<sub>rev</sub>(2)
|Σ<sub>rev</sub>(2)
|<math> \geq 2 </math>
|≥ 2
|{{TM|0RB1RZ_1LA1RB|halt}}
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|-
|Σ<sub>rev</sub>(3)
|Σ<sub>rev</sub>(3)
|<math> \geq 4 </math>
|≥ 4
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|-
|Σ<sub>rev</sub>(4)
|Σ<sub>rev</sub>(4)
|<math> \geq 6 </math>
|≥ 6
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|-
|Σ<sub>rev</sub>(5)
|Σ<sub>rev</sub>(5)
|<math> \geq 16 </math>
|≥ 16
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|-
|Σ<sub>rev</sub>(6)
|Σ<sub>rev</sub>(6)
|<math> \geq 1161 </math>
|≥ 1161
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|}
|}
='''Blanking Busy Beaver (BLB)'''=
 
There are currently no known/available Champions for this function.
== Blanking Busy Beaver ([[Blanking Busy Beaver|BLB(n,m)]]) ==
='''Period-oriented Busy Beavers'''=
{| class="wikitable"
==Busy Preperiodic Beaver ([[BBS]])==
|+
'''2 Symbols:'''
!'''2 Symbols:'''
!Steps
!Champions
|-
|BLB(1)
|nonexistent
|nonexistent
|-
|BLB(2)
|8<ref name=":3">Nick Drozd. "[https://nickdrozd.github.io/2021/02/14/blanking-beavers.html Blanking Beavers]". Accessed 15 August 2025.</ref>
|{{TM|1RB0RA_1LB1LA}}
|-
|BLB(3)
|≥ 34<ref name=":4">Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(4)
|≥ 32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5)
|≥ 32,810,047<ref>Comment #71 "https://scottaaronson.blog/?p=5661". Accessed 26 September 2025.</ref>
|{{TM|1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE}}
|-
|BLB(6)
|≥ 65,538,549<ref>Comment #62 "https://scottaaronson.blog/?p=5661". Accessed 26 September 2025.</ref>
|{{TM|1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Steps
!Champions
|-
|BLB(1,3)
|nonexistent
|nonexistent
|-
|BLB(2,3)
|≥ 77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(3,3)
|> 10<sup>42,745</sup>
|{{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps
!Champions
|-
|BLB(1,4)
|nonexistent
|nonexistent
|-
|BLB(2,4)
|≥ 1,367,361,263,049<ref name=":4" />
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}
 
== Lazy Beaver ==
=== Shifts Function ([[Lazy Beaver|LB]](n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!
!1 State
!2 States
!3 States
!4 States
!5 States
!6 States
|-
|2 Symbols
|2
|7
|22
|72
|427
|8407
|-
|3 Symbols
|2
|23
|351
|189,270
|
|
|-
|4 Symbols
|2
|93
|242,789
|
|
|
|-
|5 Symbols
|2
|956
|
|
|
|
|-
|6 Symbols
|2
|33,851
|
|
|
|
|}
== Period-oriented Busy Beavers ==
=== Busy Preperiodic Beaver ([[BBS]](n,m)) ===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Preperiod
!Preperiod
!Champions
!Champions
|-
|-
|BBS(1,2)
|BBS(1,2)
|
|0
|
|{{TM|1RA---}}
|-
|-
|BBS(2,2)
|BBS(2,2)
|
|≥ 9
|
|{{TM|1RB0LB_1LA0RB}} proven winner?
|-
|-
|BBS(3,2)
|BBS(3,2)
|<math> 101 </math>
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|-
|BBS(4,2)
|BBS(4,2)
|<math> \geq 119\,120\,230\,102 </math>
|≥ 119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(5,2)
|> 10<sup>14,006</sup>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,3)
|0
|{{TM|1RA------}}
|-
|BBS(2,3)
|≥ 165<ref>Nick Drozd. [https://nickdrozd.github.io/2021/02/14/blanking-beavers.html Blanking Beavers]</ref>
|{{TM|1RB0LA---_1LB2LA0RB}}
|-
|BBS(3,3)
|> 10 ↑↑ 6
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|-
|BBS(4,3)
|> <math>10 \uparrow^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD}}
|}
|}
'''3 Symbols:''' It seems that currently no information is available for this domain.
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Preperiod
!Preperiod
!Champions
!Champions
|-
|-
|BBS(1,4)
|BBS(1,4)
|
|0
|
|{{TM|1RA---------}}
|-
|-
|BBS(2,4)
|BBS(2,4)
|<math> \geq 293\,225\,660\,896 </math>
|≥ 205,770,076,433,044,242,247,860
|{{TM|1RB2LA0RA3LA_1LA1LB3RB1RA}}
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|}
|}
==Busy Periodic Beaver ([[BBP]])==
 
'''2 Symbols:'''
=== Busy Periodic Beaver ([[BBP]](n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Period
!Period
!Champions
!Champions
|-
|-
|BBP(1,2)
|BBP(1,2)
|
|1
|
|{{TM|1RA---}}
|-
|-
|BBP(2,2)
|BBP(2,2)
|
|≥ 9
|
|{{TM|1RB0RB_1LB1RA}} proven winner?
|-
|-
|BBP(3,2)
|BBP(3,2)
|<math> 92 </math>
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|-
|BBP(4,2)
|BBP(4,2)
|<math> \geq 212\,081\,736 </math>
|≥ 212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|}
|}
'''3 Symbols:''' It seems that currently no information is available for this domain.
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''3 Symbols:'''
!Period
!Period
!Champions
!Champions
|-
|-
|BBP(1,4)
|BBP(1,3)
|1
|{{TM|1RA------}}
|-
|BBP(2,3)
|
|
|
|
|-
|BBP(3,3)
|≥ 1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Period
!Champions
|-
|BBP(1,4)
|1
|{{TM|1RA---------}}
|-
|-
|BBP(2,4)
|BBP(2,4)
|<math> \geq 33\,209\,131 </math>
|≥ 33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|}
|}
='''Busy Beaver for Lambda Calculus ([[BBλ]])'''=
 
==Regular Busy Beaver for Lambda Calculus==
= '''Instruction-Limited Busy Beaver''' =
== Instruction-Limited Classical Busy Beaver Functions ==
=== Instruction-Limited Maximum Shifts Function ([[BBi]](n)) ===
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|BBi(1)
|1
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|-
|BBi(2)
|3
|{{TM|0RB---_1LA---|halt}}
|-
|BBi(3)
|5
|{{TM|1RB1LB_1LA---|halt}}
|-
|BBi(4)
|16
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|BBi(5)
|37
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|BBi(6)
|123
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|BBi(7)
|3,932,963
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|BBi(8)
|<math>>6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>>10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|}
 
=== Instruction-Limited Maximum Score Function ([[Σi]](n)) ===
{| class="wikitable"
|+
!
!Score
!Champions
|-
|Σi(1)
|1
|{{TM|1RH---|halt}}
|-
|Σi(2)
|2
|{{TM|1RB---_1LA---|halt}}
|-
|Σi(3)
|4
|{{TM|1RB1LB_1LA---|halt}}
|-
|Σi(4)
|5
|{{TM|1RB0LB---_1LA2RA---|halt}}
|-
|Σi(5)
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|Σi(6)
|14
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|Σi(7)
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|Σi(8)
|<math>>1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|Σi(9)
|<math>>10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|}
 
== Instruction-Limited Blanking Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|BLBi(n)]]) ==
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|BLBi(1)
|nonexistent
|nonexistent
|-
|BLBi(2)
|nonexistent
|nonexistent
|-
|BLBi(3)
|4
|{{TM|1RB0RA_1LA---}}
|-
|BLBi(4)
|12
|{{TM|1RB---_1RC---_1LC0RC}}
|-
|BLBi(5)
|30
|{{TM|1RB------_1RC------_2LC2RC0RC}}
|-
|BLBi(6)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLBi(7)
|808
||{{TM|1RB------_1RC------_0RD2LC---_1LD2RD0RC}}
|-
|BLBi(8)
|≥ 1,367,361,263,049
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|-
|BLBi(9)
|> 10<sup>42,745</sup>
|{{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}
|}
 
== Instruction-Limited Greedy Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|gBBi(n)]]) ==
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|gBBi(1)
|1
|
|-
|gBBi(2)
|3
|
|-
|gBBi(3)
|5
|
|-
|gBBi(4)
|13
|
|-
|gBBi(5)
|19
|
|-
|gBBi(6)
|25
|
|-
|gBBi(7)
|41
|
|-
|gBBi(8)
|55
|
|-
|gBBi(9)
|238
|
|-
|gBBi(10)
|941
|
|-
|gBBi(11)
|1341
|
|-
|gBBi(12)
|10465
|
|-
|gBBi(13)
|10675
|
|-
|gBBi(14)
|≥ 9,874,580
|{{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}
|}
 
= '''Program-Limited Busy Beaver''' =
== Busy Beaver for Lambda Calculus ==
=== Regular Busy Beaver for Lambda Calculus ([[BBλ]](n)) ===
For n = 0,1,2,3,5 BBλ(n) is undefined, while for the rest of <math> 20 \geq n </math> BBλ(n) = n.
For n = 0,1,2,3,5 BBλ(n) is undefined, while for the rest of <math> 20 \geq n </math> BBλ(n) = n.
{| class="wikitable"
{| class="wikitable"
Line 579: Line 954:
|-
|-
|BBλ(21)
|BBλ(21)
|<math> 22 </math>
|22
|<code> \(\1 1) (1 (\2)) </code>
|<code>\(\1 1) (1 (\2))</code>
|-
|-
|BBλ(22)
|BBλ(22)
|<math> 24 </math>
|24
|<code> \(\1 1) (1 (\\1))\(\1 1 1) (1 1) </code>
|<code>\(\1 1) (1 (\\1))\(\1 1 1) (1 1)</code>
|-
|-
|BBλ(23)
|BBλ(23)
|<math> 26 </math>
|26
|<code> \(\1 1) (1 (\\2)) </code>
|<code>\(\1 1) (1 (\\2))</code>
|-
|-
|BBλ(24)
|BBλ(24)
|<math> 30 </math>
|30
|<code> \(\1 1 1) (1 (\1)) </code>
|<code>\(\1 1 1) (1 (\1))</code>
|-
|-
|BBλ(25)
|BBλ(25)
|<math> 42 </math>
|42
|<code> \(\1 1) (\1 (2 1)) </code>
|<code>\(\1 1) (\1 (2 1))</code>
|-
|-
|BBλ(26)
|BBλ(26)
|<math> 52 </math>
|52
|<code> (\1 1) (\\2 (1 2)) </code>
|<code>(\1 1) (\\2 (1 2))</code>
|-
|-
|BBλ(27)
|BBλ(27)
|<math> 44 </math>
|44
|<code> \\(\1 1) (\1 (2 1)) </code>
|<code>\\(\1 1) (\1 (2 1))</code>
|-
|-
|BBλ(28)
|BBλ(28)
|<math> 58 </math>
|58
|<code> \(\1 1) (\1 (2 (\2)))) </code>
|<code>\(\1 1) (\1 (2 (\2))))</code>
|-
|-
|BBλ(29)
|BBλ(29)
|<math> 223 </math>
|223
|<code> \(\1 1) (\1 (1 (2 1))) </code>
|<code>\(\1 1) (\1 (1 (2 1)))</code>
|-
|-
|BBλ(30)
|BBλ(30)
|<math> 160 </math>
|160
|<code> (\1 1 1) (\\2 (1 2)) </code> and <code> (\1 (1 1)) (\\2 (1 2)) </code>
|<code>(\1 1 1) (\\2 (1 2))</code> and <code>(\1 (1 1)) (\\2 (1 2))</code>
|-
|-
|BBλ(31)
|BBλ(31)
|<math> 267 </math>
|267
|<code> (\1 1) (\\2 (2 (1 2))) </code>
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|-
|BBλ(32)
|BBλ(32)
|<math> 298 </math>
|298
|<code> \(\1 1) (\1 (1 (2 (\2)))) </code>
|<code>\(\1 1) (\1 (1 (2 (\2))))</code>
|-
|-
|BBλ(33)
|BBλ(33)
|<math> 1812 </math>
|1812
|<code> \(\1 1) (\1 (1 (1 (2 1)))) </code>
|<code>\(\1 1) (\1 (1 (1 (2 1))))</code>
|-
|-
|BBλ(34)
|BBλ(34)
|<math> 327\,686 </math>
|327,686
|<code> (\1 1 1 1) (\\2 (2 1)) </code> and <code> (\1 (1 1) 1) (\\2 (2 1)) </code>
|<code>(\1 1 1 1) (\\2 (2 1))</code> and <code>(\1 (1 1) 1) (\\2 (2 1))</code>
|-
|-
|BBλ(35)
|BBλ(35)
|<math> 5 \times 3^{3^{3}} +6 > 3.8 \times 10^{13} </math>
|<math>5 \times 3^{3^{3}} +6 = 38\,127\,987\,424\,941 > 3.8 \times 10^{13}</math>
|<code> (\1 1 1) (\\2 (2 (2 1))) </code>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|-
|BBλ(36)
|BBλ(36)
|<math> 5 \times 2^{2^{2^{3}}} +6 > 5.7 \times 10^{77} </math>
|<math>5 \times 2^{2^{2^{3}}} +6 > 5.7 \times 10^{77}</math>
|<code> (\1 1) (\1 (1 (\\2 (2 1)))) </code>
|<code>(\1 1) (\1 (1 (\\2 (2 1))))</code>
|-
|-
|BBλ(37)
|BBλ(37)
|BBλ(35) +2 = <math> 5 \times 3^{3^{3}} +8 > 3.8 \times 10^{13} </math>
|<math>BB \lambda(35) +2 =5 \times 3^{3^{3}} +8 = 38\,127\,987\,424\,943 > 3.8 \times 10^{13}</math>
|<code> \(\1 1 1) (\\2 (2 (2 1))) </code>
|<code>\(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|-
|BBλ(38)
|BBλ(38)
|<math> \geq 5 \times 2^{2^{2^{2^{2}}}} +6 > 10^{10^{4}} </math>
|<math>\geq 5 \times 2^{2^{2^{2^{2}}}} +6 > 10^{19729}</math>
|<code> (\1 1 1 1 1) (\\2 (2 1)) </code> and <code> (\1 (1 1) 1 1) (\\2 (2 1)) </code>
|<code>(\1 1 1 1 1) (\\2 (2 1))</code> and <code>(\1 (1 1) 1 1) (\\2 (2 1))</code>
|-
|-
|BBλ(39)
|BBλ(39)
|<math> \geq 5 \times 3^{3^{3^{3}}} +6 > 10^{10^{12}} </math>
|<math>\geq 5 \times 3^{3^{3^{3}}} +6 > 10^{3\,638\,334\,640\,024}</math>
|<code> (\1 1 1 1) (\\2 (2 (2 1))) </code>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|-
|BBλ(40)
|BBλ(40)
|<math> > (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16 </math>
|<math>> (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<code> (\1 1 1) (\1 (\\2 (2 1)) 1) </code>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|-
|BBλ(41)
|BBλ(41)
|<math> \geq 5 \times 3^{3^{85}} +6 > 10^{10^{40}} </math>
|<math>\geq 5 \times 3^{3^{85}} +6 > 10^{1.7 \times 10^{40}}</math>
|<code> (\1 (\1 1) 1) (\\2 (2 (2 1))) </code>
|<code>(\1 (\1 1) 1) (\\2 (2 (2 1)))</code>
|-
|-
|BBλ(42)
|BBλ(42)
|<math> \geq </math> BBλ(40) + 2 <math> > (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16 </math>
|<math>\geq BB \lambda(40) + 2 > (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<code> \(\1 1 1) (\1 (\\2 (2 1)) 1) </code>
|<code>\(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|-
|BBλ(43)
|BBλ(43)
|<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8 </math>
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code> (\1 1) (\1 (\1 (\\2 (2 1)) 2)) </code>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|-
|BBλ(44)
|BBλ(44)
|<math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16 </math>
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<code> (\1 1 1 1) (\1 (\\2 (2 1)) 1) </code>
|<code>(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|-
|BBλ(45)
|BBλ(45)
|<math> \geq </math> BBλ(43) + 2 <math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8 </math>
|<math>\geq BB \lambda(43) + 2 > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code> \(\1 1) (\1 (\1 (\\2 (2 1)) 2)) </code>
|<code>\(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|-
|BBλ(46)
|BBλ(46)
|<math> \geq </math> BBλ(44) + 2 <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16 </math>
|<math>\geq BB \lambda(44) + 2 > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<code> \(\1 1 1 1) (\1 (\\2 (2 1)) 1) </code>
|<code>\(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|-
|BBλ(47)
|BBλ(47)
Line 687: Line 1,062:
|-
|-
|BBλ(48)
|BBλ(48)
|<math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16 </math>
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<code> (\1 1 1 1 1) (\1 (\\2 (2 1)) 1) </code>
|<code>(\1 1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|-
|BBλ(49)
|BBλ(49)
|<math> > f_{\omega+1}(\frac{2 \uparrow\uparrow 6}{2}) > \text{Grahams Number} </math>
|<math>> f_{\omega+1}(\frac{2 \uparrow\uparrow 6}{2}) > \text{Graham's Number}</math>
|<code> (\1 1) (\1 (1 (\\1 2 (\\2 (2 1))))) </code>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|BBλ(63)
|<math>> f_{\omega^3}\left(2\right)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|BBλ(91)
|<math>> f_{\varepsilon_0 + 1}\left(3\right)</math>
|<code>(\1 1 (\1 1) (\\1 2 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))</code>
|-
|-
|BBλ(1850)
|BBλ(1850)
|<math> > \text{Loaders Number} </math>
|<math>> \text{Loader's Number}</math>
|<code> Too large for this list </code>
|<code>Too large for this list</code>
|}
 
=== Oracle Busy Beaver for Lambda Calculus ([[Busy Beaver for lambda calculus#Oracle Busy Beaver|BBλ<sub>1</sub>(n)]]) ===
Note that <math>f(n) = 6 + 5 \times BB \lambda(n)</math>.
{| class="wikitable"
|+
!
!BBλ<sub>1</sub>(n)
!Champions
|-
|BBλ<sub>1</sub>(1)
|0
|
|-
|BBλ<sub>1</sub>(2)
|1
|<code>1</code>
|-
|BBλ<sub>1</sub>(3)
|0
|
|-
|BBλ<sub>1</sub>(4)
|4
|<code>\1</code>
|-
|BBλ<sub>1</sub>(5)
|5
|<code>\2</code>
|-
|BBλ<sub>1</sub>(6)
|6
|<code>\\1</code>
|-
|BBλ<sub>1</sub>(7)
|7
|<code>\\2</code>
|-
|BBλ<sub>1</sub>(8)
|26
|<code>1 (\1)</code>
|-
|BBλ<sub>1</sub>(9)
|9
|<code>\\2</code>
|-
|BBλ<sub>1</sub>(10)
|36
|<code>1 (\\1)</code>
|-
|BBλ<sub>1</sub>(11)
|41
|<code>1 (\\2)</code>
|-
|BBλ<sub>1</sub>(12)
|266
|<code>1 (1 (\1))</code>
|-
|BBλ<sub>1</sub>(13)
|51
|<code>1 (\\2)</code>
|-
|BBλ<sub>1</sub>(14)
|<math>f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|<code>1 (1 (\\1))</code>
|-
|BBλ<sub>1</sub>(15)
|<math>f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{1.7 \times 10^{40}}</math>
|<code>1 (1 (\\2))</code>
|-
|BBλ<sub>1</sub>(16)
|<math>f(266)</math>
|<code>1 (1 (1 (\1)))</code>
|-
|BBλ<sub>1</sub>(17)
|<math>f(51)</math>
|<code>1 (1 (\\\2))</code>
|-
|BBλ<sub>1</sub>(18)
|<math>f^{4}(4) = f(f(266))</math>
|<code>1 (\1) 1 (\1)</code>
|-
|BBλ<sub>1</sub>(19)
|<math>f^{3}(7) = f(f(41))</math>
|<code>1 (1 (1 (\\2)))</code>
|-
|BBλ<sub>1</sub>(20)
|<math>f^{6}(4) = f^{4}(266)</math>
|<code>1 (\\1) 1 (\1)</code>
|-
|BBλ<sub>1</sub>(21)
|<math>f^{7}(4) = f^{5}(266)</math>
|<code>1 (\\2) 1 (\1)</code>
|-
|BBλ<sub>1</sub>(22)
|<math>f^{52}(4) = f^{50}(266)</math>
|<code>1 (1(\1)) 1(\1)</code>
|-
|BBλ<sub>1</sub>(28)
|<math>\geq f^{BB \lambda(f^{3}(4))}(4)</math>
|<code>1 (\1) 1 (\1) 1 (\1)</code>
|-
|BBλ<sub>1</sub>(29)
|<math>\geq f^{BB \lambda(f^{BB \lambda(f^{4}(4))+4}(4))+BB \lambda(f^{4}(4))+5}(4)</math>
|<code>1(\1)(\1 2 1)(\1)</code>
|}
 
=== Busy Beaver for De Bruijn Lambda Calculus ===
{| class="wikitable"
!n
!Value
!Champion
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|10
|<math>> 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|11
|<math>> 10^{7.625 \times 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>
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|14
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|15
|<math>> f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|18
|<math>f_{\omega^3}(2)</math>
|<code>(\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))</code>
|-
|26
|<math>f_{\epsilon_0+1}(3)</math>
|<code>(\1 1 (\1 1) (\\1 2 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))</code>
|}
|}
='''Doodle Function ([[Doodle function|doodle]])'''=
doodle(1,n) = 1 and doodle(2,n) = n


'''2 Symbols:'''
== [[Busy Beaver for SKI calculus]] ==
=== Busy Beaver for SKI calculus ===
{| class="wikitable"
! n !! Value !! Champion
|-
| 1 || 1 || S
|-
| 2 || 2 || SS
|-
| 3 || 3 || SSS
|-
| 4 || 4 || SSSS
|-
| 5 || 6 || SSS(SS)
|-
| 6 || 17 || SSS(SI)S
|-
|7
|≥ 18
|S(SSS(SI)S)
|-
|8
|≥ 19
|SS(SSS(SI)S)
|-
|9
|≥ 519
|SSI((S(SS)S)S)K
|-
|10
|≥ 1041
|SSI((S(SS)S)S)KS
|}
 
=== Busy Beaver for SK calculus ===
{| class="wikitable"
! n !! Value !! Champion
|-
| 1 || = 1 || S
|-
| 2 || = 2 || SS
|-
| 3 || = 3 || SSS
|-
| 4 || = 4 || <code>SSSS</code>
|-
| 5 || = 6 || <code>SSS(SS)</code>
|-
| 6 || ≥ 8 || <code>SSS(SSS)</code>
|-
|7
|≥ 10
|<code>SSS(SSSS)</code>
|-
|8
|≥ 23
|<code>SSS(S(SKS))S</code>
|}
 
== Fractran ([[Fractran|BBf]](n)) ==
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| 2 || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
  -1
\end{bmatrix}</math>
|-
| 3 || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
  -1 & 1
\end{bmatrix}</math>
|-
| 4 || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
  -1 & 2
\end{bmatrix}</math>
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  1 \\
  0 & -1
\end{bmatrix}</math>
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  2 \\
  0 & -1
\end{bmatrix}</math>
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  3 \\
  0 & -1
\end{bmatrix}</math>
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  4 \\
  0 & -1
\end{bmatrix}</math>
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  5 \\
  0 & -1
\end{bmatrix}</math>
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  6 \\
  0 & -1
\end{bmatrix}</math>
|-
| 11 || 10 || <code>[27/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
  -1 &  3 &  0 \\
  0 & -1 &  2 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
| 12 || 13 || <code>[81/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
  -1 &  4 &  0 \\
  0 & -1 &  2 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
| 13 || 17 || <code>[81/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
  -1 &  4 &  0 \\
  0 & -1 &  3 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
| 14 || 21 || <code>[243/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
  -1 &  5 &  0 \\
  0 & -1 &  3 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
| 15 || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|<math>\begin{bmatrix}
  0 & -2 & -1 \\
  2 &  0 & -1 \\
  -1 &  1 &  0 \\
  0 & -1 &  2
\end{bmatrix}</math>
|-
| 16 || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|<math>\begin{bmatrix}
  0 & -2 & -1 \\
  2 &  0 & -1 \\
  -1 &  1 &  0 \\
  0 & -1 &  3
\end{bmatrix}</math>
|-
| 17 || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|<math>\begin{bmatrix}
  -1 & -1 &  1 &  0 \\
  -1 &  0 &  0 &  2 \\
    0 &  1 & -1 &  0 \\
    3 &  0 &  1 & -1
\end{bmatrix}</math>
|-
| 18 || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|<math>\begin{bmatrix}
  -1 & -1 &  1 &  0 \\
  -1 &  0 &  0 &  2 \\
    0 &  1 & -1 &  0 \\
    4 &  0 &  1 & -1
\end{bmatrix}</math>
|-
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math>\begin{bmatrix}
  -1 & -1 &  1 &  0 \\
  -1 &  0 &  0 &  2 \\
    0 &  1 & -1 &  0 \\
    5 &  0 &  1 & -1
\end{bmatrix}</math>
|-
|20
|≥ 746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
      &    -1 &    -1 &    +1 &      \\
  +1 &    -1 &      &      &    +1 \\
  +1 &    +1 &      &    -1 &    -1 \\
  -1 &      &    +1 &      &      \\
      &    +2 &    -1 &      &
\end{bmatrix}</math>
|-
|21
|≥  31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
    0 &    -1 &    -1 &    1 \\
    2 &    -1 &    0 &    0 \\
  -1 &    3 &    0 &    -1 \\
  -1 &    0 &    1 &    0 \\
    0 &    2 &    -1 &    0
\end{bmatrix}</math>
|-
|22
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
  -2 &    -1 &    0 &    0 &    0 \\
  -1 &    2 &    -1 &    0 &    0 \\
    1 &    -1 &    0 &    1 &    0 \\
  -1 &    0 &    0 &    0 &    1 \\
    0 &    0 &    1 &    -1 &    0 \\
    0 &    1 &    0 &    0 &    -1
\end{bmatrix}</math>
|}
 
== Cyclic Tag ([[Cyclic Tag|CTBB(n)]]) ==
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!
!Runtime
!Champions
|-
|CTBB(2)
|5<ref>https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233</ref>
|
|-
|CTBB(3)
|<u>></u> 38<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517</ref>
|
|-
|CTBB(4)
|≥ 672<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043</ref>
|
|-
|CTBB(5)
|≥ 2^2^2^2^182<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063</ref>
|
|-
|CTBB(6)
|<u>></u> 2↑↑↑131<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479</ref>
|
|-
|CTBB(7)
|<u>></u> 4↑↑↑↑(4↑↑↑3)<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281</ref> <ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217</ref>
|
|}
== Minsky Machines ([[Register machine|MBB(n)]]) ==
{| class="wikitable"
!Domain
!Halting Time
!Champion
|-
|MBB(1)
|1
|<code>0+Z</code>
|-
|MBB(2)
|3
|<code>0+B_0-B*</code>
|-
|MBB(3)
|5
|<code>0+B_0+C_0-C*</code>
|-
|MBB(4)
|10
|<code>0+B_1+C_0-BD_1-C*</code>
|-
|MBB(5)
|24
|<code>0-DB_0+C_1-ED_1+A_1-B*</code>
|-
|MBB(6)
|≥ 49
|<code>0+B_1-FC_1+D_0-CE_0+A_1-A*</code>
|}
= '''Turmites''' =
== Terminating Turmites ([[TT]](n,k), 1D Turmites) ==
Where n is the amount of states and k is the amount of symbols.
{| class="wikitable"
!Domain
!Runtime
!Champions
|-
|TT(2)
|≥ 13
|<code>1TB---_1PA0PB</code>
|-
|TT(3)
|≥ 82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4)
|≥ 48,186
|<code>1TB1PA_1PC0PA_1TA0PD_---1TA</code>
|-
|TT(2,3)
|≥ 223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3)
|≥ 45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|-
|TT(2,4)
|> 3.467*10<sup>15</sup>
|<code>1TA2PB3TB---_3TA1PB1TA1PA</code>
|}
== 2D Turmites ([[Terminating Turmite|turNing machines]]) ==
There are currently no known/available Champions for this function.
= '''Doodle Function ([[Doodle function|doodle(c,n)]])''' =
doodle(1,n) = 1 and doodle(2,n) = n. Also note that doodle(c) = doodle(c,2).
{| class="wikitable"
|+
!'''2 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|-
|doodle(3,2)
|doodle(3,2)
|<math> \geq 487 </math>
|≥ 487
|
|
|}
|}
='''Terminating Turmites ([[TT]])'''=
 
There are currently no known/available Champions for this function.
= '''Bug Function''' =
Bug(2,2) = 2
<pre>
####
#S-#
#.F#
####
</pre>
 
Bug(3,3) = 8
<pre>
#####
#S..#
#.#.#
#.#F#
#####
</pre>
 
Bug(4,4) = 20
<pre>
######
#S...#
#..###
#....#
#..#F#
######
</pre>
 
Bug(5,5) = 42
<pre>
#######
#S...##
#.....#
#.....#
#.#.###
#.#..F#
#######
</pre>
 
Bug(6,6) = 96
<pre>
########
#S.....#
#.###.##
#.#...##
#.#....#
#..#.###
#..#..F#
########
</pre>
 
Bug(7,7) = 218
<pre>
#########
#S.###..#
#......##
#.#.##..#
#..#...##
#..#....#
#...#.###
#..#-..F#
#########
</pre>
 
Bug(8,8) = 506
<pre>
##########
#S.#.....#
#.#..##.##
#.#.##..##
#....##..#
#.#.#...##
#..##....#
#....#.###
#....#..F#
##########
</pre>
 
= '''References''' =

Latest revision as of 22:37, 9 January 2026

A collection of Busy Beaver Champions including Champions for BB-Adjacent functions. Note that for all functions with the input format f(n,m), n denotes the number of states and m denotes the number of symbols of the relevant Busy Beaver domain. Note that for functions with this input format f(n) = f(n,2).

Note: highest ref name in use: 4

State-and-Symbol-Limited Busy Beaver functions

Busy Beaver functions where the programs size is limited by the amount of states and symbols.

Original Busy Beaver Functions

Maximum Shifts Function (S(n,m), also commonly called BB(n,m))

2 Symbols: Runtime Champions
BB(1) 1 1RZ--- (bbch)
BB(2) 6 1RB1LB_1LA1RZ (bbch) 1RB0LB_1LA1RZ (bbch) 1RB1RZ_1LB1LA (bbch) 1RB1RZ_0LB1LA (bbch) 0RB1RZ_1LA1RB (bbch)
BB(3) 21 1RB1RZ_1LB0RC_1LC1LA (bbch)
BB(4) 107 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch)
BB(5) 47,176,870 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch)
BB(6) > 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
BB(7) >2↑112↑113 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)
BB(8)
BB(9) >fω(f9(2)) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH (bbch)
BB(10) >fω2(25) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ (bbch)
BB(11) >fω2(2↑↑12)>fω2(f3(9)) 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE (bbch)
BB(12) >fω4(2↑↑↑4−3)>fω4(f4(2)) 0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL (bbch)
BB(14) >fω+1(65536)>g64 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ--- (bbch)
BB(15) >fω+1(fω(1057)) 0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN (bbch)
BB(16) >fω+12(101057)
BB(18) >fω+2(fω+13(fω2(60)))
BB(20) >fω+22(21)
BB(21) >fω22(4↑↑341)
BB(40) >fωω(75500)
BB(41) >fωω4(32)
BB(51) >fε0+1(8)
3 Symbols: Runtime Champions
BB(1,3) 1 1RZ------ (bbch)
BB(2,3) 38 1RB2LB1RZ_2LA2RB1LB (bbch)
BB(3,3) ≥119112334170342541>1017 0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(4,3) >2↑↑↑2↑↑↑2↑↑↑(7.92×1028) 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)
4 Symbols: Runtime Champions
BB(1,4) 1 1RZ--------- (bbch)
BB(2,4) 3,932,964 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
BB(3,4) >(2↑155)+14[1] 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
5 Symbols: Runtime Champions
BB(1,5) 1 1RZ------------ (bbch)
BB(2,5) >1010103314360 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
BB(3,5) >fω(2↑155)>fω2(15) 1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch)
6 Symbols: Runtime Champions
BB(1,6) 1 1RZ--------------- (bbch)
BB(2,6) >10↑↑10↑↑1010115[2] 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)

Maximum Score Function (Σ(n,m))

2 Symbols: Score Champions
Σ(1) 1 1RZ--- (bbch)
Σ(2) 4 1RB1LB_1LA1RZ (bbch)
Σ(3) 6 1RB1RZ_0RC1RB_1LC1LA (bbch) 1RB1RC_1LC1RZ_1RA0LB (bbch) 1RB1LC_1LA1RB_1LB1RZ (bbch) 1RB1RA_1LC1RZ_1RA1LB (bbch) 1RB1LC_1RC1RZ_1LA0LB (bbch)
Σ(4) 13 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch) 1RB0RC_1LA1RA_1RZ1RD_1LD0LB (bbch)
Σ(5) 4098 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch) 1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC (bbch)
Σ(6) > 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
Σ(7) >2↑112↑113 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)
3 Symbols: Score Champions
Σ(1,3) 1 1RZ------ (bbch)
Σ(2,3) 9 1RB2LB1RZ_2LA2RB1LB (bbch)
Σ(3,3) ≥ 374,676,383 0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
Σ(4,3) >2↑↑↑2↑↑↑2↑↑↑(7.92×1028) 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)
4 Symbols: Score Champions
Σ(1,4) 1 1RZ--------- (bbch)
Σ(2,4) 2050 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
Σ(3,4) ≥(2↑155)+14[1] 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
5 Symbols: Score Champions
Σ(1,5) 1 1RZ------------ (bbch)
Σ(2,5) >1010103314360 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
6 Symbols: Score Champions
Σ(1,6) 1 1RZ--------------- (bbch)
Σ(2,6) >10↑↑10↑↑1010115[2] 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)

Beeping Busy Beavers

Beeping Busy Beaver (BBB(n,m))

2 Symbols: Steps taken Champions
BBB(1) 1
BBB(2) 6 1RB1LB_1LB1LA (bbch)
BBB(3) 55 1RB0LB_1LA0RC_1LC1LA (bbch)
BBB(4) ≥ 32,779,478 1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)
BBB(5) ≥1014006 1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch)
3 Symbols: Steps taken Champions
BBB(1,3)
BBB(2,3) 59[3] 1RB2LB1LA_2LB2RA0RA (bbch)
BBB(3,3) ≥ 10 ↑↑ 6 1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch)
4 Symbols: Steps taken Champions
BBB(1,4)
BBB(2,4) ≥205770076433044242247859>2×1023[4] 1RB2LA1RA1LB_0LB2RB3RB1LA (bbch)

Beeping Booping Busy Beaver (BBBB(n,m))

2 Symbols: Steps taken Champions
BBBB(1) 2
BBBB(2) 17

Maximum Consecutive Ones Function (Num(n,m))

Domain Number of Ones Champions
num(1) 1 1RZ--- (bbch)
num(2) 4 1RB1LB_1LA1LZ (bbch)
num(3) 6 1RB1LC_1RC1LZ_1LA0LB (bbch)
num(4) 12 1RB0LA_1RC1LB_1LB1RD_1RZ0RA (bbch)
num(5) 165 1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB (bbch) 0RB1LD_1LC1RB_1LD1RE_1LA1LE_1LZ0RC (bbch)
num(2,3) 6 1RB1LA1LB_0LA2RA1RZ (bbch)
num(3,3) ≥ 12 1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA (bbch)
num(4,3) >10↑44 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)

Maximum Space Function (BBspace(n,m))

2 Symbols: Cells visited Champions
BBspace(1,2) 1 1RZ--- (bbch)
BBspace(2,2) 4 1RB1LB_1LA1RZ (bbch) and 1RB0LB_1LA1RZ (bbch)
BBspace(3,2) 7 1RB1RC_1LC1RZ_1RA0LB (bbch) and 1RB0RC_1LC1RZ_1RA0LB (bbch)
BBspace(4,2) 16 1RB0RA_1LC0RD_0LD0LB_1RA1RZ (bbch)
BBspace(5,2) 12289 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch)
3 Symbols: Cells visited Champions
BBspace(1,3)
BBspace(2,3) 9 1RB2LB1RZ_2LA2RB1LB (bbch)
BBspace(2,4) 2050 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)

Size of the runtime spectrum (R(n,m))

There currently doesn't seem to be any available information about values of this function.

Reversible Turing Machines

Maximum Shifts Function (BBrev(n,m))

2 Symbols: Steps Champions
BBrev(1)
BBrev(2) 6 0RB1RZ_1LA1RB (bbch)
BBrev(3) 17 0RB1RZ_0LC1RA_1RB1LC (bbch)
BBrev(4) 48 1RB0LD_0LC0RB_1LA1LD_1LC1RZ (bbch)
BBrev(5) 388 1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA (bbch)
BBrev(6) ≥ 537,556 1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA (bbch)
BBrev(7) >1019 1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ (bbch)

Maximum Score Function (Σrev(n,m))

2 Symbols: Score Champions
Σrev(1)
Σrev(2) ≥ 2 0RB1RZ_1LA1RB (bbch)
Σrev(3) ≥ 4 0RB1RZ_0LC1RA_1RB1LC (bbch)
Σrev(4) ≥ 6 1RB0LD_0LC0RB_1LA1LD_1LC1RZ (bbch)
Σrev(5) ≥ 16 1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA (bbch)
Σrev(6) ≥ 1161 1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA (bbch)

Blanking Busy Beaver (BLB(n,m))

2 Symbols: Steps Champions
BLB(1) nonexistent nonexistent
BLB(2) 8[4] 1RB0RA_1LB1LA (bbch)
BLB(3) ≥ 34[5] 1RB1LB_1LA1LC_1RC0LC (bbch)
BLB(4) ≥ 32,779,477 1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)
BLB(5) ≥ 32,810,047[6] 1RB1LC_1RD0LE_0RD0RC_1LD1LA_1RB1RE (bbch)
BLB(6) ≥ 65,538,549[7] 1RB1LE_1RD1RB_0RD0RE_1LD1LA_0RF1RF_0LC1LC (bbch)
3 Symbols: Steps Champions
BLB(1,3) nonexistent nonexistent
BLB(2,3) ≥ 77[5] 1RB2LA0RB_1LA0LB1RA (bbch)
BLB(3,3) > 1042,745 1RB2RB1LA_2LC0LB2LB_2RC2RA0LC (bbch)
4 Symbols: Steps Champions
BLB(1,4) nonexistent nonexistent
BLB(2,4) ≥ 1,367,361,263,049[5] 1RB2RA1RA2RB_2LB3LA0RB0RA (bbch)

Lazy Beaver

Shifts Function (LB(n,m))

1 State 2 States 3 States 4 States 5 States 6 States
2 Symbols 2 7 22 72 427 8407
3 Symbols 2 23 351 189,270
4 Symbols 2 93 242,789
5 Symbols 2 956
6 Symbols 2 33,851

Period-oriented Busy Beavers

Busy Preperiodic Beaver (BBS(n,m))

2 Symbols: Preperiod Champions
BBS(1,2) 0 1RA--- (bbch)
BBS(2,2) ≥ 9 1RB0LB_1LA0RB (bbch) proven winner?
BBS(3,2) 101 1RB1LB_0RC0LA_1LC0LA (bbch)
BBS(4,2) ≥ 119,120,230,102 1RB1LC_0LA1RD_0RB0LC_1LA0RD (bbch)
BBS(5,2) > 1014,006 1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch)
3 Symbols: Preperiod Champions
BBS(1,3) 0 1RA------ (bbch)
BBS(2,3) ≥ 165[8] 1RB0LA---_1LB2LA0RB (bbch)
BBS(3,3) > 10 ↑↑ 6 1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch)
BBS(4,3) > 10↑44 1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD (bbch)
4 Symbols: Preperiod Champions
BBS(1,4) 0 1RA--------- (bbch)
BBS(2,4) ≥ 205,770,076,433,044,242,247,860 1RB2LA1RA1LB_0LB2RB3RB1LA (bbch)

Busy Periodic Beaver (BBP(n,m))

2 Symbols: Period Champions
BBP(1,2) 1 1RA--- (bbch)
BBP(2,2) ≥ 9 1RB0RB_1LB1RA (bbch) proven winner?
BBP(3,2) 92 1RB0LA_0RC1LA_1LC0RB (bbch)
BBP(4,2) ≥ 212,081,736 1RB0LA_0RC1RD_1LD0RB_1LA1RB (bbch)
3 Symbols: Period Champions
BBP(1,3) 1 1RA------ (bbch)
BBP(2,3)
BBP(3,3) ≥ 1,195 1RB2RC1LC_0RC0RB1LA_2LA2RC1LB (bbch)
4 Symbols: Period Champions
BBP(1,4) 1 1RA--------- (bbch)
BBP(2,4) ≥ 33,209,131 1RB0RA3LB1RB_2LA0LB1RA2RB (bbch)

Instruction-Limited Busy Beaver

Instruction-Limited Classical Busy Beaver Functions

Instruction-Limited Maximum Shifts Function (BBi(n))

Steps Champions
BBi(1) 1 0RH (bbch) 1RH--- (bbch)
BBi(2) 3 0RB---_1LA--- (bbch)
BBi(3) 5 1RB1LB_1LA--- (bbch)
BBi(4) 16 1RB---_0RC---_1LC0LA (bbch)
BBi(5) 37 1RB2LB---_2LA2RB1LB (bbch)
BBi(6) 123 1RB3LA1RA0LA_2LA------3RA (bbch)
BBi(7) 3,932,963 1RB2LA1RA1RA_1LB1LA3RB--- (bbch) 1RB2LA1RA_1LC1LA2RB_---1LA--- (bbch)
BBi(8) >6.889×101565 1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)
BBi(9) >1010103314360 1RB3LA4RB0RB2LA_1LB2LA3LA1RA--- (bbch)

Instruction-Limited Maximum Score Function (Σi(n))

Score Champions
Σi(1) 1 1RH--- (bbch)
Σi(2) 2 1RB---_1LA--- (bbch)
Σi(3) 4 1RB1LB_1LA--- (bbch)
Σi(4) 5 1RB0LB---_1LA2RA--- (bbch)
Σi(5) 9 1RB2LB---_2LA2RB1LB (bbch)
Σi(6) 14 1RB3LA1RA0LA_2LA------3RA (bbch)
Σi(7) 2050 1RB2LA1RA1RA_1LB1LA3RB--- (bbch) 1RB2LA1RA_1LC1LA2RB_---1LA--- (bbch)
Σi(8) >1.355×10783 1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)
Σi(9) >1010103314360 1RB3LA4RB0RB2LA_1LB2LA3LA1RA--- (bbch)

Instruction-Limited Blanking Busy Beaver (BLBi(n))

Steps Champions
BLBi(1) nonexistent nonexistent
BLBi(2) nonexistent nonexistent
BLBi(3) 4 1RB0RA_1LA--- (bbch)
BLBi(4) 12 1RB---_1RC---_1LC0RC (bbch)
BLBi(5) 30 1RB------_1RC------_2LC2RC0RC (bbch)
BLBi(6) 77 1RB2LA0RB_1LA0LB1RA (bbch)
BLBi(7) 808 1RB------_1RC------_0RD2LC---_1LD2RD0RC (bbch)
BLBi(8) ≥ 1,367,361,263,049 1RB2RA1RA2RB_2LB3LA0RB0RA (bbch)
BLBi(9) > 1042,745 1RB2RB1LA_2LC0LB2LB_2RC2RA0LC (bbch)

Instruction-Limited Greedy Busy Beaver (gBBi(n))

Steps Champions
gBBi(1) 1
gBBi(2) 3
gBBi(3) 5
gBBi(4) 13
gBBi(5) 19
gBBi(6) 25
gBBi(7) 41
gBBi(8) 55
gBBi(9) 238
gBBi(10) 941
gBBi(11) 1341
gBBi(12) 10465
gBBi(13) 10675
gBBi(14) ≥ 9,874,580 0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA--- (bbch)

Program-Limited Busy Beaver

Busy Beaver for Lambda Calculus

Regular Busy Beaver for Lambda Calculus (BBλ(n))

For n = 0,1,2,3,5 BBλ(n) is undefined, while for the rest of 20≥n BBλ(n) = n.

BBλ(n) Champions
BBλ(21) 22 \(\1 1) (1 (\2))
BBλ(22) 24 \(\1 1) (1 (\\1))\(\1 1 1) (1 1)
BBλ(23) 26 \(\1 1) (1 (\\2))
BBλ(24) 30 \(\1 1 1) (1 (\1))
BBλ(25) 42 \(\1 1) (\1 (2 1))
BBλ(26) 52 (\1 1) (\\2 (1 2))
BBλ(27) 44 \\(\1 1) (\1 (2 1))
BBλ(28) 58 \(\1 1) (\1 (2 (\2))))
BBλ(29) 223 \(\1 1) (\1 (1 (2 1)))
BBλ(30) 160 (\1 1 1) (\\2 (1 2)) and (\1 (1 1)) (\\2 (1 2))
BBλ(31) 267 (\1 1) (\\2 (2 (1 2)))
BBλ(32) 298 \(\1 1) (\1 (1 (2 (\2))))
BBλ(33) 1812 \(\1 1) (\1 (1 (1 (2 1))))
BBλ(34) 327,686 (\1 1 1 1) (\\2 (2 1)) and (\1 (1 1) 1) (\\2 (2 1))
BBλ(35) 5×333+6=38127987424941>3.8×1013 (\1 1 1) (\\2 (2 (2 1)))
BBλ(36) 5×2223+6>5.7×1077 (\1 1) (\1 (1 (\\2 (2 1))))
BBλ(37) BBλ(35)+2=5×333+8=38127987424943>3.8×1013 \(\1 1 1) (\\2 (2 (2 1)))
BBλ(38) ≥5×22222+6>1019729 (\1 1 1 1 1) (\\2 (2 1)) and (\1 (1 1) 1 1) (\\2 (2 1))
BBλ(39) ≥5×3333+6>103638334640024 (\1 1 1 1) (\\2 (2 (2 1)))
BBλ(40) >(2↑↑)1533>10↑↑↑16 (\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(41) ≥5×3385+6>101.7×1040 (\1 (\1 1) 1) (\\2 (2 (2 1)))
BBλ(42) ≥BBλ(40)+2>(2↑↑)1533>10↑↑↑16 \(\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(43) >2↑↑↑2↑↑↑2↑↑8 (\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(44) >10↑↑↑10↑↑↑16 (\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(45) ≥BBλ(43)+2>2↑↑↑2↑↑↑2↑↑8 \(\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(46) ≥BBλ(44)+2>10↑↑↑10↑↑↑16 \(\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(47)
BBλ(48) >10↑↑↑10↑↑↑10↑↑↑16 (\1 1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(49) >fω+1(2↑↑62)>Graham's Number (\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))
BBλ(63) >fω3(2) (\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))
BBλ(91) >fε0+1(3) (\1 1 (\1 1) (\\1 2 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))
BBλ(1850) >Loader's Number Too large for this list

Oracle Busy Beaver for Lambda Calculus (BBλ1(n))

Note that f(n)=6+5×BBλ(n).

BBλ1(n) Champions
BBλ1(1) 0
BBλ1(2) 1 1
BBλ1(3) 0
BBλ1(4) 4 \1
BBλ1(5) 5 \2
BBλ1(6) 6 \\1
BBλ1(7) 7 \\2
BBλ1(8) 26 1 (\1)
BBλ1(9) 9 \\2
BBλ1(10) 36 1 (\\1)
BBλ1(11) 41 1 (\\2)
BBλ1(12) 266 1 (1 (\1))
BBλ1(13) 51 1 (\\2)
BBλ1(14) f(36)=25×2223+36>2.85×1078 1 (1 (\\1))
BBλ1(15) f(41)≥25×3385+36>101.7×1040 1 (1 (\\2))
BBλ1(16) f(266) 1 (1 (1 (\1)))
BBλ1(17) f(51) 1 (1 (\\\2))
BBλ1(18) f4(4)=f(f(266)) 1 (\1) 1 (\1)
BBλ1(19) f3(7)=f(f(41)) 1 (1 (1 (\\2)))
BBλ1(20) f6(4)=f4(266) 1 (\\1) 1 (\1)
BBλ1(21) f7(4)=f5(266) 1 (\\2) 1 (\1)
BBλ1(22) f52(4)=f50(266) 1 (1(\1)) 1(\1)
BBλ1(28) ≥fBBλ(f3(4))(4) 1 (\1) 1 (\1) 1 (\1)
BBλ1(29) ≥fBBλ(fBBλ(f4(4))+4(4))+BBλ(f4(4))+5(4) 1(\1)(\1 2 1)(\1)

Busy Beaver for De Bruijn Lambda Calculus

n Value Champion
7 ≥ 7 \1 1 1 1 1 1
8 ≥ 16 (\1 1) (\\2 (1 2))
9 ≥ 68 (\1 1) (\\2 (2 (1 2)))
10 >7.625×1012 (\1 1 1) (\\2 (2 (2 1)))
11 >107.625×1012 (\1 1 1 1) (\\2 (2 (2 1)))
12 >10↑316 (\1 1 1) (\1 (\\2 (2 1)) 1)
13 >10↑310↑310↑26 (\1 1) (\1 (\1 (\\2 (2 1)) 2))
14 >10↑310↑310↑316 (\1 1 1 1 1) (\1 (\\2 (2 1)) 1)
15 >fω+1(101019,727) (\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))
18 fω3(2) (\1 1 (\\\1 3 2 1) 1 1 1) (\\2 (2 1))
26 fϵ0+1(3) (\1 1 (\1 1) (\\1 2 1) (\1 (\2 1 2))) (\\\2 3 (\3 1 2))

Busy Beaver for SKI calculus

Busy Beaver for SKI calculus

n Value Champion
1 1 S
2 2 SS
3 3 SSS
4 4 SSSS
5 6 SSS(SS)
6 17 SSS(SI)S
7 ≥ 18 S(SSS(SI)S)
8 ≥ 19 SS(SSS(SI)S)
9 ≥ 519 SSI((S(SS)S)S)K
10 ≥ 1041 SSI((S(SS)S)S)KS

Busy Beaver for SK calculus

n Value Champion
1 = 1 S
2 = 2 SS
3 = 3 SSS
4 = 4 SSSS
5 = 6 SSS(SS)
6 ≥ 8 SSS(SSS)
7 ≥ 10 SSS(SSSS)
8 ≥ 23 SSS(S(SKS))S

Fractran (BBf(n))

n BBf(n) Example Champion Vector Representation
2 1 [1/2] [−1]
3 1 [3/2] [−11]
4 1 [9/2] [−12]
5 2 [3/2, 1/3] [−110−1]
6 3 [9/2, 1/3] [−120−1]
7 4 [27/2, 1/3] [−130−1]
8 5 [81/2, 1/3] [−140−1]
9 6 [243/2, 1/3] [−150−1]
10 7 [729/2, 1/3] [−160−1]
11 10 [27/2, 25/3, 1/5] [−1300−1200−1]
12 13 [81/2, 25/3, 1/5] [−1400−1200−1]
13 17 [81/2, 125/3, 1/5] [−1400−1300−1]
14 21 [243/2, 125/3, 1/5] [−1500−1300−1]
15 28 [1/45, 4/5, 3/2, 25/3] [0−2−120−1−1100−12]
16 53 [1/45, 4/5, 3/2, 125/3] [0−2−120−1−1100−13]
17 107 [5/6, 49/2, 3/5, 40/7] [−1−110−100201−10301−1]
18 211 [5/6, 49/2, 3/5, 80/7] [−1−110−100201−10401−1]
19 370 [5/6, 49/2, 3/5, 160/7] [−1−110−100201−10501−1]
20 ≥ 746 [7/15, 22/3, 6/77, 5/2, 9/5] [−1−1+1+1−1+1+1+1−1−1−1+1+2−1]
21 ≥ 31,957,632 [7/15, 4/3, 27/14, 5/2, 9/5] [0−1−112−100−130−1−101002−10]
22 >1.146×1062 [1/12, 9/10, 14/3, 11/2, 5/7, 3/11] [−2−1000−12−1001−1010−10001001−100100−1]

Cyclic Tag (CTBB(n))

Runtime Champions
CTBB(2) 5[9]
CTBB(3) > 38[10]
CTBB(4) ≥ 672[11]
CTBB(5) ≥ 2^2^2^2^182[12]
CTBB(6) > 2↑↑↑131[13]
CTBB(7) > 4↑↑↑↑(4↑↑↑3)[14] [15]

Minsky Machines (MBB(n))

Domain Halting Time Champion
MBB(1) 1 0+Z
MBB(2) 3 0+B_0-B*
MBB(3) 5 0+B_0+C_0-C*
MBB(4) 10 0+B_1+C_0-BD_1-C*
MBB(5) 24 0-DB_0+C_1-ED_1+A_1-B*
MBB(6) ≥ 49 0+B_1-FC_1+D_0-CE_0+A_1-A*

Turmites

Terminating Turmites (TT(n,k), 1D Turmites)

Where n is the amount of states and k is the amount of symbols.

Domain Runtime Champions
TT(2) ≥ 13 1TB---_1PA0PB
TT(3) ≥ 82 1PB0PA_1TA0PC_1PA---
TT(4) ≥ 48,186 1TB1PA_1PC0PA_1TA0PD_---1TA
TT(2,3) ≥ 223 1TB0PA2PA_2PA---1PA
TT(3,3) ≥ 45,153 1PB1PA1TA_2TB2PB2PC_---2PA1TC
TT(2,4) > 3.467*1015 1TA2PB3TB---_3TA1PB1TA1PA

2D Turmites (turNing machines)

There are currently no known/available Champions for this function.

Doodle Function (doodle(c,n))

doodle(1,n) = 1 and doodle(2,n) = n. Also note that doodle(c) = doodle(c,2).

2 Symbols: Runtime Champions
doodle(3,2) ≥ 487

Bug Function

Bug(2,2) = 2 

####
#S-#
#.F#
####

Bug(3,3) = 8 

#####
#S..#
#.#.#
#.#F#
#####

Bug(4,4) = 20 

######
#S...#
#..###
#....#
#..#F#
######

Bug(5,5) = 42 

#######
#S...##
#.....#
#.....#
#.#.###
#.#..F#
#######

Bug(6,6) = 96 

########
#S.....#
#.###.##
#.#...##
#.#....#
#..#.###
#..#..F#
########

Bug(7,7) = 218 

#########
#S.###..#
#......##
#.#.##..#
#..#...##
#..#....#
#...#.###
#..#-..F#
#########

Bug(8,8) = 506 

##########
#S.#.....#
#.#..##.##
#.#.##..##
#....##..#
#.#.#...##
#..##....#
#....#.###
#....#..F#
##########

References

  1. ↑ 1.0 1.1 S. Ligocki. "BB(3,4) > Ack(14)." https://www.sligocki.com/2024/05/22/bb-3-4-a14.html Blog post, 2024. Accessed 15 August 2025.
  2. ↑ 2.0 2.1 S. Ligocki, "BB(2,6) > 10↑↑10↑↑10↑↑3". Blog post, 2023. Accessed 15 August 2025.
  3. ↑ Nick Drozd. "BBB(3,3) > 10↑↑6". Accessed 15 August 2025.
  4. ↑ 4.0 4.1 Nick Drozd. "Blanking Beavers". Accessed 15 August 2025.
  5. ↑ 5.0 5.1 5.2 Nick Drozd. "Latest Beeping Busy Beaver Results". Accessed 15 August 2025.
  6. ↑ Comment #71 "https://scottaaronson.blog/?p=5661". Accessed 26 September 2025.
  7. ↑ Comment #62 "https://scottaaronson.blog/?p=5661". Accessed 26 September 2025.
  8. ↑ Nick Drozd. Blanking Beavers
  9. ↑ https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233
  10. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517
  11. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043
  12. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063
  13. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479
  14. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281
  15. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217
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