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

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A collection of Busy Beaver [[Champions]] including Champions for BB-Adjacent functions.
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'''=
='''State-and-Symbol-Limited Busy Beaver functions'''=
==Maximum Shifts Function (BB)==
=='''Original Busy Beaver Functions'''==
'''2 Symbols:'''
===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>
|> 2 ↑↑ 2 ↑↑ 2 ↑↑ 10
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|-
Line 42: Line 42:
|-
|-
|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)
|<math> 1 </math>
|1
|{{TM|1RZ------|halt}}
|{{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^{2^{32}+1})</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|}
|}
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|-
|BB(1,4)
|BB(1,4)
|<math> 1 </math>
|1
|{{TM|1RZ---------|halt}}
|{{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)
|<math> 1 </math>
|1
|{{TM|1RZ------------|halt}}
|{{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)
|<math> 1 </math>
|1
|{{TM|1RZ---------------|halt}}
|{{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)==
===Maximum Score Function ([[Busy Beaver Functions|Σ(n,m)]])===
'''2 Symbols:'''
{| 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>
|> 2 ↑↑ 2 ↑↑ 2 ↑↑ 10
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|-
Line 205: Line 200:
|{{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)
|<math> 1 </math>
|1
|{{TM|1RZ------|halt}}
|{{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^{2^{32}+1})</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|}
|}
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|-
|Σ(1,4)
|Σ(1,4)
|<math> 1 </math>
|1
|{{TM|1RZ---------|halt}}
|{{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)
|<math> 1 </math>
|1
|{{TM|1RZ------------|halt}}
|{{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 308:
|-
|-
|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"
|+
|+
!
!'''2 Symbols:'''
!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}}
|}
|}
='''Instruction-Limited Busy Beaver'''=
=='''Maximum Space Function ([[Maximum Space Function|BB<sub>space</sub>]](n,m))'''==
==Maximum amount of steps ([[BBi]])==
{| 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)
|BB<sub>space</sub>(2,2)
|<math> 3 </math>
|4
|{{TM|0RB---_1LA---|halt}}
|{{TM|1RB1LB_1LA1RZ|halt}} and {{TM|1RB0LB_1LA1RZ|halt}}
|-
|BBi(3)
|<math> 5 </math>
|{{TM|1RB1LB_1LA---|halt}}
|-
|-
|BBi(4)
|BB<sub>space</sub>(3,2)
|<math> 16 </math>
|7
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|{{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} and {{TM|1RB0RC_1LC1RZ_1RA0LB|halt}}
|-
|-
|BBi(5)
|BB<sub>space</sub>(4,2)
|<math> 37 </math>
|16
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|{{TM|1RB0RA_1LC0RD_0LD0LB_1RA1RZ|halt}}
|-
|-
|BBi(6)
|BB<sub>space</sub>(5,2)
|<math> 123 </math>
|12289
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|-
|BBi(7)
|<math> 3\,932\,963 </math>
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|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 434:
|-
|-
|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>)==
===Maximum Score Function (Σ<sub>rev</sub>(n,m))===
'''2 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Score
!Score
!Champions
!Champions
Line 465: Line 469:
|-
|-
|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)'''=
=='''Blanking Busy Beaver ([[Busy Beaver Functions#Other Busy Beaver functions|BLB(n,m)]])'''==
There are currently no known/available Champions for this function.
{| class="wikitable"
='''Period-oriented Busy Beavers'''=
|+
==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}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Steps
!Champions
|-
|BLB(1,3)
|nonexistent
|nonexistent
|-
|BLB(2,3)
|≥ 77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|}
{| 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}}
|}
|}
'''3 Symbols:''' It seems that currently no information is available for this domain.
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''3 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,3)
|0
|{{TM|1RA------}}
|}
{| 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>
|≥ 293,225,660,896
|{{TM|1RB2LA0RA3LA_1LA1LB3RB1RA}}
|{{TM|1RB2LA0RA3LA_1LA1LB3RB1RA}}
|}
|}
==Busy Periodic Beaver ([[BBP]])==
===Busy Periodic Beaver ([[BBP]](n,m))===
'''2 Symbols:'''
{| 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.
{| class="wikitable"
|+
!'''3 Symbols:'''
!Period
!Champions
|-
|BBP(1,3)
|1
|{{TM|1RA------}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Period
!Champions
|-
|BBP(1,4)
|1
|{{TM|1RA---------}}
|-
|BBP(2,4)
|≥ 33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|}


'''4 Symbols:'''
='''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"
{| class="wikitable"
|+
|+
!
!
!Period
!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
|
|-
|BLBi(4)
|12
|
|-
|BLBi(5)
|30
|
|-
|BLBi(6)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLBi(7)
|808
|
|-
|BLBi(8)
|≥ 1,367,361,263,049
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}
 
=='''Instruction-Limited Greedy Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|gBBi(n)]])'''==
{| class="wikitable"
|+
!
!Steps
!Champions
!Champions
|-
|-
|BBP(1,4)
|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
|
|
|-
|-
|BBP(2,4)
|gBBi(14)
|<math> \geq 33\,209\,131 </math>
|≥ 9,874,580
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|{{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}
|}
|}
='''Busy Beaver for Lambda Calculus ([[BBλ]])'''=
='''Program-Limited Busy Beaver'''=
==Regular Busy Beaver for Lambda Calculus==
=='''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 892:
|-
|-
|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 1,000:
|-
|-
|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(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>
|}
|}
='''Doodle Function ([[Doodle function|doodle]])'''=
doodle(1,n) = 1 and doodle(2,n) = n


'''2 Symbols:'''
='''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"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|-
|doodle(3,2)
|doodle(3,2)
|<math> \geq 487 </math>
|≥ 487
|
|
|}
|}
='''Terminating Turmites ([[TT]])'''=
='''Turmites'''=
==Terminating Turmites ([[TT]](n,k), 1D Turmites)==
Where n is the amount of states and k is the amount of symbols. There are currently no known/available Champions for this function.
==2D Turmites ([[Terminating Turmite|turNing machines]])==
There are currently no known/available Champions for this function.
There are currently no known/available Champions for this function.
=References=

Latest revision as of 18:03, 29 September 2025

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).

State-and-Symbol-Limited Busy Beaver functions

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) > 2 ↑↑ 2 ↑↑ 2 ↑↑ 10 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
BB(7) 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)
BB(8)
BB(9) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH (bbch)
BB(10) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ (bbch)
BB(11) 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE (bbch)
BB(12) 0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL (bbch)
BB(14) 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ--- (bbch)
BB(15) 0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN (bbch)
BB(16)
BB(18)
BB(20)
BB(21)
BB(40)
BB(41)
BB(51)
3 Symbols: Runtime Champions
BB(1,3) 1 1RZ------ (bbch)
BB(2,3) 38 1RB2LB1RZ_2LA2RB1LB (bbch)
BB(3,3) 0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(4,3) 0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch)
4 Symbols: Runtime Champions
BB(1,4) 1 1RZ--------- (bbch)
BB(2,4) 3,932,964 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
BB(3,4) [1] 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
5 Symbols: Runtime Champions
BB(1,5) 1 1RZ------------ (bbch)
BB(2,5) 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
BB(3,5) 1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch)
6 Symbols: Runtime Champions
BB(1,6) 1 1RZ--------------- (bbch)
BB(2,6) [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) > 2 ↑↑ 2 ↑↑ 2 ↑↑ 10 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
Σ(7) 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) 0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch)
4 Symbols: Score Champions
Σ(1,4) 1 1RZ--------- (bbch)
Σ(2,4) 2050 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
Σ(3,4) [1] 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
5 Symbols: Score Champions
Σ(1,5) 1 1RZ------------ (bbch)
Σ(2,5) 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
6 Symbols: Score Champions
Σ(1,6) 1 1RZ--------------- (bbch)
Σ(2,6) [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) 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) [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))

2 Symbols: 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)

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) 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)
3 Symbols: Steps Champions
BLB(1,3) nonexistent nonexistent
BLB(2,3) ≥ 77[5] 1RB2LA0RB_1LA0LB1RA (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)
3 Symbols: Preperiod Champions
BBS(1,3) 0 1RA------ (bbch)
4 Symbols: Preperiod Champions
BBS(1,4) 0 1RA--------- (bbch)
BBS(2,4) ≥ 293,225,660,896 1RB2LA0RA3LA_1LA1LB3RB1RA (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)
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) 1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)
BBi(9) 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) 1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)
Σi(9) 1RB3LA4RB0RB2LA_1LB2LA3LA1RA--- (bbch)

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

Steps Champions
BLBi(1) nonexistent nonexistent
BLBi(2) nonexistent nonexistent
BLBi(3) 4
BLBi(4) 12
BLBi(5) 30
BLBi(6) 77 1RB2LA0RB_1LA0LB1RA (bbch)
BLBi(7) 808
BLBi(8) ≥ 1,367,361,263,049 1RB2RA1RA2RB_2LB3LA0RB0RA (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 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) (\1 1 1) (\\2 (2 (2 1)))
BBλ(36) (\1 1) (\1 (1 (\\2 (2 1))))
BBλ(37) \(\1 1 1) (\\2 (2 (2 1)))
BBλ(38) (\1 1 1 1 1) (\\2 (2 1)) and (\1 (1 1) 1 1) (\\2 (2 1))
BBλ(39) (\1 1 1 1) (\\2 (2 (2 1)))
BBλ(40) (\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(41) (\1 (\1 1) 1) (\\2 (2 (2 1)))
BBλ(42) \(\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(43) (\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(44) (\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(45) \(\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(46) \(\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(47)
BBλ(48) (\1 1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(49) (\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))
BBλ(1850) Too large for this list

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

Note that .

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) 1 (1 (\\1))
BBλ1(15) 1 (1 (\\2))
BBλ1(16) 1 (1 (1 (\1)))
BBλ1(17) 1 (1 (\\\2))
BBλ1(18) 1 (\1) 1 (\1)
BBλ1(19) 1 (1 (1 (\\2)))
BBλ1(20) 1 (\\1) 1 (\1)
BBλ1(21) 1 (\\2) 1 (\1)
BBλ1(22) 1 (1(\1)) 1(\1)
BBλ1(28) 1 (\1) 1 (\1) 1 (\1)
BBλ1(29) 1(\1)(\1 2 1)(\1)

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

Turmites

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

Where n is the amount of states and k is the amount of symbols. There are currently no known/available Champions for this function.

2D Turmites (turNing machines)

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

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.
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