User:Polygon/Collection of BB Champions

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).

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)	$>2\uparrow \uparrow 2\uparrow \uparrow 2\uparrow \uparrow 10$	1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
BB(7)	$>2\uparrow ^{11}2\uparrow ^{11}3$	1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)
BB(8)
BB(9)	$>f_{\omega }(f_{9}(2))$	1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH (bbch)
BB(10)	$>f_{\omega }^{2}(25)$	1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ (bbch)
BB(11)	$>f_{\omega }^{2}(2\uparrow \uparrow 12)>f_{\omega }^{2}(f_{3}(9))$	1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE (bbch)
BB(12)	$>f_{\omega }^{4}(2\uparrow \uparrow \uparrow 4-3)>f_{\omega }^{4}(f_{4}(2))$	0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL (bbch)
BB(14)	$>f_{\omega +1}(65\,536)>g_{64}$	1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ--- (bbch)
BB(15)	$>f_{\omega +1}(f_{\omega }(10^{57}))$	0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN (bbch)
BB(16)	$>f_{\omega +1}^{2}(10^{10^{57}})$
BB(18)	$>f_{\omega +2}(f_{\omega +1}^{3}(f_{\omega }^{2}(60)))$
BB(20)	$>f_{\omega +2}^{2}(21)$
BB(21)	$>f_{\omega ^{2}}^{2}(4\uparrow \uparrow 341)$
BB(40)	$>f_{\omega ^{\omega }}(75\,500)$
BB(41)	$>f_{\omega ^{\omega }}^{4}(32)$
BB(51)	$>f_{\varepsilon _{0}+1}(8)$

3 Symbols:

	Runtime	Champions
BB(1,3)	$1$	1RZ------ (bbch)
BB(2,3)	$38$	1RB2LB1RZ_2LA2RB1LB (bbch)
BB(3,3)	$\geq 119\,112\,334\,170\,342\,541>10^{17}$	0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(4,3)	$>2\uparrow \uparrow \uparrow (2^{2^{32}+1}-1)$	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)	$>2\uparrow ^{15}5$	1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)

5 Symbols:

	Runtime	Champions
BB(1,5)	$1$	1RZ------------ (bbch)
BB(2,5)	$>10^{10^{10^{3\,314\,360}}}$	1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
BB(3,5)	$>f_{\omega }(2\uparrow ^{15}5)>f_{\omega }^{2}(15)$	1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch)

6 Symbols:

	Runtime	Champions
BB(1,6)	$1$	1RZ--------------- (bbch)
BB(2,6)	$>10\uparrow \uparrow 10\uparrow \uparrow 10^{10^{115}}$	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)	$>2\uparrow \uparrow 2\uparrow \uparrow 2\uparrow \uparrow 10$	1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
Σ(7)	$>2\uparrow ^{11}2\uparrow ^{11}3$	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)	$\geq 374\,676\,383$	0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
Σ(4,3)	$>2\uparrow \uparrow \uparrow 2^{2^{32}+1}-1$	0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch)

4 Symbols:

	Score	Champions
Σ(1,4)	$1$	1RZ--------- (bbch)
Σ(2,4)	$2050$	1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)

5 Symbols:

	Score	Champions
Σ(1,5)	$1$	1RZ------------ (bbch)
Σ(2,5)	$>10^{10^{10^{3\,314\,360}}}$	1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)

Beeping Busy Beavers

Beeping Busy Beaver (BBB(n,m))

2 Symbols:

	Steps taken	Champions
BBB(1)	$1$
BBB(2)	$6$
BBB(3)	$55$	1LB0RB_1RA0LC_1RC1RA (bbch)
BBB(4)	$\geq 32\,779\,478$
BBB(5)	$\geq 10^{14006}$

3 Symbols:

	Steps taken	Champions
BBB(1,3)
BBB(2,3)
BBB(3,3)	$\geq 10\uparrow \uparrow 6$

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

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

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 (BB_space(n,m))

2 Symbols:

	Cells visited	Champions
BBspace(1,2)	$2$
BBspace(2,2)	$4$
BBspace(3,2)	$7$
BBspace(4,2)	$16$

Instruction-Limited Busy Beaver

Maximum amount of steps (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)	$>6.889\times 10^{1565}$	1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)

Maximum Score (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)	$>1.355\times 10^{783}$	1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)

Reversible Turing Machines

Maximum Shifts Function (BB_rev(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)	$\geq 537\,556$	1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA (bbch)
BBrev(7)	$>10^{19}$	1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ (bbch)

Maximum Score Function (Σ_rev(n,m))

2 Symbols:

	Score	Champions
Σrev(1)
Σrev(2)	$\geq 2$	0RB1RZ_1LA1RB (bbch)
Σrev(3)	$\geq 4$	0RB1RZ_0LC1RA_1RB1LC (bbch)
Σrev(4)	$\geq 6$	1RB0LD_0LC0RB_1LA1LD_1LC1RZ (bbch)
Σrev(5)	$\geq 16$	1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA (bbch)
Σrev(6)	$\geq 1161$	1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA (bbch)

Blanking Busy Beaver (BLB(n,m))

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

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)
BBS(2,2)
BBS(3,2)	$101$	1RB1LB_0RC0LA_1LC0LA (bbch)
BBS(4,2)	$\geq 119\,120\,230\,102$	1RB1LC_0LA1RD_0RB0LC_1LA0RD (bbch)

3 Symbols: It seems that currently no information is available for this domain.

4 Symbols:

	Preperiod	Champions
BBS(1,4)
BBS(2,4)	$\geq 293\,225\,660\,896$	1RB2LA0RA3LA_1LA1LB3RB1RA (bbch)

Busy Periodic Beaver (BBP(n,m))

2 Symbols:

	Period	Champions
BBP(1,2)
BBP(2,2)
BBP(3,2)	$92$	1RB0LA_0RC1LA_1LC0RB (bbch)
BBP(4,2)	$\geq 212\,081\,736$	1RB0LA_0RC1RD_1LD0RB_1LA1RB (bbch)

3 Symbols: It seems that currently no information is available for this domain.

4 Symbols:

	Period	Champions
BBP(1,4)
BBP(2,4)	$\geq 33\,209\,131$	1RB0RA3LB1RB_2LA0LB1RA2RB (bbch)

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 $20\geq n$ 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)	$5\times 3^{3^{3}}+6>3.8\times 10^{13}$	(\1 1 1) (\\2 (2 (2 1)))
BBλ(36)	$5\times 2^{2^{2^{3}}}+6>5.7\times 10^{77}$	(\1 1) (\1 (1 (\\2 (2 1))))
BBλ(37)	$BB\lambda (35)+2=5\times 3^{3^{3}}+8>3.8\times 10^{13}$	\(\1 1 1) (\\2 (2 (2 1)))
BBλ(38)	$\geq 5\times 2^{2^{2^{2^{2}}}}+6>10^{10^{4}}$	(\1 1 1 1 1) (\\2 (2 1)) and (\1 (1 1) 1 1) (\\2 (2 1))
BBλ(39)	$\geq 5\times 3^{3^{3^{3}}}+6>10^{10^{12}}$	(\1 1 1 1) (\\2 (2 (2 1)))
BBλ(40)	$>(2\uparrow \uparrow )^{15}33>10\uparrow \uparrow \uparrow 16$	(\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(41)	$\geq 5\times 3^{3^{85}}+6>10^{10^{40}}$	(\1 (\1 1) 1) (\\2 (2 (2 1)))
BBλ(42)	$\geq BB\lambda (40)+2>(2\uparrow \uparrow )^{15}33>10\uparrow \uparrow \uparrow 16$	\(\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(43)	$>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow 8$	(\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(44)	$>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	(\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(45)	$\geq BB\lambda (43)+2>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow 8$	\(\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(46)	$\geq BB\lambda (44)+2>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	\(\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(47)
BBλ(48)	$>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	(\1 1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(49)	$>f_{\omega +1}({\frac {2\uparrow \uparrow 6}{2}})>{\text{Graham's Number}}$	(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))
BBλ(1850)	$>{\text{Loader's Number}}$	Too large for this list

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

Note that $f(n)=6+5\times BB\lambda (n)$.

	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)	$f(36)=25\times 2^{2^{2^{3}}}+36>2.85\times 10^{78}$	1 (1 (\\1))
BBλ1(15)	$f(41)\geq 25\times 3^{3^{85}}+36>10^{10^{40}}$	1 (1 (\\2))
BBλ1(16)	$f(266)$	1 (1 (1 (\1)))
BBλ1(17)	$f(51)$	1 (1 (\\\2))
BBλ1(18)	$f^{4}(4)=f(f(266))$	1 (\1) 1 (\1)
BBλ1(19)	$f^{3}(7)=f(f(41))$	1 (1 (1 (\\2)))
BBλ1(20)	$f^{6}(4)=f^{4}(266)$	1 (\\1) 1 (\1)
BBλ1(21)	$f^{7}(4)=f^{5}(266)$	1 (\\2) 1 (\1)
BBλ1(22)	$f^{52}(4)=f^{50}(266)$	1 (1(\1)) 1(\1)
BBλ1(28)	$\geq f^{BB\lambda (f^{3}(4))}(4)$	1 (\1) 1 (\1) 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)	$\geq 487$

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.