Champions: Difference between revisions

Revision as of 19:01, 9 August 2025

Busy Beaver Champions are the current record holding Turing machines which maximize a Busy Beaver function. In this article we focus specifically on the longest running TMs. Some have been proven to be the longest running of all (and so are the ultimate champion) while others are only current champions and may be usurped in the future. For smaller domains, Pascal Michel's website is the canonical source for Busy Beaver champions and the History of Previous Champions.

2-Symbol TMs

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


	Runtime	Champions	Discovered By	Verification
BB(2)	$6$	`1RB1LB_1LA1RZ` (bbch) `1RB0LB_1LA1RZ` (bbch) `1RB1RZ_1LB1LA` (bbch) `1RB1RZ_0LB1LA` (bbch) `0RB1RZ_1LA1RB` (bbch)	Tibor Radó	Direct Simulation
BB(3)	$21$	`1RB1RZ_1LB0RC_1LC1LA` (bbch)	Proven by Shen Lin	Direct Simulation
BB(4)	$107$	`1RB1LB_1LA0LC_1RZ1LD_1RD0RA` (bbch)	Allen Brady	Direct Simulation
BB(5)	$47\,176\,870$	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch)	Heiner Marxen & Jürgen Buntrock in 1989	Direct Simulation
BB(6)	$>2\uparrow \uparrow \uparrow 5$	`1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE` (bbch)	mxdys in 2025	See mxdys's analysis on the TM page
BB(7)	$>2\uparrow ^{11}2\uparrow ^{11}3$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)	Pavel Kropitz in 2025	Analyzed by Shawn Ligocki (see TM page)
BB(8)
BB(9)	$>f_{\omega }(f_{9}(2))$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH` (bbch)	Jacobzheng in 2024
BB(10)	$>f_{\omega }^{2}(25)$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ` (bbch)	Racheline in 2024
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)	Racheline in 2024
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)	Racheline in 2024
BB(13)
BB(14)	$>f_{\omega +1}(65\,536)>g_{64}$	`1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---` (bbch)	Racheline in 2024
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)	Jacobzheng in 2025
BB(16)	$>f_{\omega +1}^{2}(10^{10^{57}})$	User:Jacobzheng/BB(16)	Jacobzheng in 2025
BB(17)
BB(18)	$>f_{\omega +2}(f_{\omega +1}^{3}(f_{\omega }^{2}(60)))$	User:Jacobzheng/BB(18)	Jacobzheng in 2025
BB(19)
BB(20)	$>f_{\omega +2}^{2}(21)$		Racheline in 2024
BB(21)	$>f_{\omega ^{2}}^{2}(4\uparrow \uparrow 341)$		Racheline in 2024
BB(40)	$>f_{\omega ^{\omega }}(75\,500)$	User:Jacobzheng/BB(40)	Jacobzheng in 2024
BB(41)	$>f_{\omega ^{\omega }}^{4}(32)$	User:Jacobzheng/BB(41)	Jacobzheng in 2024
BB(51)	$>f_{\varepsilon _{0}+1}(8)$		Racheline in 2024

3-Symbol TMs


	Runtime	Champions	Discovered By	Verification
BB(2,3)	$38$	`1RB2LB1RZ_2LA2RB1LB` (bbch)	Allen Brady in 1988	Direct Simulation
BB(3,3)	$>10^{17}$	`0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC` (bbch)	Terry & Shawn Ligocki in 2007	Analysis by Pascal Michel
BB(4,3)	$>2\uparrow \uparrow \uparrow 2^{2^{32}}$	`0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD` (bbch)	Pavel Kropitz in 2024

4-Symbol TMs


	Runtime	Champions	Discovered By	Verification
BB(2,4)	$3\,932\,964$	`1RB2LA1RA1RA_1LB1LA3RB1RZ` (bbch)	Terry & Shawn Ligocki in 2005	Pascal Michel, Heiner Marxen, Allen Brady
BB(3,4)	$>2\uparrow ^{15}5$	`1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC` (bbch)	Pavel Kropitz in 2024	Analysis by Shawn Ligocki

5-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,5)	$>10^{10^{10^{3\,314\,360}}}$	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ` (bbch)	Daniel Yuan in 2024	mxdys in Coq
BB(3,5)	$>f_{\omega }(2\uparrow ^{15}5)>f_{\omega }^{2}(15)$	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC` (bbch)	Racheline in 2024

6-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,6)	$>10\uparrow \uparrow 10\uparrow \uparrow 10^{10^{115}}$	`1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA` (bbch)	Pavel Kropitz in 2023	Analysis by Shawn Ligocki

@@ Line 164: / Line 164: @@
 |<math> 38 </math>
 |{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
-|
+|Allen Brady in 1988
-|
+|Direct Simulation
 |-
 |[[BB(3,3)]]
 |<math> > 10^{17} </math>
 |{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
-|
+|Terry & Shawn Ligocki in 2007
-|
+|[https://bbchallenge.org/~pascal.michel/beh#tm33h Analysis by Pascal Michel]
 |-
 |[[BB(4,3)]]
 |<math> > 2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
 |{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
-|
+|Pavel Kropitz in 2024
 |
 |}
@@ Line 192: / Line 192: @@
 |<math> 3\,932\,964 </math>
 |{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
-|Shawn & Terry Ligocki in 2005
+|Terry & Shawn Ligocki in 2005
 |Pascal Michel, Heiner Marxen, Allen Brady
 |-
-|BB(3,4)
+|[[BB(3,4)]]
 |<math> > 2 \uparrow^{15} 5 </math>
 |{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
@@ Line 213: / Line 213: @@
 |<math> > 10^{10^{10^{3\,314\,360}}} </math>
 |{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
-|
+|Daniel Yuan in 2024
-|
+|[https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 mxdys in Coq]
 |-
-|BB(3,5)
+|[[BB(3,5)]]
 |<math> > f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15) </math>
 |{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
-|
+|Racheline in 2024
 |
 |}
@@ Line 231: / Line 231: @@
 !Verification
 |-
-|BB(2,6)
+|[[BB(2,6)]]
 |<math> > 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}} </math>
 |{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
-|
+|Pavel Kropitz in 2023
-|
+|[https://www.sligocki.com/2023/05/20/bb-2-6-p3.html Analysis by Shawn Ligocki]
 |}