User:Azerty/Busy Beaver Functions: Difference between revisions

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This is a list of Busy Beaver functions for Turing machines and other simple Turing-complete systems and their current lower bounds.
3This is a list of Busy Beaver functions for Turing machines and other simple Turing-complete systems and their current lower bounds.


All functions here grow uncomputably fast like the original Busy Beaver function. There is no function that grow much faster like the Beeping Busy Beaver function here.
All functions here grow uncomputably fast like the original Busy Beaver function. There is no function that grow much faster like the Beeping Busy Beaver function here.


== Turing Machines ==
== Turing machines ==


=== Normal ===
=== Normal ===
Line 11: Line 11:
!Champion
!Champion
|-
|-
|[[BB(2)]]
|BB(2,2,1)
|4
|{{TM|0RB---_1LA---|halt}}
|-
|[[BB(2)|BB(2,2)]]
|6
|6
|{{TM|1RB1LB_1LA1RZ|halt}}
|{{TM|1RB1LB_1LA1RZ|halt}}
|-
|-
|[[BB(3)]]
|BB(3,2,1)
|17
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|[[BB(3)|BB(3]][[BB(2)|,2]])
|21
|21
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|-
|BB(2[[BB(2)|,3]],1)
|
|
|-
|-
|[[BB(2,3)]]
|[[BB(2,3)]]
Line 23: Line 35:
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|-
|[[BB(4)]]
|[[BB(4)|BB(4]][[BB(2)|,2]])
|107
|107
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|-
|BB(2[[BB(2)|,4]],1)
|124
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|-
|[[BB(2,4)]]
|[[BB(2,4)]]
Line 31: Line 47:
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|-
|[[BB(5)]]
|[[BB(5)|BB(5]][[BB(2)|,2]])
|47,176,870
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
Line 47: Line 63:
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|-
|-
|[[BB(6)]]
|[[BB(6)|BB(6]][[BB(2)|,2]])
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|<math>10 {\uparrow}^{2} 10 {\uparrow}^{2} 10 {\uparrow}^{2} 8</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
Line 55: Line 71:
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|-
|-
|[[BB(7)]]
|[[BB(7)|BB(7]][[BB(2)|,2]])
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|<math>10 \uparrow^{11} 10 \uparrow^{10} 4</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
Line 78: Line 94:
!Champion
!Champion
|-
|-
|BLB(2)
|BLB(2[[BB(2)|,2]])
|8
|8
|{{TM|1RB0RA_1LB1LA}}
|{{TM|1RB0RA_1LB1LA}}
|-
|-
|BLB(3)
|BLB(3[[BB(2)|,2]])
|34
|34
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
Line 90: Line 106:
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|-
|BLB(4)
|BLB(4[[BB(2)|,2]])
|32,779,477
|32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
Line 105: Line 121:
!Champion
!Champion
|-
|-
|BBr(2)
|BBr(2[[BB(2)|,2]])
|6
|6
|{{TM|0RB1RZ_1LA1RB|halt}}
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|-
|BBr(3)
|BBr(3[[BB(2)|,2]])
|17
|17
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|-
|BBr(4)
|BBr(4[[BB(2)|,2]])
|48
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|-
|BBr(5)
|BBr(5[[BB(2)|,2]])
|388
|388
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|-
|BBr(6)
|BBr(6[[BB(2)|,2]])
|537,556
|537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|-
|-
|BBr(7)
|BBr(7[[BB(2)|,2]])
|<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}}
Line 136: Line 152:
!Champion
!Champion
|-
|-
|BBn(2)
|BBn(2[[BB(2)|,2]])
|4
|4
|{{TM|1RB1LB_1LA1LZ|halt}}
|{{TM|1RB1LB_1LA1LZ|halt}}
|-
|-
|BBn(3)
|BBn(3[[BB(2)|,2]])
|6
|6
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|-
|-
|BBn(4)
|BBn(4[[BB(2)|,2]])
|12
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
Line 156: Line 172:
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|-
|BBn(5)
|BBn(5[[BB(2)|,2]])
|165
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}}
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}}
|}
|}


== Lambda Calculus ==
== Lambda calculus ==


=== SK Calculus ===
=== SK Calculus ===
Line 264: Line 280:
|}
|}


== Cyclic Tag ==
== Cyclic tag ==
{| class="wikitable"
{| class="wikitable"
!Domain
!Domain
!Lower Bound
!Lower Bound
!Champion
!Champion
|-
|CTBB(1)
|1
|Δ: () P: () > _
|-
|-
|CTBB(2)
|CTBB(2)
Line 478: Line 498:
|17
|17
|<code>>>>>>>>><[<]</code>
|<code>>>>>>>>><[<]</code>
|}
== Register machines ==
{| class="wikitable"
!Domain
!Lower Bound
!Champion
|-
|RBB(1)
|1
|<code>A- > H, 1</code>
|-
|RBB(2)
|3
|<code>A- > H, 2 / A+ > 1</code>
|-
|RBB(3)
|5
|<code>A+ > 2 / B- > H, 3 / B+ > 1</code>
|-
|RBB(4)
|8
|<code>B+ > 2 / B+ > 3 / A+ > 4 / B- > 3, H</code>
|-
|RBB(5)
|11
|<code>B+ > 2 / B+ > 3 / A+ > 4 / A+ > 5 / B- > 3, H</code>
|-
|RBB(6)
|15
|<code>B+ > 2 / B+ > 3 / B+ > 4 / A+ > 5 / A+ > 6 / B- > 4, H</code>
|}
|}

Revision as of 09:08, 5 December 2025

3This is a list of Busy Beaver functions for Turing machines and other simple Turing-complete systems and their current lower bounds.

All functions here grow uncomputably fast like the original Busy Beaver function. There is no function that grow much faster like the Beeping Busy Beaver function here.

Turing machines

Normal

Domain Lower Bound Champion
BB(2,2,1) 4 0RB---_1LA--- (bbch)
BB(2,2) 6 1RB1LB_1LA1RZ (bbch)
BB(3,2,1) 17 1RB---_0RC---_1LC0LA (bbch)
BB(3,2) 21 1RB1RZ_1LB0RC_1LC1LA (bbch)
BB(2,3,1)
BB(2,3) 38 1RB2LB1RZ_2LA2RB1LB (bbch)
BB(4,2) 107 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch)
BB(2,4,1) 124 1RB3LA1RA0LA_2LA------3RA (bbch)
BB(2,4) 3,932,964 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
BB(5,2) 47,176,870 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch)
BB(3,3) 1.191×1017 0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(2,5) 10↑23|3,314,360 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
BB(2,6) 10↑210↑21010115 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)
BB(6,2) 10↑210↑210↑28 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
BB(4,3) 10↑310↑310↑31028 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)
BB(7,2) 10↑1110↑104 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)
BB(3,4) 10↑154 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
BB(3,5) fω(10↑154) 1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch)
BB(2,7)

Blanking

Domain Lower Bound Champion
BLB(2,2) 8 1RB0RA_1LB1LA (bbch)
BLB(3,2) 34 1RB1LB_1LA1LC_1RC0LC (bbch)
BLB(2,3) 77 1RB2LA0RB_1LA0LB1RA (bbch)
BLB(4,2) 32,779,477 1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)
BLB(2,4) 1.367×1012 1RB2RA1RA2RB_2LB3LA0RB0RA (bbch)

Reversible

Domain Lower Bound Champion
BBr(2,2) 6 0RB1RZ_1LA1RB (bbch)
BBr(3,2) 17 0RB1RZ_0LC1RA_1RB1LC (bbch)
BBr(4,2) 48 1RB0LD_0LC0RB_1LA1LD_1LC1RZ (bbch)
BBr(5,2) 388 1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA (bbch)
BBr(6,2) 537,556 1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA (bbch)
BBr(7,2) 1019 1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ (bbch)

Consecutive Ones

Domain Lower Bound Champion
BBn(2,2) 4 1RB1LB_1LA1LZ (bbch)
BBn(3,2) 6 1RB1LC_1RC1LZ_1LA0LB (bbch)
BBn(4,2) 12 1RB0LA_1RC1LB_1LB1RD_1RZ0RA (bbch)
BBn(2,3)
BBn(3,3) 12 1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA (bbch)
BBn(5,2) 165 1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB (bbch)

Lambda calculus

SK Calculus

Domain Lower Bound Champion
SKBB(4) 4 SSSS
SKBB(5) 6 SSS(SS)
SKBB(6) 8 SSS(SSS)
SKBB(7) 10 SSS(SSSS)
SKBB(8) 23 SSS(S(SKS))S

BCKW Calculus

Domain Lower Bound Champion
BCKWBB(3) 3 BBB
BCKWBB(4) 5 WB(BB)
BCKWBB(5) 7 WB(BBB)
BCKWBB(6) 11 WB(WB(BB))
BCKWBB(7) 15 WB(WB(BBB))
BCKWBB(8) 23 WB(WB(WB(BB)))

De Bruijn

Domain Lower Bound Champion
BBL(6) 6 \1 1 1 1 1 1
BBL(7) 15 (\1 1) (\\2 (1 2))
BBL(8) 67 (\1 1) (\\2 (2 (1 2)))
BBL(9) 7.625×1012 (\1 1 1) (\\2 (2 (2 1)))
BBL(10) 107×1012 (\1 1 1 1) (\\2 (2 (2 1)))
BBL(11) 10↑316 (\1 1 1) (\1 (\\2 (2 1)) 1)
BBL(12) 10↑310↑310↑26 (\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBL(13) 10↑310↑310↑316 (\1 1 1 1 1) (\1 (\\2 (2 1)) 1)
BBL(14) fω+1(101019,727) (\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))

Cyclic tag

Domain Lower Bound Champion
CTBB(1) 1 Δ: () P: () > _
CTBB(2) 5 Δ: (()) P: ()() > _, () > ()()
CTBB(3) 38 Δ: ()(()) P: (()) > ((())), ()()() > _, () > ()()
CTBB(4) 672 Δ: ()()()() P: ()() > (()())(), ()() > ()()(()), () > (()), (((()))) > _
CTBB(5) 10↑24|54 Δ: (())(()()) P: (()()) > (()())(()), (()()) > (()())(()), ()(()) > (())((())), ((())) > ((()))()(), () > _
CTBB(6) 10↑3129 Δ: ((()))((())) P: ((())) -> ((()))(()()), (()()) -> (()())(()), (()) -> (())(()()()), (()()()) -> (()()())()(), ()()() -> _, () -> _
CTBB(7) 10↑410↑210↑24 Δ: ((()))((()())) P: ((()())) > ((()()))((())), ((())) > ((()))(()()()), ((())) > ((()))(()()()), (()()()) > (()()())(()()), (()()) > (()())(())(()), (()) > (())()()()()(), () > _

Fractran

Domain Lower Bound Champion
BBf(10) 7 [729/2, 1/3]
BBf(11) 10 [27/2, 25/3, 1/5]
BBf(12) 13 [81/2, 25/3, 1/5]
BBf(13) 17 [81/2, 125/3, 1/5]
BBf(14) 21 [243/2, 125/3, 1/5]
BBf(15) 28 [1/45, 4/5, 3/2, 25/3]
BBf(16) 53 [1/45, 4/5, 3/2, 125/3]
BBf(17) 107 [5/6, 49/2, 3/5, 40/7]
BBf(18) 211 [5/6, 49/2, 3/5, 80/7]
BBf(19) 370 [5/6, 49/2, 3/5, 160/7]
BBf(20) 746 [7/15, 22/3, 6/77, 5/2, 9/5]
BBf(21) 31,957,632 [7/15, 4/3, 27/14, 5/2, 9/5]

Brainfuck

Brainfuck

Domain Lower Bound Champion
BFBB(6) 6 ++++++
BFBB(7) 8 ++++[-]
BFBB(8) 12 +++[--+]
BFBB(9) 16 ++++[--+]
BFBB(10) 20 +++++[--+]
BFBB(11) 24 ++++++[--+]
BFBB(12) 30 +++++[---++]

Boolfuck

Domain Lower Bound Champion
BLBB(7) 7 >>>>>>>
BLBB(8) 9 *>*[<<>]
BLBB(9) 10 *>*[<<>]>
BLBB(10) 14 *>*>*[<<>]
BLBB(11) 15 *>*>*[<<>]>
BLBB(12) 20 *>*>*[<<<>>]

BitChanger

Domain Lower Bound Champion
BCBB(7) 7 }}}}}}}
BCBB(8) 9 }}}}<[<]
BCBB(9) 13 }}}<[}<<]
BCBB(10) 17 }}}}<[}<<]
BCBB(11) 21 }}}}}<[}<<]
BCBB(12) 25 }}}}}}<[}<<]

Bitter

Domain Lower Bound Champion
BTBB(7) 7 >>>>>>>
BTBB(8) 9 >>>><[<]
BTBB(9) 11 >>>>><[<]
BTBB(10) 13 >>>>>><[<]
BTBB(11) 15 >>>>>>><[<]
BTBB(12) 17 >>>>>>>><[<]

Register machines

Domain Lower Bound Champion
RBB(1) 1 A- > H, 1
RBB(2) 3 A- > H, 2 / A+ > 1
RBB(3) 5 A+ > 2 / B- > H, 3 / B+ > 1
RBB(4) 8 B+ > 2 / B+ > 3 / A+ > 4 / B- > 3, H
RBB(5) 11 B+ > 2 / B+ > 3 / A+ > 4 / A+ > 5 / B- > 3, H
RBB(6) 15 B+ > 2 / B+ > 3 / B+ > 4 / A+ > 5 / A+ > 6 / B- > 4, H
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